Post on 05-Apr-2018
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important signal features. In the recent years there has been a fair amount of research on
wavelet thresholding and threshold selection for signal de-noising, because wavelet
provides an appropriate basis for separating noisy signal from the image signal. The
motivation is that as the wavelet transform is good at energy compaction, the small
coefficient is more likely due to noise and large coefficient due to important signal
features. These small coefficients can be threshold without affecting the significant features
of the image.
Wavelets are nonlinear functions and do not remove noise by low-pass filtering like
many traditional methods. Low-pass filtering approaches, which are linear time invariant,
can blur the sharp features in a signal and sometimes it is difficult to separate noise from
the signal where their Fourier spectra overlap. For wavelets the amplitude, instead of the
location of the Fourier spectra, differs from that of the noise. This allows for threshold of
the wavelet coefficients to remove the noise. These localizing properties of the wavelet
transform allow the filtering of noise from a signal to be very effective. While linear
methods trade-off suppression of noise for broadening of the signal features, noise
reduction using wavelets allows features in the original signal to remain sharp. This works
very well and even overcomes pseudo-Gibbs phenomena that are often seen due to lack of
shift invariance.
Thresholding is a simple non-linear technique, which operates on one wavelet
coefficient at a time. In its most basic form, each coefficient is thresholded by comparing
against threshold, if the coefficient is smaller than threshold, set to zero; otherwise it is kept
or modified. Replacing the small noisy coefficients by zero and inverse wavelet transform
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on the result may lead to reconstruction with the essential signal characteristics and with
less noise.
This project is implemented on MATLAB. In this project, we first discuss the
features that a practical digital image denoising. Second, we present wavelet-based
denoising algorithm. Experimental results and analyses are then given to demonstrate that
the proposed algorithm is effective and can be used in a practical system.
Acknowledgement
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We are very grateful to our head of the Department of Electronics and communication
Engineering, Mr.-------, ------------ College of Engineering & Technology for having
provided the opportunity for taking up this project.
We would like to express our sincere gratitude and thanks to Mr. ---------Department of
Electronics & Communication Engineering, -------College of Engineering & Technology
for having allowed doing this project.
Special thanks to Deccan Embedded Solutions Pvt. Ltd., for permitting us to do this
project work in their esteemed organization, and also for guiding us through the entire
project.
We also extend our sincere thanks to our parents and friends for their moral support
throughout the project work. Above all we thank god almighty for his manifold mercies
in carrying out the project successfully
CONTENTS
1. Introduction
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1.1 Images in MATLAB
1.2 IMAGE REPRESENTATION
1.3 Digital Image File Types
1.4 Image Coordinate Systems
2.0 Digital Image Processing2.1 Image digitization
2.2 Image Pre-processing
2.3Image Segmentation
3.0 Image Denoising3.1 Introduction to wavelet representation
A. Fourier analysisB. Short-Time Fourier Analysis
C. Wavelet Analysis
4.0 Introduction To Matlab
1. INTRODUCTION
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Image:
A digital image is a computer file that contains graphical information
instead of text or a program. Pixels are the basic building blocks of all digital images.
Pixels are small adjoining squares in a matrix across the length and width of your digital
image. They are so small that you dont see the actual pixels when the image is on your
computer monitor.
Pixels are monochromatic. Each pixel is a single solid color that is blended from
some combination of the 3 primary colors of Red, Green, and Blue. So, every pixel has a
RED component, a GREEN component and BLUE component. The physical dimensions of
a digital image are measured in pixels and commonly called pixel or image resolution.
Pixels are scalable to different physical sizes on your computer monitor or on a photo print.
However, all of the pixels in any particular digital image are the same size. Pixels as
represented in a printed photo become round slightly overlapping dots.
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Pixel Values: As shown in this bitonal image, each pixel is assigned a tonal value,
in this example 0 for black and 1 for white.
PIXEL DIMENSIONS are the horizontal and vertical measurements of an image
expressed in pixels. The pixel dimensions may be determined by multiplying both the
width and the height by the dpi. A digital camera will also have pixel dimensions,
expressed as the number of pixels horizontally and vertically that define its resolution (e.g.,
2,048 by 3,072). Calculate the dpi achieved by dividing a document's dimension into the
corresponding pixel dimension against which it is aligned.
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Fig: Pixel Values in a Binary Image
Grayscale Images:
A grayscale image (also called gray-scale, gray scale, or gray-level) is a data matrix
whose values represent intensities within some range. MATLAB stores a grayscale image
as an individual matrix, with each element of the matrix corresponding to one image pixel.
By convention, this documentation uses the variable name I to refer to grayscale images.
The matrix can be of class uint8, uint16, int16, single, or double. While grayscale
images are rarely saved with a color map, MATLAB uses a color map to display them.
For a matrix of class single or double, using the default grayscale color map, the
intensity 0 represents black and the intensity 1 represents white. For a matrix of type uint8,
uint16, or int16, the intensity intmin (class (I)) represents black and the intensity intmax
(class (I)) represents white.
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The figure below depicts a grayscale image of class double.
Fig: Pixel Values in a Grayscale Image Define Gray Levels
Color Images:
A color image is an image in which each pixel is specified by three values one
each for the red, blue, and green components of the pixel's color. MATLAB store color
images as an m-by-n-by-3 data array that defines red, green, and blue color components for
each individual pixel. Color images do not use a color map. The color of each pixel is
determined by the combination of the red, green, and blue intensities stored in each color
plane at the pixel's location.
Graphics file formats store color images as 24-bit images, where the red, green, and
blue components are 8 bits each. This yields a potential of 16 million colors. The precision
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with which a real-life image can be replicated has led to the commonly used term color
image.
