DEBLURRING OF IMAGES USING BLIND SCHEMES

DEBLURRING OF IMAGES USING BLIND SCHEMES
DEBLURRING OF IMAGES USING
BLIND SCHEMES
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
Master of Technology
in
Computer Science and Engineering
By
Pushpanjali Sahu
DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA
2007
DEBLURRING OF IMAGES USING
BLIND SCHEMES
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
Master of Technology
in
Computer Science and Engineering
By
Pushpanjali Sahu
Under the Guidance of
Prof. B. Majhi
DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
ROURKELA
2007
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the thesis entitled, ”Deblurring of Images Using Blind Schemes”
submitted by Ms. Pushpanjali Sahu in partial fulfillment of the requirements for the
award of Master of Technology in “Computer Science & Engineering ”at the National
Institute of Technology, Rourkela (Deemed University) is an authentic work carried
out by her under my supervision and guidance.
To the best of my knowledge, the matter embodied in the thesis has not been submitted to any other University / Institute for the award of any Degree or Diploma.
Place : NIT Rourkela
Date :
Dr. B. MAJHI
Professor
Dept. of Computer Science & Engg.
National Institute of Techonology
Rourkela - 769008.
ACKNOWLEDGEMENT
A blend of gratitude, pleasure and great satisfaction is what I feel to convey my indebtedness to all
those who directly or indirectly contributed to the successful completion of this Thesis. I express
my profound and sincere gratitude to my Guide, Prof. B.Majhi, Prof., CSE, whose Persistence
guidance and support helped me in the successful completion of the work in stipulated time.
His expert knowledge and scholarly suggestion help me to visualize the difficulty aspects of the
thesis and to find their design solution.
I am grateful to Dr. S. K. Jena, Head of the Department, Computer Science and
Engineering, NIT Rourkela for his support during my work. My sincere thanks goes
to Pankaj Kumar Sa, Lecturer, Computer Science and Engineering for his constant
motivation. I am thankful to all my Professors and Lecturers and members of the
department for their generous help in various ways for the completion of the thesis
work.
I am thankful to all my batchmates and friends for their effective cooperation.
Thank you Tony, Anamika, Mita, Kalpana, Madhu.
Lastly, but not the least, I am deeply indebted to my parents and my brothers for
their constant support, encouragement and understanding during the voyage. I can
offer here only an inadequate acknowledgement of my appreciation to my family.
Pushpanjali Sahu
Contents
List of Figures
vi
List of Tables
vii
1
2
INTRODUCTION
1
1.1
Formulating Image Restoration Problem . . . . . . . . . . . . . . . . . . .
3
1.2
Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Key Motivation for Blind Image Deconvolution . . . . . . . . . . . . . . .
5
1.4
Problem Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.5
Blind Image Deconvolution Techniques at a Glance . . . . . . . . . . . .
7
1.6
Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
INTRODUCTION TO IMAGE BLUR
2.1
The Point Spread Function,PSF . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1
3
11
Types of Blur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2
Modeling space-invariant Blur . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3
Motion Blur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4
Gaussian Blur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5
Simulated Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
BLIND DECONVOLUTION USING EVOLUTIONARY ALGORITHM
19
3.1
Applications of Evolutionary Algorithm, EA . . . . . . . . . . . . . . . . 21
3.2
Multiobjective Optimization using EA . . . . . . . . . . . . . . . . . . . . 22
3.3
EA for Blind Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4
Underlying Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5
Underlying Constraint on PSF and Image . . . . . . . . . . . . . . . . . . 24
3.6
Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
iii
4
3.7
Realizing BID using EA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.8
Image fusion using pseudo-wigner distribution, PWD . . . . . . . . . . . . 26
3.9
Simulation Results of EA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
TWO PASS HEURISTIC APPROACH FOR MOTION BLUR PARAMETER
IDENTIFICATION
5
30
4.1
Motivation for Blur Identification . . . . . . . . . . . . . . . . . . . . . . . 31
4.2
Theoritical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3
Implementing the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4
Simulation Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
CONCLUSIONS AND FUTURE WORKS
Bibliography
41
43
iv
Abstract
The thesis presents two blind deconvolution schemes for image blur removal. The two
major types of blur has been worked out, namely, the gaussian blur and the motion blur.
The image corrupted by the gaussian blur is reconstructed by Evolutionary algorithm
using pseudo-wigner distribution. The second method deals with heuristically estimating
the blur parameter of the image undergone motion blur. The gaussian effect is mostly
