Copyright 2010, Lin Cong

Joint Solution of Urban Structure Detection from Hyperion Hyperspectral Images

by

Lin Cong, B.S.

A THESIS

IN

ELECTRICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE IN

ELECTRICAL ENGINEERING

Approved

Dr. Brian Nutter Chair of Committee

Dr. Daan Liang

Dr. Sunanda Mitra

Peggy Gordon Miller Dean of the Graduate School

December, 2010

Texas Tech University, Lin Cong, December 2010

ii

ACKNOWLEDGMENTS I would like to express my sincere gratitude to my advisors Dr. Brian Nutter

and Dr. Daan Liang for giving me an opportunity to work in the field of hyperspectral

satellite imaging. This thesis wouldn’t be possible without their continuous support

during my entire Masters program at Texas Tech University. Dr. Nutter was always

available to guide me to the correct method for digital signal/image processing, and

Dr. Liang provided me with the necessary hurricane damage information and study

tools. I sincerely thank them for patiently revising my thesis and other publications

and providing many valuable suggestions. Also, I want to thank my committee

member Dr. Sunanda Mitra for coming to my defense and providing valuable

suggestions.

I would also like to thank my fellow students Enrique Corona, Dr. Zheng Liu

and Jingqi Ao. Enrique provided me with Jump code for model order estimation and

Dr. Liu also helped me in the image processing. Those guys are valuable resources in

the lab whenever I have problems.

Last but not least, I would like to thank my parents back in China and all my

friends for their support and encouragement.


iii

TABLE OF CONTENTS ACKNOWLEDGMENTS .................................................................................................... ii ABSTRACT ..................................................................................................................... iv LIST OF TABLES ............................................................................................................. v LIST OF FIGURES .......................................................................................................... vi 1. INTRODUCTION .......................................................................................................... 1

1.1 Remote Sensing Based Damage Assessment ....................................................... 2 1.2 Hyperspectral Sensors .......................................................................................... 3 1.3 Existing Approaches for Hyperspectral Image Processing .................................. 4 1.4 Research Objective............................................................................................... 5 1.5 Organization of the Thesis ................................................................................... 6

2. PREPROCESSING OF HYPERION IMAGERY ................................................................ 7 2.1 Study Areas and Datasets ..................................................................................... 7 2.2 Destriping ............................................................................................................. 9 2.3 Atmospheric Correction ..................................................................................... 11 2.4 Chapter Conclusions .......................................................................................... 15

3. FEATURE EXTRACTION ........................................................................................... 17 3.1 Spectral Feature Extraction ................................................................................ 17

3.1.1 Normalized Correlation ........................................................................................... 17 3.1.2 Principal Component Analysis (PCA) ..................................................................... 20 3.1.3 Comparison of Normalized Correlation and PCA ................................................... 27

3.2 Spatial Feature Extraction .................................................................................. 28 3.2.1 Hierarchical Fourier Transform – Co-occurrence Approach ................................... 28 3.2.2 Texture Measures ................................................................................................... 38 3.2.3 Separability Assessment of Texture Measures....................................................... 42

3.3 Feature Selection ................................................................................................ 48 3.4 Chapter Conclusion ............................................................................................ 52

4. CLASSIFICATION AND RESULTS ............................................................................... 53 4.1 Supervised Classification ................................................................................... 53

4.1.1 Bayes Classifier ...................................................................................................... 53 4.1.2 Classification Results of Bayes Classifier ............................................................... 53

4.2 Unsupervised Classification ............................................................................... 59 4.2.1 Model Order Estimation .......................................................................................... 59 4.2.2 K-means Clustering ................................................................................................. 63 4.2.3 Clustering Results of K-means ............................................................................... 63

4.3 Chapter Conclusion ............................................................................................ 66 5. CONCLUSIONS AND FUTURE WORK ........................................................................ 67 BIBLIOGRAPHY ............................................................................................................ 69 APPENDIX ..................................................................................................................... 73


iv

ABSTRACT Hyperspectral remote sensing has shown great potential for disaster analysis.

In post-disaster urban damage assessment, residential areas and buildings must be

accurately identified in the images before and after the disaster. However, the

traditional spectral-only or spatial-only solutions prove ineffective for residence

detection from low resolution hyperspectral images, such as Hyperion data. To solve

this problem, a joint solution of residential area classification, based on both spectral

signature and spatial texture, is proposed in this thesis. Correlations between every

pixel spectrum and the selected endmembers’ spectra and the most significant PCA

(Principle Component Analysis) components of the spectral data provide spectral

features of every pixel. A hierarchical Fourier Transform – Co-occurrence Matrix

approach is designed to help capture spatial textures. Eight second order texture

measures are calculated based on the co-occurrence matrix, and K-fold cross

validation is performed on the training data to select the best combination of features

for the proposed algorithm.

Compared with most existing methods that focus exclusively on spectral or

spatial information and rely on high spatial resolution hyperspectral images that are

usually taken by airborne sensors, our solution makes use of both spectral signature

and macroscopic grid patterns of the residential areas and hence works well for low

resolution Hyperion imagery.


v

LIST OF TABLES 1.1. Technical specifications of hyperspectral sensors .................................................. 3

2.1. Subset of bands used in the study ........................................................................... 9

3.1. Cumulative percentage of variance for top few principal components of New Orleans data ........................................................ 22

3.2. Cross validation of feature combinations for Lubbock dataset (page 1) .......................................................................................................... 50

3.3. Cross validation of feature combinations for New Orleans dataset (page 1)................................................................................................. 51

4.1. Bayes classification result of Lubbock dataset ..................................................... 55

4.2. Bayes classification result of New Orleans dataset............................................... 57

4.3. K-means clustering result of New Orleans dataset ............................................... 66


vi

LIST OF FIGURES 1.1. Impact of Hurricane Katrina ................................................................................... 1

2.1. Lubbock dataset ...................................................................................................... 8

2.2. New Orleans dataset ................................................................................................ 9

2.3. An example of destriping ...................................................................................... 11

2.4. Solar energy distribution and atmospheric absorption .......................................... 12

2.5. Radiance versus reflectance of some major materials from the image .................................................................................................... 15

3.1. Plots of endmembers’ spectra ............................................................................... 18

3.2. Correlation images with three endmembers .......................................................... 19

3.3. Plot of eigenvalues of New Orleans data .............................................................. 22

3.4. PCA results of New Orleans dataset ..................................................................... 24

3.5. Histogram of the PCA bands ................................................................................ 26

3.6. Fourier transform of a residential area with clear grid patterns ............................ 31

3.7. Fourier transform of a residential area containing lakes ....................................... 32

3.8. Fourier transform of a disordered residential ........................................................ 33

3.9. Fourier transform of a nonresidential region ........................................................ 34

3.10. Fourier transform of a grassland region with a country road .............................. 35

3.11. Co-occurrence matrices of the five example regions .......................................... 38

3.12. Testing regions for Fisher separability and feature selection (next section) ................................................................................................. 43

3.13. Separability of all texture measures .................................................................... 44

3.14. Texture image of Lubbock data .......................................................................... 47

3.15. Texture images of New Orleans data .................................................................. 48

4.1. Bayes classification of Lubbock dataset ............................................................... 56

4.2. Bayes classification of New Orleans dataset ........................................................ 58

4.3. Results of cluster number estimation .................................................................... 62

4.4. K-means clustering result by using purely spectral features ................................. 64

4.5. K-means clustering result by using joint solution ................................................. 65


1

CHAPTER I

INTRODUCTION Hurricanes cause an extraordinary level of property losses and human

suffering, with an estimated annualized cost of $6.3 billion in the United States [1].

Recent experiences with Hurricane Katrina and Rita of 2005 once again underscore

this nation’s increasing vulnerability to hurricane disasters in spite of significant

progress made in weather forecasting and hazard preparation. According to [2], 1577

people were killed in the state of Louisiana, and almost 900,000 people in the state lost

power as a result of the Hurricane Katrina. More and more attention has been paid to

the implementation of advanced technologies to produce information that will help

reduce future hurricane losses. Remote sensing technology can be the keystone of an

integrated support system for disaster response, recovery and mitigation.

(a) (b)

Figure 1.1. Impact of Hurricane Katrina (a): Flooded I-10/I-610/West End Blvd interchange and surrounding area of northwest New Orleans and Metairie, Louisiana [3]; (b): Damage to Long Beach, Mississippi, following Hurricane Katrina [4].

http://en.wikipedia.org/wiki/Interchange_(road)�


2

1.1 Remote Sensing Based Damage Assessment Remote sensing and Geographical Information System (GIS) data have

become an increasingly important resource for the disaster management community.

Although hurricane disasters are inevitable, it is possible to minimize the potential risk

by developing disaster early warning strategies, to prepare and implement

developmental plans that provide resilience to such disasters, and to help in

rehabilitation and post-disaster reduction [5]. Remote sensing and GIS play a major

role in efficient mitigation and management of hazards.

A variety of satellites carrying different sensors have been launched in past

decades. Although none of the sensors has been designed solely for the purpose of

observing natural disasters, the variety of spectral bands in VIS (visible), NIR (near

infrared), SWIR (short wave infrared), TIR (thermal infrared) and SAR (synthetic

aperture radar) covers adequate electromagnetic spectrum and provides different

information about properties of the surface. For instance, measurements of the

reflected solar radiation give information on albedo (fraction of light that is reflected

by a body or surface), thermal sensors measure surface temperature, and microwave

sensors measure the dielectric properties, and hence the moisture content, of surface

soil or snow [6]. Compared with VIS, thermal and SAR sensors, multispectral sensors

that usually have 5 to 10 spectral bands from VIS to SWIR (or even TIR) are more

widely used for general disaster management, including hurricane assessment.

The crude spectral categorization of the reflected and emitted energy from the

earth is a limiting factor of multispectral sensor systems [7]. Over the past decades,

advances in sensor technology have overcome this limitation with the development of

hyperspectral sensor technologies. Hyperspectral sensors collect spectral information

as a set of “images”. Each image represents a narrow range of the electromagnetic

spectrum known as a spectral band. These images are combined to form a three

dimensional hyperspectral cube for the surface of the Earth. One significant feature

separating a hyperspectral imager from other optical sensors such as a multispectral

imager is that a hyperspectral spectrometer measures radiation in a series of narrow

and contiguous wavelength bands, usually from the ultraviolet to infrared. A


3

multispectral spectrometer, on the other hand, measures radiation at a few widely

separated wavelength bands [8]. Due to the sufficiency of spectral information,

hyperspectral imagery provides an opportunity for more comprehensive and detailed

representation of surface material.

1.2 Hyperspectral Sensors A number of private companies, academic institutions, and government

agencies have capabilities to acquire hyperspectral imagery, including the European

Space Agency (ESA) and the National Aeronautics and Space Administration

(NASA). Table 1.1 lists some of the currently active hyperspectral imagers along with

their technical specifications. Based on the EO – 1 satellite, Hyperion has a preferable

availability immediately after a hurricane and it is completely free to public access.

With the consideration of availability and access, this study focuses on images from

the Hyperion sensor.

Table 1.1. Technical specifications of hyperspectral sensors Sensor Operator Spectral

Resolution

Spatial

Resolution

Swath

Width

Platform

AVIRIS NASA/JPL 224 bands (400

– 2500 nm)

20 m (h. alt.)

4 m (l. alt.)

11 km

1.9 km

ER – 2

aircraft

Hyperion NASA/USGS 242 bands (356

– 2577 nm)

30 m 7.7 km EO – 1

CHRIS ESA 200 bands (415

– 1050 nm)

20 m 14 km Proba

HyMap Integrated

Spectronics

(Australia)

128 bands (450

– 2480 nm)

5 m (2km alt.) 2.3 km

(2km alt.)

Cessna

aircraft

CASI ITRES (Canada) 288 bands (380

– 1050 nm)

1.5 m 2.25 km aircraft

HYDICE Hughes Danbury

Optical Systems

210 bands (400

– 2500 nm)

1 – 4 m 270 m

(lowest.)

CV – 580

aircraft


4

1.3 Existing Approaches for Hyperspectral Image Processing Many new image processing techniques have been developed for hyperspectral

imagery since the 1990s, most of which can be categorized into one of the two general

categories, pixel level classification and sub-pixel level classification. Among pixel

level methods, Spectral Angle Mapper (SAM) computes a spectral angle between each

pixel spectrum and each endmember spectrum [9] [10]. Endmembers represent

spectral signatures considered macroscopically pure. The smaller the spectral angle,

the more similar the pixel and the endmember are. One of many advantages of SAM is

that it is not sensitive to illumination conditions, because only the directions of

spectral vectors are considered, not the magnitude. Another widely used pixel level

classification method is by matching absorption features at specific positions in the

spectra [11] - [14]. This method is implemented in the ENVI© software application as

Spectral Feature Fitting [15].