A color array can be of class uint8, uint16, single, ordouble. In a color array of
class single ordouble, each color component is a value between 0 and 1. A pixel whose
color components are (0, 0, 0) is displayed as black, and a pixel whose color components
are (1, 1, 1) is displayed as white. The three color components for each pixel are stored
along the third dimension of the data array. For example, the red, green, and blue color
components of the pixel (10,5) are stored in RGB(10,5,1), RGB(10,5,2), and
RGB(10,5,3), respectively.
The following figure depicts a color image of class double.
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Fig: Color Planes of a True color Image
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Indexed Images:
An indexed image consists of an array and a colormap matrix. The pixel values in
the array are direct indices into a colormap. By convention, this documentation uses the
variable name X to refer to the array and map to refer to the colormap.
The colormap matrix is an m-by-3 array of class double containing floating-point
values in the range [0, 1]. Each row ofmap specifies the red, green, and blue components
of a single color. An indexed image uses direct mapping of pixel values to colormap
values. The color of each image pixel is determined by using the corresponding value of X
as an index into map.
A colormap is often stored with an indexed image and is automatically loaded with
the image when you use the imread function. After you read the image and the colormap
into the MATLAB workspace as separate variables, you must keep track of the association
between the image and colormap. However, you are not limited to using the default
colormap--you can use any colormap that you choose.
The relationship between the values in the image matrix and the colormap depends
on the class of the image matrix. If the image matrix is of class single or double, it
normally contains integer values 1 through p, where p is the length of the colormap. The
value 1 points to the first row in the colormap, the value 2 points to the second row, and so
on. If the image matrix is of class logical, uint8 oruint16, the value 0 points to the first
row in the colormap, the value 1 points to the second row, and so on.
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The following figure illustrates the structure of an indexed image. In the figure, the image
matrix is of class double, so the value 5 points to the fifth row of the colormap.
Fig: Pixel Values Index to Colormap Entries in Indexed Images
1.3 Digital Image File Types:
The 5 most common digital image file types are as follows:
1. JPEG is a compressed file format that supports 24 bit color (millions of colors). This is
the best format for photographs to be shown on the web or as email attachments. This is
because the color informational bits in the computer file are compressed (reduced) and
download times are minimized.
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2. GIF is an uncompressed file format that supports only 256 distinct colors. Best used
with web clip art and logo type images. GIF is not suitable for photographs because of its
limited color support.
3. TIFFis an uncompressed file format with 24 or 48 bit color support. Uncompressed
means that all of the color information from your scanner or digital camera for each
individual pixel is preserved when you save as TIFF. TIFF is the best format for saving
digital images that you will want to print. Tiff supports embedded file information,
including exact color space, output profile information and EXIF data. There is a lossless
compression for TIFF called LZW. LZW is much like 'zipping' the image file because there
is no quality loss. An LZW TIFF decompresses (opens) with all of the original pixel
information unaltered.
4. BMP is a Windows (only) operating system uncompressed file format that supports 24
bit color. BMP does not support embedded information like EXIF, calibrated color space
and output profiles. Avoid using BMP for photographs because it produces approximately
the same file sizes as TIFF without any of the advantages of TIFF.
5. Camera RAW is a lossless compressed file format that is proprietary for each digital
camera manufacturer and model. A camera RAW file contains the 'raw' data from the
camera's imaging sensor. Some image editing programs have their own version of RAW
too. However, camera RAW is the most common type of RAW file. The advantage of
camera RAW is that it contains the full range of color information from the sensor. This
means the RAW file contains 12 to 14 bits of color information for each pixel. If you shoot
JPEG, you only get 8 bits of color for each pixel. These extra color bits make shooting
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element (5, 2). You use normal MATLAB matrix subscripting to access values of
individual pixels.
For example, the MATLAB code
I (2, 15)
Returns the value of the pixel at row 2, column 15 of the image I.
Spatial Coordinates:
In the pixel coordinate system, a pixel is treated as a discrete unit, uniquely
identified by a single coordinate pair, such as (5, 2). From this perspective, a location such
as (5.3, 2.2) is not meaningful.
At times, however, it is useful to think of a pixel as a square patch. From this
perspective, a location such as (5.3, 2.2) is meaningful, and is distinct from (5, 2). In this
spatial coordinate system, locations in an image are positions on a plane, and they are
described in terms ofx and y (not rand c as in the pixel coordinate system).The following
figure illustrates the spatial coordinate system used for images. Notice that y increases
downward.
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2.0 Digital Image Processing
Digital image processing is the use of computer algorithms to perform image
processing on digital images. As a subfield of digital signal processing, digital image
processing has many advantages over analog image processing; it allows a much wider
range of algorithms to be applied to the input data, and can avoid problems such as the
build-up of noise and signal distortion during processing.
2.1 Image digitization:
An image captured by a sensor is expressed as a continuous function f(x,y) of two
co-ordinates in the plane. Image digitization means that the function f(x,y) is sampled into
a matrix with M rows and N columns. The image quantization assigns to each continuous
sample an integer value. The continuous range of the image function f(x,y) is split into K
intervals. The finer the sampling (i.e., the larger M and N) and quantization (the larger K)
the better the approximation of the continuous image function f(x,y).
2.2 Image Pre-processing:
Pre-processing is a common name for operations with images at the lowest level of
abstraction -- both input and output are intensity images. These iconic images are of the
same kind as the original data captured by the sensor, with an intensity image usually
represented by a matrix of image function values (brightness). The aim of pre-processing is
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an improvement of the image data that suppresses unwanted distortions or enhances some
image features important for further processing. Four categories of image pre-processing
methods according to the size of the pixel neighborhood that is used for the calculation of
new pixel brightness:
o Pixel brightness transformations.
o Geometric transformations.
o Pre-processing methods that use a local neighborhood of the processed
pixel.
o Image restoration that requires knowledge about the entire image.
2.3 Image Segmentation:
Image segmentation is one of the most important steps leading to the analysis of
processed image data. Its main goal is to divide an image into parts that have a strong
correlation with objects or areas of the real world contained in the image.Two kinds of
segmentation
1. Complete segmentation: This results in set of disjoint regions uniquely
corresponding with objects in the input image. Cooperation with higher
processing levels which use specific knowledge of the problem domain is
necessary.
