image processing and recognition

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PROJECT REPORT, JUNE 2012 IMAGE DENOISING AND COMPRESSION USING ADAPTIVE

WAVELET THRESHOLDING

Dept. OF ECE , SBCE 1

CHAPTER 1

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 signalfeatures. Traditionally, this is achieved by linear processing such as Wiener filtering. From a

historical point of view, wavelet analysis is a new method, though its mathematical date back to

the work of Joseph Fourier in the nineteenth century. Fourier laid the foundations with his

theories of frequency analysis, which proved to be enormously important and influential. The

attention of researchers gradually turned from frequency-based analysis to scale-based analysis

when it started to become clear that an approach measuring average fluctuations at different

scales might prove less sensitive to noise. The first recorded mention of what we now call a

wavelet seems to be in 1909, in a thesis by Alfred Haar. In the late nineteen-eighties, when

Daubechies and Mallat first explored and popularized the ideas of wavelet transforms, skeptics

described this new field as contributing additional useful tools to a growing toolbox of

transforms.The inquiring skeptic, however maybe reluctant to accept these claims based on

asymptotic theory without looking at real-world evidence.Fortunately, there is an increasing

amount of literature now addressing these concerns that help us appraise of the utility of wavelet

shrinkage more realistically.Wavelet denoising attempts to remove the noise present in the signal

while preserving the signal characteristics,regardless of its frequency content. It involves three

steps: a linear forward wavelet transform, nonlinear thresholding step and a linear inverse

wavelet transform.Wavelet denoising must not be confused with smoothing; smoothing only

removes the high frequencies and retains the lower ones.Wavelet shrinkage is a non-linear

process and iswhat distinguishes it from entire linear denoising technique such as least squares.

As will be explained later, wavelet shrinkage depends heavily on the choice of a thresholding

parameter and the choice of this threshold determines, to a great extent the efficacy of denoising.

Researchers have developed various techniques for choosing denoising parameters and so far

there is no best universal threshold determination technique.A more precise explanation of the

wavelet denoising procedure can be given as follows. Assume that the observed data is X(t) =

S(t) + N(t) where S(t) is the uncorrupted signal with additivenoise N(t).Let W(.) and W-1

(.)

denotes the forward and inverse of wavelet transform transform. Let D(Y, ) denote the


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denoising operator with threshold. We intend to denoise X(t) to recover (t) as an estimate ofS(t). The procedure can be summarized in three steps.

Y=W(X)

Z=D(Y,)

=W-1(Z)D(Y,) being the threshold operator and being the threshold. Thresholding is a simple non-

lineartechnique, 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 on the result may lead to reconstruction with the essential

signal characteristics and with less noise.

The aim of this project is to study various thresholding techniques such as Sureshrink,

Visushrink and BayeShrink and determine the best one for image denoising. In the course of the

project,we also aimed to use wavelet denoising as a means of compression and were successfully

able to implement a compression technique based on a unified denoising and compression

principle. Two main issues regarding image denoising were addressed in this project. Firstly, an

adaptive threshold for wavelet thresholding images was proposed, based on the

GGD(Generalized Gaussian Response) modeling of subband coefficients, and test results

showed excellent performance. Secondly, a coder was designed specifically for simultaneouscompression and denoising.


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CHAPTER 2

LITERATURE SURVEY

A vast literature has emerged recently on signal denoising using nonlinear techniques, inthe setting of additive white Gaussian noise.. The seminal work on signal denoising via wavelet

thresholding or shrinkage of Donoho and Johnstone ([13],[14],[15],[16]) 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 in Besov

spaces ([13], [14]). 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,[1], [4], [8],[9],[10], [12], [18], [19], [24], [27],[28],[29],

[32], [33],[35], and references therein. Both theoretical and experimental results indicate that our

choice of shrinkage parameters yields uniformly better results than Donoho and Johnstone's

VisuShrink procedure; an example suggests, however, that Donoho and Johnstone's (1994, 1995,

1996) SureShrink method, which uses a different shrinkage parameter for each dyadic level,

achieves a lower error.

On a seemingly unrelated front, lossy compression has been proposed for denoising in

several works [6], [5], [21], [25],[28]. Concerns regarding the compression rate were explicitly

addressed. This is important because any practical coder must assume a limited resource (such as

bits) at its disposal for representing the data. Building on this works, explain about why

compression (via coefficient quantization) is appropriate for filtering noise from signal by

making the connection that quantization of transform coefficients approximates the operation of

wavelet thresholding for denoising. That is, denoising is mainly due to the zero-zone and that the

full precision of the thresholded coefficients is of secondary importance. The method of

quantization is facilitated by a criterion similar to Rissanen's minimum description length

principle. An important issue is the threshold value of the zero-zone (and of wavelet

thresholding). For a natural image, it has been observed that its subband coefficients can be well

modeled by a Laplacian distribution. With this assumption, we derive a threshold which is easy

to compute and is intuitive. Experiments show that the proposed threshold performs close to


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optimal thresholding.Other works [4],[12],[13],[14],[15],[16] also addressed the connection

between compression and denoising, especially with nonlinear algorithms such as wavelet

thresholding in a mathematical framework. However, these latter works were not concerned with

quantization and bitrates: compression results from a reduced number of nonzero wavelet

coefficients, and not from an explicit design of a coder. 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. The emergence of wavelets has led to a convergence of

linear expansion methods used in signal processing and applied mathematics. In particular,

subband coding methods and their associated filters are closely related to wavelet constructions

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.

The theoretical formalization of filtering additive Gaussian noise (of zero-mean and

standard deviation) via thresholding wavelet coefficients was pioneered by Donoho and

Johnstone[14]. 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


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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 are

considered negligible. Outside of this region, the thresholded coefficients are kept to full

precision (that is, without quantization).Their most well-known thresholding methods include

VisuShrink [14] and SureShrink [15]. These threshold choices enjoy asymptotic minimax

optimalities over function spaces such as Besov spaces.

For image denoising, however, VisuShrink is known to yield overly smoothed images.

This is because its threshold choice (called the universal threshold and 2 is the noisevariance), can be unwarrantedly large due to its dependence on the number of samples,M ,which

is more than 105

for a typical test image of size 512*512 SureShrink uses a hybrid of the

universal threshold and the SURE threshold, derived from minimizing Steins unbiased risk

estimator [30], and has been Shown toper formwell.SureShrink will be themain comparison to

the method proposed here, and, as will be seen later in this paper,our proposed threshold often

yields better result.

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 groundedon the empirical observation that the wavelet coefficients in a subband of a natural image can be

summarized adequately by a Generalized Gaussian Distribution (GGD). This observation is well-

accepted in the image processing community (for example, see [20], [22], [23],[29], [34], [36])

and is used for state-of-the-art image coding [20], [22], [36]. It follows from this observation that

the average MSE (in a subband) can be approximated by the corresponding Bayesian squared

error risk with the GGD. 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

aredraws from a GGD. The goal is to find the soft-threshold that minimizes this Bayesian risk,

and we call our method BayesShrink.

