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CHAPTER 1

IMAGE PROCESSING

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1.1 INTRODUCTION

The limitations of commonly used transforms like FFT, Wavelet in

case of edge preservations and also in denoising with the noises such as

Additive White Gaussian Noise (AWGN), Speckle are well-known and

therefore we have used a transformation named Contourlet which is

better in edge preservations and denoising.

The approach in this transformation starts with the discrete domain

construction and then sparse expansion in the continuous domain. A discrete

domain multi-resolution and multi-directional expansion using non-

separable filter banks are constructed and this results in expansion in contour

segments and thus, the name. The main difference between Contourlet and

other transformations is that, in this new transformation Laplacian pyramid

along with the Directional Filter Banks are used. As a result, this not only

detects the edge discontinuities, but also converts all these discontinuities

into continuous domain. This is the advantage of this transformation when

compared with other transformations in case of decomposition process.

In denoising, a new algorithm is proposed based on the transformation

named Contourlet. A threshold value is being introduced and the resultant of

the decomposition process are compared with this threshold and determined

whether the image value (pixel) is corrupted or not. It is accordingly

modified or preserved and thus finally a conclusion is arrived that the

Contourlet transformation is better than the existing algorithms in image

processing in case of edge preservations and denoising even with the noises

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1.2 DESCRIPTION OF THE PROJECT

FIG 1.1 Description of the Project

START

LOAD THE IMAGE

ADDING MULTIPLICATIVE NOISE

DENOISING

Wavelet Decomposition Contourlet Decomposition

Wavelet coefficients

Denoising Algorithm

Contourlet coefficients

Thresholding Algorithm

Wavelet Reconstruction Contourlet Reconstruction

A3

COMPARATIVE RESULTS

STOP

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1.3 IMAGE PROCESSING

An image may be defined as a two-dimensional function f(x, y), where

x and y are spatial coordinates, and the amplitude of f at any pair of

coordinates (x, y) is called the intensity orgray levelof the image at that

point. Whenx, y, and the amplitude values of f are all finite, discrete

quantities, we call the image a digital image.

A digital image is composed of a finite number of elements, each of

which has a particular location and value. These elements are referred to as

picture elements,image elementsand pixels. Pixelis the term most widely

used to denote the elements of a digital image. An image is an array, or a

matrix, of square pixels (picture elements) arranged in columns and rows.

Image processing modifies pictures to improve them (enhancement,

restoration), extract information (analysis, recognition), and change their

structure (composition, image editing). Images can be processed by optical,

photographic, and electronic means, but image processing using digital

computers is the most common method because digital methods are fast,

flexible, and precise.

An image can be synthesized from a micrograph of various cell

organelles by assigning a light intensity value to each cell organelle. The

sensor signal is digitized, and then converted to an array of numerical

values, each value representing the light intensity of a small area of the cell.

The digitized values are called picture elements, or pixels, and are stored

in computer memory as a digital image. A typical size for a digital image is

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an array of 512 by 512 pixels, where each pixel has value in the range of 0 to

255. The digital image is processed by a computer to achieve the desired

result.

Image enhancement improves the quality (clarity) of images for human

viewing. Removing blurring and noise, increasing contrast, and revealing

details are examples of enhancement operations. For example, an image

might be taken of an endothelial cell, which might be of low contrast and

somewhat blurred. Reducing the noise and blurring and increasing the

contrast range could enhance the image. The original image might have

areas of very high and very low intensity, which mask details. An adaptive

enhancement algorithm reveals these details. Adaptive algorithms adjust

their operation based on the image information (pixels) being processed. In

this case the mean intensity, contrast, and sharpness (amount of blur

removal) could be adjusted based on the pixel intensity statistics in various

areas of the image.

The various types of the image formats are

GIF Graphics Interchange Format; an 8-bit (256 colour), non-

destructively compressed bitmap format. Mostly used for web. Have several

sub-standards one of which is the animated GIF.

JPEG Joint Photographic Experts Group; a very efficient (i.e. much

information per byte) destructively compressed 24 bit (16 million colours)

bitmap format. Widely used, especially for web and Internet (bandwidth-

limited).

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TIFF Tagged Image File Format; the standard 24 bit publication

bitmap format. Compresses non-destructively with, for instance, Lempel-

Ziv-Welch (LZW) compression.

PS Postscript, a standard vector format. Has numerous sub-standards

and can be difficult to transport across platforms and operating systems.

PSD a dedicated Photoshop format that keeps all the information in an

image including all the layers.

BMP Windows Bitmap;1-bit, 8-bit, 24-bit uncompressed images

In electrical and computer science engineering, image processing is

any form of signal processing for which the input is an image, such as

photographs or frames of video; the output of image processing can be either

an image or a set of characteristics or parameters related to the image. Most

image-processing techniques involve treating the image as a two-

dimensional signal and applying standard signal-processing techniques to it.

Image processing usually refers to digital image processing, but optical

and analog image processing are also possible.

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Other image processing operations are

Geometric transformations, such as enlargement, reduction, androtation.

Color corrections such as brightness and contrast adjustments,quantization, or conversion to a different color space.

Digital compositing orOptical compositing (combination of two ormore images). Used in filmmaking to make a "matte".

Interpolation and recovery of a full image from a raw image formatusing a Bayer filter pattern.

Image editing (e.g., to increase the quality of a digital image). Image registration (alignment of two or more images), differencing

and morphing.

Image segmentation, Extending dynamic range by combiningdifferently exposed images.

2-D object recognition with affine invariance.