2. Partial segmentation: in which regions do not correspond directly with image
objects. Image is divided into separate regions that are homogeneous with
respect to a chosen property such as brightness, color, reflectivity, texture, etc.
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Unfortunately, there is no general theory for determining what `good image enhancement
is when it comes to human perception. If it looks good, it is good! However, when image
enhancement techniques are used as pre-processing tools for other image processing
techniques, then quantitative measures can determine which techniques are most
appropriate.
3.0 Image Denoising
Introduction:
An Image is often corrupted by noise in its acquisition or transmission. The goal of
denoising is to remove the noise while retaining as much as possible the important signal
features. Traditionally, this is achieved by linear processing such as Wiener filtering. A
vast literature has emerged recently on signal denoising using nonlinear techniques, in the
setting of additive white Gaussian noise. The seminal work on signal denoising via wavelet
thresholding have shown that various wavelet thresholding schemes for denoising have
near-optimal properties in the minimax sense and perform well in simulation studies of
one-dimensional curve estimation. It has been shown to have better rates of convergence
than linear methods for approximating functions. Thresholding is a nonlinear technique, yet
it is very simple because it operates on one wavelet coefficient at a time. Alternative
approaches to nonlinear wavelet-based denoising can be found in, for example and
references therein.
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The intuition behind using lossy compression for denoising may be
explained as follows. A signal typically has structural correlations that a good coder can
exploit to yield a concise representation. White noise, however, does not have structural
redundancies and thus is not easily compressable. Hence, a good compression method can
provide a suitable model for distinguishing between signal and noise. The discussion will
be restricted to wavelet-based coders, though these insights can be extended to other
transform-domain coders as well. A concrete connection between lossy compression and
denoising can easily be seen when one examines the similarity between thresholding and
quantization, the latter of which is a necessary step in a practical lossy coder. That is, the
quantization of wavelet coefficients with a zero-zone is an approximation to the
thresholding function. Thus, provided that the quantization outside of the zero-zone does
not introduce significant distortion, it follows that wavelet-based lossy compression
achieves denoising. With this connection in mind, this paper is about wavelet thresholding
for image denoising and also for lossy compression. The threshold choice aids the lossy
coder to choose its zero-zone, and the resulting coder achieves simultaneous denoising and
compression if such property is desired.
Denoising i.e. restoration of electronically distorted images is an old but
also still a relevant problem. There are many different cases of distortions. One of the most
prevalent cases is distortion due to additive white Gaussian noise which can be caused by
poor image acquisition or by transferring the image data in noisy communication channels.
Early methods to restore the image used linear filtering or smoothing methods. These
methods where simple and easy to apply but their effectiveness is limited since this often
leads to blurred or smoothed out in high frequency regions.
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All denoising methods use images artificially distorted with well defined white
Gaussian noise to achieve objective test results. Note however that in real world images, to
discriminate the distorting signal from the true image is an ill posed problem since it is
not always well defined whether a pixel value belongs to the image or it is part of
unwanted noise.
Newer and better approaches perform some thresholding in the wavelet domain
of an image. The idea of wavelet thresholding relies on the assumption that the signal
magnitudes dominate the magnitudes of the noise in a wavelet representation, so that
wavelet coefficients can be set to zero if their magnitudes are less than a predetermined
threshold. More recent developments focus on more sophisticated methods, like local or
context-based thresholding in the wavelet domain. Some methods are inspired by wavelet-
based image compression methods.
The theoretical formalization of filtering additive iidGaussian noise (of zero-mean
and standard deviation) via thresholding wavelet coefficients was pioneered by Donoho
and Johnstone. A wavelet coefficient is compared to a given threshold and is set to zero if
its magnitude is less than the threshold; otherwise, it is kept or modified (depending on the
thresholding rule). The threshold acts as an oracle which distinguishes between the
insignificant coefficients likely due to noise, and the significant coefficients consisting of
important signal structures.
Thresholding rules are especially effective for signals with sparse or near-sparse
representations where only a small subset of the coefficients represents all or most of the
signal energy. Thresholding essentially creates a region around zero where the coefficients
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are considered negligible. Outside of this region, the thresholded coefficients are kept to
full precision (that is, without quantization).
Since the works of Donoho and Johnstone, there has been much research on
finding thresholds for nonparametric estimation in statistics. However, few are specifically
tailored for images. In this project, we propose a framework and a near-optimal threshold
in this framework more suitable for image denoising. This approach can be formally
described as Bayesian, but this only describes our mathematical formulation, not our
philosophy. The formulation is grounded on the empirical observation that the wavelet
coefficients in a sub band of a natural image can be summarized adequately by a
generalized Gaussiandistribution (GGD). This observation is well-accepted in the image
processing community and is used for state-of-the-art image coders. It follows from this
observation that the average MSE (in a sub band) can be approximated by the
corresponding Bayesian squared error risk with the GGD as the prior applied to each in an
iid fashion. That is, a sum is approximated by an integral. We emphasize that this is an
analytical approximation and our framework is broader than assuming wavelet coefficients
are iid draws from a GGD. The goal is to find the soft-threshold that minimizes this
Bayesian risk, and we call our methodBayesShrink.
Adaptive Threshold for BayesShrink
The GGD, following is
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ADAPTIVE WAVELET THRESHOLDING FOR IMAGE DENOISING AND
COMPRESSION
Histogram of the wavelet coefficients of four test images. For each image, from top to
bottom it is fine to coarse scales: from left to right, they are the HH, HL, and LH sub
bands, respectively.
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3.1 Introduction to wavelet representation:
The wavelet concept and its origins
The central idea to wavelets is to analyze (a signal) according toscale. Imagine a function
that oscillates like a wave in a limited portion of time or space and vanishes outside of it.