The proposed Bayesian risk minimization is subband-dependent.Given the signal being

generalized Gaussian distributed and the noise being Gaussian, via numerical calculation a


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nearly optimal threshold for soft-thresholding is found to be TB=2/x(where 2

is the noise

variance and x the signal variance). This threshold gives a risk within 5% of the minimal risk

over a broad range of parameters in the GGD family. To make this threshold data-driven, the

parameters 2and x are estimated from the observed data, one set for each subband. To achieve

simultaneous denoising and compression, the nonzero thresholded wavelet coefficients need to

be quantized.Uniform quantizer and centroid reconstruction is used on the GGD. The design

parameters of the coder, such as the number of quantization levels and binwidths,are decided

based on a criterion derived from Rissanens minimum description length(MDL) principle [26].

While achieving mean-squared -error performance comparable with other popular thresholding

schemes,the MDL procedure tends to keep far fewer coefficients. From this property, we

demonstrate that our method is an excellent tool for simultaneous denoising and compression .

This criterion balances the tradeoff between the compression rate and distortion, and yields a

nice interpretation of operating at a fixed slope on the rate-distortion curve.

.


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CHAPTER 3

INTRODUCTION TO IMAGE DENOISING

3.1 THRESHOLDING

3.1.1 Motivation for Wavelet thresholding

The plot of wavelet coefficients in Fig.3.2 suggests that small coefficients are dominated

by noise, while coefficients with a large absolute value carry more signal information than noise.

Replacing noisy coefficients (small coefficients below a certain thresholdvalue) by zero and an

inverse wavelet transform may lead to a reconstruction that has lesser noise. Stated more

precisely, we are motivated to this thresholding idea based on the following assumptions:

The decorrelating property of a wavelet transform creates a sparse signal: most untouched

coefficients are zero or close to zero.

Noise is spread out equally along all coefficients.

The noise level is not too high so that we can distinguish the signal wavelet coefficients from

the noisy ones.

Fig.3.1 A noisy signal in time domain


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Fig 3.2.The same signal in wavelet domain.

As it turns out, this method is indeed effective and thresholding is a simple and efficient

method for noise reduction. Further, inserting zeros creates more sparsity in the wavelet domain

and here we see a link between wavelet denoising and compression which has been described in

sources.

3.1.2 Hard and soft thresholding

Hard and soft thresholding with threshold are defined as follows.The hard thresholding

operator is defined as

D(U,)=U for all |U|>

=0 otherwise

The soft thresholding operator on the other hand is defined as

D(U,)=sgn(U) max(0,|U|-)

Hard threshold is a keep or kill procedure and is more intuitively appealing. The transfer

function of the same is shown in Fig 3.3. The alternative, soft thresholding (whose transfer

function is shown in Fig 3.4 ), shrinks coefficients above the threshold in absolute value. While

at first sight hard thresholding may seem to be natural, the continuity of soft thresholding has

some advantages.


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Fig 3.3 Hard thresholding

Fig 3.4 Soft thresholding

It makes algorithms mathematically more tractable . Moreover, hard thresholding does not even

work with some algorithms. Sometimes, pure noise coefficients may pass the hard threshold and

appearas annoying blips in the output. Soft thesholding shrinks these false structures.

3.1.3 Threshold determination

As one may observe, threshold determination is an important question when denoising. A

small threshold may yield a result close to the input, but the result may still be noisy. A large

threshold on the other hand, produces a signal with a large number of zero coefficients. This

leads to a smooth signal.Paying too much attention to smoothness, however,destroys details and

in image processing may cause blur and artifacts.

3.1.4 Comparison with Universal threshold

The threshold univ= (N being the signal length, 2 being the noise variance) iswell known in wavelet literature as the Universal threshold. It is the optimal threshold in the


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asymptotic sense and minimises the cost function of the difference between the function and the

soft thresholded version of the same in the L2norm sense. In our case, N=2048,=1 therefore

theoretically

univ=

= 3.905 Eqn.3.1

It seems that the universal threshold is not useful to determine a threshold. However,it is useful

for obtain a starting value when nothing is known of the signal condition. One can surmise that

the universal threshold may give a better estimate for the soft threshold if the number of samples

are larger (since the threshold is optimal in the asymptotic sense).

3.2 EDGE PRESERVING DENOISING

Denoising is a fundamental step in many image processing tasks. Linear methods have

been very popular for their simplicity and speed but their usage is limited since they tend to blur

images. Nonlinear methods are more time consuming but they perform much better in general.

There are different nonlinear denoising methods. The key idea is to perform anisotropic

diffusion as oppose to isotropic diffusion done by linear methods. Nonlinear methods behave

differently depending on the image content. Close to edges they diffuse along the edges but not

across and in smooth areas perform standard isotropic diffusion. Thus nonlinear methods remove

noise and simultaneously preserve edges.

3.3 NOISE MODELS

The principle sources of noise in digital images arise during image acquisition and/or

transmission.The perfomance of imaging sensors is affected by a veriety of factors,such as

environment conditions during image acuisition, and by the quality of the sencing elements

themselves.For instance, in acquiring images with CCD camera,light levels and sensor

temperature are major factors affecting the amount of noise in the resulting image.Images are

corrupted during transmission principally due to interference in the channel used for

transmission.For example,an image transmitted using a wireless network might be corrupted as a

result of lighting or other atmospheric disturbance.


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3.3.1 Spatial and Frequency Properties of Noise

Relevant to our discussion are parameter that define the spatial characteristics of noise,

and whether the noise is correlated with the image. Frequency properties refer to the frequency

content of noise in the Fourier sense. For example, when the Fourier spectrum of noise is

constant, the noise usually is called white noise. This terminology is a carryover from the

physical properties of white light, which contains nearly all frequency in the visible spectrum in

equal proportions. It is not difficult to show that the Fourier spectrum of a function containing all

frequencies in equal proportions is a constant.

3.3.2 Some Important Noise Probability Density Functions

Based on the assumptions in the previous section, The spatial noise descriptor with which

we shall be concerned is the statistical behavior of the intensity values in the noise component.

These may be considered random variable, characterized by a probability density

function(PDF).The following are among the most common PDFs found in image processing

applications.

Gaussian noise

Because of its mathematical tractability in both spatial and frequency domains, Gaussian(also

called normal) noise models are used frequently in practice. In fact. this tractability is so

convenient that it often results in Gaussian models being used in situations in which they

marginally applicable at best.The PDF of a Gaussian random variable,z,is given by

P(z) =

2/22 Eqn.3.1Where z represents intensity, is the mean(average) value of z and is its standard deviation.The standard deviation squared,2is called the variance of z.

Reyleigh noise

The PDF of Reyleigh noise is given by

P(z) { Eqn.3.2The mean and variance of this density are given by


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a+ Eqn.3.3and

2 =

Eqn.3.4

Erlang (gamma) noise

The PDF of Erlang noise is given by

P(z) =

Eqn.3.5

Where the parameter are such that a>0,b is a positive integer, and ! indicates factorial. The

mean and variance of this density are given by

b/aAnd

2 = b/a2

Exponential noise

The PDF of exponential noise is given by

P(z) { Eqn.3.6Where a >0,The mean and variance of this density function are

1/aAnd

2 = 1/a2

Note that this PDF is a special case of the erlang PDF,with b=1.