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1.4 ABOUT MATLAB

MATLAB [6] is a software package for high-performance numerical

computation and visualization. It provides an interactive environment with

hundreds of built-in functions for technical computation, graphics and

animation. Best of all, it also provides easy extensibility with its own high-

level programming language. The name MATLAB stands for MATrix

LABoratory.

MATLABs built in functions provide excellent tools for linear algebra

computations, data analysis, signal processing, optimization, numerical

solution of ordinary differential equations and many other scientific

computations. There are also numerous functions for 2-D and 3-D graphics

as well as animation. Also, for those who cannot do with their FORTRAN or

C codes, MATLAB provides an external interface to run those programs

from within. MATLABs language is very easy to learn and to use.

There are also several optional Toolboxes available from the

developers of MATLAB. These toolboxes are collections of functions

written for special applications such as Symbolic computation, Image

processing, Statistics, Control system design and Neural networks etc.

The basic building block of MATLAB is the matrix. The fundamental

data-type is the array. Vectors, scalars, real matrices and complex matrices

are all automatically handled as special cases of the basic data-type. The

built in functions are optimized for vector operations and consequently

vectorized. Commands or codes run faster in MATLAB.

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1.5 NOISES IN IMAGE PROCESSING

In image processing, considering fields like remote sensing and

medical applications, we come across many noises like AWGN (Additive

White Gaussian Noise), Speckle (Multiplicative), Impulse etc which

affects the valuable features and important information.

1.5.1 ADDITIVE WHITE GAUSSIAN NOISE

In communications, the additive white Gaussian noise (AWGN)

channel model is one in which the information is given a single impairment

a linear addition of wideband or white noise with a constant spectral density

(expressed as watts per hertz of bandwidth) and a Gaussian distribution of

noise samples. The model does not account for the phenomena of fading,

frequency selectivity, interference, nonlinearity or dispersion. However, it

produces simple and tractable mathematical models which are useful for

gaining insight into the underlying behavior of a system before these other

phenomena are considered. Wideband Gaussian noise comes from many

natural sources, such as the thermal vibrations of atoms in antennas (referred

to as thermal noise or Johnson-Nyquist noise), shot noise, black body

radiation from the earth and other warm objects, and from celestial sources

such as the Sun.

The AWGN channel is a good model for many satellite and deep space

communication links. It is not a good model for most terrestrial links

because of multipath, terrain blocking, interference, etc. However for

terrestrial path modeling, AWGN is commonly used to simulate background

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noise of the channel under study, in addition to multipath, terrain blocking,

interference, ground clutter and self interference that modern radio systems

encounter in terrestrial operation.

1.5.1.1 EFFECTS OF NOISE IN TIME DOMAIN

FIG 1.2 Zero-Crossings of a Noisy Cosine

In serial data communications, the AWGN mathematical model is used

to model the timing error caused by random jitter (RJ).

The graph shown above shows an example of timing errors associated

with AWGN. The variable t represents the uncertainty in the zero crossing.

As the amplitude of the AWGN is increased, the Signal-to-noise ratio

decreases. This results in increased uncertainty t.

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When affected by AWGN, The average number of either positive

going or negative going zero-crossings per second at the output of a narrow

band pass filter when the input is a sine wave is:

Where,

f0 = the center frequency of the filter

B = the filter bandwidth

SNR = the signal-to-noise power ratio in linear terms

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1.5.1.2 EFFECTS OF NOISE IN PHASOR DOMAIN

1.5.1.2.1 AWGN CONTRIBUTIONS IN THE PHASOR DOMAIN

In modern communication systems, band limited AWGN cannot be

ignored. When modeling band limited AWGN in the phasor domain,

statistical analysis reveals that the amplitudes of the real and imaginary

contributions are independent variables which follow the Gaussian

distribution model. When combined, the resultant phasor's magnitude is a

Rayleigh distributed random variable while the phase is uniformly

distributed from 0 to 2.

FIG 1.3 The graph shows how band limited AWGN can affect a

coherent carrier signal

The instantaneous response of the Noise Vector cannot be precisely

predicted; however its time-averaged response can be statistically predicted.

As shown in the graph, we confidently predict that the noise phasor will

reside inside the 1 circle about 38% of the time; the noise phasor will

reside inside the 2 circle about 86% of the time; and the noise phasor will

reside inside the 3 circle about 98% of the time.

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1.5.2 MULTIPLICATIVE NOISE

Multiplicative noise [5] is a type of signal dependent noise. In this case

variance of the noise is a function of the signal amplitude. The figure below

shows a comparison between a one dimensional (1-D) step-like signal

corrupted with an additive white Gaussian noise AWGN and a multiplicative

noise. In the case of multiplicative noise, the noise variance is higher when

amplitude of the signal is higher. In relation to images, noise in bright

regions have higher variations and could be interpreted wrongly as features

in the original image. Thus it is harder and more complicated to smooth the

noise without degrading true image features.

FIG 1.4 Step-like signals (a) Original signal (b) Signal corrupted with

AWGN and (c) Signal corrupted with multiplicative noise

Multiplicative noise degrades the quality of the image and affects the

performance of important image processing techniques such as detection,

segmentation, and classification. Therefore an effective preprocessing filter

is desirable in these cases.

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Objectives of any filtering approach are

To effectively suppress the noise in uniform regions. To preserve and enhance edges and other similar image features. To provide a visually natural appearance.

Multiplicative noise is commonly found in many real world signal

processing applications. Unlike additive noise, this kind of noise is much

more difficult to be removed from the corrupted signal, mainly because of its

multiplicative nature. When such noise is present in a bright area in an

image, it will be multiplied by high intensity values, thus its random

variation will increase, or be magnified. On the other hand, if the noise is

introduced into a dark area, the change of the random variation may be much

less significant.