The wavelets are such functions: wave-like but localized. One chooses a particular wavelet,
stretches it (to meet a given scale) and shifts it, while looking into its correlations with the
analyzed signal. This analysis is similar to observing the displayed signal (e.g., printed or
shown on the screen) from various distances. The signal correlations with wavelets
stretched to large scales reveal gross (rude) features, while at small scales fine signal
structures are discovered. It is therefore often said that the wavelet analysis is to see both
the forest andthe trees.
In such a scanning through a signal, the scale and the position can vary
continuously or in discrete steps. The latter case is of practical interest in this thesis. From
an engineering point of view, the discrete wavelet analysis is a two channel digital filter
bank (composed of the low pass and the high pass filters), iterated on the low pass output.
The low pass filtering yields an approximation of a signal (at a given scale), while the high
pass (more precisely, band pass) filtering yields the details that constitute the difference
between the two successive approximations. A family of wavelets is then associated with
the band pass and a family of scaling functions with the low pass filters.
A)Fourier analysis:
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Signal analysts already have at their disposal an impressive arsenal of tools. Perhaps
the most well-known of these is Fourier analysis, which breaks down a signal into
constituent sinusoids of different frequencies. Another way to think of Fourier analysis is
as a mathematical technique for transformingour view of the signal from time-based to
frequency-based.
Figure 2
For many signals, Fourier analysis is extremely useful because the signals frequency
content is of great importance. So why do we need other techniques, like wavelet analysis?
Fourier analysis has a serious drawback. In transforming to the frequency domain,
time information is lost. When looking at a Fourier transform of a signal, it is impossible to
tell whena particular event took place. If the signal properties do not change much overtime that is, if it is what is called a stationary signalthis drawback isnt very
important. However, most interesting signals contain numerous non stationary or transitory
characteristics: drift, trends, abrupt changes, and beginnings and ends of events. These
characteristics are often the most important part of the signal, and Fourier analysis is not
suited to detecting them.
B)Short-Time Fourier Analysis
In an effort to correct this deficiency, Dennis Gabor (1946) adapted the Fourier
transform to analyze only a small section of the signal at a timea technique called
windowingthe signal.Gabors adaptation, called the Short-Time FourierTransform(STFT),
maps a signal into a two-dimensional function of time and
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frequency.
Figure 3
The STFT represents a sort of compromise between the time- and frequency-based
views of a signal. It provides some information about both when and at what frequencies a
signal event occurs. However, you can only obtain this information with limited precision,
and that precision is determined by the size of the window. While the STFT compromise
between time and frequency information can be useful, the drawback is that once you
choose a particular size for the time window, that window is the same for all frequencies.
Many signals require a more flexible approachone where we can vary the window size to
determine more accurately either time or frequency.
C.Wavelet Analysis
Wavelet analysis represents the next logical step: a windowing technique with
variable-sized regions. Wavelet analysis allows the use of long time intervals where we
want more precise low-frequency information, and shorter regions where we want high-
frequency information.
Figure 4
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Heres what this looks like in contrast with the time-based, frequency-based,
and STFT views of a signal:
Figure 5
You may have noticed that wavelet analysis does not use a time-frequency region, but
rather a time-scaleregion. For more information about the concept of scale and the link
between scale and frequency, see How to Connect Scale to Frequency?
What Can Wavelet Analysis Do?
One major advantage afforded by wavelets is the ability to perform local analysis,
that is, to analyze a localized area of a larger signal. Consider a sinusoidal signal with a
small discontinuity one so tiny as to be barely visible. Such a signal easily could be
generated in the real world, perhaps by a power fluctuation or a noisy switch.
Figure 6
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A plot of the Fourier coefficients (as provided by the fft command) of this signal shows
nothing particularly interesting: a flat spectrum with two peaks representing a single
frequency. However, a plot of wavelet coefficients clearly shows the exact location in time
of the discontinuity.
Figure 7
Wavelet analysis is capable of revealing aspects of data that other
signal analysis techniques miss, aspects like trends, breakdown points, discontinuities in
higher derivatives, and self-similarity. Furthermore, because it affords a different view of
data than those presented by traditional techniques, wavelet analysis can often compress or
de-noise a signal without appreciable degradation. Indeed, in their brief history within the
signal processing field, wavelets have already proven themselves to be an indispensable
addition to the analysts collection of tools and continue to enjoy a burgeoning popularity
today.
What Is Wavelet Analysis?
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Now that we know some situations when wavelet analysis is useful, it is worthwhile
asking What is wavelet analysis? and even more fundamentally,
What is a wavelet?
A wavelet is a waveform of effectively limited duration that has an average value of zero.
Compare wavelets with sine waves, which are the basis of Fourier analysis.
Sinusoids do not have limited duration they extend from minus to plus
infinity. And where sinusoids are smooth and predictable, wavelets tend to be
irregular and asymmetric.
Figure 8Fourier analysis consists of breaking up a signal into sine waves of various
frequencies. Similarly, wavelet analysis is the breaking up of a signal into shifted and
scaled versions of the original (ormother) wavelet. Just looking at pictures of wavelets and
sine waves, you can see intuitively that signals with sharp changes might be better analyzed
with an irregular wavelet than with a smooth sinusoid, just as some foods are better
handled with a fork than a spoon. It also makes sense that local features can be described
better with wavelets that have local extent.
The Continuous Wavelet Transform:
Mathematically, the process of Fourier analysis is represented by the Fourier
transform:
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which is the sum over all time of the signal f(t) multiplied by a complex exponential.
(Recall that a complex exponential can be broken down into real and imaginary sinusoidal
components.) The results of the transform are the Fourier coefficients F(w), which when
multiplied by a sinusoid of frequency w yields the constituent sinusoidal components of the
original signal. Graphically, the process looks like:
Figure 9
Similarly, the continuous wavelet transform (CWT) is defined as the sum over all
time of the signal multiplied by scaled, shifted versions of the wavelet function
The result of the CWT is a series many wavelet coefficientsC, which are a function
of scale and position.
