Lecture 3 Intensity Transformations and Spatial Filtering 03...Lin ZHANG, SSE, 2016 Lecture 3...

Lin ZHANG, SSE, 2016

Lecture 3Intensity Transformations

and Spatial Filtering

Lin ZHANG, PhDSchool of Software Engineering

Tongji UniversityFall 2016


Outline

• About image enhancement• Some basic intensity transformation functions• Histogram processing• Fundamentals of spatial filtering• Smoothing spatial filters• Sharpening spatial filters


About Image Enhancement

• Image enhancement is the process of making images more useful

• The reasons for doing this include:• Highlighting interesting detail in images• Removing noise from images• Making images more visually appealing


About Image Enhancement—Several Examples

original image result of image enhancement


About Image Enhancement

• Spatial domain VS frequency domain• Spatial domain techniques directly manipulate image pixels

• Frequency domain techniques manipulate Fourier transform or wavelet transform of an image


Outline

• About image enhancement• Some basic intensity transformation functions

• Image negatives• Thresholding• Log transformations• Power‐law (Gamma) transformations• Piecewise‐linear transformation functions

• Histogram processing• Fundamentals of spatial filtering• Smoothing spatial filters• Sharpening spatial filters


Image Negatives

• Most spatial domain enhancement operations can be reduced to the form

where f(x, y) is the input image, g(x, y) is the processed image and T is some operator defined over some neighborhood of (x, y)

( , ) ( , )g x y T f x y


Image Negatives

• Most spatial domain enhancement operations can be reduced to the form

( , ) ( , )g x y T f x yOrigin x

y Image f (x, y)

(x, y)


Image Negatives

• When neighborhood is simply the pixel itself ( , ) ( , )g x y T f x y

( )s T rwhere s refers to the processed image pixel value and rrefers to the original image pixel value. In this case T is referred to as a grey level transformation function or a point processing operation


Image Negatives

• The negative of an image with intensity levels [0, L‐1] is

• Used for enhancing white or gray details embedded in dark regions, especially when the black areas are dominant in size

( ) 1s T r L r

An X‐ray image of two hands (a) Negative image of (a)


Image Negatives

• The negative of an image with intensity levels [0, L‐1] is

• Used for enhancing white or gray details embedded in dark regions, especially when the black areas are dominant in size

( ) 1s T r L r

Implementation Tips

1. You can write your own code since it is straightforward2. Or you can use the routine imcomplement3. Or you can use the routine imadjust


Thresholding

• Thresholding transformations are particularly useful for segmentation in which we want to isolate an object of interest from a background

1,0,

r thresholds

otherwise

An X‐ray image of two hands (a) Thresholding result of (a)


Thresholding

• Thresholding transformations are particularly useful for segmentation in which we want to isolate an object of interest from a background

Implementation Tips

1. You can write your own code since it is straightforward2. Or you can use the routine im2bw

1,0,

r thresholds

otherwise


Log Transformations

• The general form of the log transformation is s = c * log(1 + r), where c is a constant, and r>=0

• It can expand the values of dark pixels in an image while compressing the higher level values

• It compresses the dynamic range of images with large variations in pixel values (such as Fourier spectrum)

(a) original image; (b) Fourier magnitude of (a) in linear scale; (c) Fourier magnitude of (a) shown in log scale

(a) (b) (c)


Power‐Law (Gamma) Transformations

• Power law transformations have the following form

• Power‐law curves with fractional values of map a narrow range of dark input values to a wider range of output values, with the opposite being true for higher values of input levels. Varying gives a whole family of curves.

s crwhere c and are positive constants



• Power law transformations have the following forms cr

where c and are positive constants

Plots of for various values of . All curves were scaled to fit in the range shown.

s cr



• Power law transformations have the following forms cr

where c and are positive constants

Implementation Tips

1. You can write your own code since it is straightforward2. Or you can use the routine imadjust



• Power law transformations are useful for general‐purpose contrast manipulation

s = r 0.6

s = r 0.4



s = r 3.0

s = r 4.0


Another Contrast Stretching Function

• The following function is another commonly used contrast stretching function

1

1 / Esm r

where m and E are two parameters

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

m


Another Contrast Stretching Function—An Exampleoriginal input constrast stretched result


Piece‐wise Linear Transformations

• The advantage of piece‐wise linear functions is that its form can be arbitrarily complex; the disadvantage is that their specification requires more user input

• Three usually used piece‐wise linear transformations• Contrast stretching• Intensity‐level slicing• Bit‐plane slicing



• Contrast stretching is a process that expands the range of intensity levels in an image so that it spans the full intensity range of the recording medium or display device