Since the noise variation greatly depends on the intensity levels of the

image pixels being corrupted, it is not easy to establish an appropriate

statistical model for the noise by simply examining the corrupted image.

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1.5.3 IMPULSE NOISE

A short burst of an acoustic energy consisting of either a single impulse

or a series of impulses. The pressure time history of a single impulse

includes a rapid rise to a peak pressure, followed by a slower decay of the

pressure envelope to ambient pressure, both occurring within 1 second.

When the intervals between impulses are less than 500 milliseconds, the

noise is considered continuous, with the exception of successive bursts of

automatic weapons fire, which is considered impulse noise.

FIG 1.5 Impulse Noise

mm pistol at shooters ear

1000

0

1000

000

0 5 10 15 0

Time, milliseconds

ressure,

ascals

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CHAPTER 2

TRANSFORMS USED IN IMAGE

PROCESSING

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2.1 VARIOUS EXISTING TRANSFORMS

To extract the information from the noisy corrupted image we use

many transforms like FFT, Wavelet etc.

2.1.1 FOURIER TRANSFORM

In mathematics, the Fourier transform is an operation that transforms

one complex-valued function of a real variable into another. The new

function, often called the frequency domain representation of the original

function, describes which frequencies are present in the original function.

This is in a similar spirit to the way that a chord of music can be described

by notes that are being played. In effect, the Fourier transform decomposes a

function into oscillatory functions.

The Fourier transform (FT) is similar to many other operations in

mathematics which make up the subject of Fourier analysis. In this specific

case, both the domains of the original function and its frequency domain

representation are continuous and unbounded. The term Fourier transform

can refer to both the frequency domain representation of a function or to the

process/formula that "transforms" one function into the other.

The expressions are:

for every real x &

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The problems with Fourier analysis arise with non-stationary signals.

The frequency components in signals can be found with Fourier techniques.

But these techniques does not tell us at what times these frequency

components occur. This is not a problem in stationary signals (signals whose

frequency content do not change in time are called stationary signals).

Because the answer is simply at all times. In non stationary signals we can

use short time Fourier transforms but it has a time-frequency resolution

problem. At this point we need another transform to solve this problem.

2.1.2 FAST FOURIER TRANSFORM (FFT)

A fast Fourier transform (FFT) is an efficient algorithm to compute the

discrete Fourier transform (DFT) and its inverse. There are many distinct

FFT algorithms involving a wide range of mathematics, from simple

complex-number arithmetic to group theory and number theory. Various

types of FFT algorithms are Cooley-Tukey algorithm, Prime-factor FFT

algorithm, Bruun's FFT algorithm, Rader's FFT algorithm, and

Bluestein's FFT algorithm.

Even though we have an advantage that FFT is very fast when

compared with other transforms, this is equally complex too and so we need

some simple but efficient transform.

2.1.3 WAVELET TRANSFORM

Fourier transform based spectral analysis is the dominant analytical

tool for frequency domain analysis. However, Fourier transform cannot

provide any information of the spectrum changes with respect to time.

Fourier transform assumes the signal is stationary, but PD signal is always

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non-stationary. To overcome this deficiency, a modified method-short time

Fourier transform allows representing the signal in both timeand frequency

domain through time windowing functions. The window length determines a

constant time and frequency resolution.

Thus, a shorter time windowing is used in order to capture the transient

behavior of a signal; we sacrifice the frequency resolution. The nature of the

real PD signals is non-periodic and transient. Such signals cannot easily be

analyzed by conventional transforms. So, an alternative mathematical tool-

wavelet transform must be selected to extract the relevant time-amplitude

information from a signal. Thus we go for wavelet.

A wavelet is a mathematical function used to divide a given function or

continuous-time signal into different scale components. Usually one can

assign a frequency range to each scale component. Each scale component

can then be studied with a resolution that matches its scale. A wavelet

transform is the representation of a function by wavelets. The wavelets arescaled and translated copies (known as "daughter wavelets") of a finite-

length or fast-decaying oscillating waveform (known as the "mother

wavelet"). Wavelet transforms have advantages over traditional Fourier

transforms for representing functions that have discontinuities and sharp

peaks, and for accurately deconstructing and reconstructing finite, non-

periodic and/or non-stationary signals.

In formal terms, this representation is a wavelet series representation of

a square integrable function with respect to either a complete, orthonormal

set of basis functions, or an over complete set of Frame of a vector space

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(also known as a Riesz basis), for the Hilbert space of square integrable

functions.

Wavelet transforms are classified into discrete wavelet transforms

(DWTs) and continuous wavelet transforms (CWTs). Note that both DWT

and CWT are continuous-time (analog) transforms. They can be used to

represent continuous-time (analog) signals. CWTs operate over every

possible scale and translation whereas DWTs use a specific subset of scale

and translation values or representation grid.

The word wavelet is due to Morlet and Grossmann in the early 1980s.

They used the French word ondelette, meaning "small wave". Soon it was

transferred to English by translating "onde" into "wave", giving "wavelet".

The WT of a signal x is calculated by passing it through a series of

filters. First the samples are passed through a low pass filter with impulse

response g resulting in a convolution of the two:

The signal is also decomposed simultaneously using a high-pass filter

h. These two filters results in wavelet coefficients.