Multiplying each coefficient by the appropriately scaled and shifted wavelet yields the
constituent wavelets of the original signal:
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Figure 10
Scaling
Weve already alluded to the fact that wavelet analysis produces a time-scale
view of a signal and now were talking about scaling and shifting wavelets.
What exactly do we mean byscale in this context?
Scaling a wavelet simply means stretching (or compressing) it.
To go beyond colloquial descriptions such as stretching, we introduce the scale factor,
often denoted by the letter a.
If were talking about sinusoids, for example the effect of the scale factor is very easy to
see:
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Figure 11
The scale factor works exactly the same with wavelets. The smaller the scale factor, the
more compressed the wavelet.
Figure 12
It is clear from the diagrams that for a sinusoid sin (w t) the scale factor a is related(inversely) to the radian frequency w. Similarly, with wavelet analysis the scale is related
to the frequency of the signal.
Shifting
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Figure 14
3. Shift the wavelet to the right and repeat steps 1 and 2 until youve covered the whole
signal.
Figure 15
4. Scale (stretch) the wavelet and repeat steps 1 through 3.
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Figure 16
5.Repeat steps 1 through 4 for all scales.
When youre done, youll have the coefficients produced at different scales by
different sections of the signal. The coefficients constitute the results of a regression of the
original signal performed on the wavelets.
How to make sense of all these coefficients? You could make a plot on which the x-
axis represents position along the signal (time), they-axis represents scale, and the color at
each x-y point represents the magnitude of the wavelet coefficient C. These are the
coefficient plots generated by the graphical tools.
Figure 17
These coefficient plots resemble a bumpy surface viewed from above.
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frequency w.
High scale a=>Stretched wavelet=>Slowly changing, coarse features=>Low
frequency w.
The Scale of Nature:
Its important to understand the fact that wavelet analysis does not produce a time-
frequency view of a signal is not a weakness, but a strength of the technique.
Not only is time-scale a different way to view data, it is a very natural way to view data
deriving from a great number of natural phenomena.
Consider a lunar landscape, whose ragged surface (simulated below) is a result of
centuries of bombardment by meteorites whose sizes range from gigantic boulders to dust
specks.
If we think of this surface in cross-section as a one-dimensional signal, then it is
reasonable to think of the signal as having components of different scaleslarge features
carved by the impacts of large meteorites, and finer features abraded by small meteorites.
Figure 20
Here is a case where thinking in terms of scale makes much more sense than thinking
in terms of frequency. Inspection of the CWT coefficients plot for this signal reveals
patterns among scales and shows the signals possibly fractal nature.
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For many signals, the low-frequency content is the most important part. It is what
gives the signal its identity. The high-frequency content on the other hand imparts flavor or
nuance. Consider the human voice. If you remove the high-frequency components, the
voice sounds different but you can still tell whats being said. However, if you remove
enough of the low-frequency components, you hear gibberish. In wavelet analysis, we
often speak of approximations and details. The approximations are the high-scale, low-
frequency components of the signal. The details are the low-scale, high-frequency
components.
The filtering process at its most basic level looks like this:
Figure 23
The original signal S passes through two complementary filters and emerges as two
signals.
Unfortunately, if we actually perform this operation on a real digital signal, we
wind up with twice as much data as we started with. Suppose, for instance that the original
signal S consists of 1000 samples of data. Then the resulting signals will each have 1000
samples, for a total of 2000.
These signals A and D are interesting, but we get 2000 values instead of the 1000
we had. There exists a more subtle way to perform the decomposition using wavelets. By
looking carefully at the computation, we may keep only one point out of two in each of the
two 2000-length samples to get the complete information. This is the notion of own
sampling. We produce two sequences called cA and cD.
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Figure 24
The process on the right which includes down sampling produces DWT
Coefficients. To gain a better appreciation of this process lets perform a one-stage discretewavelet transform of a signal. Our signal will be a pure sinusoid with
high- frequency noise added to it.
Here is our schematic diagram with real signals inserted into it:
Figure 25
The MATLAB code needed to generate s, cD, and cA is:
s = sin(20*linspace(0,pi,1000)) + 0.5*rand(1,1000);
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[cA,cD] = dwt(s,'db2');
where db2 is the name of the wavelet we want to use for the analysis.
Notice that the detail coefficients cD is small and consist mainly of a high-frequency noise,
while the approximation coefficients cA contains much less noise than does the original
signal.
[length(cA) length(cD)]
ans = 501 501
You may observe that the actual lengths of the detail and approximation coefficient
vectors are slightly more than half the length of the original signal. This has to do with the
filtering process, which is implemented by convolving the signal with a filter. The
convolution smears the signal, introducing several extra samples into the result.
Multiple-Level Decomposition:
The decomposition process can be iterated, with successive approximations being
decomposed in turn, so that one signal is broken down into many lower resolution
components. This is called the wavelet decomposition tree.
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Figure 26
Looking at a signals wavelet decomposition tree can yield valuable information.
Figure 27
Number of Levels:
Since the analysis process is iterative, in theory it can be continued indefinitely. In
reality, the decomposition can proceed only until the individual details consist of a single
sample or pixel. In practice, youll select a suitable number of levels based on the nature of
the signal, or on a suitable criterion such as entropy.
Wavelet Reconstruction:
Weve learned how the discrete wavelet transform can be used to analyze or
decompose, signals and images. This process is called decomposition or analysis. The other
half of the story is how those components can be assembled back into the original signal
without loss of information. This process is called reconstruction, or synthesis. The
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mathematical manipulation that effects synthesis is called the inverse discrete wavelet
transforms(IDWT). To synthesize a signal in the Wavelet Toolbox, we reconstruct it from
the wavelet coefficients:
Figure 28
Where wavelet analysis involves filtering and down sampling, the wavelet
reconstruction process consists of up sampling and filtering. Up sampling is the process of
lengthening a signal component by inserting zeros between samples:
Figure 29
The Wavelet Toolbox includes commands like idwt and waverec that perform
single-level or multilevel reconstruction respectively on the components of one-
dimensional signals. These commands have their two-dimensional analogs, idwt2 and
waverec2.