Input image Form of piece‐wise function Output image



• Intensity level slicing highlights a specific range of intensities in an image

(a) This transformation highlights intensity range [A, B] and reduces all the other intensities to a lower level

(b) This transformation highlights intensity range [A, B] and preserves all the other intensities



• Intensity level slicing highlights a specific range of intensities in an image

(a) Aortic angiogram; (b) result of using a slicing transformation of the type (a) (previous slide); (c) result of using a slicing transformation of the type (b) (previous slide);

(a) (b) (c)



• Bit‐plane slicing isolates particular bits of the pixel values and can highlight interesting aspects of that image• Higher‐order bits usually contain most of the significant visual information

• Lower‐order bits contain subtle details


Bit‐plane slicing—an example

Image of our school



bp1bp0

bp2 bp3 bp4

bp5 bp6 bp7



Original image reconstruction by highest 3 bitplanes

reconstruction by highest 4 bitplanes reconstruction by highest 5 bitplanes



• Bit‐plane slicing• Decomposing an image into its bitplanes is useful for analyzing the relative importance of each bit in the image, a process that aids in determining the adequacy of the number of bits used to quantize the image

• Such a decomposition is also useful for compression


Outline

• About image enhancement• Some basic intensity transformation functions• Histogram processing

• A taste of probability• Introduction of histogram• Histogram computation• Properties of the histogram• Histogram equalization• Using histogram statistics for image enhancement

• Fundamentals of spatial filtering• Smoothing spatial filters• Sharpening spatial filters


I and my histogram!


If is continuous at x, we have

A Taste of Probability

• Probability density function (PDF)• PDF f(x) of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value

( ) 0, ( ) 1f x f t dt

It satisfies the following requirements,

(1)

2

11 2 2 1( ) ( ) ( ) ( )

x

xP x X x F x F x f t dt (2)

(3) ( )f x ' ( ) ( )F x f xwhere F(x) is the cumulative distribution function of X



• Cumulative distribution function (CDF)• F(x) describes the probability that a real‐valued random variable X with a given probability distribution will be found at a value less than or equal to x.

( ) ( ) ( )x

P X x F x f t dt

( ) ( ) ( ) ( )

b

aP a X b F b F a f t dt



• PDF of the uniform distribution1 ,

( )0,

a x bf x b a

otherwise

Corresponding CDF?



• PDF of the Gaussian (Normal) distribution2

2( )

21( ) ,2

x

f x e x

where and are constants, and 0

Johann Carl Friedrich Gauss (30 April 1777 – 23 February 1855) was a German mathematician and physical scientist who contributed significantly to many fields



• PDF of the Gaussian (Normal) distribution2

2( )

21( ) ,2

x

f x e x

where and are constants, and 0



• Expectation of the random variable X with f(x) as its probability density function

( ) ( )E X xf x dx

It can be regarded as a weighted mean or average

Theorem If the random variable Y is the function of the random variable X, Y=g(X), its expectation is

( ) ( ( )) ( ) ( )E Y E g X g x f x dx



• Variance• The variance is a measure of how far a set of numbers is spread out

2

2

( ) ( )

( ( )) ( )

D X E X E X

x E X f x dx

Variance of a random variable X is defined as

( )D X is called the standard deviation of X



• Variance

22( ) ( ) ( )D X E X E X Theorem

Proof:

2

22

22

22

( ) [ ( )]

2 ( ) ( )

( ) 2 ( ) ( ) ( )

( ) ( )

D X E X E X

E X XE X E X

E X E X E X E X

E X E X



• Expectation and variance of Gaussian distribution

If , that means 2~ ( , )X N 2

2( )

21( )2

x

f x e

Then,

2

( )( )

E XD X

How to prove it? Assignment



• Covariance• covariance is a measure of how much two random variables change together

cov( , ) ( ) ( )X Y E X E X Y E Y X and Y are two random variables, their covariance is defined as

Correlation coefficient

cov( , )( ) ( )XY

X YD X D Y


Introduction of Histogram

• The histogram h of an image represents the frequency with which different grey‐levels occur

• Suppose f is an image represented by a total of b bits, in which case . Then h(x) is equal to the number of pixels in f, whose value is x

: [0,2 1]bf


Introduction of HistogramExamplef is a 3‐bit 44 image

0 31 2 4 5 6 7intensity levels

pixel occurrence5 5 3 3

33340 61

77 2 2

2


Introduction of HistogramExample

0 31 2 4 5 6 7intensity levels

pixel occurrence

1 1

3

5

1 122

2 1

0

( )b

x

h x area of the image

f is a 3‐bit 44 image

5 5 3 33334

0 61

77 2 2

2


Histogram Computation

Algorithm: hist_comp (im) //im is a b-bit image1. Assign zeros to all elements of the array

[ ] : [0,2 1]bh i i

2. Loop all the pixels (x, y) of the image im, and

[ ( , )] [ ( , )] 1h im x y h im x y

Implementation Tips

1. You can use the routine imhist


Properties of the Histogram

• h(x) does not depend on the locations of image pixels. Thus, it is impossible to reproduce f from its histogram.