Uniform noise

The PDF of uniform is given by

P(z) { Eqn.3.7The mean and variance is given by

a+b/2And


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2 = (b-a)2/12

Impulse(salt and pepper)noise

The PDF of (bipolar) impulse noise is given by

P(z) = Eqn.3.8If b>a, intensity b will appear as a light dot in the image.Conversely,level a will appear

like a dark dot. If either pa or pb is zero, the impulse noise is called unipolar.If neither

probability is zero and especially if they are approximately equal, impulse noise values will

resemble salt and pepper noise. Data dropout and Spike noise also terms used to refer to this type

of noise. We use terms impulse or salt and pepper noise interchangeably.

3.4 NOISE IN NATURAL COLOR PHOTOS

With the surging popularity of digital cameras, digital photography is rapidly replacing

the traditional photography as the photography of choice for virtually all but a few devoted

professionals. In digital photography, post image processing is an integral part for obtaining

better images even for the casual picture takers. Post image processing is especially important for

people who are willing to go beyond point-and-shoot, and one of the key steps in image

processing is denoising.All digital cameras today take color photos. (Some cameras allow for

black-and-white images, but these are converted from color images using in-cameras.) Noise is

present in virtually all digital photos, and there are several sources for it. When light (photons)

strike the image sensor, electrons are produced. These \photoelectrons" give rise to analog

signals which are then converted into digital pixels by an Analog to Digital (A/D)Converter. The

random nature of photons striking the image sensor is an important source for noise. This type of

noise, known as photon shot noise, is roughly proportional to the square root of the signal level

as a result of the Central Limit Theorem. Thus the lower the signal is the higher the noise

becomes relative to the signal. As a result noise in color images can be very pronounced in

images shot under low light conditions because the signals must be amplified more. In general,

noise level is very low for photos shot outdoor using low ISO (ISO 100 or less). But with most

consumer compact cameras noise becomes visible at ISO 200, and it becomes unacceptable at

ISO 400 or higher. With more advanced and expensive digital SLRs noise remains low even at


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ISO 400, and becomes unacceptable at ISO 1600 or higher. Noise is in general much worse

under artificial lighting, especially under fluorescent lighting. One of the important

characteristics of digital noise is that they are not uniform across all channels. Very often noise is

concentrated in the blue channel while the green and the red channels are relatively clean. For

photos taken under artificial lighting (without a ash), the blue channel can be so noisy that it is

often unrecognizable

Fig.3.5 The original natural color image without artificial noise.

Another significant source of noise is the so called leakage current. Semiconductor image

sensors work by converting energy from photons into electrical energy, in the form of a current

or voltage signal. Unfortunately thermal energy present in the semiconductor can also generate

an electrical signal that is indistinguishable from the optical signal. As temperature increases, so

does leakage current in the circuit. The effects of leakage current are most apparent in long

exposures in which the light signal is very low.


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Fig.3.6 A zoom-in (upper-left) of the color image in 3.5 and its RGB

channels. Color noise is more evident. The red (upper-right) and green (lower-left) channels are

much cleaner than the blue (lower-right) channel. Modeling noise in digital color photos can be a

difficult task. The photon shot noise is clearly signal dependent and thus not uniform from pixel

to pixel. Nearly all digital cameras today use the so-called Bayer Pattern in their photo sensors,

where half of their pixels are used to capture the green channel and the other half are divided

evenly to capture the red and the blue channels. These partial data are then interpolated to

complete the RGB channels of a color photo. So unless we access the raw data (most consumer

digital cameras do not have this feature) it is clear that noise is not independent from pixel to

pixel in any channel. Most digital photos are in JPEG format, which degrade images throughquantization and artifacts. Furthermore, all cameras employ proprietary in-camera sharpening,

denoising and anti-aliasing. These factors combine to make effective modeling of noise, at least

in the images taken by consumer cameras exported in JPEG format, virtually impossible. For this


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reason, any noise model assuming independent and identically distributed noise from pixel to

pixel can be unrealistic.

3.5 PEAK SIGNAL-TO-NOISE RATIO

The phrase peak signal-to-noise ratio, often abbreviated PSNR, is an engineering term for

the ratio between the maximum possible power of a signal and the power of corrupting noise that

affects the fidelity of its representation. Because many signals have a very wide dynamic range,

PSNR is usually expressed in terms of the logarithmic decibel scale.

The PSNR is most commonly used as a measure of quality of reconstruction of lossy

compression codecs (e.g., for image compression). The signal in this case is the original data,

and the noise is the error introduced by compression. When comparing compression codecs it is

used as an approximation to human perception of reconstruction quality, therefore in some cases

one reconstruction may appear to be closer to the original than another, even though it has a

lower PSNR (a higher PSNR would normally indicate that the reconstruction is of higher

quality). One has to be extremely careful with the range of validity of this metric; it is only

conclusively valid when it is used to compare results from the same codec (or codec type) and

same content.

It is most easily defined via the mean squared error (MSE) which for two mn

monochrome images I and K where one of the images is considered a noisy approximation of the

other is defined as:

MSE =

2The PSNR is defined as:

PSNR = 10 . log10(MAX2

I/MSE)

=20 . log10(MAXI/)Here, MAXI is the maximum possible pixel value of the image. When the pixels are represented

using 8 bits per sample, this is 255. More generally, when samples are represented using linear

PCM with B bits per sample, MAXI is 2B1. Forcolor images with three RGB values per pixel,

the definition of PSNR is the same except the MSE is the sum over all squared value differences
http://en.wikipedia.org/wiki/Signal_%28information_theory%29http://en.wikipedia.org/wiki/Noisehttp://en.wikipedia.org/wiki/Dynamic_rangehttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Codechttp://en.wikipedia.org/wiki/Image_compressionhttp://en.wikipedia.org/wiki/Mean_squared_errorhttp://en.wikipedia.org/wiki/255_%28number%29http://en.wikipedia.org/wiki/PCMhttp://en.wikipedia.org/wiki/Color_imagehttp://en.wikipedia.org/wiki/Color_imagehttp://en.wikipedia.org/wiki/RGBhttp://en.wikipedia.org/wiki/RGBhttp://en.wikipedia.org/wiki/Color_imagehttp://en.wikipedia.org/wiki/PCMhttp://en.wikipedia.org/wiki/255_%28number%29http://en.wikipedia.org/wiki/Mean_squared_errorhttp://en.wikipedia.org/wiki/Image_compressionhttp://en.wikipedia.org/wiki/Codechttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Dynamic_rangehttp://en.wikipedia.org/wiki/Noisehttp://en.wikipedia.org/wiki/Signal_%28information_theory%29


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divided by image size and by three. Alternately, for color images the image is converted to a

different color space and PSNR is reported against each channel of that color space, e.g., YCbCr

orHSL.

Typical values for the PSNR in lossy image and video compression are between 30 and

50 dB, where higher is better. Acceptable values for wireless transmission quality loss are

considered to be about 20 dB to 25 dB. When the two images are identical, the MSE will be zero.

Q=90, PSNR 45.53dB Q=30, PSNR 36.81dB

Fig.3.7 Image at different PSNR values
http://en.wikipedia.org/wiki/Color_spacehttp://en.wikipedia.org/wiki/YCbCrhttp://en.wikipedia.org/wiki/HSL_and_HSVhttp://en.wikipedia.org/wiki/Lossy_compressionhttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Lossy_compressionhttp://en.wikipedia.org/wiki/HSL_and_HSVhttp://en.wikipedia.org/wiki/YCbCrhttp://en.wikipedia.org/wiki/Color_space


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CHAPTER 4

IMAGE DENOISING AND COMPRESSION

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 compressible. 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 (see Fig.4.1). Thus, provided that the

quantization outside of the zero-zone does not introduce significant distortion, it follows that

wavelet-based lossy compression achieves denoising.