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2.1.4 CONTOURLET TRANSFORM

The contourlet transform [2] is a directional transform which is capable

of capturing contour and fine details in an image. For one-dimensional

piecewise smooth signals, like scan lines of an image, wavelets have been

established as the right tool, because they provide an optimal representation

for these signals in a certain sense. In addition, the wavelet representation is

amenable to efficient algorithms; in particular it leads to fast transforms and

convenient tree data structures. These are the key reasons for the success of

wavelets in many signal processing and communication applications.

Although wavelet was good in 2-D in isolating the edge

discontinuities, itll not see the smoothness along the contours. In addition,

separable wavelets can capture only limited directional information an

important and unique feature of multidimensional signals. These

disappointing behaviors indicate that more powerful representations are

needed in higher dimensions.

To see how one can improve the 2-D separable wavelet transform for

representing images with smooth contours, the following scenario is

considered. Imagine that there are two painters, one with a wavelet-style

and the other with a new style, both wishing to paint a natural scene. Both

painters apply a refinement technique to increase resolution from coarse to

fine. Here, efficiency is measured by how quickly, that is with how few

brush strokes, and one can faithfully reproduce the scene.

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FIG 2.1 Wavelet versus contourlet: the successive refinement by the two

systems near a smooth contour, which is shown as a thick curve

separating two smooth regions

Consider the situation when a smooth contour is being painted, as

shown in Figure above. Because 2-D wavelets are constructed from tensor

products of 1-D wavelets, the wavelet- style painter is limited to using

square-shaped brush strokes along the contour, using different sizes

corresponding to the multiresolution structure of wavelets. As the resolution

becomes finer, we can clearly see the limitation of the wavelet style painter

who needs to use many fine dots to capture the contour. The new style

painter, on the other hand, explores effectively the smoothness of the

contour by making brush strokes with different elongated shapes and in a

variety of directions following the contour.

For the human visual system, it is well-known that the receptive fields

in the visual cortex are characterized as being localized, oriented, and band

pass. Furthermore, experiments in searching for the sparse components of

natural images produced basis images that closely resemble the

aforementioned characteristics of the visual cortex. This result supports the

hypothesis that the human visual system has been tuned so as to capture the

essential information of a natural scene using a least number of visual active

cells. More importantly, this result suggests that for a computational image

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representation to be efficient, it should be based on a local, directional, and

multiresolution expansion.

Inspired by the painting scenario and studies related to the human

visual system and natural image statistics, we identify a wish list for new

image representations:

Multi-resolution: The representation should allow images to besuccessively approximated, from coarse to fine resolutions.

Localization: The basis elements in the representation should belocalized in both the spatial and the frequency domains.

Critical sampling: For some applications (e.g., compression), therepresentation should form a basis, or a frame with small redundancy.

Directionality: The representation should contain basis elementsoriented at a variety of directions, much more than the few directions that

are offered by separable wavelets.

Anisotropy: To capture smooth contours in images, therepresentation should contain basis elements using a variety of elongated

shapes with different aspect ratios.

Among these properties, the first three are successfully provided by

separable wavelets, while the last two require new constructions. Moreover,

a major challenge in capturing geometry and directionality in images comes

from the discrete nature of the data: the input is typically sampled images

defined on rectangular grids. For example, directions other than horizontal

and vertical look very different on a rectangular grid. Because of

pixelization, the notion of smooth contours on sampled images is not

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obvious. For these reasons, unlike other transforms that were initially

developed in the continuous domain and then discretized for sampled data,

our approach starts with a discrete-domain construction and then studies its

convergence to an expansion in the continuous domain.

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CHAPTER 3

WAVELET AND CONTOURLET

DECOMPOSITION PROCESS

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3.1 WAVELET DECOMPOSITION

The Wavelet Transform is nothing but a system of filters. There are

two filters involved, one is the wavelet filter, and the other is the scaling

filter. The wavelet filter is a high pass filter, while the scaling filter is a

low pass filter.

FIG 3.1 Wavelet Transform Decomposition Process

Here both the outputs from the high and low pass filters are down

sampled and the output at each level result in wavelet coefficients.

The disadvantage of this method is that since the outputs are down

sampled, the lower frequency overlaps with the down sampled higher

frequency and this results in Frequency Scrambling.

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The general wavelet denoising procedure is as follows:

Apply wavelet transform to the noisy signal to produce the noisywavelet coefficients to the level which we can properly distinguish the PD

occurrence.

Select appropriate threshold limit at each level and threshold method(hard or soft thresholding) to best remove the noises.

Inverse wavelet transforms of the threshold wavelet coefficients toobtain a denoised signal.

3.2 CONTOURLET DECOMPOSITION

Comparing the wavelet scheme with the new scheme, the improvement

of the new scheme can be attributed to the grouping of nearby wavelet

coefficients, since they are locally correlated due to the smoothness of the

contours. Therefore, we can obtain a sparse expansion for natural images by

first applying a multiscale transform, followed by a local directional

transform to gather the nearby basis functions at the same scale into linear

structures. In essence, we first use a wavelet-like transform for edge

detection, and then a local directional transform for contour segment

detection.

With this insight, we considered a double filter bank [1] structure for

obtaining sparse expansions for typical images having smooth contours. In

this double filter bank, the Laplacian pyramid [4] is first used to capture the

point discontinuities, and then followed by a directional filter bank [1] to

link point discontinuities into linear structures. The overall result is an image

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expansion using basic elements like contour segments, and thus is named

contourlets. In particular, contourlets have elongated supports at various

scales, directions, and aspect ratios. This allows contourlets to efficiently

approximate a smooth contour at multiple resolutions in much the same way

as the new scheme.

