## Chapter x,y and intensity value of f are all

Chapter – 1                                          Introduction

1

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Image
denoising plays a great significance role in the digital image processing. We
use denoising process to achieve final noise free estimates by removing the
unnecessary noise from the image. Various fields in which Image denoising plays
an important role are as Astronomy filed, Medical field, SAR Images etc. to
analyze and extracting various features of images. There are various approaches
to perform denoising and each approach have their own merits and demerits. In the
following research work, we have discussed and analyzed some of the denoising
techniques on the MRI images.

Topics
explored in this chapter are as follows:

·
Introduction to Digital Image and Noise

o
Noise

o
Types of noises

·
Image Denoising

·
Denoising Techniques

1.
Introduction
to Digital Images and Noise

An
image is a Two-Dimensional Function f(x,y) where x and y are spatial
coordinates, and the amplitude of f at any pair of coordinate (x,y) is called
grey level or intensity value of image at that point. When x,y and intensity
value of f are all discrete finite quantities , we  call that image as digital image .

1.1.
Noise

Noise is something which distorts the digital image. Noise
can cause changes in brightness or color in an image. There are many reasons by
which a noise can occur in an image such as variation in brightness of
surrounding or environmental conditions. It
is unwanted information that lower the quality of an image. Noise adds
irrelevant information and also depicts undesired information in an image.
Digital images are liable to suffer from various types of  noises. Different sources of noises are
below  :

a)      Gaussian
images occurs in digital images by image acquisition i.e noise is produced by
poor illuminations and by high temperature .

b)      Inference
in transmission channel leads to noise in images .

c)      Atmospheric
disturbance causes noise in images while transmission through wireless network.

1.

1.1.

1.2.
Types of noises

a)
noise: Unwanted
signal are called as an additive noises. Here the noise is added to original image.
The additive noise of an image is defined as:

g(x,y)=f(x,y)+n(x,y)

Where g(x,y) is referred to as an noisy image, f(x,y)
is referred to as an original given image and n(x,y) is referred to as additive
noise of an image. Gaussian noise is an example of the additive noise

Multiplicative noise:
Multiplicative noise is an unwanted noise which multiplies original signals
while transmission, capturing or any other processing. The multiplicative noise
of an image is defined as:

g(x,y)=f(x,y)×n(x,y)

Where
g(x,y) is referred to as an noisy image, f(x,y) is referred to as an original
given image and n(x,y) is referred to as function which is multiplicative

Gaussian Noise : The
Gaussian noise has additive as a standard model. In this the probability
density function is equal to normal distribution or it is called as Gaussian
distribution. This noise is found in the image acquisition. At each point the
intensity of pixel value is independent by the noise. It can be calculated as

Where
z is referred as grey level, µ is called mean value and ? is the standard
deviation.

Salt and pepper noise:
Impulse noise is also sometimes called as a salt and pepper noise. In a gray
scale the bright pixels are contained in the dark regions and the black and
white pixels in the bright region. This noise is mostly caused by a converter
errors or bit errors of transmission. This type of noise is eliminated by dark
or bright pixels in a large part. In this only pixel parts are corrupted but
rest is noise free.

The Probability Density Function
(PDE) is given by :

If
b>a intensity then b will appear as light pixel in a image, or intensity a
will appears as a dark pixel in image. If either

or

is zero, then it is unipolar.

Film grain:
The grain of photographic film is same as statistical distribution. It is a
signal dependent noise. The number of dark grains in an area is random with
binomial distribution if film grains are distributed uniformly and if every grain
has same independent probability to develop to a dark silver grain by photon
absorption. We usually regard film grain as a non-oriented noise source.

Shot noise:
shot noise is a classification of the electronic noise which is modeled by
process of Poisson. It usually originates by electronic charge of discrete
nature. It has root mean square which is proportional to square root of density
of image. The noises at distinct pixels are not dependent upon one another.
Additional shot noise is also present in image due to dark leakage of current
in the sensor of the image which is called dark shot noise.

Quantization noise:
The noise caused by quantizing the pixels of a detected image to various
discrete levels is has known as quantization noise, around uniform dispersion,
and it can be signal dependence, however it will be signal free if other
available noise sources are sufficiently enormous to cause dithering, or if the
dithering is expressly connected. This blunder is either because of adjusting
or truncation. The error signal is once in a while considered as an additional
random signal is referred as quantization noise due to its stochastic conduct.

Anisotropic noise:
Some of the noise sources appear with a critical introduction in images. For
instance, image sensors are at times subject to push noise or section noise.
Anisotropic noise surfaces are intriguing for some perception and graphics
applications. The spot tests can be utilized as contribution for surface age,
e.g., Line Integral Convolution (LIC), yet can likewise be utilized
specifically for representation without anyone else’s input. They are
particularly reasonable for the perception of tensor fields that can be
utilized to characterize a metric for the anisotropic thickness field. We display
a novel strategy for producing stochastic examples to make anisotropic noise
surfaces comprising of non-covering circles, whose size and thickness
coordinate a given metric. Our technique bolsters a programmed pressing of the
circular examples bringing about surfaces like those created by anisotropic
response dispersion.