Fig.4.1 Thresholding function can be approximated by quantization with a zero-zone.

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 likelydue 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.


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Thresholding essentially creates a region around zero where the coefficients are considered

negligible.

4.1 THE WAVELET TRANSFORM

The Wavelet transform is a transform which provides the time-frequency representation.

(There are other transforms which give this information too, such as short time Fourier

transform,Wigner distributions,etc.).Often times a particular spectral component occurring at any

instant can be of particular interest. In these cases it may be very beneficial to know the time

intervals these particular spectral components occur. For example, in EEGs, the latency of an

event-related potential is of particular interest (Event-related potential is the response of the brain

to a specific stimulus like flash-light, the latency of this response is the amount of time elapsed

between the onset of the stimulus and the response).Wavelet transform is capable of providing

the time and frequency information simultaneously, hence giving a time-frequency

representation of the signal. How wavelet transform works is completely a different fun story,

and should be explained after short time Fourier Transform (STFT) . The WT was developed as

an alternative to the STFT. The STFT will be explained in great detail in the second part of this

tutorial. It suffices at this time to say that the WT was developed to overcome some resolution

related problems of the STFT.To make a real long story short, we pass the time-domain signal

from various high pass and low pass filters, which filters out either high frequency or low

frequency portions of the signal. This procedure is repeated, every time some portion of the

signal corresponding to some frequencies being removed from the signal. Here is how this

works: Suppose we have a signal which has frequencies up to 1000 Hz. In the first stage we split

up the signal in to two parts by passing the signal from a high pass and a low pass filter (filters

should satisfy some certain conditions, so-called admissibility condition) which results in two

different

versions of the same signal: portion of the signal corresponding to 0-500 Hz (low pass portion),

and 500-1000 Hz (high pass portion). Then, we take either portion (usually low pass portion) or

both, and do the same thing again. This operation is called decomposition .Assuming that we

have taken the low pass portion, we now have 3 sets of data, each corresponding to the same

signal at frequencies 0-250 Hz, 250-500 Hz, 500-1000 Hz. Then we take the lowpass portion

again and pass it through low and high pass filters; we now have 4 sets of signals corresponding


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to 0-125 Hz, 125-250 Hz,250-500 Hz, and 500-1000 Hz. We continue like this until we have

decomposed the signal to a pre-defined certain level. Then we have a bunch of signals, which

actually represent the same signal, but all corresponding to different frequency bands. We know

which signal corresponds to which frequency band, and if we put all of them together and plot

them on a 3-D graph, we will have time in one axis, frequency in the second and amplitude in the

third axis. The uncertainty principle, originally found and formulated by Heisenberg, states that,

the momentum and the position of a moving particle cannot be known simultaneously. This

applies to our subject as follows:The frequency and time information of a signal at some certain

point in the time-frequency plane cannot be known. In other words: We cannot know what

spectral component exists at any given time instant. The best we can do is to investigate what

spectral components exist at any given interval of time. This is a problem of resolution, and it is

the main reason why researchers have switched to WT from STFT.

4.1.1 Discrete wavelet transform

The foundations of the DWT go back to 1976 when Croiser, Esteban, and Galand devised

a technique to decompose discrete time signals. Crochiere, Weber, and Flanagan did a similar

work on coding of speech signals in the same year. They named their analysis scheme as

subband coding. In 1983, Burt defined a technique very similar to subband coding and named it

pyramidal codingwhich is also known as multiresolution analysis. Later in 1989, Vetterli and Le

Gall made some improvements to the subband coding scheme, removing the existing redundancy

in the pyramidal coding scheme. Subband coding is explained below

Subband coding and Multiresolution

The main idea is the same as it is in the CWT. A time-scale representation of a digital

signal is obtained using digital filtering techniques. Recall that the CWT is a correlation between

a wavelet at different scales and the signal with the scale (or the frequency) being used as a

measure of similarity. The continuous wavelet transform was computed by changing the scale of

the analysis window, shifting the window in time, multiplying by the signal, and integrating over

all times. In the discrete case, filters of different cutoff frequencies are used to analyze the signal

at different scales. The signal is passed through a series of high pass filters to analyze the high

frequencies, and it is passed through a series of low pass filters to analyze the low frequencies.

The resolution of the signal, which is a measure of the amount of detail information in the signal,


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is changed by the filtering operations, and the scale is changed by upsampling and

downsampling (subsampling) operations.

Subsampling a signal corresponds to reducing the sampling rate, or removing some of the

samples of the signal. For example, subsampling by two refers to dropping every other sample of

the signal.Subsampling by a factor n reduces the number of samples in the signal n

times.Upsampling a signal corresponds to increasing the sampling rate of a signal by adding new

samples to the signal. For example, upsampling by two refers to adding a new sample, usually a

zero or an interpolated value, between every two samples of the signal. Upsampling a signal by a

factor ofn increases the number of samples in the signal by a factor of n. The procedure starts

with passing this signal (sequence) through a half band digital low pass filter with impulse

response h[n]. Filtering a signal corresponds to the mathematical operation of convolution of the

signal with the impulse response of the filter. The convolution operation in discrete time is

defined as follows:

Eqn. 4.1A half band lowpass filter removes all frequencies that are above half of the highest

frequency in

the signal.

For example, if a signal has a maximum of 1000 Hz component, then half band low pass

filtering removes all the frequencies above 500 Hz.The unit of frequency is of particular

importance at this time. In discrete signals, frequency is expressed in terms of radians.

Accordingly, the sampling frequency of the signal is equal to 2p radians in terms of radial

frequency. Therefore, the highest frequency component that exists in a signal will be p radians, if

the signal is sampled at Nyquists rate (which is twice the maximum frequency that exists in the

signal); that is, the Nyquists rate corresponds to p rad/s in the discrete frequency domain.

Therefore using Hz is not appropriate for discrete signals. However, Hz is used whenever it is

needed to clarify a discussion, since it is very common to think of frequency in terms of Hz. It

should always be remembered that the unit of frequency for discrete time signals is radians.

After passing the signal through a half band lowpass filter, half of the samples can be

eliminated according to the Nyquists rule, since the signal now has a highest frequency of p/2


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radians instead of p radians. Simply discarding every other sample will subsamplethe signal by

two, and the signal will then have half the number of points. The scale of the signal is now

doubled. Note that the lowpass filtering removes the high frequency information, but leaves the

scale unchanged. Only the subsampling process changes the scale.Resolution, on the other hand,

is related to the amount of information in the signal, and therefore, it is affected by the filtering

operations. Half band lowpass filtering removes half of the frequencies, which can be interpreted

as losing half of the information. Therefore, the resolution is halved after the filtering operation.

Note, however, the subsampling operation after filtering does not affect the resolution, since

removing half of the spectral components from the signal makes half the number of samples

redundant anyway.