observed in astronomical imaging. The image deblurring for motion blurred image is
required due to hardware incapability of capturing the exact information of moving
object or with moving camera. In this thesis, an observed image is assumed to be
the two dimensional convolution of the true image with a linear-shift invariant blur,
known as point-spread function, psf, and the additive noise is assumed to be zero.
The Evolutionary algorithm has been implemented to remove gaussian blur. The
atmospheric turbulence is mostly modelled by the gaussian psf. The algorithm proceeds
by randomly generating the psf’s at each generation. The psf’s at each generation are
used to estimate the true image. The best fitted images are then given as input to the
next generation. After few generation, the most feasible images are chosen. These
closer estimated images are fused using pseudo-wigner distribution to reconstruct the
final required image.
The inherent dynamic characteristic of the nature gives rise to motion blur. Whenever there is relative motion between the object to be captured and the imaging system,
the image captured at that instant is suffered by the type of blur known as motion blur.
A new heuristic approach has been framed out with the purpose of estimating the
motion blur parameter. This type of blur is characterised by its length and the motion
direction. These parameters are then used to restore the image. The motion direction is
estimated from the fourier domain of the observed motion blurred image. The length
is then iteratively computed using Entropy and the RMSE as the quality metrics.
v
List of Figures
1.1
Image Degradation and Restoration Process . . . . . . . . . . . . . . . . .
3
2.1
Image formation with PSF . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2
Result for different blur effect. . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1
General scheme of EA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2
General scheme of BID using EA . . . . . . . . . . . . . . . . . . . . . . . 26
3.3
Gaussian Blurred Lena Image and Restored Lena Image
3.4
More Gaussian Blurred Image Lena Image and Restored Image . . . . . 28
3.5
Gaussian effected Lena Image . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6
Gaussian blurred lake image with varying value of standard deviation. . 29
4.1
Motion Blurred Lena Image with varying angle and its corresponding
. . . . . . . . . 28
Fourier Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2
Plotted Entropy of images. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3
Entropy plot of images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4
(a)Lena image blurred with psf length=11 and angle=23. (b) Restored
image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
vi
List of Tables
3.1
Algorithm for PSF generation . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2
Results of Observed Blur Parameter . . . . . . . . . . . . . . . . . . . . . 40
vii
Chapter 1
INTRODUCTION
1
Vision is the foremost trusted source of information compare to other human perceptions. And Image is the basic container of any pictorial information . The process of
retrieving and analyzing the pictorial information by a digital computer is known as
digital image processing. The improvement of pictorial information for human interpretation and processing of scene data for autonomous machine perception are the root
application areas that had sown the interest in image processing field decades ago[6].
Ideally, when an image is generated from a physical process, its values are propotional to energy radiated by a physical source. And hence, the resultant image, i(x,y),
is nonzero and finite[5], i.e.,
i(x, y) ∈ Z
(1.1)
where Z is a finite set of integers, and x, y denote spatial coordinates. Hence, an
image is interpreted as a two-dimensional light intensity function, i(x, y), and the value
of i, at any point (x, y) is propotional to the brightness(or gray level) of the image at
that point[5]. A digital image can be considered as a matrix whose row and column
indices represent point in the image and the corresponding matrix element known
as picture element, pixels value identifies the gray level at that point. The digital image
processing takes as input an image always but the output can be an image or some
relevant information retrieved after applying some function on the given input image.
The various methods included as fundamental techniques of digital image processing
are[5]:
1. Image Representation and Modeling
2. Image Enhancement
3. Image Restoration
4. Image Analysis
5. Image Reconstruction
6. Image Data Compression
2
Figure 1.1: Image Degradation and Restoration Process
The image processing technique dealt in this thesis is image restoration. Image
restoration has been explored a lot till date but much has to be mined of still. From
the wide broad spectrum of image restoration, only a part is worked out in this thesis,
reconstruction of the true image and blur parameter estimation.
1.1 Formulating Image Restoration Problem