Among several sub-pixel classification approaches, the “Spectral Hourglass”

was assessed first to extract endmembers from hyperspectral images and then to use

endmembers to linearly unmix every pixel [16]. This method was successfully applied

to detect minerals at mining sites. However, finding all the endmembers for complex

environments, such as urban areas, is difficult, because most land cover objects in

urban regions are smaller than the spatial resolution of many hyperspectral sensors.

Also, the very large number of endmembers in an urban region usually makes linear

spectral unmixing difficult to implement. In order to simplify heterogeneous urban

surfaces, Ridd proposed the Vegetation – Impervious surface – Soil (VIS) model, by

which every pixel in an urban environment can be explained through proportions of

vegetation, impervious surface and soil [17-18]. As a combination and extension of

both [16] and [17], [19] and [20] developed the Multiple Endmember Spectral Mixture

Analysis (MESMA) technique, which models spectra as a linear sum of spectrally

pure endmembers that vary on a per-pixel basis, rather than using the same set of

endmembers to unmix every pixel. Thus, the full set of endmembers is divided into

three subsets, i.e. vegetation, impervious surface and soil, and every model used to

unmix pixels contains three endmembers, one from each subset.


5

Generally, the spectral analysis methods listed above achieved great success in

geology applications but were not as successful in urban remote sensing. It was

reported that even the sophisticated MESMA technique had signature confusion

problems between dry exposed soil and bright impervious surface [21] and that a

spatial resolution of at least 5 m is required in order to adequately capture urban

structures [20] [22].

In addition to spectral processing, spatial analysis is also widely used in remote

sensing. Algorithms based on Fourier transform [23], wavelet transform [24] - [26],

co-occurrence matrices [27] - [29], Gauss-Markov models [30] etc. have been

developed to capture the land cover textures. Microscopic textures have been

successfully utilized in agricultural classification, iceberg detection and urban

structure detection with (very) high resolution remote sensors (usually not

hyperspectral imagers). However, less work has been done for macroscopic urban

textures in low resolution data.

1.4 Research Objective One of the key premises for post-disaster remote sensing-based damage

assessment is the separation of built environment from the natural environment.

However, the traditional spectral-only or spatial-only approaches prove ineffective for

residence detection from low resolution hyperspectral images. Therefore, a joint

solution of residential area classification, based on both spectral signature and spatial

texture, is proposed in this thesis. Correlations between every pixel spectrum and the

selected endmembers’ spectra and the most significant PCA (Principle Component

Analysis) components of the spectral data provide spectral features of every pixel. A

hierarchical Fourier Transform – Co-occurrence Matrix approach is designed to help

capture spatial textures. Eight second order texture measures are calculated based on

the co-occurrence matrix, and K-fold cross validation is performed on the training data

to select the best combination of features for the proposed algorithm.

Compared with existing methods that rely on hyperspectral images in high

spatial resolution, the proposed joint solution makes use of both spectral signature and


6

macroscopic grid patterns of the residential areas and works well for low resolution

Hyperion imagery.

1.5 Organization of the Thesis In this chapter, the background of remote sensing-based disaster assessment

and many spectral and spatial analysis techniques of hyperspectral image processing

are reviewed. The objectives of this thesis are also explained. In Chapter II, the data

used in the research is first introduced. Preprocessing to remove the vertical stripe

noise from sensor errors and convert the radiance captured by sensors to the surface

reflectance is then explained. The spectral and spatial feature extractions for the joint

solution are explained in Chapter III. In Chapter IV, the results of both supervised and

unsupervised classification are presented and discussed. Conclusions are made, and

possible future work is previewed in Chapter V.


7

CHAPTER II

PREPROCESSING OF HYPERION IMAGERY In this chapter, the image quality of the original Hyperion data is enhanced by

performing histogram normalization for each band. Also, the on-sensor radiance is

converted to surface reflectance by the Fast Line-of-sight Atmospheric Analysis of

Spectral Hypercubes (FLAASH) module in the ENVI© software application.

2.1 Study Areas and Datasets This work selected two study areas. The first area is within the City of

Lubbock, TX, centered around 33°34.5’N, 101°53’W, with a population about

210,000. One important and relevant attribute of Lubbock is that most roads in the city

are built from asphalt or concrete, with only a few exceptions built from bricks. An

EO1 Hyperion scene acquired on January 5th, 2003, was used. A 150 km2 region

centered on Lubbock was subset from Hyperion L1R data (not georegistered format)

as shown in Fig. 2.1a. The second study area is within the City of New Orleans, LA,

centered around 29°58.5’N, 90°12.7’W. An EO1 Hyperion scene was acquired on

April 24th, 2005, about four months before the Hurricane Katrina. A 125 km2 region

centered on New Orleans was subset from Hyperion L1R data as shown in Fig. 2.2a.

Both hyperspectral images have a 30 m spatial resolution.

To approximate ground truth, two images (Fig. 2.1b and Fig. 2.2b) were

delineated based on visual observation and additional sources (e.g. Google Earth). The

residential and natural training datasets used for supervised classifications in Chapter

IV are overlaid in Fig. 2.1c and Fig. 2.2c.


8

(a) (b) (c)

Figure 2.1. Lubbock dataset (a) Original Lubbock data shown in true color (R: 640.5 nm; G: 548.92 nm; B: 457.34 nm); (b) Manually made ground truth (white: residential areas; black: natural areas); (c) Original data with training ROI overlaid (red: residential training area; green: natural training area).


9

(a) (b) (c)

Figure 2.2. New Orleans dataset (a) Original New Orleans data shown in true color (R: 640.5 nm; G: 548.92 nm; B: 457.34 nm); (b) Manually made ground truth (white: residential areas; gray: construction areas with saturated reflectance and without strong spatial texture; black: natural areas); (c) Original data with training ROI overlaid (red: residential training area; green: natural training area; blue: river training data).

The Hyperion hyperspectral sensor has 242 bands, from 356 nm to 2577 nm,

with a spectral resolution of about 10 nm. Of these, 158 bands have acceptable signal

to noise ratios and calibration.

Table 2.1. Subset of bands used in the study Band index Wavelength

1 – 91 426 – 1336 nm 92 – 123 1477 – 1790 nm 124 - 158 1981 – 2355 nm

2.2 Destriping As a pushbroom spectrometer, the onboard Hyperion imager is comprised of

256 individual sensors arranged in a line perpendicular to the flight direction of the


10

spacecraft [31]. Different areas of the surface are imaged as the spacecraft flies

forward. While the advantage of pushbroom design is longer exposure time, image

quality of Hyperion data is vulnerable to varying sensitivity of individual sensors. As a

result of different sensitivity, obvious vertical stripes exist in certain bands that have a

lower signal to noise ratio.

In order to remove the stripes, histogram normalization is applied to each

vertical column of each color band, based on the assumption that the image

background is homogeneous overall. In other words, all vertical samples are

normalized to the same variation and offset to the same mean value within a single

band. For each spectral band, the global mean and standard deviation are calculated as

reference values. Then, local mean and standard deviation for every column and every

color are calculated. By comparing the global and local statistics, a scale factor α and

an offset β are calculated and applied to the original data.

Let mik and σik denote local mean and standard deviation of sample k in band i.

Also, let im and iσ denote global mean and standard deviation, respectively, of band

i. The algorithm can be expressed as finding a scale factor αik and an offset βik so that

the measurement at the position of sample k, line j in band i (xijk) can be normalized to

x*ijk, as equations 2.1 - 2.3 interpret. As shown in Fig. 2.3, artificial stripes are visually

removed by using histogram normalization.

*ijk ik ijk ikx xα β= + (2.1)

iik

ik

σασ

= (2.2)

ik i ik ikm mβ α= − (2.3)


11

(a) (b)

Figure 2.3. An example of destriping (a) Band 119 (1336.15 nm) before vertical destriping; (b) Band 119 after vertical destriping.

2.3 Atmospheric Correction Because solar radiation passes through the atmosphere before it is collected by

the remote sensor, remotely sensed images include information about both the

atmosphere and the earth surface [32]. To be more specific, the solar energy

distribution across all wavelengths, and the absorption features of water vapor, O2,

CO2 and aerosols in the atmosphere all affect the radiation measured by sensors. As a

result, removing the influence of the atmosphere is a critical preprocessing step.


12

(a)

(b)

Figure 2.4. Solar energy distribution and atmospheric absorption (a) Solar spectrum, image courtesy of [33]; (b) Atmosphere transmittance, image courtesy of [34].

The FLAASH module developed by Spectral Sciences, Inc., in collaboration

with the U.S. Air Force Research Laboratory (AFRL) and Spectral Information

Technology Application Center (SITAC) was applied in this project to compensate for

atmosphere effects.


13

According to [35] - [36], FLAASH processes radiance images with spectral

coverage from the mid-IR through UV wavelengths, where the thermal emission can

be neglected. For this situation, the spectral radiance L at a sensor pixel can be

parameterized as:

1 1

ea

e e

BAL LS S

ρρρ ρ

= + +− −

, (2.4)

where:

ρ is the pixel surface reflectance;

ρe is an average surface reflectance for the surrounding region of a pixel, which

accounts for the adjacency effect;

S is the spherical albedo of the atmosphere (capturing the backscattered surface-

reflected photons);

La is the radiance backscattered into the sensor by the atmosphere that did not reach

the surface.

As surface-independent coefficients, A and B vary with atmospheric and

geometric conditions and model the atmospheric transmission and zenith angles. Note

that all of the variables in equation 2.4 are wavelength-dependant. For simplicity, all

of the wavelength indices are omitted. The first term in the equation represents the

radiance directly reflected from the surface to the sensor (including the photons that

left the surrounding surface once, then backscattered to the target surface by the

atmosphere, and then reflected to the sensor directly). The second term interprets the

radiance from the surrounding surface that is re-scattered into the sensor by the

atmosphere. And finally, the last term La represents the solar radiance reflected by the

atmosphere without reaching the surface.

Variables in equation 2.4 are dependent on such parameters as sensor altitude,

ground elevation, solar and viewing angle, spatial and spectral resolution, water vapor

amount in the atmosphere, etc. While most of those parameters can be provided by the

users, column water vapor, clouds and aerosol are not well known and may vary


14

across the scene. FLAASH first retrieves preliminary column water vapor on a pixel-

by-pixel basis. For simplicity, the adjacency effect is ignored in this step, so that ρe

and ρ are taken as equal in equation 2.4. The basis of the retrieval algorithm is the

strong correlation of column water vapor and the ratio of reference radiance (the

shoulders of the water absorption band) to absorption radiance (center of the same

water absorption band) [31]. MODTRAN4 radiation transfer code incorporated within

FLAASH generates values of A+B, S and La for different column water vapor at

reference and absorption wavelengths. Then, a set of reference and absorption

radiances L are simulated for a series of reflectances (0, 0.01, 0.02, … 0.99, 1). The

results are transformed into a 2-dimensional Look-Up Table (LUT), for which the

reference radiance and ratio are the two independent variables, and the water vapor is

the dependent variable. This 2-dimensional LUT is then searched to retrieve the

column water vapor for each pixel.

Cirrus or other kinds of high altitude clouds are identified by using radiance

around the 1.38 um water absorption band [31]. Under clear sky conditions, this band

is very dark, because most of its photons are absorbed by water vapor when they

traverse the atmosphere. However, when high altitude clouds are present, photons do

not complete the full path through the atmosphere. Instead, they are reflected to the

sensor by the clouds before they can be absorbed. As a result, the radiances around

1.38 um are used in FLAASH to detect cirrus, and the cloud pixels are removed when

the adjacency effect is analyzed in the next step.

FLAASH convolves the data cube with a point spread function to describe the

photons reflected from adjacent pixels and scattered to the sensor by atmosphere.

Currently, FLAASH assumes that the point spread function is a modified radial

exponential function, independent of wavelength. The result is a new data cube of

spatially convolved spectral radiance Le. Then, ρe can be approximately solved by

equation 2.5.

( )1

ee a

e

A BL LSρ

ρ+

= +−

(2.5)


15

After the column water vapor retrieval, variables such as A, B, S and La are

generated from the MODTRAN radiation simulation. Then, average surface

reflectance ρe can be solved from equation 2.5 using the above variables and Le from

the spatially convolved radiance image. Finally, reflectance ρ can be solved for each

band of each pixel by equation 2.4. Because the FLAASH module is built in the

ENVI© software package, it can be implemented easily given that all the parameters

required by the software are provided correctly. For more detailed information of the

FLAASH algorithm, please refer to [35] - [39].

(a) (b)

(c) (d)

Figure 2.5. Radiance versus reflectance of some major materials from the image (a) Relatively pure soil; (b) Relatively pure vegetation; (c) Relatively pure road; (d) A mixture of vegetation, construction material and soil from a residential neighborhood. Yellow: radiance; White: reflectance.