Half the samples can be discarded without any loss of information. In summary, the

lowpass filtering halves the resolution, but leaves the scale unchanged. The signal is then

subsampled by 2 since half of the number of samples are redundant. This doubles the scale.This

procedure can mathematically be expressed as Having said that, we now look how the DWT is

actually computed: The DWT analyzes the signal at different frequency bands with different

resolutions by decomposing the signal into a coarse approximation and detail information. DWT

employs two sets of functions, called scaling functions and wavelet functions,which are

associated with low pass and highpass filters, respectively. The decomposition of the signal into

different frequency bands is simply obtained by successive high pass and low pass filtering of

the time domain signal. The original signal x[n] is first passed through a half band high pass

filter g[n] and a lowpass filter h[n]. After the filtering, half of the samples can be eliminated

according to the Nyquists rule, since the signal now has a highest frequency of p /2 radians

instead of p . The signal can therefore be subsampled by 2, simply by discarding every other

sample. This constitutes one level of decomposition and can mathematically be expressed as

follows

yhigh[k]=

2k-n) Eqn. 4.2

ylow[k]= 2k-n) Eqn.4.3where yhigh[k] and ylow[k] are the outputs of the highpass and lowpass filters, respectively, after

subsampling by 2. Figure 3.5 illustrates this procedure, where x[n] is the original signal to be

decomposed, and h[n] and g[n] are lowpass and highpass filters, respectively.


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Fig.4.2 The Subband Coding Algorithm

As an example, suppose that the original signal x[n] has 512 sample points, spanning a

frequency band of zero to p rad/s. At the first decomposition level, the signal is passed through

the highpass and lowpass filters, followed by subsampling by 2. The output of the high pass filter

has 256 points (hence half the time resolution), but it only spans the frequencies p/2 to p rad/s

(hence double the frequency resolution). These 256 samples constitute the first level of DWT

coefficients. The output of the lowpass filter also has 256 samples, but it spans the other half of

the frequency band, frequencies from 0 to p/2 rad/s. This signal is then passed through the same


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low pass and high pass filters for further decomposition. The output of the second low pass filter

followed by subsampling has 128 samples spanning a frequency band of 0 to p/4 rad/s, and the

output of the second highpass filter followed by subsampling has 128 samples spanning a

frequency band of p/4 to p/2 rad/s.

The second high pass filtered signal constitutes the second level of DWT coefficients.

This signal has half the time resolution, but twice the frequency resolution of the first level

signal. In other words, time resolution has decreased by a factor of 4, and frequency resolution

has increased by a factor of 4 compared to the original signal. The lowpass filter output is then

filtered once again for further decomposition. This process continues until two samples are left.

For this specific example there would be 8 levels of decomposition, each having half the number

of samples of the previous level. The DWT of the original signal is then obtained by

concatenating all coefficients starting from the last level of decomposition (remaining two

samples, in this case). The DWT will then have the same number of coefficients as the original

signal. The frequencies that are most prominent in the original signal will appear as high

amplitudes in that region of the DWT signal that includes those particular frequencies.

The difference of this transform from the Fourier transform is that the time localization of

these frequencies will not be lost. However, the time localization will have a resolution that

depends on which level they appear. If the main information of the signal lies in the high

frequencies, as happens most often, the time localization of these frequencies will be more

precise, since they are characterized by more number of samples. If the main information lies

only at very low frequencies, the time localization will not be very precise, since few samples are

used to express signal at these frequencies. This procedure in effect offers a good time resolution

at high frequencies, and good frequency resolution at low frequencies. Most practical signals

encountered are of this type. The frequency bands that are not very prominent in the original

signal will have very low amplitudes, and that part of the DWT signal can be discarded without

any major loss of information, allowing data reduction.


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Fig.4.3 1-D wavelet decomposition

The wavelet decomposition of an image is done as follows: In the first level of decomposition,

the image is split into 4 subbands,namely the HH,HL,LH and LL subbands. The HH subband

gives the diagonal details of the image;the HL subband gives the horizontal features while the

LH subband represent the vertical structures. The LL subband is the low resolution residual

consisiting of low frequency components and it is this subband which is further split at higher

levels of decomposition.

Fig.4.4 Subbands of 2D orthogonal wavelet transform


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Fig.4.5 Original image used for demonstrating the 2-D wavelettransform.

Fig.4.6 A one-level (K =1), 2-D wavelet transform using the symmetric wavelet transform with

the 9/7 Daubechies coefficients (the high-frequency bands have been enhanced to show detail).


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4.2 WAVELET THRESHOLDING AND THRESHOLD SELECTION

Let the signal be {ij,i,j=1,....,N},where N is some integer power of 2.It has been

corrupted by additive noise and one observes

gij=ij+ij, where i,j=1,.....N Eqn.4.4

Where {ji} are independent and identically distributed(iid) as normal N(0,2) and independent

of {ij}.The goal is to remove the noise,or denoise {gij},and obtain an estimate{ij} of {ij}

which minimizes the mean square error.

MSE()= 2 Eqn.4.5We can use the same principles of thresholding and shrinkage to achieve denoising as in 1-D

signals. The problem again boils down to finding an optimal threshold such that the mean

squared error between the signal and its estimate is minimized.

The different methods for denoising we investigate differ only in the selection of the

threshold. The basic procedure remains the same :

Calculate the DWT of the image.

Threshold the wavelet coefficients.(Threshold may be universal or subband adaptive)

Compute the IDWT to get the denoised estimate.

Soft thresholding is used for all the algorithms due to the following reasons: Soft thresholding

has been shown to achieve near minimax rate over a large number of Besov spaces. Moreover, it

is also found to yield visually more pleasing images. Hard thresholding is found to introduce

artifacts in the recovered images.We now study three thresholding techniques-

VisuShrink,SureShrink and BayesShrinkand investigate their performance for denoising various

standard images.

4.2.2 VisuShrink

Visushrinkis thresholding by applying the Universal threshold proposed by Donoho and

Johnstone. This threshold is given by where is the noise variance and M is thenumber of pixels in the image. It is proved that the maximum of any M values iidas N(0,

2)will

be smaller than the universal threshold with high probability, with the probability approaching 1


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as M increases. Thus, with high probability, a pure noise signal is estimated as being identically

zero.

This is because the universal threshold(UT) is derived under the constraint that with high

probability the estimate should be at least as smooth as the signal. So the UT tends to be high for

large values of M, killing many signal coefficients along with the noise. Thus, the threshold does

not adapt well to discontinuities in the signal.

4.2.3 SureShrink

What is SURE ?Let = (i : i = 1,.. d) be a length-d vector, and letx = {xi} (withxi

distributed as N(i,1)) be multivariate normal observations with mean vector. Let ^=

^(x) be

an fixed estimate of based on the observations. SURE(Steins unbiased Risk Estimator) is a

method for estimating the loss ||^- ||

2in an unbiased fashion.

SURE(t;x)=d-2.#{i:|xi|


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and

C(x, )=.(x, )/2(1/ ) Eqn.4.9

and

(t )= -uut-1du Eqn.4.10The parameter x is the standard deviation and is the shape parameter.It has been

observed that with a shape parameter ranging from 0.5 to 1, we can describe the the

distribution of coefficients in a subband for a large set of natural images.Assuming such a

distribution for the wavelet coefficients, we empiricallyestimate and X for each subband and

try to find the threshold T which minimizes the Bayesian Risk, i.e, the expected value of the

mean square error.