• Usually the histogram is normalized by the area of the image (number of pixels). This gives the histogram an interpretation of a (discrete) probability density function (PDF) with properties:

1

0

0 , ( ) 0, ( ) 1L

x

x L h x h x

where L is the number of possible intensity levels


Contrast difference?

Properties of the Histogram


Histogram Equalization

• Motivation• An image whose pixels tend to occupy the entire range of possible intensity levels and tend to be distributed uniformly, will have an appearance of high contrast



• Motivation

f T(f)Intensity transformation T

0 L‐1intensity levels

occurrence probability

0 L‐1intensity levels

occurrence probability

11L

T

• T can make f have a uniform histogram• Applying T to f is called the histogram equalization of f

How to find such a T?


Histogram EqualizationLet’s consider the continuous case first for simplicity

Take the intensity values of f as a random variable X

Its normalized histogram is the PDF of X( )Xh x

Such a T:

0( ) ( 1) ( )

X

XY T X L h t dt can satisfy the requirements How to

verify?


Histogram Equalization( )Y T X

0

1( ) ( ) ( )( 1) ( )

1 1( )( 1) ( ) 1

Y X X x

X

XX

dxh y h x h xdy d L h t dt

dx

h xL h x L

That means after applying T to f, the resulting image has a uniform histogram



Let’s consider the discrete case

For a b‐bit image with size m n, the corresponding histogram equalization transform T maps intensity p to

0 0

2 1( ) (2 1)bp p

b kk

k k

nT p nm n m n

where denotes the number of pixels with intensities kkn



• Notes on discrete case for HE• Histogram is an approximation to a PDF, thus, perfectly flat histograms are rare in practical applications

• HE tends to spread the histogram of the input image to a wider range of intensity scale. The net result is contrast enhancement

• HE is a fully automatic procedure; no parameters


Histogram Equalization—Examples

0

500

1000

1500

2000

2500

3000

3500

4000

0 50 100 150 200 250

0

500

1000

1500

2000

2500

3000

3500

4000

0 50 100 150 200 250



0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 50 100 150 200 250

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 50 100 150 200 250



0

1000

2000

3000

4000

5000

6000

7000

8000

0 50 100 150 200 250

0

1000

2000

3000

4000

5000

6000

7000

8000

0 50 100 150 200 250


Summary about Histogram

• Histogram represents the intensity occurrences of a given image

• Histogram is a global descriptor; reflect no structural information; two totally different images can have identical histograms

• Histogram has many potential applications in DIP• Histogram equalization can enhance the image’s contrast


Outline



Fundaments of Spatial Filtering

• Linear VS nonlinear operations

( , ) ( , )H f x y g x y

Consider a general operator H, that produces an output image g(x, y), for a given input image f(x, y):

H is said to be a linear operator if

( , ) ( , )

( , ) ( , )

( , ) ( , )

i i j j

i i j j

i i j j

H a f x y a f x y

a H f x y a H f x y

a g x y a g x y

Otherwise, H is said to be a nonlinear operator



• Linear VS nonlinear operations• Filters can be categorized as linear and nonlinear ones based on the properties of the associated operations



• The mechanics of linear spatial filtering• The filtering result at a specific position is the sum of the products between the filter kernel and the image patch

• Traverse the whole image to get the final result

Given an image f(x, y)

Form an image g(x, y)

g(x, y) is a weighted average of a patch near (x, y) in f . And the weight pattern is called as filter kernel h



• The mechanics of linear spatial filtering

123 127 128 119 115 130

140 145 148 153 167 172

133 154 183 192 194 191

194 199 207 210 198 195

164 170 175 162 173 151

Original Image x

y

Enhanced Image x

y



• The mechanics of linear spatial filteringr s t

u v w

x y z

Origin x

y Image f (x, y)

eprocessed = v*e + r*a + s*b + t*c + u*d + w*f + x*g + y*j + z*i

Filter hSimple 3*3

Neighbourhoode 3*3 Filter

a b c

d e f

g j iOriginal Image

Pixels

*

The above is repeated for every pixel in the original image to generate the filtered image