FIG 3.2 Contourlet Transform Decomposition Process

In the frequency domain, the contourlet transform [2] provides a

multiscale and directional decomposition. We would like to point out that

the decoupling of multiscale and directional decomposition stages offers a

simple and flexible transform, but at the cost of a small redundancy (up to

33%, which comes from the Laplacian pyramid). The above figure illustrates

the Contourlet decomposition process, where LL (Low Low), LH (Low

High), HL (High Low) and HH (High High) are frequency components of

the input noisy image. Laplacian Pyramid [4] separates the Low frequencycomponents (LL) and the High frequency components using Low pass filters

and Band pass filters respectively. The output of the band pass filters are

processed by Directional filter banks [1].

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3.2.1 PYRAMID FRAMES

FIG 3.3 Laplacian pyramids (a) one level of decomposition. The outputs

are the coarse approximation a[n] and difference b[n] between the

original signal and the prediction (b) a new reconstruction scheme for

the Laplacian pyramid

One way to obtain a multiscale decomposition is to use the Laplacian

pyramid (LP) [4]. The LP decomposition at each level generates a down

sampled low pass version of the original and the difference between the

original and the prediction, resulting in a band pass image. The Laplacian

pyramid is for the multiscale decomposition [1], where H and G are called

(low pass) analysis and synthesis filters, respectively, and M is the sampling

matrix. The process can be iterated on the coarse (down sampled low pass)

signal. In multidimensional filter banks, sampling is represented by sampling

matrices; for example, down sampling x[n] by M yields xd[n] = x[Mn],

where M is an integer matrix. A drawback of the LP is the implicit over

sampling. However, in contrast to the critically sampled wavelet scheme, the

LP has the distinguishing feature that each pyramid level generates only one

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band pass image (even for multidimensional cases), and this image does not

have scrambled frequencies. This frequency scrambling happens in the

wavelet filter bank when a high pass channel, after down sampling, is folded

back into the low frequency band, and thus its spectrum is reflected. In the

LP, this effect is avoided by down sampling the low pass channel only.

3.2.2 ITERATED DIRECTIONAL FILTER BANKS

FIG 3.4 2-D Directional Filter Banks

Considering a 2-D directional filter bank (DFB) [3] that can be

maximally decimated while achieving perfect reconstruction. The DFB is

efficiently implemented via an l-level binary tree decomposition that leads to

2l sub bands with wedge-shaped frequency partitioning. The original

construction of the DFB involves modulating the input image and using

quincunx filter banks [3] with diamond-shaped filters. To obtain the desired

frequency partition, a complicated tree expanding rule has to be followed for

finer directional sub bands.

LAPLACIAN PYRAMID

INPUT IMAGE

DIRECTIONAL FILTER BANK

QUINCUNX FILTER

BANK

SHEARING OPERATOR

CONTOURLET

TRANSFORMED IMAGE

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FIG 3.5 Directional filter bank. Frequency partitioning where l = 3 and

there are 2 3 = 8 real wedge shaped frequency bands. Subbands 0-3

correspond to the mostly horizontal directions, while subbands 4-7

correspond to the mostly vertical directions

Considering a new construction for the DFB which avoids modulation

of input image and uses a simpler rule for expanding the decomposition tree.

This simplified DFB is intuitively constructed from two building blocks.

The first building block is a two-channel quincunx filter bank with fan filters

that divides a 2-D spectrum into two directions: horizontal and vertical.

FIG 3.6 Two-dimensional spectrum partition using quincunx filter

banks with fan filters. The black regions represent the ideal frequency

supports of each filter. Q is the quincunx sampling matrix

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The second building block of the DFB is a shearingoperator, which

amounts to just reordering of image samples. Figure below shows an

application of a shearing operator where a 45 direction edge becomes a

vertical edge.

FIG 3.7 Example of shearing operation that is used like a rotation

operation for the DFB decomposition. (a) The Cameraman image (b)

the Cameraman image after shearing operation

By adding a pair of shearing operator and its inverse (unshearing) to

before and after, respectively, a two channel filter bank, we obtain a

different directional frequency partition while maintaining perfect

reconstruction. Thus, the key in the DFB is to use an appropriate

combination of shearing operators together with two-direction partition of

quincunx filter banks at each node in a binary tree-structured filter bank, to

obtain the desired 2-D spectrum division.

Using multirate identities, it is instructive to view an l-level tree-

structured DFB equivalently as a 2l parallel channel filter bank with

equivalent filters and overall sampling matrices. These equivalent

(directional) synthesis filters are denoted as Dk(l)

, 0 k < 2l, which

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correspond to the sub bands. The corresponding overall sampling matrices

were shown to have the following diagonal forms

Which means sampling is separable. The two sets correspond to the

mostly horizontaland mostly verticalset of directions, respectively.

3.2.3 MULTISCALE AND DIRECTIONAL DECOMPOSITION

THE DISCRETE CONTOURLET TRANSFORM

Combining the Laplacian pyramid and the directional filter bank, the

combination into a double filter bank structure [3] can be described. Since

the directional filter bank (DFB) was designed to capture the high frequency

(representing directionality) of the input image, the low frequency content is

poorly handled. In fact, with the frequency partition, low frequency would

leak into several directional sub bands, hence the DFB alone does not

provide a sparse representation for images. This fact provides another reason

to combine the DFB with a multiscale decomposition, where low

frequencies of the input image are removed before applying the DFB.

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FIG 3.8 The contourlet filter bank: first, a multiscale decomposition

into octave bands by a Laplacian pyramid is computed, and then a

directional filter bank is applied to each band pass channel

This shows a multiscale and directional decomposition using a

combination of a Laplacian pyramid (LP) and a directional filter bank

(DFB). Band pass images from the LP are fed into a DFB so that directional

information can be captured. The scheme can be iterated on the coarse

image. The combined result is a double iterated filter bank structure, named

contourlet filter bank, which decomposes images into directional sub bands

at multiple scales.