(T)=E(-X)2=EXEY|X(-X)2 Eqn.4.11Where= T(Y),Y|X ~N(x,2 ) and X ~ GGx,.The optimal threshold T* is given byT*(x,)=arg min r(T) Eqn.4.12

This is a function of the parameter x and .Since there is no closed form solution for

T*,numerical calculation is used to find its value.It is observed that threshold value is set by

TB(x)=2/ x Eqn.4.13

It is very close to T*.The estimated threshold TB=2/ x is not only nearly optimal but

also has intuitive appeal. The normalized threshold, TB/. is inversely proportional to , the

standard deviation of X, and proportional to X, the noise standard deviation. When /X>1, the noise dominates and the

normalized threshold is chosen to be large to remove the noise which has overwhelmed the

signal. Thus, this threshold choice adapts to both the signal and the noise characteristics as

reflected in the parameters and X.


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Parameter Estimation to determine the Threshold

The GGD parameters x, need to be estimated to compute TB(x).The noise variance 2

is estimated from the subband HH1 by the robust median estimator.

=

, Yij subband HH1 Eqn.4.14

The parameter does not explicitly enter to the expression ofTB(x).Therefore it suffices to the

estimate directly the signal standard x.The observation model is Y=X+V,with X and V

independent of each other,hence

y2= x2+2 Eqn.4.15Where y2is the variance of Y.since Y is modelled as zero-mean, y2can be found empirically by

y2=

Eqn.4.16

Where n x n is the size of the subband under consideration.Thus

B( )=2/x Eqn.4.17Where

x= )To summarize,Bayes Shrink performs soft thresholding,with the data-driven, subband dependent

threshold, B( )=

2

/

x

4.3 MDL PRINCIPLE FOR COMPRESSION-BASED DENOISING

Recall our hypothesis is that compression achieves denoising because the zero-zone in

the quantization step (typical in compression methods) corresponds to thresholding in denoising.

Forthe purpose of compression, after using the adaptive threshold B( ) for the zero-zone,there still remains the questions of how to quantize the coefficients outside of the zero-zoneand

how to code them. Fig. 4.2 illustrates the block diagram of the compression method. It shows that

the coder needs to decide on the design parameters m, (the number of quantization bins and thebinwidth, respectively), in addition to the zero-zone threshold.


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Fig.4.9 Block diagram for compression based denoising.

Compression is Process of reducing the amount of data required to represent a given quantity ofinformation.Data are the means by which is information is conveyed. Various amounts of data

can be used to represent the same amount of information, representation that contain irrelevant

and repeated information are said to contain redundant data. Quantization means Digitizing the

amplitude values.

Denoising is achieved in the wavelet transform domain by lossy-compression, which

involves the design of parameters T,m and , relating to the zero-zone width, the number of

quantization levels, and the quantization binwidth, respectively.The choice of these parametersis discussed next.When compressing a signal, two important objectives are to be kept in mind.

On the one hand, the distortion between the compressed signal and the original should be kept

low; on the other hand, the description of the compressed signal should use as few bits as

possible to code.

Typically, these two objectives are conflicting, thus a suitable criterion is needed to reach

a compromise.Rissanens MDL principle allows a tradeoff between these two objectives . Our

MDL procedures achieve comparable MSE performance, while keeping far fewer (nonzero)

coefficients. All the MDL procedures and their comparative counterparts in this paper are based

on the assumption that the wavelet coefficients from a given subband are a simple random

sample from some distribution. Within the MDL paradigm, more elaborate dependence

structures (both within and between subbands) could be incorporated. According to the MDL


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principle, given a sequence of observations, the best model is the one that yields the shortest

description length for describing the data using the model, where the description length can be

interpreted as the number of bits needed for encoding. This description can be accomplished by a

two-part code: one part to describe the model and the other the description of the data using the

model.

In the original thresholding scheme, the thresholded coefficients are then inverse

transformed. In this work, to show that quantization approximates thresholding, there is an

additional step of quantizing the thresholded coefficients before inverse transform. More

precisely, given the set of observations , we wish to find a model to describe it. The MDL

principle chooses which minimizes the two-part code-length,L(Y,

) = L(Y|

) + L(

) Eqn.4.18

Where L(Y|) is the code length for Y based on X, and L() is the code length for.In Saitossimultaneous compression and denoising method for a length-M one-dimensional signal, the

hard-threshold function was used to generate the models T(Y), where the number ofnonzero coefficients to retain is determined by minimizing the MDL criterion. The first term

L(Y|) is the idealized code-length with the normal distribution and the second term L() istaken to be 3/2Klog2M , of which Klog2M are the bits needed to indicate the location of each

nonzero coefficient (assuming an uniform indexing) and (1/2)log log2M for the value of each of

the coefficients for justification on using (1/2)log2M bits to store the coefficient value. Although

compression has been achieved in the sense that a fewer number of nonzero coefficients are kept,

does not address the quantization step necessary in a practical compression setting.

In the following section, an MDL-based quantization criterion will be developed by

minimizing L(Y,) with the restriction that belongs to the set of quantized signals. ThisMDLQ compression with BayesShrink zero-zone selection is applied to each subband

independently. The steps discussed are summarized as follows.

Estimate the noise variance2 , and the GGD standard Deviation x.

Calculate the threshold B , and soft-threshold the wavelet Coefficients. To quantize the nonzero coefficients, minimize thrshold over m and and to find the

corresponding quantized coefficient which is the compressed, denoised estimate of X.


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The coarsest subband LLJ is quantized differently in that it is not thresholded, and the

quantization. assumes the uniform distribution. The coefficients LLJ are essentially local

averages of the image, and are not characterized by distributions with a peak at zero, thus the

uniform distribution is used for generality. With the mean subtracted, the uniform distribution is

assumed to be symmetric about zero. Every quantization bin(including the zero-zone) is of width

, and the reconstruction values are the midpoints of the intervals.

4.4 IMAGE DENOISING ALGORITHM

This section describes the image denoising algorithm, which achieves near optimal soft

thresholding in the wavelet domain for recovering original signal from the noisy one. The

algorithm is very simple to implement and computationally more efficient. It has following steps:

Perform multiscale decomposition of the image corrupted by Gaussian noise using wavelet

transform.

Estimate the noise variance 2

.

For each level, compute the scale parameter .

For each subband (except the low pass residual)

a) Compute the standard deviation y.

b) Compute threshold TN.

c) Apply soft thresholding to the noisy coefficients

Invert the multiscale decomposition to reconstruct the denoised imagef .