Reconstruction Filters:
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The filtering part of the reconstruction process also bears some discussion, because it
is the choice of filters that is crucial in achieving perfect reconstruction of the original
signal. The down sampling of the signal components performed during the decomposition
phase introduces a distortion called aliasing. It turns out that by carefully choosing filters
for the decomposition and reconstruction phases that are closely related (but not identical),
we can cancel out the effects of aliasing.
The low- and high pass decomposition filters (L and H), together with their
associated reconstruction filters (L' and H'), form a system of what is called quadrature
mirror filters:
Figure 30
Reconstructing Approximations and Details:
We have seen that it is possible to reconstruct our original signal from the
coefficients of the approximations and details.
Figure31
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It is also possible to reconstruct the approximations and details themselves from
their coefficient vectors.
As an example, lets consider how we would reconstruct the first-level
approximation A1 from the coefficient vector cA1. We pass the coefficient vector cA1
through the same process we used to reconstruct the original signal. However, instead of
combining it with the level-one detail cD1, we feed in a vector of zeros in place of the
detail coefficients
vector:
Figure 32
The process yields a reconstructed approximationA1, which has the same length as
the original signal S and which is a real approximation of it. Similarly, we can reconstruct
the first-level detail D1, using the analogous process:
Figure 33
The reconstructed details and approximations are true constituents of the original
signal. In fact, we find when we combine them that:
A1 +D1 = S
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Note that the coefficient vectors cA1 and cD1because they were produced by
Down sampling and are only half the length of the original signal cannot directly be
combined to reproduce the signal.
It is necessary to reconstruct the approximations and details before combining
them. Extending this technique to the components of a multilevel analysis, we find that
similar relationships hold for all the reconstructed signal constituents.
That is, there are several ways to reassemble the original signal:
Figure 34
Relationship of Filters to Wavelet Shapes:
In the section Reconstruction Filters, we spoke of the importance of choosing the
right filters. In fact, the choice of filters not only determines whether perfect reconstruction
is possible, it also determines the shape of the wavelet we use to perform the analysis. To
construct a wavelet of some practical utility, you seldom start by drawing a waveform.
Instead, it usually makes more sense to design the appropriate quadrature mirror filters, and
then use them to create the waveform. Lets see
how this is done by focusing on an example.
Consider the low pass reconstruction filter (L') for the db2 wavelet.
Wavelet function position
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plot(H2)
Figure 36
If we iterate this process several more times, repeatedly up sampling and
convolving the resultant vector with the four-element filter vector Lprime, a pattern begins
to emerge:
Figure 37
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The curve begins to look progressively more like the db2 wavelet. This means that
the wavelets shape is determined entirely by the coefficients of the reconstruction filters.
This relationship has profound implications. It means that you cannot choose just any
shape, call it a wavelet, and perform an analysis. At least, you cant choose an arbitrary
wavelet waveform if you want to be able to reconstruct the original signal accurately. You
are compelled to choose a shape determined by quadrature mirror decomposition filters.
The Scaling Function:
Weve seen the interrelation of wavelets and quadrature mirror filters. The wavelet
function is determined by the high pass filter, which also produces the details of the
wavelet decomposition.
There is an additional function associated with some, but not all wavelets. This is
the so-called scaling function . The scaling function is very similar to the wavelet function.
It is determined by the low pass quadrature mirror filters, and thus is associated with the
approximations of the wavelet decomposition. In the same way that iteratively up-
sampling and convolving the high pass filter produces a shape approximating the wavelet
function, iteratively up-sampling and convolving the low pass filter produces a shape
approximating the scaling function.
Multi-step Decomposition and Reconstruction:
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A multi step analysis-synthesis process can be represented as:
Figure 38
This process involves two aspects: breaking up a signal to obtain the wavelet
coefficients, and reassembling the signal from the coefficients. Weve already discussed
decomposition and reconstruction at some length. Of course, there is no point breaking up a
signal merely to have the satisfaction of immediately reconstructing it. We may modify the
wavelet coefficients before performing the reconstruction step. We perform wavelet
analysis because the coefficients thus obtained have many known uses, de-noising and
compression being foremost among them. But wavelet analysis is still a new and emerging
field. No doubt, many uncharted uses of the wavelet coefficients lie in wait. The Wavelet
Toolbox can be a means of exploring possible uses and hitherto unknown applications of
wavelet analysis. Explore the toolbox functions and see what you discover.
WAVELET DECOMPOSITION:
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Images are treated as two dimensional signals, they change horizontally and
vertically, thus 2D wavelet analysis must be used for images. 2D wavelet analysis uses the
same mother wavelets but requires an extra step at every level of decomposition. The 1D
analysis filtered out the high frequency information from the low frequency information at
every level of decomposition; so only two sub signals were produced at each level.
In 2D, the images are considered to be matrices with N rows and M columns. At
every level of decomposition the horizontal data is filtered, then the approximation and
details produced from this are filtered on columns.
Fig 1: Decomposition of an Image
At every level, four sub-images are obtained; the approximation, the vertical
detail, the horizontal detail and the diagonal detail. Below the Saturn image has been
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decomposed to one level. The wavelet analysis has found how the image changes
vertically, horizontally and diagonally.
Fig 2:2-D Decomposition of Saturn Image to level 1
To get the next level of decomposition the approximation sub-image is decomposed, this
idea can be seen in figure 3.
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Fig 3: Saturn Image decomposed to Level 3. Only the 9 detail sub-images and the final
sub-image is required to reconstruct the image perfectly.