Specifically, let a0 [n] be the input image. The output after the LP stage

is J band pass images bj [n], j = 1, 2. . . J (in the fine-to-coarse order) and a

low pass image aj [n]. That means, the j-th level of the LP decomposes the

image aj1 [n] into a coarser image aj [n] and a detail image bj [n]. Each

band pass image bj [n] is further decomposed by an lj -level DFB into 2lj

band pass directional images c(lj ) j, k [n], k = 0, 1, . . . , 2lj 1.

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3.2.4 THEOREM

In a contourlet filter bank, the following hold:

1) If the LP and the DFB use perfect-reconstruction filters, then the

discrete contourlet transform achieves perfect reconstruction, which means it

provides a frame operator.

2) If the LP and the DFB use orthogonal filters, then the discrete

contourlet transform provides a tight frame with frame bounds equal to 1.

3) The discrete contourlet transform has a redundancy ratio that is less

than 4/3.

4) Suppose an lj -level DFB is applied at the pyramidal level j of the

LP, then the basis images of the discrete contourlet transform (i.e. the

equivalent filters of the contourlet filter bank) have an essential support size

of width C2j

and length C2j+lj2

.

5) Using FIR filters, the computational complexity of the discrete

contourlet transform is O (N) for N-pixel images.

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3.2.5 PROOF

1) This is obvious as the discrete contourlet transform [2] is a

composition of perfect-reconstruction blocks.

2) With orthogonal filters, the LP [4] is a tight frame with frame

bounds equal to 1, which means it preserves the l2-norm,

or . Similarly, with orthogonal filters the DFB

is an orthogonal transform, which means .

Combining these two stages, the discrete contourlet transform satisfies thenorm preserving or tight frame condition.

3) Since the DFB is critically sampled, the redundancy of the discrete

contourlet transform is equal to the redundancy of the LP, which is

4) Using multirate identities, the LP band pass channel corresponding

to the pyramidal level j is approximately equivalent to filtering by a filter of

size about C12j

C12j, followed by down sampling by 2j1 in each

dimension. For the DFB, it is noted that after lj levels (lj 2) of tree-

structured decomposition, the equivalent directional filters have support of

width about C22 and length about C22lj1

. Combining these two stages, again

using multirate identities, into equivalent contourlet filter bank channels, we

see that contourlet basis images have support of width about C2j

and length

about C2j+lj2

.

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5) Let LP and Ld be the number of taps of the pyramidal and

directional filters used in the LP and the DFB, respectively (without loss of

generality we can suppose that low pass, high pass, analysis and synthesis

filters have same length). With a polyphase implementation, the LP filter

bank requires Lp/2+1 operation per input sample. Thus, for an N- pixel

image, the complexity of the LP stage in the contourlet filter bank is:

For the DFB, its building block two-channel filter banks require Ld

operations per input sample. With l-level full binary tree decomposition, the

complexity of the DFB multiplies by l. This holds because the initial

decomposition block in the DFB is followed by two blocks at half rate, four

blocks at quarter rate and so on. Thus, the complexity of the DFB stage foran N-pixel image is:

Thus we obtain the desired result. Since the multiscale and directionaldecomposition stages are decoupled in the discrete contourlet transform, we

can have a different number of directions at different scales, thus offering a

flexible multiscale and directional expansion. Moreover, the full binary tree

decomposition of the DFB in the contourlet transform can be generalized to

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arbitrary tree structures, similar to the wavelet packets generalization of the

wavelet transform. The result is a family of directional multiresolution

expansions, which we call contourlet packets. The examples of possible

frequency decompositions by the contourlet transform and contourlet packets

are shown. In particular, contourlet packets allow finer angular resolution

decomposition at any scale or direction, at the cost of spatial resolution. In

addition, from Theorem 1, it is noted that by altering the depth of the DFB

decomposition tree at different scales (and even at different orientations in a

contourlet packets transform), a rich set of contourlets with variety of support

sizes and aspect ratios are obtained. This flexibility allows the contourlet

transform and the contourlet packets to fit smooth contours of various

curvatures well.

FIG 3.9 Examples of possible frequency decomposition by the

contourlet transform and contourlet packets

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CHAPTER 4

CONTOURLET DENOISING

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4.1 CONTOURLET DENOISING

A common approach for image denoising is to convert the noisy image

into a transform domain such as the wavelet and Contourlet domain, and

then compare the transform coefficients with a fixed threshold. This

algorithm performs a hypothesis test to determine if a pixel is corrupted or

not, thus not depending on a fixed threshold.

As the first step, a noisy image is transformed into the contourlet

domain by the contourlet decomposition [5]. Then, for each coefficient,

variance for the associated Gaussian distribution is estimated. Subsequently,

hypothesis tests are performed with the variance noisy image. The

coefficients that are obtained are then compared with a given threshold and

then determined whether it is corrupted or not. If it is, then it is determined

to fall into a smooth region and may be processed for noise suppression. If it

is not, then it should stand for an image feature pixel and be preserved.

Afterwards, the processed contourlets are utilized to reconstruct the

image, which is the final denoised output.