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CHAPTER 5

RESULTS AND DISCUSSIONS

5.1 COMPARISON OF PERFORMANCE OF VARIOUS IMAGE DENOISING

METHODS

5.1.1 Image denoising using various method

Image denoising at =20


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Image Denoising at =35


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Fig. 5.1 Image denoising at various values

5.2 LIST OF PSNR VALUES OF VARIOUS METHODS IN VARIOUS TEST IMAGES

LENA Softthresholding Hardthresholding Bayes shrink Visu shrink

=20 30.924 33.8907 38.4678 24.7025

=25 30.1202 32.4858 36.7674 24.5026

=35 29.6335 30.5089 34.3431 24.0487

=40 28.8147 30.2401 33.4924 23.7769

Table 5.1 PSNR values of Lena image for various thesholding methods


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BARBARA Softthresholding

Hardthresholding

Bayes shrink Visu shrink

=20 22.81 23.5 25.5 19.15

=25 21.21 22.81 24.89 18.45

=35 21.25 21.92 23.90 17.1421

=40 21.02 21.53 23.41 16.52

Table 5.2 PSNR values of Barbara image various thesholding methods

5.3 CONVERSION OF MATLAB FUNCTION PROGRAM OF VISUSHRINK TO CPROGRAM

5.3.1 Procedure

Use MATLAB CODER in MATLAB .If we type >>coder in command window to create

new project for MATLAB CODER

Fig.5.2 MATLAB Coder Project window A MATLAB CODER window will be opened and here we can add files for C conversion


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Fig.5.3 MATLAB coder option to add files

C code can generate by using the option Build and here also have settings to select hardware

implementation details.

Fig.5.4 MATLAB CODER option to generate C program


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Fig.5.5 Code generation report

5.4 IMLEMENTATION OF VISU SHRINK METHOD IN DSK6713 DSP KIT

C program for Visu shrink thresholding method has been created in Code Composer

Studio and build it.An out file has been generated and loaded it in DSK6713 DSP kit. Address

location for input and output as follows.Here y be the input and x be the output.


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Fig.5.6 Address locations of input and output


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CHAPTER 6

CONCLUSION

A simple subband adaptive threshold has been proposed in this project to address theissue of image recovery from its noisy counterpart. It is based on the generalized Guassian

distribution modeling of subband coefficients.The proposed BayesShrinkthreshold specifies the

zero-zone of the quantization step of thiscoder, and this zero-zone is the main agent in the coder

which removes the noise. The image denoising algorithm uses soft thresholding to provide

smoothness and better edge preservation at the same time. In this project comparison of several

thresholding calculation methods have been done. Analysis of comparison shows that Bayes

shrink method shows the better PSNR value than other methods.C code for visu shrink

thresholding method has been generated and build in Code Composer Studio.


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APPENDIX

Matlab program for image denoising

clear;

clc;clearall;

close all;

display('select the image');display(' 1:lena.png');

display(' 2:barbara.png');

display(' 3:boat.png');

display(' 4:shuttle.jpg');ss1 =input('enter your choice: ');

switch ss1

case 1

f=imread('lena.png');case 2

f=imread('barbara.jpg');

case 3f=imread('boat.png');

case 4

f=imread('index.jpg');end

s = double(f);

sigma=input('GAUSSIAN NOISE STANDARD DEVIATION ='); % Gaussian noise standard

deviationIn = randn(size(s))*sigma; % White Gaussian noise

g = s + In; %noisy image2g=uint8(g);x=g;

% find default values (see ddencmp).

[thr,sorh,keepapp] = ddencmp('den','wv',x);display('');

display('select wavelet');

display('enter 1 for haar wavelet');

display('enter 2 for db2 wavelet');

display('enter 3 for db4 wavelet');display('enter 4 for sym wavelet');

display('enter 5 for sym wavelet');display('enter 6 for bior wavelet');display('enter 7 for bior wavelet');

display('enter 8 for mexh wavelet');

display('enter 9 for coif wavelet');display('enter 10 for meyr wavelet');


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display('enter 11 for morl wavelet');

display('enter 12 for rbio wavelet');

display('press any key to quit');display('');

ww=input('enter your choice: ');

switch wwcase 1wv='haar';

case 2

wv='db2';case 3

wv='db4' ;

case 4

wv='sym2'case 5

wv='sym4';

case 6wv='bior1.1';

case 7

wv='bior6.8';

case 8wv='mexh';

case 9

wv='coif5';case 10

wv='dmey';

case 11

wv='mor1';case 12

wv='jpeg9.7';

otherwisequit;

end

display('');display('enter 1 for soft thresholding');

display('enter 2 for hard thresholding');

display('enter 3 for bayes soft thresholding');

display('enter 4 for sure shrink');display('enter 5 for wiener filtering');

sorh=input('sorh: ');

display('enter the level of decomposition');

level=input(' enter level 1 or 2 or 3 etc: ');switch sorh

case 1

sorh='s';


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xd = wdencmp('gbl',x,wv,level,thr,sorh,keepapp);

case 2

sorh='h';xd = wdencmp('gbl',x,wv,level,thr,sorh,keepapp);

case 3

sig=20;V=(sig/256)^2;npic=g;

filtertype=wv;

levels=level;%Doing the wavelet decomposition

[C,S]=wavedec2(npic,levels,filtertype);

st=(S(1,1)^2)+1;

bayesC=[C(1:st-1),zeros(1,length(st:1:length(C)))];var=length(C)-S(size(S,1)-1,1)^2+1;

%Calculating sigmahat

sigmahat=median(abs(C(var:length(C))))/0.6745;forjj=2:size(S,1)-1

%for the H detail coefficients

coefh=C(st:st+S(jj,1)^2-1);

thr=bayes(coefh,sigmahat);bayesC(st:st+S(jj,1)^2-1)=sthresh(coefh,thr);

st=st+S(jj,1)^2;

% for the V detail coefficientscoefv=C(st:st+S(jj,1)^2-1);

thr=bayes(coefv,sigmahat);

bayesC(st:st+S(jj,1)^2-1)=sthresh(coefv,thr);

st=st+S(jj,1)^2;%for Diag detail coefficients

coefd=C(st:st+S(jj,1)^2-1);

thr=bayes(coefd,sigmahat);bayesC(st:st+S(jj,1)^2-1)=sthresh(coefd,thr);

st=st+S(jj,1)^2;

endbayespic=waverec2(bayesC,S,filtertype);

xd=bayespic;

figure, imagesc(uint8(bayespic));colormap(gray);

case 4xd = VisuThresh(g);

case 5

[n m]= size(f);

xd = wiener2(g,[m n]);end

subplot(2,2,1), imshow(f);title('original image');

subplot(2,2,2),imshow(g);


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title('noisy image');

subplot(2,2,3),xd=uint8(xd);

imshow(xd);title('denoised image');subplot(2,2,4),sub=f-xd;

sub=abs(1.2*sub);

imshow(im2uint8(sub));title('difference image');ff=im2double(f);xdd=im2double(xd);display(' ');

display(' ');

psnr_value=WPSNR(ff,xdd)display(' ');

display(' ');

mse=compare11(ff,xdd)

function to perform soft thresholding

function op=sthresh(X,T);ind=find(abs(X)T);X(ind)=0;

X(ind1)=sign(X(ind1)).*(abs(X(ind1))-T);op=X;

Function to calculate Threshold for BayesShrink

function threshold=bayes(X,sigmahat)len=length(X);

sigmay2=sum(X.^2)/len;

sigmax=sqrt(max(sigmay2-sigmahat^2,0));ifsigmax==0 threshold=max(abs(X));

else threshold=sigmahat^2/sigmax;

end

Function for visu thresholding methodfunction [x] = VisuThresh(y,type)

ifnargin < 2,

type = 'Soft';end

thr = sqrt(2*log(length(y))) ;

ifstrcmp(type,'Hard'),

x = HardThresh(y,thr);else

x = SoftThresh(y,thr);

endFunction for PSNR

function f = WPSNR(A,B,varargin)

ifA == B


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error('Images are identical: PSNR has infinite value')

end

max2_A = max(max(A));max2_B = max(max(B));

min2_A = min(min(A));

min2_B = min(min(B));ifmax2_A > 1 | max2_B > 1 | min2_A < 0 | min2_B < 0error('input matrices must have values in the interval [0,1]')

end

e = A - B;ifnargin


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Fmat = Z;

