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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.

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