Speckle noise:
the multiplicative noise is called as an speckle noise. This noise is
multiplied to the original image. It is present in the ultrasound image. This
noise occurs in all coherent systems like acoustics or laser imagery. This
noise decreases the contrast of an image and it also observes beneficial
details of an ultrasound image. This noise is an granular noise which
inherently exist and also degrade quality of SAR , active radar, coherence
tomography and medical ultrasound images.

g(x,y)=f(x,y)×h(x,y)

Where
g(x,y) is called noisy image, f(x,y) is the original image and h(x,y) is a multiplicative
degrade. Its Raleigh Distribution is given by equation as:

2.
Image
Denoising

Image denoising is used for the analysation of a
image. It is used for recovery of the digital image which is impure by the
noise. A image restoration or denoising method is used to decrease the noise
and also to preserve the edges of image, sharpen the image details or
significant features. It is referred to as a recovery from digital image which
is been degraded from the noise. The methods in this are orientations towards
the degradation. Here we preserve the details of an image.

Restoration filter

+

g(x,y)

f(x,y)                                                                                                                       f'(x,y)

Noise  n(x,y)

Here in this diagram given above f(x,y)
is referred to as an original image. n(x,y) is referred to as an noise added to
make an degraded image. The g(x,y) is referred to as an degraded image. Here
restoration filter is applied to form a approximate image f(x,y) which is
represented as f'(x,y).

3.
Denoising
Techniques

Anisotropic diffusion
filter: In image processing the most explode
topic is image denoising. There are many methods purposed for denoising such as
wiener filter, wavelet thresholding, PDE(Partial Differential equation), total
variation minimization method, non-local methods and bilateral filtering.

Noise
reduction using PDE: a paradigm used for noise reduction is by using non-linear
diffusions to remove the noise from images. The Gaussian filter to denoise the
image is achieved by convolving the Gaussian kernel K? with noisy
image u0.

With
standard deviation  ? , is equivalent to
the solution of the diffusion equation in two dimensions.

Where

is
the 2-D image. In general form this can be written as:

,
where I0 is the original image. In general form this can be written
as

,

Where c(x,y,t) is the diffusion
conductance or diffusivity of the equation,

Wiener
filter: wiener filter is a filter in a signal
processing. This filter is used to create approximate of a target arbitrary
process by linear time-invariant (LTI) filtering of a observed process of an
noise image, thinking of known noise spectra, stationary signal and also
additive noise. This filter also minimizes mean square error in desired process
and in estimated random processes.

Total
Variation Minimizing: this method was used to
minimize a total variation norm for removing noise in an image. A new
constrained minimization algo was derived like time dependent nonlinear where
noise statistics were used to determine the constraints. It is used to reduce
incompatible local oscillations from images. It is a quantity used to measure
the oscillations form functions.

Non-local
Means: Non-local means
is an algorithm in image preparing for image denoising. Not at all like local
mean filter, which take mean estimation of a gathering of pixels surrounding an
objective pixel for smoothning of the image, non-local referred to filtering
takes a mean of pixels in the image, weighted by how comparative these pixels
are to the objective pixel. This outcomes in substantially has high post
filtering display, and lesser loss of detail from the image contrasted and from
local mean algorithm.

Chapter – 2                                            Literature

2

The Literature Survey passes on clarity and focus to
the research issue, upgrades the technique to be proposed and extends the basic
data about issue close by. It gives a structure to setting up the criticalness
of the examination. Inside the setting of a quantitative research approach, the
literature survey has a critical measure of time and effort. Distinctive
critical journals, research papers and other related papers are used for
increasing start to finish learning. These papers are referenced underneath
with a short introduction :

Perona and Malik
(1990)1
developed a iterative approach of edge detection and multiscale
smoothing which is known as Anisotropic diffusion filter. This filter relies
upon second order partial differential equation (PDE) in anisotropic medium. It
method is that it upgrades picture quality by protecting article limits, edge
sharpening. It beats the drawbacks of spatial filters and its obstruction is
that it produces stair case affect.

Tomasi and
Manduchi (1998)2   shown Bilateral filter, This filter is the
mix of range filtering and domain filtering. It is non-iterative and
fundamental filter. The advantage of bilateral filter is that it jam edges and
gives best results over center filter yet it prompts all the all the more
smoothing or blurring of a photo.

J.Wang et al
(2006)3 proposed fast non-local means algorithm.
As the non-local means denoising algorithm proposed by Buades et al., is
computationally over the top thusly, another algorithm that abatements the cost
of computation for calculating the similarity of neighboring window is
proposed. Regardless introducing a deduced measure about the resemblance of
neighborhood windows, by then a profitable Summed square Image (SSI) plan and
Fast Fourier Transform (FFT) to revive all the calculation of this measure.
This algorithm is around fifty times faster than the original non-local
Algorithm both theoretically and experimentally, yet conveys equivalent results
of the extent which mean-squared error (MSE) and perceptual image quality.