C code for visu shrink method

#include"rt_nonfinite.h"#include"VisuThresh.h"

#include"VisuThresh_initialize.h"


#include"rtGetInf.h"

#define NumBitsPerChar 8U

#include"rtGetNaN.h"#include"rt_nonfinite.h"

#include"rtGetNaN.h"

#include"rtGetInf.h"


#include"VisuThresh_terminate.h"

real_T rtInf;real_T rtMinusInf;

real_T rtNaN;

real32_T rtInfF;real32_T rtMinusInfF;

real32_T rtNaNF;

void main(void) {

real_T x[63];real_T value=1;

size_t realSize=1;

VisuThresh_initialize();VisuThresh(x);

rtGetInf();

rtGetInfF();

rtGetMinusInf();rtGetMinusInfF();

rt_InitInfAndNaN(realSize);

rtIsInf(value);rtIsInfF(value);

rtIsNaN(value);

rtIsNaNF(value);

VisuThresh_terminate();}

void VisuThresh(real_T x[63])

{int32_T k;

staticconstreal_T dv0[63] = { 18.0, 8.5, 2.0, -6.5, -6.0, 34.0, -17.5, -9.0,


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-9.5, 0.0, -7.0, 9.0, 24.5, -6.0, -8.0, 12.0, 13.5, 9.0, 15.5, 28.5, 7.0,

16.0, 0.0, 6.0, -12.5, -6.0, 37.5, -44.0, 6.0, 13.0, -3.0, -16.0, 23.5, 24.5,

-15.5, -18.0, 17.5, 17.0, 16.0, 12.5, 19.0, -28.0, 22.5, 8.5, -15.0, 0.0,3.0, 0.0, -37.5, -4.0, 15.0, 9.0, -6.5, -6.0, 37.0, -30.5, -8.5, -8.5, -1.0,

6.0, 21.5, 28.5, 3.0 };

real_T b_x;real_T y[63];staticconstreal_T b_y[63] = { 18.0, 8.5, 2.0, -6.5, -6.0, 34.0, -17.5, -9.0,

-9.5, 0.0, -7.0, 9.0, 24.5, -6.0, -8.0, 12.0, 13.5, 9.0, 15.5, 28.5, 7.0,

16.0, 0.0, 6.0, -12.5, -6.0, 37.5, -44.0, 6.0, 13.0, -3.0, -16.0, 23.5, 24.5,-15.5, -18.0, 17.5, 17.0, 16.0, 12.5, 19.0, -28.0, 22.5, 8.5, -15.0, 0.0,

3.0, 0.0, -37.5, -4.0, 15.0, 9.0, -6.5, -6.0, 37.0, -30.5, -8.5, -8.5, -1.0,

6.0, 21.5, 28.5, 3.0 };

for(k = 0; k < 63; k++) {

b_x = fabs(dv0[k]) - 2.09629414793641;

y[k] = fabs(b_x);x[k] = b_y[k];

res[k] = b_x;

}

for(k = 0; k < 63; k++) {

b_x = x[k];

if(x[k] > 0.0) {b_x = 1.0;

} elseif(x[k] < 0.0) {

b_x = -1.0;

} else {if(x[k] == 0.0) {

b_x = 0.0;

}}

x[k] = b_x;}

for(k = 0; k < 63; k++) {

x[k] *= (res[k] + y[k]) / 2.0;}

}

void VisuThresh_initialize(void)

{rt_InitInfAndNaN(8U);

}

real_T rtGetInf(void)


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{

size_t bitsPerReal = sizeof(real_T) * (NumBitsPerChar);

real_T inf = 0.0;if(bitsPerReal == 32U) {

inf = rtGetInfF();

} else {uint16_T one = 1U;enum {

LittleEndian,

BigEndian} machByteOrder = (*((uint8_T *) &one) == 1U) ?LittleEndian :BigEndian;

switch (machByteOrder) {

caseLittleEndian:

{union {

LittleEndianIEEEDoublebitVal;

real_TfltVal;} tmpVal;

tmpVal.bitVal.words.wordH = 0x7FF00000U;

tmpVal.bitVal.words.wordL = 0x00000000U;inf = tmpVal.fltVal;

break;

}

caseBigEndian:

{

union {BigEndianIEEEDoublebitVal;

real_TfltVal;

} tmpVal;

tmpVal.bitVal.words.wordH = 0x7FF00000U;

tmpVal.bitVal.words.wordL = 0x00000000U;inf = tmpVal.fltVal;

break;

}

}}

return inf;

}

real32_T rtGetInfF(void){

IEEESingle infF;

infF.wordL.wordLuint = 0x7F800000U;


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return infF.wordL.wordLreal;

}

real_T rtGetMinusInf(void){

size_t bitsPerReal = sizeof(real_T) * (NumBitsPerChar);

real_T minf = 0.0;if(bitsPerReal == 32U) {minf = rtGetMinusInfF();

} else {

uint16_T one = 1U;enum {

LittleEndian,

BigEndian

} machByteOrder = (*((uint8_T *) &one) == 1U) ?LittleEndian :BigEndian;switch (machByteOrder) {

caseLittleEndian:

{union {

LittleEndianIEEEDoublebitVal;

real_TfltVal;

} tmpVal;

tmpVal.bitVal.words.wordH = 0xFFF00000U;

tmpVal.bitVal.words.wordL = 0x00000000U;minf = tmpVal.fltVal;

break;

}

caseBigEndian:

{

union {BigEndianIEEEDoublebitVal;

real_TfltVal;

} tmpVal;

tmpVal.bitVal.words.wordH = 0xFFF00000U;

tmpVal.bitVal.words.wordL = 0x00000000U;

minf = tmpVal.fltVal;break;

}

}

}

return minf;

}


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real32_T rtGetMinusInfF(void)

{

IEEESingle minfF;minfF.wordL.wordLuint = 0xFF800000U;

return minfF.wordL.wordLreal;

}

/* End of code generation (rtGetInf.c) */

void rt_InitInfAndNaN(size_t realSize)

{(void) (realSize);

rtInf = rtGetInf();

rtInfF = rtGetInfF();

rtMinusInf = rtGetMinusInf();rtMinusInfF = rtGetMinusInfF();

}

boolean_T rtIsInf(real_T value){

return ((value==rtInf || value==rtMinusInf) ? 1U : 0U);

}

boolean_T rtIsInfF(real32_T value)

{

return(((value)==rtInfF || (value)==rtMinusInfF) ? 1U : 0U);}

boolean_T rtIsNaN(real_T value)

{

#ifdefined(_MSC_VER) && (_MSC_VER


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}

Header files

rt_nonfinite.h#ifndef __RT_NONFINITE_H__

#define __RT_NONFINITE_H__#if defined(_MSC_VER) && (_MSC_VER


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rtGetInf.h

#ifndef __RTGETINF_H__#define __RTGETINF_H__

#include

image processing and recognition

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