When compressing with orthogonal wavelets the energy retained is:
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The number of zeros in percentage is defined by:
Wavelet based denoising schemes:
The idea of wavelet thresholding relies on the assumption that the signal
magnitudes dominate the magnitudes of the noise in a wavelet representation, so that
wavelet coefficients can be set to zero if their magnitudes are less than a predetermined
threshold. Donoho and Johnstone proposed hard- and soft-thresholding methods for
denoising, where the former leaves the magnitudes of coefficients unchanged if they are
larger than a given threshold, while the latter just shrinks them to zero by the threshold
value.
However, the major problem with both methods and most of its variants is the
choice of a suitable threshold value. Most signals show a spatially non-uniform energy
distribution, which motivates the choice of a non-constant threshold. Since a given noisy
signal may consist of some parts where the magnitudes of the signal are below the globally
defined threshold and other parts where the noise magnitudes exceed that given threshold,
methods relying on a globally defined threshold cut of parts of the signal, on the one hand,
and leave some noise untouched, on the other hand. This observation led to the idea of a
spatially adaptive threshold choice depending on the relationship of local energy (variance)
of the observed signal and the noise variance.
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Chang et al. 3, 4 were the first to propose this kind of spatially adaptive wavelet
thresholding for image denoising. Their method of selecting a spatially adaptive threshold
is based on a context model, which involves neighboring coefficients of the wavelet
decomposition for the estimation of the local variance. The authors extended this idea by
using a more elaborate context model and by iterating the context-based thresholding
process in the denoised wavelet representation, which led to significantly improved.
Denoising by wavelet thresholding:
Wavelet thresholding is a popular approach for denoising due to its simplicity. In its most
basic form, this technique operates in the orthogonal wavelet domain, where each
coefficient is thresholdedby comparing against a threshold; if the coefficient is smaller
than the threshold it is set to zero, otherwise, it is kept or modified. One of the first reports
about this approach was by Weaver et al[Weaver92].
Hard and soft thresholding:
Two standard thresholding policies are: hard-thresholding, (keep or kill), and
soft-thresholding(shrink or kill). In both cases, the coefficients that are below a certain
threshold are set to zero. In hard thresholding, the remaining coefficients are left
unchanged.
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Shrinkage factors that multiply the wavelet coefficients in
(a) Hard-thresholding and (b) soft-thresholding.
Most methods for estimating the threshold assume AWGN noise and an orthogonal wavelet
transform. Among those, well known is the universalthresholdof Donoho and Johnstone
where n is the estimate of the standard deviation of additive white noise and n is the total
number of the wavelet coefficients in a given detail image. The rationale behind this
threshold is to remove all the coefficients that are smaller than the expected maximum of
i.i.d. normal noise: if {ui} is a sequence of n i.i.d. random variables with normal
distributionN(0, 1), then the maximum maxi{|ui|} is smaller than 2log(n) with a probability
approaching one when n tends to infinity. Moreover, the probability that maxi {|ui|}
exceeds 2log (n) by a value t is smaller than et2/2 [Donoho92a, Vidakovic94]. At
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different resolution scales, the threshold (2.4.3) differs only in the constant factor that is
related to the number of the coefficients in a given sub band.
Other thresholds that are estimated in an adaptive way for each level were
proposed, e.g., in [Donoho95b, Hilton97, Jansen97, Nason94, Weyrich98]. Among those,
well known is the SURE threshold of [Donoho95b], derived from minimizing the Steins
unbiased risk estimate [Stein81] when soft-thresholding is used. Nason [Nason94]
proposed a threshold selection based on a cross-validation procedure, which is further
extended in [Jansen97, Weyrich98] and applied to correlated noise. Other methods, like
[Chang00b, Ruggeri99], derive the optimum threshold by minimizing the mean squared
error in a thresholded signal under an assumed prior distribution of the wavelet
coefficients. Hiltons data analytic threshold [Hilton97] takes into account the spatial
clustering properties of wavelet coefficients. However, this threshold as well as all the
others mentioned above isspatially uniform, i.e., of the constant value for the whole detail
image.
It is obvious that spatially uniform thresholding is not the best thing one can do.
Instead of applying a constant threshold to all the coefficients (in a given sub band) it
would be better to decide for each coefficient separately what is better: keeping or killing (a
nice discussion is in [Jansen01b, p.102]). It was shown in Sec. 2.3.5 that the mean squared
error would be minimized by selecting the coefficients the signal component of which is
above noise standard deviation and removing the others. A spatially varying threshold
selection can better approach this unrealistic dream. In this respect, spatially adaptive
thresholding with context modelingof wavelet coefficients [Chang98, Chang00a] is a state-
of-the art approach for image denoising. Briefly, this approach applies a soft thresholding
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with the threshold equal to 2n /X, where n is the noise standard deviation and Xis the
standard deviation of thesignal; to estimate Xat a given position, the coefficients with
similarcontext are clustered; actually the context variable in [Chang00a] isa weighted
average of the coefficient magnitudes in a moving window. This method appears as a
reference method in Table 5.1. Other approaches, which rely on the decay of individual
coefficients across scales, will be reviewed in the next Section.
Wavelet domain Bayes estimation:
Bayesian approaches to wavelet shrinkage are less ad-hoc than earlier proposals
and were shown to be effective. In general, Bayes rules are shrinkers and their shape in
many cases has a desirable property: it can heavily shrink small arguments and only
slightly shrink large arguments. The resulting actions on wavelet coefficients can be very
close to thresholding.
4.0 INTRODUCTION TO MATLAB
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What Is MATLAB?