2.4 Chapter Conclusions The preprocessing of the Hyperion hyperspectral data was explained in this

chapter. In the first step, histogram normalization was implemented to remove the


16

visual stripes caused by different sensitivity of specific pushbroom sensors. In the

second step, the FLAASH algorithm was applied to compensate for the atmospheric

absorption effect and convert the measured radiance to the reflectance that is

necessary for the material recognition in the following procedures. After the two steps

of preprocessing, the resulting data was ready for spectral-spatial feature extraction in

Chapter III.


17

CHAPTER III

FEATURE EXTRACTION In this chapter, a method of spectral – spatial feature extraction from low

resolution Hyperion data is explained. In general, the spectral feature elements will

derive from the spectral similarity of each pixel with man-made material endmembers

or spectrum of each pixel in a reduced dimensional space. A hierarchical Fourier

transform – co-occurrence matrix method is designed and implemented within the

sliding window centered at each pixel. Eight second order texture measures are

calculated based on the co-occurrence matrix, and K-fold cross validation is

performed on the training data to select the best combination of features for the

proposed joint solution.

3.1 Spectral Feature Extraction

3.1.1 Normalized Correlation Based on the prior knowledge that most roads in Lubbock are built from

asphalt or concrete with only a few exceptions built from bricks, the normalized

correlation of a pixel’s spectrum against asphalt and/or concrete spectra is used as the

measurement of spectral similarity of the pixel to manmade materials. Because streets

are less than one pixel wide in the low resolution Hyperion data, extraction of pure

construction material from the data is extremely difficult. Instead, three different

asphalt and concrete spectra are selected from the USGS digital spectral library

(speclib06a [40]) as endmembers. Spectral plots of these three endmembers are shown

in Fig. 3.1.

Let xij denote spectrum at position (i, j) and yk denote spectrum of endmember

k. The normalized correlation coefficient at position (i, j) against endmember k is

defined as:

,

,

,, ,

[( )( )]i j k

i j k

i j x k yi j k

x y

E x yc

µ µ

σ σ

− −= , (3.1)


18

where E is the expected value operator, and μ and σ denote mean and standard

deviation, respectively. The calculated correlation coefficient is between -1 and 1. The

higher the correlation is, the stronger the similarity between the pixel and the specific

street material endmember. The correlation is 1 if the endmember matches the pixel

exactly, and it is near zero or even negative if the endmember does not fit the pixel at

all.

Figure 3.1. Plots of endmembers’ spectra White: construction asphalt (paving asphalts); Red: asphaltic concrete (paving concretes); Green: concrete light gray road.


19

(a) (b) (c)

Figure 3.2. Correlation images with three endmembers

The three resulting correlation images for paving asphalt, paving concrete and

concrete light gray road are shown in Fig. 3.2. Note that the large areas of soil on the

north and south of the city are brighter than some residential areas, which indicates

that those natural areas have higher spectral similarity to construction materials than

some residential areas do. This phenomenon is due to the spectral confusion between

pure soil and some urban materials such as asphalt (Fig. 3). As a complex mixture of

vegetation, soil and manmade material, a pixel in a residential area often tends to be

less spectrally similar to construction materials than a pixel of a bare soil region. For

example, pixels with a very high proportion of construction material (selected from a

7-lane road in the image) and pixels from pure soil field typically have similar

correlation coefficients against asphalt or concrete, both about 0.9; while pixels from


20

neighborhood blocks, which contain relatively high percentage of vegetation, usually

just have a correlation value between 0.3 and 0.7 against those construction materials.

Thus, a single pixel is difficult to classify by only using spectrum, especially in pixel

level classification. However, if many spectrally similar pixels are identified in a line

or even in a grid pattern, the likelihood of streets becomes high. Thus, macroscopic

spatial texture is desired for accurate residential area classification, especially for low

resolution hyperspectral images.

3.1.2 Principal Component Analysis (PCA) PCA is a mathematical procedure that transforms a number of possibly

correlated variables into a smaller number of uncorrelated variables called principal

components. The first principal component accounts for as much of the variability in

the data as possible, and each succeeding component accounts for as much of the

remaining variability as possible [41]. The most significant components are selected to

serve as spectral features.

The first step of PCA is to calculate the covariance matrix as follows:

1

1cov ( ) ( )N

ti i

ix u x u

N =

= − × −∑ , (3.2)

where ix denotes any spectrum vector and u denotes the mean value vector of all

spectra. The covariance matrix is the average of outer product matrices of spectral

samples. In the second step, eigenvalues of the covariance matrix are calculated and

sorted in descending order. The transform matrix (T) is formed by concatenating

transposed eigenvectors ( tie ) row by row in the same order as the eigenvalues. Finally,

all the spectra are transformed into PCA space by left-multiplying the transform

matrix as in equation 3.3.

1

* 2

...

t

t

i i i

tl

ee

x T x x

e

= ⋅ = ⋅

(3.3)


21

Subscript l in equation 3.3 represents the dimensionality of new data, which depends

on the number of eigenvectors used to build the transform matrix. Generally, PCA is

used to reduce the dimensionality of spectral data, while, at the same time, keeping as

much information as possible.

Because the New Orleans dataset contains large areas of river and the spectral

difference between water and other materials are more dramatic than the difference

between construction and natural materials, the first few principle components tend to

be dominated by the large spectral variance between water and other materials. In

order to make the PCA transform more sensitive to the spectral variance between

construction and natural materials, water pixels are detected and labeled as “water”,

then the remaining pixels are PCA transformed. After classification, the “water”

segmentation is merged back into the “natural” class.

Based on the distinct spectral signature of water, such as the reflectance peak

at about 548 nm (green band) and the near-zero reflectance at about 1205 nm, which is

very different from most other materials, an algorithm similar to Normalized

Difference Vegetation Index (NDVI) is designed to assess whether the target is (or

contains) liquid water or not. The water index is defined in equation 3.4. By definition,

the water index is a scalar between -1 and 1, unless the NIR band around 1205 nm is

very noisy and the reflectance is negative, which does happen in a rare case. Generally

the higher the index value is, the higher the possibility that the target has liquid water.

In practice, 0.5 has been found as a reliable threshold between water and non-water.

The water mask built for the New Orleans dataset is shown in Fig. 3.4a.

GREEN NIRwater indexGREEN NIR

−=

+ (3.4)


22

Figure 3.3. Plot of eigenvalues of New Orleans data

Table 3.1. Cumulative percentage of variance for top few principal components of New Orleans data

Principal Component

Index

Ordinary PCA Water-removed PCA

1 72.56% 59.79% 2 96.82% 94.34% 3 98.63% 97.54% 4 99.08% 98.36% 5 99.26% 98.70%

In Fig. 3.3, for both PCA transforms, each of the first two principal bands has

at least 10 times stronger variance than all the other components. As shown in Table

3.1, the cumulative percentage of variance of the first two components is around 95%,

which means that the combination of the first two principal components is a good

representation of the complete dataset. By comparing Fig. 3.4b and 3.4c, one can

observe that the first component image of the water-removed PCA transform has a

higher visual contrast between residential areas and natural areas than the first

component image of the traditional PCA transform. Furthermore, as shown in the

histograms of the first bands of the ordinary PCA and the water-removed PCA

0 20 40 60 80 100 120 140 1607

8

9

10

11

12

13

14

principle component index

log1

0 of

eig

enva

lues

water removed PCAordinary PCA


23

transform (Fig. 3.5a and 3.5b), the peak of the residential training data and the peak of

the natural training data coincide for the ordinary PCA transform, but well separate for

the water-removed PCA transform. The second bands of both PCA transforms have

similar separability as shown in Fig. 3.5c and 3.5d. According to the comparison

between the histograms, the conclusion that water-removed PCA transform generates

preferable spectral features to the traditional PCA transform can be made.

(a) (b) (c)


24

(d) (e)

Figure 3.4. PCA results of New Orleans dataset (a) Water mask; (b) The first band of the traditional PCA transform; (c) The first band of water-removed PCA transform; (d) The second band of the traditional PCA transform; (e) The second band of water-removed PCA transform.


25

(a)

(b)

-15000 -12000 -9000 -6000 -3000 0 3000 6000 9000 12000 15000 0

10

20

30

40

50

60

70

80

90histogram of first component of ordinary PCA

data range

hist

ogra

m

residential training datanatural training data

-15000 -12000 -9000 -6000 -3000 0 3000 6000 9000 12000 15000 0

20

40

60

80

100

120

140

160

180

200

data range

hist

ogra

m

histogram of first component of water-removed PCA



26

(c)

(d)

Figure 3.5. Histogram of the PCA bands (a) Histogram of the first component of the traditional PCA; (b) Histogram of the first component of water-removed PCA; (c) Histogram of second component of the traditional PCA; (d) Histogram of the second component of water-removed PCA.

-15000 -12000 -9000 -6000 -3000 0 3000 6000 9000 12000 15000 0

20

40

60

80

100

120

140

data range

hist

ogra

m

histogram of second component of ordinary PCA


-15000 -12000 -9000 -6000 -3000 0 3000 6000 9000 12000 15000 0

10

20

30

40

50

60

70

80

data range

hist

ogra

m

histogram of second component of water-removed PCA



27

3.1.3 Comparison of Normalized Correlation and PCA Two alternative spectral feature extraction methods used in this study were

introduced in the previous two sections. Generally, both methods have pros and cons.

In this section, a detailed comparison between the two methods is presented.

The advantage of the correlation method is that as a measurement of spectral

similarity, the value of normalized correlation coefficient is proportional to the degree

to which the pixel is spectrally similar to the endmember. In theory, the residential

areas have higher spectral similarity to construction materials than natural areas do,

and as a result, the correlation value should be larger in residential areas than in

natural areas. To this extent, the correlation coefficient has a clearer physical meaning

as a spectral feature than PCA components do. However, the correlation method

requires prior knowledge about the prevailing man-made materials in the study site,

which is sometimes not easily available. Furthermore, considering the coarse spatial

resolution, the physical meaning of correlation coefficient may be undermined by the

spectral confusion between soil and neighborhood mixture, as explained in section

3.1.1.

On the other hand, the PCA method does not require prior knowledge of road

materials and is able to efficiently reduce the volume of the dataset based on the

inherent structure of the data. However, the PCA transform is not discrimination-

oriented. Sometimes, just having the largest variance does not mean that the particular

direction is the best suited to classify different clusters. It is possible that the directions

discarded by PCA might be exactly the directions that are required to distinguish

between classes. One of these awkward cases is described in [42]. PCA might

discover, for example, the gross features that characterize (uppercase letters) Os and

Qs, but might ignore the tail that distinguishes an O from a Q. Similarly, in presence

of large areas of water pixels or reflectance-saturated pixels, the huge spectral

difference between water (or saturated) pixels and the other pixels is characterized as

the gross features by the traditional PCA transform, and the difference between

residential and natural pixels is characterized as less important features. By removing


28

the water pixels before calculating the covariance matrix, the performance of PCA

transform can be improved to some extent.

In this thesis, correlation coefficients and PCA components are both used to

extract spectral information from the Lubbock dataset and the New Orleans dataset.

3.2 Spatial Feature Extraction The selected datasets represent two different styles of city arrangement. The

city of Lubbock is built on clear grid patterns and is relatively easy to distinguish from

the surrounding rural areas. The city of New Orleans is less structured: the streets and

avenues are not orthogonal to each other, and the direction of roads varies from

neighborhood to neighborhood. The complexity of urban arrangement in New Orleans

makes the classification quite challenging but not impossible. To capture the periodic

textures oriented in different directions in different neighborhoods, a hierarchical

Fourier Transform – Co-occurrence Matrix approach is developed. The Fourier

transform is first implemented to detect the angle of local periodic texture, and the co-

occurrence matrix is calculated with the dominant angle detected from the Fourier

transform. Finally, spatial features are extracted from a series of second order texture

measures calculated from co-occurrence matrices. Both the Fourier transform and the

co-occurrence matrix are calculated on a sliding window basis, and the extracted

spatial features are assigned to the center pixel of the window.

3.2.1 Hierarchical Fourier Transform – Co-occurrence Approach 1. Two-dimensional Fourier Transform

The two-dimensional Fourier transform (FT) is defined as equation 3.5 and

3.6:

1 1 2 ( )

0 0

1[ , ] [ , ]mk nlN M jM N

n mF k l f m n e

MNπ− − − +

= =

= ×∑∑ , and (3.5)

1 1 2 ( )

0 0

1[ , ] [ , ]mk nlN M jM N

l kf m n F k l e

MNπ− − +

= =

= ×∑∑ , (3.6)


29

where f denotes the original image and F denotes the Fourier image. M and N are the

dimensions of original and transformed images. Every component in the Fourier

domain, i.e. F[k,l], represents the magnitude and phase of the 2D harmonic wave, with

frequency of k in one dimension and frequency of l in the other dimension. For our

study images and typical road spacing, a tile of 31-by-31 to 51-by-51 is found

appropriate for reliable detection in the Fourier analysis. If the tile is too large, it tends

to cover both residential and nonresidential areas; if the tile is too small, the Fourier

components associated with grid patterns tend to be close to DC and are vulnerable to

misclassification as low frequency components. In practice, a 41-by-41 tile is used for

all applications.