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The general block of the denoising part can be represented as,

FIG 4.1 Denoising Process Using Contourlet Transformation

C T U T D C P ITI

I C TI TI

T DI G

F TU

P TI

I

UPP I

C T U T

C T UCTI

I I G

D I D I G

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4.2 ALGORITHM DESCRIPTION

1. The process starts with the noisy image.

2. This image is converted into CT domain, using the decomposition process. In Wavelet, we determine the coefficients using scaling and a

wavelet filter. But, in Contourlet, we construct discrete-domain

multiresolution and multi direction using non-separable filter banks, in the

same way wavelets are obtained from the filter banks. This construction

results in flexible multiresolution, local and directional image expansion

using contour sectors and so named Contourlet transform.

3. Thus from the decomposition process the coefficients are determined.

4. Then for each noisy image pixels, the variance is estimated.

5. The resultant values are then compared with a threshold value andthen determine whether the pixel is corrupted or not.

6. If the pixel is corrupted, it is suppressed or modified. Else it is notcorrupted, the pixels then preserved.

7. Then all the resultant coefficients are reconstructed which results indenoised image.

8. After reconstruction the SNR and the IEF values are calculated forthis algorithm.

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4.3 THRESHOLDING

Generally for denoising, the coefficients of the noisy image are compared

with the threshold value. These threshold values are either obtained by the trial

and error method, or by considering some standard method. Since human eyes

are very sensitive to intensity of neighboring pixel values, in image denoising

techniques the variance between the considered pixel and the neighboring pixel

values must be less. Considering the threshold values depending on the

variance, the noise level in the corrupted image still decreases.

In this algorithm, a threshold value is set based upon the variance of the

corrupted image. Based upon the results from various variance levels (nvar),

the threshold is fixed. The intensity of the noise being added to the image (th)

and the standard deviation of the noise less image (sigma) are also the deciding

factors in fixing the threshold values.

Based upon the various results obtained, we deduce that the threshold

values must be fixed depending upon the high noise level and low noise level.

In Speckle noise, the default variance level is 0.04, so considering Speckle

noise variance (nvar) above 0.05 as high noise level and below 0.05 as low

noise level, we introduced two threshold values separately.

Now to reconstruct the image, the coefficients above the threshold

values are retained for Contourlet reconstruction and the remaining noisy

coefficients are suppressed. The retained coefficients are reconstructed to

obtain the Denoised image. This algorithm is very simpler and effective

compared to other algorithms.

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CHAPTER 5

RESULTS AND DISCUSSION

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5.1 RESULTS AND DISCUSSION

5.1.1 RESULTS FOR MULTIPLICATIVE (SPECKLE) NOISE

Images SNR(dB) IEF

Wavelet Proposed

algorithm using

Contourlet

Transform

Wavelet Proposed

algorithm using

Contourlet

Transform

NOISE LEVEL=0.03dB

LENA 9.44 11.77 2.01 3.45

PEPPER 10.73 11.36 2.10 2.43

SATELLITE 8.15 10.22 1.83 2.95

MEDICAL 13.36 15.73 3.02 5.25

BARBARA 10.04 9.81 1.65 1.57

NOISE LEVEL=0.04dB

LENA 7.94 10.95 1.87 3.75

PEPPER 9.21 10.68 1.97 2.75

SATELLITE 6.65 9.34 1.69 3.16

MEDICAL 11.55 14.81 2.66 5.62

BARBARA 8.84 9.39 1.66 1.88NOISE LEVEL=0.06dB

LENA 5.75 10.01 1.66 4.44

PEPPER 7.02 9.65 1.75 3.19

SATELLITE 4.47 8.40 1.52 3.76

MEDICAL 8.84 14.56 2.13 8.03

BARBARA 6.99 8.67 1.58 2.33

NOISE LEVEL=0.1dB

LENA 3.15 8.20 1.47 4.75

PEPPER 4.44 8.56 1.55 3.97

SATELLITE 1.95 6.48 1.37 3.88

MEDICAL 5.65 12.00 1.71 7.36

BARBARA 4.56 7.66 1.45 2.96

TABLE 5.1 Comparison between Wavelet and Contourlet Transforms

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5.1.2 QUALITATIVE RESULTS

FIG 5.1 Results of Lena image for lower speckle noise level (.03)

The input image being selected here is the Lena image and when the

noise level of the speckle is 0.03, the results obtained are such that, the SNR

of the corrupted image is 6.41dB, while for wavelet denoising the SNR and

IEF obtained are 9.44dB and 2.01 respectively. But for the proposed

algorithm, the SNR and the IEF obtained are 11.74dB and 3.42 respectively,

which clearly proves that this new algorithm is more efficient comparatively.

If the SNR (Signal to Noise Ratio) value is high, it means that more

amount of information is extracted from the noisy image and from the

results we obtained for this corrupted image, SNR is high for the new

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algorithm using Contourlet transformation, and hence it extracts more

information.

If the IEF (Image Enhancement Factor) value is high, it means that

more edges are preserved and the information I those edge regions also can

be extracted and here too the new algorithm out performs the wavelet.

We can also infer that the image we got after Contourlet

transformation is better than Wavelet in Visual perspective as well.

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FIG 5.2 Results of Lena image for speckle noise level 0.04

The image to be denoised is the Lena image with the speckle noise

level 0.04 that is added and the SNR value of noise is 5.21dB. The SNR

and IEF values for wavelet denoising are 7.91dB and 1.86 respectively and

for the proposed algorithm, results are 10.09dB and 3.79 respectively

proving that Contourlet based new algorithm is better than the wavelet

algorithm.

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FIG 5.3 Results of Lena image for speckle noise level 0.06


level 0.06 that is added and the SNR value of noise is 3.53dB. The SNR

and IEF values for wavelet denoising are 5.73dB and 1.66 respectively and

for the proposed algorithm, results are 9.99dB and 4.44 respectively

proving that Contourlet based new algorithm is better than the wavelet

algorithm.