MATLAB is a high-performance language for technical computing. It integrates
computation, visualization, and programming in an easy-to-use environment where
problems and solutions are expressed in familiar mathematical notation. Typical uses
include
1. Math and computation
2. Algorithm development
3. Data acquisition
4. Modeling, simulation, and prototyping
5. Data analysis, exploration, and visualization
6. Scientific and engineering graphics
7. Application development, including graphical user interface building.
MATLAB is an interactive system whose basic data element is an array that does not
require dimensioning. This allows you to solve many technical computing problems,
especially those with matrix and vector formulations, in a fraction of the time it would take
to write a program in a scalar non interactive language such as C or FORTRAN.
The name MATLAB stands for matrix laboratory. MATLAB was originally written to
provide easy access to matrix software developed by the LINPACK and EISPACK
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projects. Today, MATLAB engines incorporate the LAPACK and BLAS libraries,
embedding the state of the art in software for matrix computation.
MATLAB has evolved over a period of years with input from many users. In
university environments, it is the standard instructional tool for introductory and advanced
courses in mathematics, engineering, and science. In industry, MATLAB is the tool of
choice for high-productivity research, development, and analysis.
MATLAB features a family of add-on application-specific solutions called
toolboxes. Very important to most users of MATLAB, toolboxes allow you to learnand
apply specialized technology. Toolboxes are comprehensive collections of MATLAB
functions (M-files) that extend the MATLAB environment to solve particular classes of
problems. Areas in which toolboxes are available include signal processing, control
systems, neural networks, fuzzy logic, wavelets, simulation, and many others.
The MATLAB System:
The MATLAB system consists of five main parts:
Development Environment:
This is the set of tools and facilities that help you use MATLAB functions and files.
Many of these tools are graphical user interfaces. It includes the MATLAB desktop and
Command Window, a command history, an editor and debugger, and browsers for viewing
help, the workspace, files, and the search path.
The MATLAB Mathematical Function:
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This is a vast collection of computational algorithms ranging from elementary
functions like sum, sine, cosine, and complex arithmetic, to more sophisticated functions
like matrix inverse, matrix eigen values, Bessel functions, and fast Fourier transforms.
The MATLAB Language:
This is a high-level matrix/array language with control flow statements, functions,
data structures, input/output, and object-oriented programming features. It allows both
"programming in the small" to rapidly create quick and dirty throw-away programs, and
"programming in the large" to create complete large and complex application programs.
Graphics:
MATLAB has extensive facilities for displaying vectors and matrices as graphs, as
well as annotating and printing these graphs. It includes high-level functions for two-
dimensional and three-dimensional data visualization, image processing, animation, and
presentation graphics. It also includes low-level functions that allow you to fully customize
the appearance of graphics as well as to build complete graphical user interfaces on your
MATLAB applications.
The MATLAB Application Program Interface (API):
This is a library that allows you to write C and Fortran programs that interact with
MATLAB. It includes facilities for calling routines from MATLAB (dynamic linking),
calling MATLAB as a computational engine, and for reading and writing MAT-files.
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MATLAB WORKING ENVIRONMENT:
MATLAB DESKTOP:-
Matlab Desktop is the main Matlab application window. The desktop contains five
sub windows, the command window, the workspace browser, the current directory
window, the command history window, and one or more figure windows, which are shown
only when the user displays a graphic.
The command window is where the user types MATLAB commands and
expressions at the prompt (>>) and where the output of those commands is displayed.
MATLAB defines the workspace as the set of variables that the user creates in a work
session. The workspace browser shows these variables and some information about them.
Double clicking on a variable in the workspace browser launches the Array Editor, which
can be used to obtain information and income instances edit certain properties of the
variable.
The current Directory tab above the workspace tab shows the contents of the current
directory, whose path is shown in the current directory window. For example, in the
windows operating system the path might be as follows: C:\MATLAB\Work, indicating
that directory work is a subdirectory of the main directory MATLAB; WHICH IS
INSTALLED IN DRIVE C. clicking on the arrow in the current directory window shows a
list of recently used paths. Clicking on the button to the right of the window allows the user
to change the current directory.
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MATLAB uses a search path to find M-files and other MATLAB related files,
which are organize in directories in the computer file system. Any file run in MATLAB
must reside in the current directory or in a directory that is on search path. By default, the
files supplied with MATLAB and math works toolboxes are included in the search path.
The easiest way to see which directories are on the search path. The easiest way to see
which directories are soon the search path, or to add or modify a search path, is to select set
path from the File menu the desktop, and then use the set path dialog box. It is good
practice to add any commonly used directories to the search path to avoid repeatedly
having the change the current directory.
The Command History Window contains a record of the commands a user has
entered in the command window, including both current and previous MATLAB sessions.
Previously entered MATLAB commands can be selected and re-executed from the
command history window by right clicking on a command or sequence of commands. This
action launches a menu from which to select various options in addition to executing the
commands. This is useful to select various options in addition to executing the commands.
This is a useful feature when experimenting with various commands in a work session.
Using the MATLAB Editor to create M-Files:
The MATLAB editor is both a text editor specialized for creating M-files and a
graphical MATLAB debugger. The editor can appear in a window by itself, or it can be a
sub window in the desktop. M-files are denoted by the extension .m, as in pixelup.m. The
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MATLAB editor window has numerous pull-down menus for tasks such as saving,
viewing, and debugging files. Because it performs some simple checks and also uses color
to differentiate between various elements of code, this text editor is recommended as the
tool of choice for writing and editing M-functions. To open the editor , type edit at the
prompt opens the M-file filename.m in an editor window, ready for editing. As noted
earlier, the file must be in the current directory, or in a directory in the search path.
Getting Help:
The principal way to get help online is to use the MATLAB help browser, opened as
a separate window either by clicking on the question mark symbol (?) on the desktop
toolbar, or by typing help browser at the prompt in the command window. The help
Browser is a web browser integrated into the MATLAB desktop that displays a Hypertext
Markup Language(HTML) documents. The Help Browser consists of two panes, the help
navigator pane, used to find information, and the display pane, used to view the
information. Self-explanatory tabs other than navigator pane are used to perform a search.