Figs. 3.6 - 3.10 demonstrate the Fourier transforms of three out of many types

of residential regions and two out of many types of natural regions in both Lubbock

and New Orleans datasets. For a clear grid pattern (Fig. 3.6), the FT components

associated with the periodic street patterns have large magnitudes, shown as very

bright pixels in Fig. 3.6b. The relationship between a spatial pattern and its Fourier

components is that all the associated components are aligned orthogonal to the spatial

pattern. If the gray level changes following an approximate sinusoidal pattern, most of

the energy will be concentrated into a few single components, and the distance from

those strong components to the origin indicates the number of periods that the spatial

pattern has within the tile. On the other hand, because no periodic pattern exists in the

natural region, the peak Fourier components are very close to the origin, as shown in

Fig. 3.9b and Fig. 3.10b. Based on the difference in the Fourier domain, the position of

peak FT components was initially designed as the spatial feature of the center pixel of

the tile.

However, for the residential region shown in Fig. 3.7, where the existence of

lakes interrupts the periodic spatial pattern, many low frequency components are

stronger than or at least comparable to the actual components associated with the street

spacing (Fig. 3.7b). For the residential areas, where more than one kind of street

pattern exists in the tile, as shown in Fig. 3.8, it is almost impossible to use a pair of

Fourier components to represent the spatial pattern, or in other words to use the


30

position of a single peak component as an effective spatial feature for the center pixel

of the tile. Even if a single peak Fourier component is not always trustworthy, the

direction of the strongest cumulative energy in Fourier domain is usually consistent

with the direction of the street pattern (subimage c and d of Fig. 3.6 – 3.10). Because

the strongest direction is more noise tolerant than strongest component, a more

complex algorithm combining the detected direction and co-occurrence matrix has

been designed.


31

(a) (b) (c)

(d)

Figure 3.6. Fourier transform of a residential area with clear grid patterns (a) Correlation image of a residential area with clear grid patterns; (b) Magnitude of Fourier transform (zoomed in and DC suppressed); (c) Direction of the maximum energy highlighted in white; (d) Cumulative energy with regard to angles.

0 20 40 60 80 100 120 140 160 1800

1000

2000

3000

4000

5000

6000

7000

angle (degree)

dire

ctio

nal e

nerg

y in

Fou

rier d

omai

n

directional energy in Fourier domaindirection of maximum energy


32

(a) (b) (c)

(d)

Figure 3.7. Fourier transform of a residential area containing lakes (a) Correlation image of a residential area containing lakes; (b) Magnitude of Fourier transform (zoomed in and DC suppressed); (c) Direction of the maximum energy highlighted in white; (d) Cumulative energy with regard to angles.

0 20 40 60 80 100 120 140 160 1800

2000

4000

6000

8000

10000

12000

angle (degree)

ener

gy

directional energy spectrumdirection of the maximum engergy


33

(a) (b) (c)

(d)

Figure 3.8. Fourier transform of a disordered residential (a) PCA image (1st band) of a disordered residential area in the city of New Orleans; (b) Magnitude of Fourier transform (zoomed in and DC suppressed); (c) Direction of the maximum energy highlighted in white; (d) Cumulative energy with regard to angles.

0 20 40 60 80 100 120 140 160 1800

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

12

angle (degree)

ener

gy

directional energy spectrumdirection of the maximum energy


34

(a) (b) (c)

(d)

Figure 3.9. Fourier transform of a nonresidential region (a) PCA image (1st band) of a nonresidential region; (b) Magnitude of Fourier transform (zoomed in and DC suppressed); (c) Direction of the maximum energy highlighted in white; (d) Cumulative energy with regard to angles.

0 20 40 60 80 100 120 140 160 1800.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

11

angle (degree)

dire

ctio

nal e

nerg

y in

Fou

rier d

omai

n

directional energy in Fourier domaindirection of the maximu energy


35

(a) (b) (c)

(d)

Figure 3.10. Fourier transform of a grassland region with a country road (a) PCA image (1st band) of a grassland region with a country road; (b) Magnitude of Fourier transform (zoomed in and DC suppressed); (c) Direction of the maximum energy highlighted in white; (d) Cumulative energy with regard to angles.

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7x 10

11

angle (degree)

ener

gy

directional engergy spectrumdirection of the maximum energy


36

2. Co-occurrence matrix

A co-occurrence matrix is a matrix that is defined over an image to be the

distribution of co-occurring gray levels at a given offset between the reference and the

neighboring pixels. First proposed by Haralick et al. in 1970s [43], the co-occurrence

matrix (aka gray-level co-occurrence matrix) is defined as equation 3.7 and 3.8:

,1 1

1, ( , ) & ( , )( , ) ( , ) & ( , )

0,

n m

x yp q

if I p q i I p x q y j orC i j if I p q j I p x q y i

otherwise∆ ∆

= =

= + ∆ + ∆ == = + ∆ + ∆ =

∑∑ , and (3.7)

,,

,,

( , )( , )

( , )x y

x yx y

i j

C i jP i j

C i j∆ ∆

∆ ∆∆ ∆

=∑

, (3.8)

where i and j denote the quantized gray levels; I, C and P denote the original image

(matrix), the framework matrix and the co-occurrence matrix, respectively; x∆ and

y∆ denote the offset between the reference and the neighbor pixels. The difference

between the framework matrix and the co-occurrence matrix is that the entries of the

framework matrix are the counts of co-occurring gray levels, while the entries of the

final co-occurrence matrix are normalized to the probabilities. By definition, co-

occurrence matrices are always symmetric square matrices, and the size is dependent

on the quantization levels.

In place of offset ( x∆ , y∆ ), a displacement and angle are often used to

describe the relationship between the reference and the neighbor pixels. In order to

achieve rotational invariance, four co-occurrence matrices were usually calculated

based on the same displacement in four orthogonal angles (i.e. 0o, 45o, 90o and 135o to

the horizontal right) [27], then the texture measures derived from different co-

occurrence matrices are averaged. The approach of calculating four co-occurrence

matrices in four orthogonal angles was called the omnidirectional method. Rather than

using the omnidirectional method, the angle in our study is determined by the prior

Fourier analysis, and the displacement is set as 1 for all applications, based on the

street spacing. Compared with the omnidirectional method, the angle of maximum


37

energy in the Fourier domain is more desirable, because that angle should be

orthogonal to the periodic street patterns, and the neighboring pixels in that angle

should have the largest contrast, and, at the same time, the angle from Fourier analysis

is also rotationally invariant. The omnidirectional method, on the other hand, includes

both correct and incorrect directions and the spatial features extracted, as an average

of four, is not as sharp as those extracted by using angles from Fourier analysis.

Fig. 3.11 demonstrates the gray-level co-occurrence matrices of the five

regions in Fig. 3.6 - 3.10. Neighboring pixels in residential areas have large contrast

and the associated co-occurrence matrices (Fig. 3.11a-c) have many high-probability

entries away from the diagonal, which is not unexpected. The natural areas are plain

enough in gray level to have most nonzero entries lie on or close to the diagonal in the

associated co-occurrence matrices (Fig. 3.11d-e).

(a) (b)

(c) (d)

gray level i

gray

leve

l j

10 20 30 40 50 60

10

20

30

40

50

60

gray level i

gray

leve

l j

10 20 30 40 50 60

10

20

30

40

50

60

gray level i

gray

leve

l j

10 20 30 40 50 60

10

20

30

40

50

60

gray level i

gray

leve

l j

10 20 30 40 50 60

10

20

30

40

50

60


38

(e)

Figure 3.11. Co-occurrence matrices of the five example regions (a) Co-occurrence matrix of the residential area with clear grid patterns; (b) Co-occurrence matrix of the residential area containing lakes; (c) Co-occurrence matrix of the disordered residential area with more than one street structure; (d) Co-occurrence matrix of the nonresidential area; (e) Co-occurrence matrix of the grassland region with a country road.

3.2.2 Texture Measures The co-occurrence matrix is used for a series of second order texture

calculations. The difference between first order and second order texture measures is

that the first order measures are statistics calculated from the original image, such as

mean and variance, and do not consider pixel neighborhood relationships, while the

second order measures consider the relationship between groups of two pixels in the

original image and are usually calculated from a co-occurrence matrix [44]. According

to the purpose, texture measures can be divided into three general groups: contrast

group, orderliness group and statistics group.

Generally, the texture measures are weighted averages of cell contents of a co-

occurrence matrix. The contrast group focuses on the contrast between neighboring

pixels in the original image and creates weights for each entry of the co-occurrence

matrix so that the resulting measures are larger (or smaller in some measures) where a

stronger contrast exists. Because entries on the diagonal in a co-occurrence matrix

represent no contrast, and contrast increases away from the diagonal, measures in this

group create an increasing (or decreasing in some measures) weight as the distance

from the diagonal increases. The orderliness group measures how regular (orderly) the

gray level i

gray

leve

l j

10 20 30 40 50 60

10

20

30

40

50

60


39

pixel values are within the sliding window. Generally an orderly image has most of the

probability concentrated in a few entries in the co-occurrence matrix, while a

disorderly image has the probability almost evenly distributed across a number of

entries. Thus, measures in this group create a weight increasing (or decreasing in some

measures) with the commonness of the gray level co-occurring combination. The

statistics group calculates the mean, variance, correlation etc. of the co-occurrence

matrix [44].

(1) Contrast (CON)

2,

, 1( )

N

i ji j

CON P i j=

= −∑ (3.10)

The weights are second norm of the gray level difference between two

neighboring pixels. The higher the contrast measure is, the higher contrast the image

has. A simple image window will have a contrast measure approaching 0.

(2) Dissimilarity (DIS)

,, 1

N

i ji j

DIS P i j=

= −∑ (3.11)

The weights are the first norm of the gray level difference between

neighboring pixels. A simple image window will have a contrast measure approaching

0.

(3) Homogeneity (HOM)

,2

, 11 ( )

Ni j

i j

PHOM

i j=

=+ −∑ (3.12)

The weights are inverse second norm of the gray level difference between two

neighboring pixels. The value of homogeneity is between 0 and 1, with 1 for a

perfectly plain window and close to 0 for a sharply contrasted window.

(4) Similarity (SIM)


40

,

, 11

Ni j

i j

PSIM

i j=

=+ −∑ (3.13)

The weights are inverse first norm of the gray level difference between two

neighboring pixels. The value of homogeneity is between 0 and 1, with 1 for a

perfectly plain window and close to 0 for a sharply contrasted window.

(5) Angular Second Moment (ASM)

2,

, 1

N

i ji j

ASM P=

= ∑ (3.14)

The weight of an entry’s content (probability) is the probability itself. The

value of ASM is between 0 and 1, and high values occur when the window is orderly.

The reason why a higher value of ASM indicates a more orderly image is simply (but

not strictly) interpreted as follows:

Assume that there are N quantized gray levels and hence N2 entries in the co-

occurrence matrix. The summation of probabilities is always 1:

, ,1 1

1, 0N N

i j i ji j

P where P= =

= ≥∑∑ . (3.15)

If both sides of (3.15) are squared, (3.16) can be found:

2 2, , , ,

1 1 1 1 , 1 , 1( ) 1 1 2

N N N N N N

i j i j i j m ni j i j i j m n

m i orn j

P P P P= = = = = =

≠≠

= ⇔ = −∑∑ ∑∑ ∑ ∑ . (3.16)

The following inequality is always true based on the Cauchy-Schwarz theorem:

2 2, , , , , ,2 , 0, 0.i j m n i j m n i j m nP P P P where P P≤ + ≥ ≥ (3.17)

The equality in (3.17) occurs if and only if:

, , , 1 ( , ) ,1 ( , ) ,i j m nP P i j N m n N m i or n j= ∀ ≤ ≤ ≤ ≤ ≠ ≠ .

Through some simple mathematical calculations, the following inequality can be

found from equation 3.16:


41

2 2 2, ,

1 1 1 11 1 ( 1)

N N N N

i j i ji j i j

P N P= = = =

≥ ≥ − − ⇔∑∑ ∑∑

2, 2

1 1

11N N

i ji j

PN= =

≥ ≥∑∑ , (3.18)

where the maximum of ASM will be found if and only if exactly one entry of the co-

occurrence matrix is 1 and all the other entries are 0; the minimum of ASM will be

found if and only if all the entries contain the same probability 2

1N

.