The image is characterized by two major components : illumination and reflectance components. But, practically apart from these two components, the image formation also
depends on the characteristicts of the object being captured, environmental conditions
during capture, and the imaging system being used. These other components produce
an ill-effect during image acquisition to produce a degraded image, c. The process of
reconstructing the original scene from a degraded version is the goal of image restoration. The ill-effect causing function, d f is known as the blur. The additive noise effect is
also considered as another cause of degradation. Thus, the image degradation model
is,
c = df i + η
(1.2)
Given c, some knowledge about the degradation function d f , and some knowledge
about the additive noise term η, the objective of restoration is to obtain an estimate î of
the original image. This estimate, î, should be as close as possible to the original input
image. In general, the more knowledge we have about d f and η, the closer î will be to
i. Figure 1.1 shows the image degradation and restoration process.
Image restoration refers to removal or minimization of known degradations in an
3
image. The process includes deblurring of images degraded by the limitations of a
sensor or its environment, noise filtering, and correction of geometric distortion or
non-linearities due to sensors.
The image restoration had found its break through application in the engineering
community in the area of astronomical imaging. Due to rapidly changing index of refraction of the atmosphere, ground-based imaging systems were subject to blurring[3].
Extraterrestrial observations of the earth and the planets were degraded by motion blur
as a result of slow camera shutter speeds relative to rapid space-craft motion. Till today,
key focus area of image restoration is astronomical imaging. Another challenging area
where image restoration play a very important part is in the area of medical imaging.
The media also, particularly movies, even remain untouched by image restoration. The
use of digital techniques to restore aging and deteriorated films is also important and
well-known application area of image restoration. Digital image restoration is being
used in many other applications as well, say, for example, lastly but not the least, to
improve federal aviation inspection procedures, restoration has been used to restore
blurry X-ray images of air craft wings[11].
1.2 Problem Statement
Image deconvolution refers to the act of discovering the original image from the observed corrupted image. The blind image deconvolution,BID, refers to the task of separating two convolved signals, i and d, when both the signals are either unknown or
partially known[1]. The recovery process or reconstructing process can be subdivided
into two categories as:
1. Classical Restoration
2. Blind Image Restoration
Classical restoration includes the techniques that utilize some prior information
regarding the degradation of image during reconstruction while Blind Image restoration
is the process of estimating both the true image and the blur from the degraded
image characteristics, using partial information or no information about the imaging
system. In classical restoration, the blurring function is given and the degradation
process is inverted using one of the many known restoration algorithms. In blind
4
image deconvolution, an observed image c(x, y), is assumed to be the two dimensional
convolution of the true image i(x, y) with a linear-shift invariant blur , known as pointspread function, PSF, d(x, y) and the additive noise is assumed zero[1]. That is,
c(x, y) = i(x, y) ∗ d(x, y)
(1.3)
The problem of reconstructing the true image i(x, y) requires the deconvolution of the
PSF, d(x, y) from the degraded image, i(x, y). A lot of research has been done exploring
the various methods for image deconvolution as blind techniques. But still, is a critical
and challenging problem for the researchers.
1.3 Key Motivation for Blind Image Deconvolution
Inspite of being difficult problem, the blind image deconvolution has enjoyed wide
application area in most of the practical scenarios. The major motivations behind blind
image deconvolution can be focused as:
1. Use of high cost adaptive-optics systems to overcome the blurring problem in
astronomical imaging is impractical for analyzing some observation. Instead the
blind deconvolution is cheapest way to retrieve the relevant information from
the degraded image as a post-processing technique.
2. Some application area such as medical imaging rely on high image quality for
close diagonsis, like X-ray imaging, which inturn demands for increased incident
X-ray beams intensity. But practically, this is hazardous for patients health and
hence blurring is inevitable. Hence, BID is used to tackle with the degradation.
3. Instant deblurring cannot be done by predeterming certain PSF in real-time
applications such as video-conferencing. Also, on-line degradation determining
technique is error prone and create artifacts in restored image.
4. Lastly, but not least, to predermine any information about any scenario is practically either too costly, or dangerous and sometimes mostly impossible. Also, the