• The mechanics of linear spatial filtering

( , ) ( , ) ( , )a b

s a t b

g x y h s t f x s y t

Filtering can be given in equation form

where a and b are the half‐width and half‐height

It is referred as correlation. (Moving a filter mask over the image and computing the sum of products at each location)


Fundaments of Spatial Filtering—An Example

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

f1 1 1

1 1 1

1 1 1

h =

g

Mean filter



0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

f

0

1 1 1

1 1 1

1 1 1

h =

g

Mean filter



0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

f

0 10

1 1 1

1 1 1

1 1 1

h =

g

Mean filter



0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

f

0 10 20

1 1 1

1 1 1

1 1 1

h =

g

Mean filter



0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

f

0 10 20 30

1 1 1

1 1 1

1 1 1

h =

g

Mean filter



0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

f

0 10 20 30 30

1 1 1

1 1 1

1 1 1

h =

g

Mean filter



0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

f

0 10 20 30 30 30

1 1 1

1 1 1

1 1 1

h =

g

Mean filter



0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

f

0 10 20 30 30 30 20

1 1 1

1 1 1

1 1 1

h =

g

Mean filter



0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

f1 1 1

1 1 1

1 1 1

h =

g

0 10 20 30 30 30 20 10

0 20 40 60 60 60 40 20

0 30 60 90 90 90 60 30

0 30 50 80 80 90 60 30

0 30 50 80 80 90 60 30

0 20 30 50 50 60 40 20

10 20 30 30 30 30 20 10

10 10 10 0 0 0 0 0

Mean filter



Filtered by

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

h =



• Correlation VS Convolution• Convolution is the same as correlation, except that the filter (or the signal) is first rotated 180 degree (for symmetric filters, they are the same)

• Convolution is an important concept in linear systems (we will revisit it later)

( , ) ( , ) ( , )a b

s a t b


Correlation of h and f

( , ) ( , ) ( , )a b

s a t b


Convolution of h and f



• A vivid example for convolution

0 0 0 0 00 0 0 0 00 0 1 0 00 0 0 0 00 0 0 0 0

Result ?

Do convolution




0 0 0 0 00 0 0 0 00 0 1 0 00 0 0 0 00 0 0 0 0

Identical copyDo convolution




0 0 0 0 00 0 0 0 00 0 0 0 10 0 0 0 00 0 0 0 0

Result ?

Do convolution




0 0 0 0 00 0 0 0 00 0 0 0 10 0 0 0 00 0 0 0 0

Do convolutionShift right by 2 pixel



Implementation Tips for Spatial Filtering

The following Matlab routines are commonly used, fspecial, imfilter, conv2, colfilt, medfilt2, and ordfilt2



From mathematical view, convolution has more good properties than correlation in theory. Thus, in most cases, for theoretical analysis, linear filtering is represented as convolution

For 1D continuous case, convolution is defined as

( )* ( ) ( ) ( )f t h t f t h d



Convolution has the following properties

1. 1 2 2 1( ) * ( ) ( ) * ( )f t f t f t f t

2. 1 2 3 1 2 1 3( ) * ( ) ( ) ( ) * ( ) ( ) * ( )f t f t f t f t f t f t f t

1 2 3 1 2 3( ) * ( ) * ( ) ( ) * ( ) * ( )f t f t f t f t f t f t3.

We will encounter it again in lecture 4 for filtering in the frequency domain


Outline



Smoothing Spatial Filters

• The output of a linear smoothing spatial filter is simply the (weighted) average of the pixels in the neighborhood defined by the filter mask; sharp transitions are reduced

• Linear smoothing filters are low‐pass filters; they will blur edges

• They can reduce irrelevant details in an image and get a gross representation of objects of interest



• Two simple square average filters

1 1 1

1 1 1

1 1 1

19

116

1 2 1

2 4 2

1 2 1



Images smoothed with square averaging filter masks of different sizes



Original Image Smoothed Image Thresholded Image

• Another example for averaging filter• By smoothing the original image we get rid of lots of the finer detail which leaves only the gross features for thresholding



• Gaussian filter• A linear smoothing spatial filter, whose coefficients are determined by a Gaussian function

• It assigns more weight to the positions near the center, and less weight to the positions far away from the center



• Gaussian filter2 2

2 2

1( , ) exp2 2

x yG x y

0.003 0.013 0.022 0.013 0.0030.013 0.059 0.097 0.059 0.0130.022 0.097 0.159 0.097 0.0220.013 0.059 0.097 0.059 0.0130.003 0.013 0.022 0.013 0.003

5 x 5, = 1

where is called standard deviation



• Gaussian filter• Gaussian filters have infinite support, but discrete filters use finite kernels

How to properly select the size of a Gaussian kernel for a given σ?