With repetitive spatial patterns, an orderly image tends to have most of the

probability concentrated in only a few entries of the co-occurrence matrix while

leaving all the other entries approaching 0, and the ASM is close to 1 as a result. On

the other hand, with different spatial patterns here and there, a disorderly image tends

to have the probability almost evenly distributed across a number of entries, which

results in a relatively small value of ASM.

(6) Maximum Probability (MAX)

,max( )i jMAX P= (3.19)

Having most of the probability concentrated in only a few entries of co-

occurrence matrix, an orderly image will have a relatively large value of MAX. A

disorderly image, on the other hand, having the probability almost evenly distributed

across many entries, will have a relatively small value of MAX.

(7) Entropy (ENT)

, 2 ,, 1

logN

i j i ji j

ENT P P=

= −∑ (3.20)

Here, we must assume that 20 log 0 0× = . As the name suggests, entropy is a

measure of disorder. It is 0 if the window is perfectly orderly and is larger for a more

disorderly window.

(8) Co-occurrence matrix Correlation (COR)


42

,

, 1

,, 1

( )

( )

N

i i ji j

N

j i ji j

i P

j P

µ

µ

=

=

=

=

∑

∑ (3.21)

2 2,

, 1

2 2,

, 1

( )

( )

N

i i j ii j

N

j i j ji j

P i

P j

σ µ

σ µ

=

=

= −

= −

∑

∑ (3.22)

,, 1

( )( )Ni j

i ji j i j

i jCOR P

µ µσ σ=

− −= ∑ (3.23)

COR measures the linear dependency of gray levels between neighboring

pixels. Intuitively, 0 means uncorrelated and 1 means perfectly correlated.

3.2.3 Separability Assessment of Texture Measures In order to further analyze the separability of all the eight texture measures,

two classes of testing data are selected from each dataset, as shown in Fig. 3.12. Note

that some imperfect testing data is deliberately selected, i.e. residential areas with

weak periodic patterns and natural areas with low uniformity because of country roads

or small houses. Fisher’s discriminant ratio is calculated, for each texture, as a

measurement of separability between residential and natural areas. As shown in

equation 3.26, the separability is defined as the ratio of between-class variance over

within-class variance. In the equations, 21 1 1, ,p µ σ and 2

2 2 2, ,p µ σ are probability, mean

and variance of the two classes, respectively. The testing result of each texture

measure is shown in Fig. 3.13.

2 2 2 21 1 2 2 1 2 1 2( ) ( ) ( )b p p p pσ µ µ µ µ µ µ= − + − = − (3.24)

2 2 21 1 2 2w p pσ σ σ= + (3.25)

1 2

2 2 2are constants1 2 1 2 1 2

2 2 2 2 21 1 2 2 1 1 2 2

( ) ( )p pb

w

p pseparabilityp p p p

σ µ µ µ µσ σ σ σ σ

− −= = →

+ + (3.26)


43

(a) (b)

Figure 3.12. Testing regions for Fisher separability and feature selection (next section) (a) Original Lubbock data overlaid with testing region; (b) Original New Orleans data overlaid with testing region.


44

Figure 3.13. Separability of all texture measures

From Fig. 3.13, three general conclusions can be made:

(1) Homogeneity and Similarity have better separability than the other texture features

in both datasets.

(2) Although the first four measures have similar definitions, Homogeneity and

Similarity generally provide better separability than Contrast and Dissimilarity, which

is not the initial expectation but is not unreasonable. A comparison between

Dissimilarity and Similarity is illustrated in equations 3.27 and 3.28. Note that the

current quantization level N is 64.

,

, 1

, , ,63 62 1

( )

63 62 ... 0

N

i ji j

i j i j i ji j i j i j i j

DIS P i j

P P P=

− = − = − = =

= −

= + + + +

∑

∑ ∑ ∑ ∑ (3.27)

CON DIS HOM SIM ASM MAX ENT COR0

1

2

3

4

5

6

7

8

Texture Measures

Sep

arab

ility

Lubbock dataNew Orleans data


45

,

, 1

, , ,,

1 62 63

1

...2 63 64

Ni j

i j

i j i j i ji j

i j i j i j i j

PSIM

i jP P P

P

=

= − = − = − =

=+ −

= + + + +

∑

∑ ∑ ∑ ∑ (3.28)

In both equations, terms are arranged in an order of decreasing effect. Dissimilarity

and Contrast are dependent on the presence of large gray-level difference (i-j) between

neighboring pixels. Even if the associated probabilities are relatively small, terms of

large gray-level difference tend to dominate these two texture measures. Similarity

and Homogeneity are dependent on the occurring probabilities of pixel pairs in low

contrast. In other words, the first two measures are sensitive to whether there are some

pixel pairs with VERY HIGH contrast in the window, while the second two measures

are sensitive to whether MOST of the pixel pairs in the window have low contrast. An

example to illustrate the performance difference of the four measures is the rightmost

natural ROI in Fig 3.12b, where the grassland is partitioned by a country road.

Because of the high contrast between the country road pixels and the surrounding

grassland pixels, Contrast and Dissimilarity provide as large values for the grassland

region as for the major residential areas, (Fig. 3.15a and 3.15b). Better than Contrast

and Dissimilarity, Homogeneity and Similarity provide reasonable values for that

grassland region (Fig. 3.15c and 3.15d), because although high contrast exists within

the window, the plain texture still prevails. Based on the separability test, linear

measures are always better than quadratic ones.

(3) The performance of the contrast group is generally better than that of the

orderliness group. Based on our observation, the inferior performance of orderliness

group is because of the fact that neither residential nor natural areas are actually

orderly. Because of the presence of noise in PCA or correlation bands, both types of

area have the probability widely distributed across many entries of co-occurrence

matrix under our current 64 quantization levels, although the residential area has a

visually repetitive pattern, and the natural area has a visually plain pattern.

Experiments have proved that a lower quantization level, such as 32 or 24, will


46

increase the degree of orderliness for both kinds of areas, which however still does no

good to the separability between the two kinds of areas.


47

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 3.14. Texture image of Lubbock data (a) CON; (b) DIS; (c) HOM; (d) SIM; (e)ASM; (f) MAX; (g) ENT; (h) COR.


48

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 3.15. Texture images of New Orleans data (a) CON; (b) DIS; (c) HOM; (d) SIM; (e) ASM; (f) MAX; (g) ENT; (h) COR.

3.3 Feature Selection Among all the 11 features (2 PCA bands, 1 correlation band and 8 texture

measures) extracted, some are highly related to each other and some may be not

contributive for discrimination between residential and natural areas. In order to find

the best combination of the features, K-fold cross validation is performed for the


49

training dataset (Fig. 3.12) for each combination of the features. The top ranked

combinations of Lubbock and New Orleans datasets are demonstrated in Table 3.1 and

3.2. The full tables are listed in the appendix. Note that the cross validation of each

combination is repeated 100 times to minimize the randomness caused by the K-fold

partition. Also, as Co-occurrence COR does not display visual coherence for

residential and natural areas (Fig. 3.14h and 3.15h), it is removed from the pool of

joint features to lessen the computation load. In Table 3.1 and 3.2, features are

arranged in the order: PCA1, PCA2, spectral correlation, CON, DIS, HOM, SIM,

ASM, MAX and ENT. A bit is associated with each feature, and it is set to one if the

feature is selected and zero if it is not. For example, the bit pattern “1000000000”

means that only PCA1 is selected in the combination.

From the tables, rather than one specific combination performing much better

than all the others, many combinations have similar best-level performance, which is

not unexpected. For the Lubbock dataset (Table 3.1): the two PCA bands, at least one

of CON and DIS, and at least one of HOM and SIM are almost always in the top 30

combinations; ASM and MAX are never selected in the top combinations; spectral

correlation and ENT do not show high significance in the top combinations. For the

New Orleans dataset (Table 3.2): the three spectral bands, at least one of HOM and

SIM, and ENT are almost always in the top 30 combinations; ASM and MAX are

never selected in the top combinations; CON and DIS are not significant.

Following the general trend described above, PCA1, PCA2, DIS, HOM, SIM

and ENT are combined as the best feature group for Lubbock dataset; PCA1, PCA2,

spectral correlation, SIM and ENT are combined as the best feature group for the New

Orleans dataset. Note that different training dataset may produce slightly different

ranks.


50

Table 3.2. Cross validation of feature combinations for Lubbock dataset (page 1) Rank Combination Error Rank Combination Error Rank Combination Error Rank Combination Error 1 1100111001 1.88% 56 1011110001 2.40% 111 1101101010 2.79% 166 1000101001 3.10% 2 1111111001 1.90% 57 1111110010 2.40% 112 1110101000 2.80% 167 1100111100 3.10% 3 1101011001 1.97% 58 1111100100 2.41% 113 1111100110 2.80% 168 1011111011 3.11% 4 1101011000 2.01% 59 1111111100 2.42% 114 1011000001 2.81% 169 1100001001 3.12% 5 1101001001 2.02% 60 1011101001 2.43% 115 0111101000 2.82% 170 1001011010 3.12% 6 1101010001 2.04% 61 1001001001 2.43% 116 0111111000 2.82% 171 1010111010 3.12% 7 1110111001 2.05% 62 1110111010 2.43% 117 0111110000 2.83% 172 1110110001 3.13% 8 1101101000 2.07% 63 1001111001 2.44% 118 1010101000 2.83% 173 1100010001 3.14% 9 1111001001 2.08% 64 1111100010 2.45% 119 1101001011 2.84% 174 1010110000 3.14% 10 1111011001 2.10% 65 1000111001 2.46% 120 1110111100 2.84% 175 1011100011 3.14% 11 1011100000 2.10% 66 1001111000 2.46% 121 1101100010 2.84% 176 1011101010 3.14% 12 1101110000 2.11% 67 1001001000 2.46% 122 1111001010 2.84% 177 1110001001 3.15% 13 1001100000 2.13% 68 1111011100 2.47% 123 1111111101 2.85% 178 1110010000 3.16% 14 1100111000 2.13% 69 1110111011 2.48% 124 0111001000 2.87% 179 1110011001 3.17% 15 1111111000 2.13% 70 1001010001 2.48% 125 1101100100 2.90% 180 1100011001 3.17% 16 1101111000 2.15% 71 1011101000 2.49% 126 1101011100 2.92% 181 1001111010 3.18% 17 1111010001 2.15% 72 1010111000 2.50% 127 1100110001 2.93% 182 1110010001 3.19% 18 1110111000 2.16% 73 1011110000 2.51% 128 1111110110 2.93% 183 1001011011 3.20% 19 1101111001 2.16% 74 0111011000 2.52% 129 1101101100 2.94% 184 1001100010 3.20% 20 1101100001 2.17% 75 1111101100 2.53% 130 1111010100 2.95% 185 1111111111 3.22% 21 1111011000 2.17% 76 1101000001 2.54% 131 1111101110 2.95% 186 1111000011 3.24% 22 1101100000 2.18% 77 1111000001 2.54% 132 1101001100 2.96% 187 1010111011 3.24% 23 1010111001 2.19% 78 1111001011 2.55% 133 1101110100 2.96% 188 1101010100 3.24% 24 1011001001 2.21% 79 1001110000 2.58% 134 1110101001 2.96% 189 1101010010 3.25% 25 1011011001 2.21% 80 0111001001 2.58% 135 1111010010 2.97% 190 1000111010 3.26% 26 1111101001 2.21% 81 1111010011 2.59% 136 1101111100 2.97% 191 1111011111 3.26% 27 1011011000 2.23% 82 1001101000 2.61% 137 1111101101 2.97% 192 0111100010 3.27% 28 1011010001 2.23% 83 1001100001 2.61% 138 1111011101 2.98% 193 1001111011 3.27% 29 1111101000 2.24% 84 1101011011 2.61% 139 1101010011 2.98% 194 1010110001 3.28% 30 1111110001 2.25% 85 1001110001 2.62% 140 1100110000 2.99% 195 1001110010 3.30% 31 1001011001 2.26% 86 1001101001 2.62% 141 1011111010 2.99% 196 1001110011 3.32% 32 1111110000 2.26% 87 0111111001 2.62% 142 1010101001 3.00% 197 1000110000 3.33% 33 1111001000 2.27% 88 1101011010 2.63% 143 0111010000 3.00% 198 0110101000 3.34% 34 1111100001 2.28% 89 0111010001 2.63% 144 0110111000 3.00% 199 1001100011 3.34% 35 1111010000 2.28% 90 1001010000 2.64% 145 1010010000 3.01% 200 1001101011 3.34% 36 1101101001 2.29% 91 1111110100 2.64% 146 1111110101 3.02% 201 1011100100 3.35% 37 1111111010 2.30% 92 1000111000 2.65% 147 1100001000 3.03% 202 1001101010 3.37% 38 1101010000 2.31% 93 1101100011 2.68% 148 1110011000 3.03% 203 1010010001 3.37% 39 1111111011 2.31% 94 1100101001 2.68% 149 1111100101 3.03% 204 0110101001 3.37% 40 1111100000 2.31% 95 0111011001 2.69% 150 1010001000 3.05% 205 0111111011 3.37% 41 1101001000 2.32% 96 1101110011 2.69% 151 1011011010 3.05% 206 1010001001 3.40% 42 1011111001 2.33% 97 0111101001 2.70% 152 1000010000 3.05% 207 1000110001 3.40% 43 1011001000 2.33% 98 1111001100 2.70% 153 1100011000 3.05% 208 1110000001 3.41% 44 1101110001 2.33% 99 1101111010 2.70% 154 1011100010 3.07% 209 1111101111 3.41% 45 1111011011 2.34% 100 1101101011 2.71% 155 1001000001 3.07% 210 1110111101 3.42% 46 1111101010 2.34% 101 1101111011 2.72% 156 1110110000 3.07% 211 1111110111 3.42% 47 0111100000 2.37% 102 1100111010 2.73% 157 1011011011 3.08% 212 1000111011 3.42% 48 1111011010 2.37% 103 0110111001 2.73% 158 1000101000 3.08% 213 1100000001 3.43% 49 1011010000 2.38% 104 1101110010 2.75% 159 1100010000 3.08% 214 1110111110 3.44% 50 1111100011 2.38% 105 1100111011 2.76% 160 1101001010 3.09% 215 1010011000 3.44% 51 1001011000 2.38% 106 1100101000 2.76% 161 1011101011 3.09% 216 1011001011 3.45% 52 1011100001 2.38% 107 0111110001 2.77% 162 1110001000 3.09% 217 1110100001 3.45% 53 1011111000 2.38% 108 0111100001 2.79% 163 1011110010 3.10% 218 0111011011 3.46% 54 1111110011 2.38% 109 1111011110 2.79% 164 1000001000 3.10% 219 1111100111 3.46% 55 1111101011 2.38% 110 1111111110 2.79% 165 1011110011 3.10% 220 0111111010 3.46%