degradation specified is not necessary true for deblurring. Hence, blind approach
is adopted to solve the problem.
5
1.4 Problem Characteristics
The blind deconvolution of image proceeds based on certain assumptions. The problem
of blind deconvolution of image has few characteristics as listed below:1. The convolved signals, i.e, the true image and the psf are irreducible[1]. This is
the most important characteristic for the unambigous deconvolution. The true
image(or the psf ) should not be resultant of convolution of two or more other
independent different component signal. Let us assume that the true image,
i(x, y) = i1 (x, y) ∗ i2 (x, y) and the psf, d(x, y) = d1 (x, y) ∗ d2 (x, y), then the image
model can be re-written as :c(x, y) = i1 (x, y) ∗ i2 (x, y) ∗ d1 (x, y) ∗ d2 (x, y)
(1.4)
Hence, the deconvolution would be ambiguous as its quite difficult to clearly
classify the true image component and the psf component.
2. The blind image deconvolution produces the scaled, shifted version of the true
image. So,
î = Ai(x − b1 , y − b2 )
(1.5)
where A, b1 , b2 are arbitrary real constants and î is the estimation of original
image. This shifted and scaled version is again treated using some other information as support constraint etc. to further improve the result[1]. But the blind
deconvolution cannot find A, b1 ,b2 .
3. Many of the blind deconvolution techniques assume noiseless condition for reconstruction. But, the noise persists in every practical scenerio. Hence, neglecting
the noise while reconstruction obiously effect the result.
4. The blind deconvolution is ill-posed problem. The solution set may entirely
change even with small variation in the assumed data sets used for reconstruction.
Moreover, modelling the data set for exact deconvolution is not an easy task. Also,
the exact deconvolution demands for consistent data set which is not applicable
for every practical scenerio.
5. The imaging system may also be noisy. So, the noisy imaging system may also
6
lead to false result. The noise information is statistical in nature, so the direct
substraction is also not possible[1].
6. The resultant is obtained based on one or the other optimal criteria, which in turn
rely on some partial information about the imaging procedure. The additional
assumptions and proper initialization is required for proper result. Hence, the
solution is not unique.
7. The three important factors namely convergence property, computational complexity and portability altogether are hard to meet in a single existing blind
deconvolution technique. There exists no clear demarcation for all three mostly
the convergence property. Also, the properties are not fixed but have to variate
as the imaging system as well as imaging procedure and application varies. Let,
say, for example, in real-time imaging application there is requirement of less
computational complexity and speedy convergence while in medical imaging,
the reliability plays an important role for good result as compare to other two.
1.5 Blind Image Deconvolution Techniques at a Glance
The process of separating two signals that have been convolved is termed as the deconvolution problem. When both the signals are unknown, it comes under the category
of blind technique. The separation of signals using blind deconvolution technique
proceeds with some distinguishing characteristics of both the signals that has to be
deconvolved. Some signal characteristics must be known and those characteristics has
to be kept as nonspecific as possible in order to achieve the solution. There are wide
applications of blind deconvolution in seismic, speech, signal processing and ofcourse,
in image processing.
The blind deconvolution in image processing first started in the area of astronomical imaging. Due to rapidly changing refractive index of the atmosphere, the
ground-based imaging system suffered from blurring. This was first scenerio for applying restoration technique but the various practical applications demand for blind
deconvolution still today. The blind deconvolution of images are done by following
either of the two approaches[1] :1. The degradation function, psf, is identified and then using any classical restoration
7
technique such as inverse filtering, weiner filtering, pseudoinverse filtering, the
true image is identified. This method is simple and less computation is required.
The algorithms in this approach are known as a priori blur identification technique.
2. The identification of the psf and the true image is done simultaneously in the
restoration algorithm. Hence the algorithms in this approach are complex.
Based on these two approaches, the various existing blind image deconvolution
techniques is classified. Lane and Bates approach for multidimensional deconvolution
is based on the concept that any degraded image, c, can be deconvolved if the individual convolved components, i1 , i2 , ...., in , have compact support and c has dimension
greater than one[14]. Their method exploit the analytical properties of Z-transform for