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FIG 5.4 Results of Lena image for speckle with high noise level 0.1


level 0.1 that is added and the SNR value of noise is 1.46dB. The SNR and

IEF values for wavelet denoising are 3.15dB and 1.47 respectively and for

the proposed algorithm, results are 8.08dB and 4.62 respectively proving

that Contourlet based new algorithm is better than the wavelet algorithm.

Visually too we can say that Contourlet Denoised image is far better than the

Wavelet transform.

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FIG 5.5 Results of Satellite image for speckle with noise level 0.03

Here we take Satellite image as the test image, in which contours

should be more clearer for gathering information and the qualitative results

are also equally important. Here also the proposed algorithm based on

Contourlet is better with the results, SNR and IEF are 10.32dB and 3.00

respectively while for wavelet SNR and IEF are 8.22dB and 1.85

respectively.

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FIG 5.7 Results of Satellite image for speckle with noise level 0.06





respectively while for wavelet SNR and IEF are 4.50dB and 1.52

respectively.

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FIG 5.8 Results of Satellite image for speckle with high noise level 0.1





respectively while for wavelet SNR and IEF are 1.94dB and 1.36respectively.

So its been evident with the results that this proposed algorithm based

on Contourlet transform is best suit for Satellite image applications.

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FIG 5.9 Results of Pepper image for speckle with noise level 0.06

Here we take Pepper image as the test image, in which the edges are

more, where the edges after denoising should be preserved. Here also the

proposed algorithm based on Contourlet is better with the results, SNR and

IEF are 9.62dB and 3.17 respectively while for wavelet SNR and IEF are

7.06dB and 1.76 respectively. Since IEF factor is high in the proposed

algorithm, we deduce that this algorithm can effectively replace the wavelet

algorithm.

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FIG 5.10 Results of Barbara image for speckle with high noise level 0.1

Here we take Barbara image as the test image, which is one of the

standard images used in Image Processing. Considering the results obtained,

the proposed algorithm based on Contourlet is better with, SNR and IEF as

7.71dB and 3.00 respectively while for wavelet SNR and IEF are 4.56dB

and 1.45 respectively.

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FIG 5.11 Results of Medical image for speckle with low noise level 0.03

Here we take Medical image as the test image, which needs more

accuracy and high SNR in medical applications for saving a life. With this

new proposed algorithm based on Contourlet which has high SNR and IEF

as 15.77dB and 5.26 respectively where as for wavelet SNR and IEF are

13.35dB and 3.01 respectively.

So its been evident that the former transform will be more helpful in

medical applications as well.

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CHAPTER 6

MATLAB-GUI

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6.1 MATLAB FUNCTIONS

1. imreadloads the image.2. imshow displays the image.3. pdfbdec performs decomposition process.4. pdfb2vec converts into vector and this command results in

coefficients.

5. pdfbrec performs the reconstruction process.

6.2 MATLAB GUI (GRAPHICAL USER INTERFACE)

6.2.1 INTRODUCTION

The main reason GUIs are used is because it makes things simple for

the end-users of the program. If GUIs were not used, people would have to

work from the command line interface, which can be extremely difficult and

frustrating. A simple GUI that will add together two numbers, displaying the

answer in a designated text field is discussed.

FIG 6.1 Simple GUI that adds 2 numbers

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6.2.2 INITIALIZING GUIDE (GUI CREATOR)

Commandguide is typed in command window.

FIG 6.2 Commandguide is typed in command window

The first option Blank GUI is selected.

FIG 6.3The first option Blank GUI is selected

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The following screen will be shown.

FIG 6.4 The GUI Editor Screen

Before adding components blindly, it is good to have a rough idea

about how you want the graphical part of the GUI to look like so that itll be

easier to lay it out. The finished GUI will look like following

FIG 6.5 Sample GUI model

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CONCLUSION

Hence the new proposed algorithm based on the Contourlet

transformation is found to be more efficient than the wavelet algorithm in

Image Denoising particularly for the removal of speckle noise. This is

concluded based on considering test images like Lena, Barbara, Peppers

along with Satellite images and Medical images after corrupting with

Speckle noise which is a multiplicative noise of various noise levels like

0.03, 0.04, 0.06 and 0.1. Thus the obtained results in qualitative and

quantitative analysis shows that this proposed algorithm outperforms the

wavelet in terms of SNR, IEF values and visual perspective as well. The

algorithms are implemented using MATLAB 7.5 R2007b.

FUTURE WORK

It has been theoretically studied and proved that new algorithm using

the Contourlet Transformation is better in Denoising and other Image

Processing operations. This can be implemented using hardware also and the

feasible hardware is using VLSI with high memory.

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REFERENCES

1. Chan W.Y, N.F.Law, W.C.Siu Multiscale Feature Analysis usingDirectional Filter Bank.

2. M. N. Do and M. Vetterli, Contourlets: a Directional Multiresolutionimage representation, in Proceedings of 2002 IEEE International

Conference on Image Processing.

3. M. N. Do and M. Vetterli, Oct. 2001 Pyramidal directional filterbanks and curvelets, in Proc. IEEE Int. Conf. on Image Proc.,

4. P. J. Burt and E. H. Adelson, April 1983 The Laplacian pyramid as acompact image code,IEEETrans. Commun., vol. 31, no. 4, pp. 532

540.

5. Zhiling Longa,b and Nicolas H. Younana Denoising of images withMultiplicative Noise Corruption

6. www.mathworks.com

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