• Gaussian filter• Gaussian filters have infinite support, but discrete filters use finite kernels

• Rule of thumb: set filter half‐width to about 3σ

2~ ( , ),if X Nthen

( 3 ) 0.9974P X

3σ‐rule from the theory of probability:



• Gaussian filter—an example Gaussian filter used for smoothing



• Gaussian filter—an example



• Gaussian filter—some properties• Remove “high‐frequency” components from the image (low‐pass filter)

• Convolution with itself is another Gaussian• Separable kernel

• Factors into product of two 1D Gaussians



• Separability of the Gaussian filter• The 2D Gaussian can be expressed as the product of two functions

• In this case, the two functions are 1D Gaussian

2 2

2 2

2 2

2 2

1( , ) exp2 2

1 1exp exp2 22 2

x yG x y

x y


Smoothing Spatial Filters—Noise Types

• Salt and pepper noise: contains random occurrences of black and white pixels

• Impulse noise: contains random occurrences of white pixels

• Gaussian noise: variations in intensity drawn from a Gaussian normal distribution

Original

Gaussian noise

Salt and pepper noise

Impulse noise


• Filtering salt‐and‐pepper noise using Gaussian filters

What’s wrong with the results?

3x3 5x5 7x7


Alternative solution: median filters



• Median filter (a nonlinear filter)• It replaces the value of a pixel by the median of the intensity values in the neighborhood of that pixel

• It can be used to remove (salt‐and‐pepper noise)


Smoothing Spatial Filters—Median Filter

• Median filter is robust to outliers



• An exampleSalt-and-pepper noise Median filtered



• Median VS Gaussian filtering

Original image Add salt&pepper noise



• Median VS Gaussian filtering

Median filter3 33 3 Gaussian filter, 0.5


Outline



Sharpening Spatial Filters

• They are used to highlight transitions in intensity• Remove blurring from images• Highlight edges

• Usually, such filters are based on discrete differentiation



• First order derivative• It’s just the difference between subsequent values and measures the rate of change of the function

)()1( xfxfxf



• Second order derivative• Simply takes into account the values both before and after the current value

2

2 ( 1) ( 1) 2 ( )f f x f x f xx



• Laplacian operator• It is based on 2nd‐order partial derivatives• It can be used for highlighting intensity discontinuities in an image

2 22

2 2

f ffx y

The Laplacian is defined as follows:

In discrete case2 ( 1, ) ( 1, ) ( , 1) ( , 1) 4 ( , )f f x y f x y f x y f x y f x y



• Laplacian operator

The Laplacian filter kernel is defined as

0 1 0

1 -4 1

0 1 0

When considering the diagonal dimensions, the Laplacian kernel can be

1 1 1

1 -8 1

1 1 1



• Laplacian operator is rotation invariant, that means2 2 ' '( , ) ( , )f x y f x y

where'

'

cos sinsin cos

x xyy

Proof:

We need to verify2 2 2 2

2 2 '2 '2

f f f fx y x y


'

'

cos sinsin cos

x xyy

From

we can have '

'

cos sinsin cos

x xy y

' '

' '

cos sinsin cos

x x yy x y

' ' ' cos ( sin )f f x f y f fx x x y x x y


2 2 2 2 2

'2 2 ' ' ' 2 '

2 2 2 2

2 2

2 2 22

2

cos ( sin )

cos ( sin ) cos cos ( sin ) ( sin )

cos sin cos sin cos

f f x f y f x f yx x x x y x y x x y x

f f f fx x y y x y

f f fx x y y x

22

2 sinfy

In a similar way, 2 2 2 2 2

2 2'2 2 2sin sin cos sin cos cosf f f f f

y x x y y x y


2 2 2 2

2 2 2 2'2 '2 2 2

2 2

2 2

cos sin sin cosf f f fx y x y

f fx y

Thus, we have

(completed)



• Laplacian filter can be used to sharpen image by2( , ) ( , ) ( , )g x y f x y c f x y

where c is a constant



• An example for Laplacian Sharpening

Input image




Laplacian image




Final result(= original image – 0.8*laplacian image)




Input image Sharpened image


Summary

• In this lecture, we have learned• About image enhancement• Some basic intensity transformation functions• Histogram processing• Fundamentals of spatial filtering• Smoothing spatial filters• Sharpening spatial filters


Thanks for your attention

Lecture 3 Intensity Transformations and Spatial Filtering 03...Lin ZHANG, SSE, 2016 Lecture 3...

Documents

Transcript of Lecture 3 Intensity Transformations and Spatial Filtering 03...Lin ZHANG, SSE, 2016 Lecture 3...