deconvolving the degraded image. The technique is known as zero sheet separation .
This method is based on assumptions such as :
• The imaging system is noiseless.
• The true image, i, and the degradation function, d are irreducible. The ZT of each
i(x, y) and d(x, y) is zero on a single continuous surface, known as zero sheet[14].
• The zero sheets of i(x, y) and d(x, y) are distinct.
• The true image and the degradation function have finite support.
This method can also be utilized for deconvolving more than two signals. But this
approach is prone to noise.
A priori blur identification techniques are the simplest blind deconvolution techniques. The degradation function is estimated first and the true image is estimated.
The approach shows the promising results once the blurring parameters are estimated.
The technique is applicable to the situations in which the true image is known to possess special features, and/or when the degradation function is known to be of special
parameteric form. One of the most popular method of this category for blur identification is use of frequency domain nulls of the degraded image to perform blind
deconvolution [15]. The algorithm is sensitive to noise, since the noise has the effect of
masking the frequency domain nulls of observed image. The algorithm is improved
by considering the noise effect by Tekalp [17] by using the bicepstrum instead of the
cepstrum. Though the cepstrum method is more popular due to its computational
8
complexity. The major limitation of the algorithms of this class is that a parameteric
form of the degradation function has to be known.
Another class of blind image deconvolution include nonparameteric deterministic
image constraints restoration techniques. The algorithms of this group assume some
deterministic constraints for estimating the true image. These deterministic constraints
are like non-negativity, known finite support and existence of blur invariant edges. The
methods apply these constraints as optimal criteria. The true image and the blur both
are identified simultaneously. The two most popular methods of this group are:• Iterative Blind Deconvolution first proposed by Ayers and Dainty [18]. This is
the simplest blind technique till date. The method uses the Fast-Fourier transform(FFT) for reconstructing the true image and the psf. The algorithm starts with
some random initial guess about the true image, i(x, y). With this initial estimate,
the algorithm proceeds iteratively computing the psf and the true image. The
technique utilizes some a priori information regarding i(x, y) and d(x, y). The observed image can be noisy or can be noiseless. The algorithm alternates between
the image domain and the fourier domain. The fourier domain constraint is imposed while dealing with fourier components and similarly, the image domain
constraint is enforced while estimating the image and the blur. The algorithm
is prefered for its low computational complexity[1]. The entire computation is
done in the fourier domain which is cheaper and faster than the spatial domain
operation.But the major drawback is that the algorithm is unstable. The algorithm is modified for robustness against noise by using weiner-filter instead of
inverse filter by Deepa Kundur [1]. The solution uniqueness and convergence
criteria are not fixed.
• Nonnegativity and Support Constraints Recursive Inverse Filtering(NAS-RIF)
was proposed as a solution to problems associated with the poor convergence
properties of the iterative blind deconvolution [1]. The method involves the
iterative minimization of a convex cost function. The image is restored by filtering
the blurred image to produce an image estimate which is restricted to lie on a
convex set representing the known deterministic constraints of the true image.
The major advantage of the method is that it does not impose any constraint on
PSF extent, the information about the PSF size is difficult to obtain as required in
9
other algorithm.
The neural network approach of image restoration is emerging research area. The
perception based adaptive approach of deconvolution by Stuart and Gaun is contribution to overcome spatial-variant type of degradation [19]. The concept of adaptive
constraint parameter (ACP) restoration is introduced in order to cope with different
statistical properties in different parts of an image.
1.6 Thesis Overview
This thesis consists of 5 chapters. The first chapter provides an introduction about
the problem and brief idea of different current approaches in the field of blind image
restoration. Chapter 2 briefly describes about the commonly occuring blur. Chapter 3
presents the Evolutionary algorithm as a solution to gaussian blur. In Chapter 4 novel
approach of motion blur estimation is presented.Chapter 6 concludes this thesis.
10
Chapter 2
INTRODUCTION TO IMAGE BLUR
11
The recognition and interpretation of actual information depends entirely on the image
quality. Something that is hazy and indistinct to the sight that restrict the sound and
clear visual perception have impact on image if captured at that instant. This ill-effect
created from the indistinctness of information deteroirating the image quality while