G.C. Gavrincea
(2007)4  shown wavelet
based denoising In this paper for image denoising four decomposition using haar
wavelet provoking arrangement of coefficients called wavelet coefficients and to
get rid of noisy bit of banner, detail coefficients from each level are
thresholded and this is a substantially more compelling strategy for managing
noisy banners as opposed to filtering.

A. Dauwe et al.
(2008)5
proposed several progressions to the original non-local means algorithm
introduced by Buades et al. As a result of the colossal measure of weight
calculations, the original algorithm has a high computational cost. A
difference in image quality towards the original algorithm is to dismiss the responsibilities
from one of a kind windows. This shocking effect of dissimilar windows can be
wiped out by setting their contrasting weights with zero. This stimulated
approach is additionally streamlined by misusing the symmetry in the weights,
which generally halves the calculation time. Appeared differently in relation
to the original algorithm, this method produces images with extended psnr and
better visual execution in less calculation time. The proposed upgrades can
also be associated in other image Processing assignments which use the
possibility of repetitive structures, for instance, intra-frame Super
resolution or detection of digital image impersonation.

V. Kamati er al
(2009)6 shown changes to non-local means (NLM)
image denoising method to reduce the computational multifaceted nature. The
proposed strategy replaces the window similarity by a changed multi-resolution
based approach with numerous less relationships rather than all pixels
examination. This approach also uses filtering out non-similar neighborhood
pixels in light of settled estimated window diminish mean regards. The proposed
approach is around 80 times speedier than interesting Baudes NLM algorithm with
close subjective and to target quality estimations.

k Bartusek et al
(2011)7
presented wavelet develop denoising frameworks focusing as for the
wavelet thresholding systems and the point of confinement estimation.
soft,  Hard, semi-soft  and non-negative garrote thresholding
techniques are delineated and associated with test images with two various
point of confinement estimators; one uses the general edge and the second is
gotten from the Bayesian risk minimization. The results are stood out
concurring from three parameters: SNR, control contrast and power slant.

D. Peter and
Ramya et al. (2012)8  proposed a novel adaptive non-local means for
image denoising. In this adaptive non local means filtering, immediately the
image is smoothed using Gaussian filter and after that the noisy image is to be
segmented in light of power level using k means amass which is especially
effective in denoising significantly noisy level images. In light of test comes
to fruition adaptive non-local means filter is had all the earmarks of being
astoundingly effective in visual quality and gives extraordinary result as
stand out from regular non-local means algorithm.

B. vijilin and
V.K. Govindan (2013)9  displayed perfect edge decision for wavelet
change in light of visual quality. With perfect utmost regard, most outrageous
dreary information is ousted from input image realizing better weight and
besides high visual quality revamped image.

Chapter – 3                           Problem Defination

3

The quality of a image is affected by noise. Noise
plays an important role in the degradation of a image. The noise which is
present in a image is removed or degraded by the image denoising process. Noise
in a image is mostly occurred due to the data transmission media, sources of
radiation which are discreate, capturing of the instruments and quantization of
a image. All these factors are responsible for the degradation of a image. The noise
in a image can be removed by various denoising techniques. In this a image
noise is investigated for terms of its causes and how an image noise can be
removed better or which denoising algo gives better results in removing noise
from a image. A better denoising algorithm will provide a better deblurring of
an image. In a MRI various denoising algorithms provide different values of the
noise removal values.

Chapter – 4                                                Objective

4

The main objective of this MRI image are as follows:

1.      In
this various algorithms are used to give different values of a single image.

2.      To
apply a denoising algorithm on an image, so as to improve the quality of a
image

3.      The
information of a image should not be lost.

4.      To
calculate the values of various algorithms like wiener, total variation, bilateral,NL-means,
wavelet denoising and anisotropic diffusion  which will show the change in the image.

5.      To
compare the values of each denoising algorithm used on a MRI image.

Chapter – 5                                  Proposed work

5

The method of removing an nose from the image
is a challenging task. Various approaches are there for the removal of a noise
in a image, and these approaches provides a different results in the denoising
of an image. The main focus of these entire denoising algorithms is to remove
the noise in an image. In this various denoising algorithms are used on a
single type of image. The various algorithms used in this are as follows:

1.      Wiener
algorithm

2.      Total
Variation algorithm

3.      Bilateral
algorithm

4.      NL-means
algorithm

5.      SureShrink
algorithm

6.      VisuShrink
algorithm

7.      BayesShrink
algorithm

8.      Anisotropic
diffusion

All these various
denoising algorithms provides there different results. And the result of all
these algorithms is compared with each other. At the end the algorithm which
removes the noise in a image in large quantity is chosen as a best algorithm. All
these different algorithms will have different values for each different image.
Here we will perform these algorithms of different MRI images.