51

Table 3.3. Cross validation of feature combinations for New Orleans dataset (page 1) Rank Combination Error Rank Combination Error Rank Combination Error Rank Combination Error 1 1110001001 5.67% 56 1111011100 7.77% 111 0111011000 8.32% 166 1010111000 8.72% 2 1110011001 5.77% 57 1111111011 7.79% 112 1100111000 8.34% 167 0001001001 8.73% 3 1110111000 5.81% 58 0111111001 7.79% 113 0100010001 8.34% 168 1110010110 8.75% 4 1110001000 5.84% 59 0010001001 7.82% 114 1110111110 8.35% 169 1111101110 8.75% 5 1110010001 5.84% 60 1111001100 7.84% 115 1010011001 8.36% 170 1110110110 8.75% 6 1110101000 5.86% 61 0110101001 7.85% 116 0001011001 8.37% 171 1011001001 8.75% 7 1110011000 5.87% 62 1111001011 7.86% 117 0011001001 8.38% 172 1111010110 8.77% 8 1110010000 5.89% 63 1110110100 7.86% 118 0000101001 8.38% 173 0101101001 8.78% 9 1110110000 5.96% 64 1111011010 7.87% 119 0111111000 8.38% 174 0111010011 8.78% 10 1110101001 6.01% 65 1111111100 7.88% 120 1101111001 8.39% 175 0011100001 8.79% 11 1110110001 6.03% 66 1111010011 7.89% 121 1010010001 8.39% 176 1101111000 8.80% 12 1110111001 6.06% 67 0110110001 7.89% 122 0000110001 8.41% 177 0010011000 8.81% 13 1111011000 6.19% 68 0110111000 7.90% 123 0100011001 8.41% 178 0110011011 8.81% 14 1111001000 6.29% 69 0010111001 7.91% 124 0101111001 8.42% 179 1110001110 8.81% 15 1111001001 6.29% 70 1111101010 7.92% 125 1011111001 8.42% 180 1110101101 8.81% 16 1111011001 6.34% 71 1100111001 7.92% 126 0100101001 8.42% 181 0111101011 8.82% 17 1111010000 6.39% 72 0010011001 7.94% 127 1111100011 8.43% 182 1011101001 8.82% 18 1111111000 6.42% 73 1110100100 7.96% 128 1101001001 8.44% 183 0101110001 8.83% 19 1111010001 6.50% 74 1111010100 7.99% 129 0111001000 8.45% 184 1111011101 8.83% 20 1110100001 6.57% 75 0011011001 8.02% 130 1010101001 8.46% 185 0001010001 8.83% 21 1111111001 6.60% 76 0000001001 8.03% 131 0011010001 8.48% 186 0111011100 8.83% 22 1111101000 6.63% 77 0110101000 8.04% 132 1000001001 8.48% 187 0010101000 8.83% 23 1111110001 6.66% 78 0010010001 8.05% 133 1101010001 8.50% 188 1110100000 8.84% 24 1111101001 6.68% 79 0100111001 8.05% 134 1111011110 8.52% 189 0010111011 8.85% 25 1110000001 6.75% 80 0110011000 8.07% 135 1111000100 8.53% 190 0010001100 8.85% 26 1111100001 6.80% 81 1111110010 8.07% 136 0011101001 8.53% 191 0001110001 8.86% 27 1111110000 6.83% 82 0010101001 8.07% 137 0011110001 8.53% 192 1000101001 8.86% 28 1111100000 6.93% 83 1100001001 8.08% 138 0110101011 8.54% 193 1011110001 8.86% 29 1111000001 7.12% 84 1111101011 8.11% 139 0111111011 8.54% 194 0010101100 8.87% 30 1110011010 7.36% 85 0111001001 8.11% 140 1000010001 8.54% 195 1111110110 8.87% 31 1110101010 7.39% 86 1100011001 8.12% 141 1010110001 8.55% 196 0110101100 8.88% 32 0110111001 7.43% 87 1101011001 8.12% 142 1000111001 8.57% 197 0010110000 8.88% 33 1110110010 7.44% 88 0010110001 8.14% 143 1101101001 8.57% 198 1100111011 8.89% 34 1110001100 7.50% 89 0110110000 8.14% 144 0100110001 8.57% 199 1101100001 8.89% 35 1110111011 7.52% 90 1100101001 8.15% 145 1000011001 8.58% 200 0101010001 8.89% 36 1110101011 7.55% 91 1100010001 8.16% 146 0111011011 8.58% 201 0111001100 8.90% 37 1110111100 7.56% 92 1111101100 8.16% 147 0110010000 8.58% 202 1000110001 8.90% 38 0110011001 7.58% 93 1010111001 8.17% 148 0111100001 8.59% 203 0011011000 8.92% 39 1110111010 7.58% 94 0011111001 8.18% 149 1110101110 8.59% 204 0010111100 8.92% 40 0110001001 7.59% 95 1111110011 8.18% 150 1111111110 8.60% 205 0001101001 8.92% 41 1110011011 7.60% 96 1111100100 8.18% 151 1111100010 8.61% 206 1110110101 8.92% 42 1111001010 7.61% 97 1100110001 8.21% 152 1110111101 8.62% 207 1011010001 8.92% 43 1110010100 7.61% 98 0000010001 8.21% 153 1101110001 8.62% 208 0111110011 8.93% 44 1110010010 7.63% 99 0101011001 8.21% 154 0010111000 8.65% 209 0100111000 8.93% 45 1111010010 7.63% 100 1111110100 8.23% 155 1101011000 8.65% 210 0111101000 8.93% 46 1110010011 7.65% 101 1110000100 8.23% 156 0111010000 8.66% 211 1111010101 8.95% 47 1110001010 7.67% 102 0000011001 8.23% 157 1110100010 8.67% 212 1010111100 8.97% 48 1110011100 7.67% 103 0110111011 8.25% 158 0110110011 8.67% 213 0010010100 8.97% 49 1110001011 7.67% 104 0111010001 8.26% 159 0001111001 8.68% 214 0110010011 8.97% 50 1110110011 7.69% 105 0111101001 8.27% 160 0111001011 8.68% 215 1111001101 8.97% 51 1110101100 7.72% 106 1010001001 8.27% 161 0101001001 8.70% 216 0010101011 8.97% 52 0110010001 7.72% 107 0000111001 8.27% 162 0110111100 8.71% 217 1110100011 8.97% 53 1111111010 7.73% 108 0100001001 8.28% 163 1111001110 8.71% 218 0100111011 8.99% 54 0111011001 7.73% 109 1011011001 8.29% 164 1001011001 8.72% 219 1110011101 8.99% 55 1111011011 7.73% 110 0111110001 8.29% 165 1110011110 8.72% 220 1111111101 8.99%


52

3.4 Chapter Conclusion The spectral-spatial feature extraction is explained in this chapter. Normalized

correlation and PCA transform are calculated based on the hyperspectral dataset and

provide two independent sources of spectral features. A hierarchical Fourier transform

– co-occurrence matrix method is designed to help extract the macroscopic spatial

patterns from both residential and natural areas. The direction of spatial repetition is

determined by sweeping the Fourier space and searching the angle of maximum

energy. The distribution of co-occurring gray levels in the direction of spatial

repetition is then calculated. Eight different second order texture measures are

calculated from the co-occurrence matrix and compared. K-fold cross validation is

performed for all of the combinations of the features, and a best joint feature vector is

selected for each dataset.

The comparison of classification results between using the purely spectral

feature, using purely spatial feature, and using the joint feature vector is presented in

the next chapter.


53

CHAPTER IV

CLASSIFICATION AND RESULTS Both the Bayes classification and K-means clustering methods are used

separately to classify the joint feature vectors into residential and natural classes. The

results of classification and comparisons between the joint solution, the spectral-only

solution and the spatial-only solution are discussed in this chapter.

4.1 Supervised Classification

4.1.1 Bayes Classifier The Bayes classifier is based on Bayes decision theory. The likelihood

function of class iω with respect to the variable x , ( | )ip x ω , is assumed to follow

multivariate Gaussian distribution as shown by equation 4.1, where k stands for the

dimensionality of the variable and iµ and iΣ stand for the mean vector and the

covariance matrix of class iω , which can be estimated from the available training

vectors. As the objective function of each class, the posterior probability, ( | )ip xω ,

can be estimated using Bayes Rule as shown by equation 4.2 [45].

11/2/2

1 1( | ) exp[ ( ) ( )]2(2 )

ti i i ik

i

p x x xω µ µπ

−= − − Σ −Σ

(4.1)

( ) ( | )( | )( )

i ii

P p xP xp x

ω ωω = (4.2)

For two-class application, Bayes classification rule can be stated as

1 2 1

1 2 2

if ( | ) ( | ), is classified asif ( | ) ( | ), is classified as

P x P x xP x P x xω ω ωω ω ω

><

.

4.1.2 Classification Results of Bayes Classifier For the Lubbock dataset, the Bayes classification results that used purely

spectral features, purely spatial features and joint features are displayed in Fig. 4.1 and

Table 4.1. For spectral solution, many small-area misclassifications were found within

residential regions, where manmade structure has low concentration and hence the


54

spectral signature is more like that of natural objects; and pixels on and around

country roads were misclassified from the natural class to the residential class. For the

spatial solution, several large-area residential misclassifications were found near Slide

road and the south Loop, where the typical street patterns do not exist. Generally,

according to the error rate, the spectral solution has a better performance for

residential classification while the spatial solution has a better performance for

classifying natural areas.

The joint solution owns the complementary advantages from both the spectral-

only solution and the spatial-only solution: within the residential segmentation, there

is almost no large-area misclassification around the Slide and the south Loop because

of the lack of typical street patterns, and much less small-area misclassification due to

the low concentration of manmade structures; within the natural segmentation,

misclassification around country roads is also lessened.


55

Table 4.1. Bayes classification result of Lubbock dataset

Spectral Solution

Classified as Residential

Classified as Natural Error Rate

Residential Region (include training) 50443 8222 14.20%

Natural Region (include training) 12281 61734 16.59%

Residential Region (exclude training) 48676 8203 14.42%

Natural Region (exclude training) 12182 59308 17.14%

Spatial Solution







Joint

Solution








56

(a) (b) (c)

Figure 4.1. Bayes classification of Lubbock dataset (a) Using purely spectral features; (b) Using purely spatial features; (c) Using joint features. Blue: Correct residential region; Green: Correct natural region; Red: Residential region classified as natural; Magenta: Natural region classified as residential.


57

For the New Orleans dataset, the Bayes classification results using purely

spectral features, purely spatial features and the joint features are displayed in Fig. 4.2

and Table 4.2. Similar to the results of Lubbock dataset, spectral solution is prone to

many small-area misclassifications within the residential segmentation, while spatial

solution made a few large-area misclassifications where periodic spatial texture is not

available. The joint solution generally performs better than both the spectral-only and

the spatial-only solutions again.

Table 4.2. Bayes classification result of New Orleans dataset

Spectral Solution







Spatial Solution







Joint

Solution








58

(a) (b) (c)

Figure 4.2. Bayes classification of New Orleans dataset (a) Using purely spectral features; (b) Using purely spatial features; (c) Using joint features. Blue: Correct residential area; Green: Correct natural and river area; Red: Residential area classified as natural or river; Magenta: Natural or river area classified as residential


59

4.2 Unsupervised Classification The K-means method is applied to cluster the joint feature vectors. A key to

the success of the K-means method is the initial estimation of the number of clusters

(K). Although it is reasonable to force the number as two (residential and natural

clusters), the complexity of data structures sometimes requires more than two clusters

in order to make each cluster compact. As a result, the “Jump method” of model order

estimation is first applied to estimate the number of clusters, and then K-means is

implemented to both joint feature vectors and purely spectral feature vectors by using

the estimated number as the initial value of K. Finally, several sub-clusters are

combined to form two general clusters, residential and natural region.

4.2.1 Model Order Estimation The “true” number of clusters in the dataset is estimated by “Jump method”

described in [46] - [47]. Below is a brief review of the “Jump method”.

The procedure is based on “distortion”, which is a measure of within-cluster

dispersion. Vector x denotes a p-dimensional variable with a mixture distribution of G

components, each with covariance ∑; let 1 2, ,..., Kc c c be a set of candidate cluster

centers, in which xc denotes the closest one to x . The minimum achievable distortion

associated with fitting K centers to the data is then

1

1

,...,

1 min [( ) ( )]K

tK x xc c

d E x c x cp

−= − Σ − , (4.3)

which is simply the average Mahalanobis distance between each sample and its

nearest center. Note that if the covariance matrix ∑ is the identity matrix, distortion is

simply mean squared error (Euclidian distance). In practice, one generally estimates

Kd using ˆKd , the minimum distortion obtained by applying the K-means clustering

algorithm to the observed data.

Although a natural and simple idea is to find the value of K associated with the

minimum of Kd , the concept is unsuccessful because the distortion function Kd is


60

proved to be nonincreasing convex with regard to K. However, when transformed to

an appropriate negative power, the distortion curve will exhibit a sharp jump at the

“true” number of clusters.

For a single p-dimensional Gaussian variable x , let pK k = , where k can be

any positive number and is the floor operator. It was proved in [39] that:

2lim Kpd k −

→∞= . (4.4)

Because k is the pth root of the number of centers, K, the following relationship holds

approximately for a large enough p:

/2p pKd k K− ∝ = , (4.5)

which means that the transformed distortion function of a single Gaussian distribution

is approximately linear to K. This distribution is observed for even relatively low

values of p.

In the case that the distribution of x is a mixture of G Gaussian clusters with

equal probability and common covariance, and if the clusters do not suffer severe

overlapping, it can be proved that:

2

,lim

,Kp

K Gd

k K G−→∞

∞ <= ≥

. (4.6)

If Kd is raised to the power of 2p

− , equation 4.6 becomes:

/20,

,p

K

K Gd Ka K G

G

−<

≈ ≥

, where 0 1a< < . (4.7)

Based on equation 4.7, the transformed distortion should theoretically be two

pair-wise lines, the first one with a slope of 0 and the second one with a slope

proportional to 1/G. A major jump is expected to happen at K=G. The “true” number


61

of clusters can be located by the application of a first order forward difference

operator as in

/2 /21

ˆ ˆarg max[ ]p pK K

KG d d− −

−= − . (4.8)

For more details about the “jump method” and some modifications to improve the

accuracy and computing speed, the reader is referred to [46] – [48].


62

(a)

(b)

Figure 4.3. Results of cluster number estimation (a) Purely spectral features for New Orleans data. (b) Joint features for New Orleans data.

1 2 3 4 5 6 7 8 9 1030

40

50

60

70

80

90Estimated Number of Clusters (Kest = 3)

Number of Clusters Tested

JUM

P Va

lue

1 2 3 4 5 6 7 8 9 1015

20

25

30Estimated Number of Clusters (Kest = 4)

Number of Clusters Tested

JUM

P V

alue


63

4.2.2 K-means Clustering K-means clustering is a method of cluster analysis that partitions N

observations into K clusters in which each observation belongs to the cluster with the

nearest mean [49]. Given a set of feature vectors ( 1 2, ,... nx x x ), K-means clustering

partitions the N observations into K sets (K<N), 1 2{ , ,... }KS S S S= , in order to

minimize the within-cluster sum of squares:

2

1arg min

j i

K

j ii x S

x µ= ∈

−∑ ∑ , (4.9)

where iµ is the centroid of points in the set iS .

4.2.3 Clustering Results of K-means In the purely spectral solution, the initial value of K is set to 3, based on the

Jump estimation, and the resulting three clusters can be broadly interpreted as

residential regions, rural regions, and the rivers, as shown in Fig. 4.4a. The rural

cluster and the river cluster are then merged into a more general natural cluster. For

the joint solution, the initial value of K is set to 4, based on the Jump optimization, and

the resulting four clusters can be generally interpreted as dark residential areas

(mixture of vegetation and some construction materials), bright construction areas

(saturation caused by some materials), rural areas, and the river, as shown in Fig. 4.5a.

The dark residential cluster and the bright construction cluster are then merged into a

more general residential cluster, and the rural cluster and the river cluster are merged

into a second more general natural cluster. The comparison of both results against the

reference image is shown in Fig. 4.4b and Fig. 4.5b. Although, for purely spectral

features, the result of K-means are worse than that of Bayes classification, K-means

and Bayes classification have comparable error rates for joint features. This result is

promising, because if K-means can always produce results comparable to Bayes

classification by adding texture features for other datasets, the unsupervised clustering

does not require training data, which can be quite subjective.

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http://en.wikipedia.org/wiki/Mean�


64

(a) (b)

Figure 4.4. K-means clustering result by using purely spectral features (a) The three clusters found by K-means. Blue: predominantly comprised of residential areas; Green: predominantly comprised of soil areas; Red: predominantly comprised of river. (b) Comparison with the reference after combining the soil and river into a general natural cluster. Blue: correct residential area; Green: correct natural area; Red: residential area classified as natural; Magenta: natural area classified as residential.


65

(a) (b)

Figure 4.5. K-means clustering result by using joint solution (a) The four clusters found by K-means. Blue: primarily comprised of residential areas; Green: primarily comprised of soil areas; Red: primarily comprised of saturated residential pixels; Magenta: primarily comprised of river. (b) Comparison with the reference after combining the soil and river into a general natural cluster, and residential and saturated residential pixels into a general residential cluster. Blue: correct residential area; Green: correct natural area; Red: residential area classified as natural; Magenta: natural area classified as residential.


66

Table 4.3. K-means clustering result of New Orleans dataset Spectral Solution Residential Area Natural Area

Classified as Residential 52718 8064

Classified as Natural 14997 33441

Error Rate 22.15% 19.43%

Joint Solution Residential Area Natural Area

Classified as Residential 62510 11476

Classified as Natural 5205 30029

Error Rate 7.69% 27.65%

4.3 Chapter Conclusion The comparison of classification results between using joint features, using

purely spectral features and using purely spatial features is presented in this chapter.

Both supervised classification (Bayes classifier) and unsupervised classification (K-

means clustering) are implemented. For Bayes classification, the joint solution

generally has preferable performance for classifying both residential and natural areas

to the purely spectral solutions. The K-means clustering result of the joint solution is

also better than that of purely spectral solutions and comparable to the performance of

supervised classification, which indicates that adding the texture features can improve

the separability between residential and natural data so that training before

classification can be avoided, at least for some datasets.


67

CHAPTER V

CONCLUSIONS AND FUTURE WORK In this thesis, a joint spectral-spatial feature extraction method is developed for

residential structure detection from low spatial-resolution Hyperion hyperspectral

datasets.

Chapter I reviewed the current hyperspectral image processing methods,

especially for purely spectral algorithms. In Chapter II, preprocessing including

destriping and atmospheric compensation is introduced. The design of joint features is

presented in Chapter III. Both normalized correlation and the most significant PCA

bands are used as spectral features (Section 3.1). The Hierarchical Fourier transform –

co-occurrence matrix method is developed in Section 3.2. The Fourier transform is

used to detect the direction of the dominant spatially repetitive pattern, and the co-

occurrence matrix is subsequently calculated by using the detected angle. Eight

different texture measures are extracted from the co-occurrence. All the combinations

of the joint features are evaluated by K-fold cross validation, and the best combination

of each dataset (containing PCA1, PCA2, DIS, HOM, SIM and ENT for Lubbock

dataset; containing PCA1, PCA2, spectral correlation, SIM and ENT for New Orleans

dataset) is selected. In Chapter IV, both Bayes classification and K-means clustering

are implemented on Lubbock and New Orleans datasets. The fact that the joint

solution generally performs better than the purely spectral and the purely spatial

solutions proves that merging spectral information with macroscopic texture

information is beneficial for the classification. More testing and verification on

additional datasets are required in the future.

Based on the results in Chapter IV, three conclusions can be received:

(1) Improved accuracy in classification between residential and natural areas was

achieved by using both spectral and macroscopic spatial information.

(2) Improved accuracy in unsupervised clustering can be also achieved by adding

spatial features.


68

The segmentations of residential and natural areas may have, but are not

limited to, the following two applications:

(1) Comparing the segmentations of residential areas before and after a hurricane in

the same position may be an effective method to analyze the disaster effect on the

urban region, as the street structures may be destroyed after a severe hurricane.

(2) The segmentations of residential and natural areas can be used for model choice in

spectral unmixing. For pixels within the segmentation of residential areas, models

containing construction materials will have higher likelihood to use for unmixing; for

pixels within the natural segmentation, models not containing construction materials

will have higher likelihood of use. The sub-pixel classification may be able to interpret

the hurricane effect on the city.


69

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73

APPENDIX The full tables of the cross validation of feature combinations are listed in this

appendix.


74

Table 3.2. Cross validation of feature combinations for Lubbock dataset (page 1)

Rank Combination Error Rank Combination Error Rank Combination Error Rank Combination Error 1 1100111001 1.88% 56 1011110001 2.40% 111 1101101010 2.79% 166 1000101001 3.10% 2 1111111001 1.90% 57 1111110010 2.40% 112 1110101000 2.80% 167 1100111100 3.10% 3 1101011001 1.97% 58 1111100100 2.41% 113 1111100110 2.80% 168 1011111011 3.11% 4 1101011000 2.01% 59 1111111100 2.42% 114 1011000001 2.81% 169 1100001001 3.12% 5 1101001001 2.02% 60 1011101001 2.43% 115 0111101000 2.82% 170 1001011010 3.12% 6 1101010001 2.04% 61 1001001001 2.43% 116 0111111000 2.82% 171 1010111010 3.12% 7 1110111001 2.05% 62 1110111010 2.43% 117 0111110000 2.83% 172 1110110001 3.13% 8 1101101000 2.07% 63 1001111001 2.44% 118 1010101000 2.83% 173 1100010001 3.14% 9 1111001001 2.08% 64 1111100010 2.45% 119 1101001011 2.84% 174 1010110000 3.14% 10 1111011001 2.10% 65 1000111001 2.46% 120 1110111100 2.84% 175 1011100011 3.14% 11 1011100000 2.10% 66 1001111000 2.46% 121 1101100010 2.84% 176 1011101010 3.14% 12 1101110000 2.11% 67 1001001000 2.46% 122 1111001010 2.84% 177 1110001001 3.15% 13 1001100000 2.13% 68 1111011100 2.47% 123 1111111101 2.85% 178 1110010000 3.16% 14 1100111000 2.13% 69 1110111011 2.48% 124 0111001000 2.87% 179 1110011001 3.17% 15 1111111000 2.13% 70 1001010001 2.48% 125 1101100100 2.90% 180 1100011001 3.17% 16 1101111000 2.15% 71 1011101000 2.49% 126 1101011100 2.92% 181 1001111010 3.18% 17 1111010001 2.15% 72 1010111000 2.50% 127 1100110001 2.93% 182 1110010001 3.19% 18 1110111000 2.16% 73 1011110000 2.51% 128 1111110110 2.93% 183 1001011011 3.20% 19 1101111001 2.16% 74 0111011000 2.52% 129 1101101100 2.94% 184 1001100010 3.20% 20 1101100001 2.17% 75 1111101100 2.53% 130 1111010100 2.95% 185 1111111111 3.22% 21 1111011000 2.17% 76 1101000001 2.54% 131 1111101110 2.95% 186 1111000011 3.24% 22 1101100000 2.18% 77 1111000001 2.54% 132 1101001100 2.96% 187 1010111011 3.24% 23 1010111001 2.19% 78 1111001011 2.55% 133 1101110100 2.96% 188 1101010100 3.24% 24 1011001001 2.21% 79 1001110000 2.58% 134 1110101001 2.96% 189 1101010010 3.25% 25 1011011001 2.21% 80 0111001001 2.58% 135 1111010010 2.97% 190 1000111010 3.26% 26 1111101001 2.21% 81 1111010011 2.59% 136 1101111100 2.97% 191 1111011111 3.26% 27 1011011000 2.23% 82 1001101000 2.61% 137 1111101101 2.97% 192 0111100010 3.27% 28 1011010001 2.23% 83 1001100001 2.61% 138 1111011101 2.98% 193 1001111011 3.27% 29 1111101000 2.24% 84 1101011011 2.61% 139 1101010011 2.98% 194 1010110001 3.28% 30 1111110001 2.25% 85 1001110001 2.62% 140 1100110000 2.99% 195 1001110010 3.30% 31 1001011001 2.26% 86 1001101001 2.62% 141 1011111010 2.99% 196 1001110011 3.32% 32 1111110000 2.26% 87 0111111001 2.62% 142 1010101001 3.00% 197 1000110000 3.33% 33 1111001000 2.27% 88 1101011010 2.63% 143 0111010000 3.00% 198 0110101000 3.34% 34 1111100001 2.28% 89 0111010001 2.63% 144 0110111000 3.00% 199 1001100011 3.34% 35 1111010000 2.28% 90 1001010000 2.64% 145 1010010000 3.01% 200 1001101011 3.34% 36 1101101001 2.29% 91 1111110100 2.64% 146 1111110101 3.02% 201 1011100100 3.35% 37 1111111010 2.30% 92 1000111000 2.65% 147 1100001000 3.03% 202 1001101010 3.37% 38 1101010000 2.31% 93 1101100011 2.68% 148 1110011000 3.03% 203 1010010001 3.37% 39 1111111011 2.31% 94 1100101001 2.68% 149 1111100101 3.03% 204 0110101001 3.37% 40 1111100000 2.31% 95 0111011001 2.69% 150 1010001000 3.05% 205 0111111011 3.37% 41 1101001000 2.32% 96 1101110011 2.69% 151 1011011010 3.05% 206 1010001001 3.40% 42 1011111001 2.33% 97 0111101001 2.70% 152 1000010000 3.05% 207 1000110001 3.40% 43 1011001000 2.33% 98 1111001100 2.70% 153 1100011000 3.05% 208 1110000001 3.41% 44 1101110001 2.33% 99 1101111010 2.70% 154 1011100010 3.07% 209 1111101111 3.41% 45 1111011011 2.34% 100 1101101011 2.71% 155 1001000001 3.07% 210 1110111101 3.42% 46 1111101010 2.34% 101 1101111011 2.72% 156 1110110000 3.07% 211 1111110111 3.42% 47 0111100000 2.37% 102 1100111010 2.73% 157 1011011011 3.08% 212 1000111011 3.42% 48 1111011010 2.37% 103 0110111001 2.73% 158 1000101000 3.08% 213 1100000001 3.43% 49 1011010000 2.38% 104 1101110010 2.75% 159 1100010000 3.08% 214 1110111110 3.44% 50 1111100011 2.38% 105 1100111011 2.76% 160 1101001010 3.09% 215 1010011000 3.44% 51 1001011000 2.38% 106 1100101000 2.76% 161 1011101011 3.09% 216 1011001011 3.45% 52 1011100001 2.38% 107 0111110001 2.77% 162 1110001000 3.09% 217 1110100001 3.45% 53 1011111000 2.38% 108 0111100001 2.79% 163 1011110010 3.10% 218 0111011011 3.46% 54 1111110011 2.38% 109 1111011110 2.79% 164 1000001000 3.10% 219 1111100111 3.46% 55 1111101011 2.38% 110 1111111110 2.79% 165 1011110011 3.10% 220 0111111010 3.46%


75

Table 3.2. (cont.) Cross validation of feature combinations for Lubbock dataset (page 2)



76




77




78


Rank Combination Error Rank Combination Error Rank Combination Error Rank Combination Error 881 0010001010 9.87% 917 0001010110 10.5% 953 0010000001 11.5% 989 0000100100 12.1% 882 0010011011 9.87% 918 0000010010 10.6% 954 0001000100 11.5% 990 0010100100 12.2% 883 0011100110 9.88% 919 0100100111 10.6% 955 0010101111 11.7% 991 0000011101 12.2% 884 0011110111 9.88% 920 0100100101 10.6% 956 0010110110 11.7% 992 0000100011 12.2% 885 0000001010 9.90% 921 0011000001 10.6% 957 0010101101 11.7% 993 0010110101 12.2% 886 0000010011 9.92% 922 0011100101 10.6% 958 0000001111 11.8% 994 0000100010 12.2% 887 0010101010 9.93% 923 0001001101 10.6% 959 0011000110 11.8% 995 0000000100 12.2% 888 0001011101 9.94% 924 0011001111 10.6% 960 0000110110 11.8% 996 0010011101 12.2% 889 0001001110 9.94% 925 0001001111 10.6% 961 0010001111 11.8% 997 0011000101 12.3% 890 0001110111 9.95% 926 0001100101 10.7% 962 0010001110 11.8% 998 0010100010 12.3% 891 0101000111 9.96% 927 0000101100 10.7% 963 0000101111 11.8% 999 0001000101 12.3% 892 0100010111 9.99% 928 0100000111 10.7% 964 0000010110 11.8% 1000 0000110101 12.4% 893 0010011010 10.0% 929 0000001100 10.7% 965 0010010110 11.8% 1001 0010100111 12.4% 894 0100110111 10.0% 930 0100000101 10.7% 966 0000001110 11.8% 1002 0010100000 12.4% 895 0011001110 10.0% 931 0010101100 10.8% 967 0010011111 11.8% 1003 0000100111 12.5% 896 0010111101 10.0% 932 0010001100 10.8% 968 0000101101 11.9% 1004 0000100000 12.5% 897 0011101101 10.1% 933 0011010111 10.9% 969 0000011111 11.9% 1005 0010000110 12.7% 898 0001101101 10.1% 934 0001000001 10.9% 970 0010110111 11.9% 1006 0010000111 12.7% 899 0000111101 10.1% 935 0000110010 10.9% 971 0001000110 11.9% 1007 0000000111 12.7% 900 0101000101 10.1% 936 0000010100 10.9% 972 0011000111 11.9% 1008 0010100110 12.8% 901 0000110011 10.1% 937 0000100001 10.9% 973 0010010111 11.9% 1009 0000000110 12.8% 902 0000101010 10.1% 938 0101000010 11.0% 974 0000000001 11.9% 1010 0000100110 12.9% 903 0011100111 10.1% 939 0001010111 11.0% 975 0000110111 11.9% 1011 0010100101 13.0% 904 0000011011 10.1% 940 0000011100 11.0% 976 0010001101 11.9% 1012 0010000101 13.0% 905 0001100111 10.2% 941 0010010100 11.1% 977 0000010111 11.9% 1013 0000100101 13.1% 906 0011110101 10.2% 942 0010100001 11.1% 978 0010011110 12.0% 1014 0000000101 13.2% 907 0001110101 10.2% 943 0011010101 11.1% 979 0001000111 12.0% 1015 0011000010 13.2% 908 0010111111 10.3% 944 0010011100 11.1% 980 0010000011 12.0% 1016 0101000000 13.7% 909 0000111111 10.3% 945 0011000011 11.1% 981 0000001101 12.0% 1017 0001000010 14.7% 910 0100100110 10.4% 946 0001010101 11.1% 982 0100000010 12.0% 1018 0010000010 15.3% 911 0100000110 10.4% 947 0010110100 11.1% 983 0010000100 12.0% 1019 0000000010 15.8% 912 0000011010 10.4% 948 0000110100 11.1% 984 0010010101 12.1% 1020 0011000000 16.1% 913 0010010010 10.4% 949 0011000100 11.4% 985 0010100011 12.1% 1021 0001000000 19.4% 914 0010110010 10.4% 950 0001000011 11.4% 986 0000011110 12.1% 1022 0010000000 24.4% 915 0011010110 10.5% 951 0010101110 11.4% 987 0000000011 12.1% 1023 0100000000 27.2% 916 0011001101 10.5% 952 0000101110 11.4% 988 0000010101 12.1%


79

Table 3.3. Cross validation of feature combinations for New Orleans dataset (page 1)



80

Table 3.3. (cont.) Cross validation of feature combinations for New Orleans dataset (page 2)



81




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83


Rank Combination Error Rank Combination Error Rank Combination Error Rank Combination Error 881 0111000110 11.9% 917 1100000011 12.7% 953 1100000101 13.6% 989 1100100000 15.7% 882 1101100111 11.9% 918 0100100011 12.7% 954 0010000111 13.6% 990 0100100000 16.2% 883 0101100111 11.9% 919 0110100000 12.7% 955 0000000110 13.6% 991 0110000000 16.4% 884 1101000100 11.9% 920 0111000101 12.7% 956 0000000011 13.6% 992 1100100010 16.5% 885 0100100100 12.0% 921 1101000110 12.8% 957 0100100101 13.6% 993 1010100000 16.6% 886 0110000110 12.0% 922 0110100111 12.8% 958 0100000101 13.6% 994 0100100010 16.8% 887 0000100100 12.0% 923 1010000011 12.8% 959 1010000101 13.6% 995 0111000000 16.9% 888 0010100110 12.0% 924 0001000001 12.8% 960 1001000110 13.6% 996 0011000010 17.1% 889 1000001111 12.1% 925 1010100101 12.8% 961 1101000101 13.7% 997 0000000010 17.2% 890 0000001111 12.1% 926 0010100101 12.8% 962 1010000111 13.7% 998 1011000010 17.2% 891 0100000100 12.1% 927 1010000110 12.8% 963 0001000110 13.7% 999 0010000010 17.5% 892 1111000111 12.1% 928 1100000110 12.8% 964 1001000011 13.8% 1000 0010100000 17.6% 893 1000000001 12.2% 929 1001000100 12.9% 965 0001000011 13.8% 1001 1010000010 17.9% 894 0000000100 12.2% 930 0111000111 13.0% 966 1100000111 13.8% 1002 1100000010 18.2% 895 1100100011 12.2% 931 0100000011 13.0% 967 0101000101 13.8% 1003 0100000010 18.3% 896 1110000111 12.2% 932 0110000111 13.0% 968 0100000111 13.8% 1004 0000100010 18.4% 897 0001100111 12.2% 933 0100100110 13.1% 969 1100100111 13.8% 1005 1101000010 18.5% 898 0011000011 12.3% 934 1010100111 13.1% 970 0100100111 13.8% 1006 1000100010 18.7% 899 0001100010 12.3% 935 1001000001 13.1% 971 1101000111 13.9% 1007 0101000010 19.1% 900 1001100111 12.3% 936 0010100111 13.1% 972 0101000111 13.9% 1008 0001000010 21.0% 901 0101000100 12.3% 937 0101000011 13.1% 973 0111000010 14.1% 1009 1000000010 21.6% 902 1011000110 12.4% 938 0110100010 13.1% 974 0110000010 14.1% 1010 1000100000 21.7% 903 0010000011 12.4% 939 1000100011 13.1% 975 0000100101 14.1% 1011 1001000010 22.2% 904 0110100101 12.4% 940 0000100011 13.2% 976 1000100101 14.2% 1012 1011000000 24.0% 905 1000100100 12.4% 941 0011000101 13.2% 977 0000100111 14.2% 1013 0011000000 24.6% 906 0011000110 12.5% 942 1000100110 13.2% 978 1000000101 14.2% 1014 0100000000 26.4% 907 1011000011 12.5% 943 0100000110 13.3% 979 1000100111 14.2% 1015 0101000000 26.5% 908 0001000100 12.5% 944 0101000110 13.3% 980 0000000111 14.2% 1016 0000100000 26.6% 909 1001100010 12.6% 945 0000100110 13.3% 981 0001000111 14.2% 1017 1010000000 27.0% 910 0110000101 12.6% 946 0011000111 13.4% 982 1001000111 14.3% 1018 1100000000 27.0% 911 1001100000 12.6% 947 1011000101 13.4% 983 0000000101 14.3% 1019 0001000000 27.7% 912 0001100000 12.6% 948 1011000111 13.4% 984 1000000111 14.3% 1020 1101000000 28.1% 913 1000000100 12.6% 949 0010000101 13.4% 985 0001000101 14.3% 1021 0010000000 28.9% 914 1101000011 12.7% 950 1000000011 13.5% 986 1001000101 14.4% 1022 1001000000 31.8% 915 0010000110 12.7% 951 1000000110 13.5% 987 1010100010 15.3% 1023 1000000000 39.8% 916 1100100110 12.7% 952 1100100101 13.5% 988 0010100010 15.3%

Copyright 2010, Lin Cong

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