image aquisition is known as blur.
The basic approach for any deblurring technique is to acquire atleast minimal information about the blur caused. The various existing techniques to revert back the effect
and reconstruct the true image depends on the quantitative yet effecive knowledge
about the cause of degradation. The fundamental method of image restoration is to
estimate the parameters of degradation and then apply any classical restoration technique for image reconstruction[6]. The convergence of many a priori deconvolution
techniques also demands for certain parameteric assumptions and estimation of image
degradation[1]. The existing blind deconvolution techniques, such as IBD, [1] too rely
on some initial estimation of the image[3].
2.1 The Point Spread Function,PSF
The degradation producing ill-effect of blur is termed as the point spread function, psf.
Any type of blur is characterized by the psf . The electromagnetic radiation or other
imaging waves propogated from a point source or point object is known as the psf. The
quality of any imaging system depends on the degree of spreading(blurring) of the
point object. The psf defines the impulse response of a point source[1]. This is shown
in Figure 2.1. When an image is captured by any recording system, the intensity of a
pixel of the recorded image is directly proportional to the intesity of the corresponding
section of the sight be captured. But this is ideal situation. Practically, the recorded
intensity either gets effected by the noise or blur.
2.1.1 Types of Blur
The psf causing degradation of any image are often two types as:
1. Spatial-variant blur
2. Spatial-invariant blur
12
Figure 2.1: Image formation with PSF
The mathematical model for image formation can be reformulated by the linear
equation[6]:
c = df i + η
(2.1)
where c is the observed blurred image, i is the true image η is the noise and d f is
the blurring function, the psf. The defocusing of any portion of the sight depends on
the intensity and other characteristics of the corresponding section of the sight being
captured. If the blurring is not distributed in a homogenous way, i.e, every pixel is
defocused differently such that blur effects every pixel with varying strength, then
the blur is known as spatial-variant blur. But if the image is blurred such that every
spatial location is effected equally then it is considered as spatial-invariant blur. The
degradation function, as stated in (1.1), is said to be spatial-invariant if for any shift
k ∈ Z2 ,
c(x) = d f [i(x)] implies that c(x − k) = d f [i(x − k)]
According to signal processing theory, this shift-invariant linear function is expressed
in the form of convolution. So the above Equation 2.1.1, can be expressed in case of
spatial-invariant blur as,
c(x, y) = i(x, y) ∗ d(x, y) + η(x, y)
13
(2.2)
where ’*’ indicates spatial convolution. The above eq, in discrete form is represented
as,
X
c(x, y) =
i(m, n)d(x − m, y − n)
(2.3)
m,n
The above equation in an frequency domain can be written as :
C(u, v) = I(u, v)D(u, v) + N(u, v)
(2.4)
Thus blurring in this case is described as the convolution of the original image with
a psf. Many restoration schemes assume that the psf, d, is spatially-invariant and spatial
dependence is often ignored. But there are certain scenerios, where spatial variation
of the degradation function has to be considered. For example, if the scene contains
two objects moving with different velocities to the recording system,then the psf is
effectively different for each [12]. Also in the case of original Hubble Space Telescope
Wide-field/Planetary camera, due to errors in the shaping of the mirrors,the psf has a
large amount of spatial variation[13].
The image restoration in space-invariant case is done by deconvolving the observed
image with psf. In space-variant case, the psf at different regions are first estimated
and the corresponding image region is deconvolved with it and finally, the different
restored image regions are joined together to reconstruct the entire true image estimate.
The process of modelling and image restoration in case of spatially-varying blur is
more challenging problem as compare to spatially-invariant blur. The various types of
spatial-invariant blur and their existing identification approaches has been described in
the thesis.
2.2 Modeling space-invariant Blur
The difficulty in solving the restoration problem with a space-varying blur motivates
the use of the stationary model of blur. The different models have been proposed in
the literature for representing the space-invariant psf. The models can be described as:
1. Motion Blur: The moving objects when captured by the camera or the stationary
object captured by moving camera cause motion blur. Many types of blur has
been framed out in the literature. This can be in the form of a translation, a
14
rotation, a sudden change of scale or to some combination of these. But, here
only the translation is considered. When the object is moving with a constant
velocity V is being captured by the camera with an exposure time interval [0 T],
the distortion is given by[4]:


1


 VT δ(y − t), 0 ≤ (x − s) ≤ VT
d(x, y; s, t) = 


 0
otherwise.
(2.5)
2. Out-of-focus Blur: When a sight is captured by the camera in a two-dimensional
field, it may happen that some parts are in focus while other parts are not. The
degree of defocus depends upon the effective lens diameter and the distance
between the object and the camera[1]. This leads to the point-spread function:


1
2
2
2


 πr2 if (x + y ) ≤ r
d(x, y) = 


 0
otherwise
where r is the radius of the circle of confusion[4].
3. Atmospheric Turbulence Blur: The blur occured due to the long-exposure through
the atmosphere in certain application is modelled as the gaussian psf [3]. The psf
in this case is given as:
d(x, y) = K exp (−
x2 + y2
)
2σ2
2.3 Motion Blur
When a photograph is taken of any moving object or the imaging system itself is
moving then the degradation caused is motion blur. Motion blur cause significant
degradation of the image. This is caused by the movement of the object relative to the
sensor in the camera. The motion blur occurs if any of the following condition persist,
1. Moving Object captured by static camera,
2. Static Object captured by camera in motion,
3. Both Object and camera are in motion,
15
4. Shutter movement, film is exposed in a camera by the movement of the shutter
across the film plane.
The two types of motion blur has been studied. They are:
• Linear-Horizontal Motion Blur: The motion blur arising either to camera moving
or the object moving horizontally is given as, L being the blur length. More
precisely, L is the number of additional points in the image resulting from a
single point in the original scene[4].


−L
1


 L if 2 ≤ x ≤
d(x) = 


 0 otherwise
L
2
• Angular Motion Blur: When the scene to be recorded translates at a constant velocity, V, with an angle of θ degrees from the horizontal axis during the exposure
interval, [0 T], then the psf observed is given as[7]:


1


 L if 0 ≤ |x| ≤ L cos θ, y = L sin θ
d(x, y) = 


 0 otherwise
There are several techniques for prevention motion blur either during image capture
or by using any postprocessing techniques to remove blur. The motion blur can be solved
by adopting any of the following measure:• Using hardware in the optical system of the camera to stabilise the motion
• Post-processing technique by estimating the camera’s motion from single image(blind deconvolution)
Hardware approach can be the solution but is effective for removing small amount
of camera shake for short exposure. But this method does not involve any image
processing method, so is not discussed. The motion debluring as a post processing
step can be done by estimating the motion blur parameters.
The post-processing techniques for motion blur estimation include Cepstral Analysis, Radon Transform, Steerable Filter [20]. Stern and Kopieka provided the analytical
method of finding optical transfer function for image motion using moments [21]. A
novel approach of identification of motion direction given only the blurred image is
16
presented by Tan and Zhang [22]. The thesis presents a heuristic approach of estimating the motion blur parameter. The algorithm exploits the fourier spectrum for
direction estimation. This is discussed in detail in the chapter 4.
2.4 Gaussian Blur
The image degradation due to atmospheric condition is modelled by the gaussian
effect. The gaussian blur is a type of image blurring filter that uses normal distribution
for calculating the transformation to apply to each pixel in the image. The visual effect
of this blurring is a smooth blur resembling that of viewing the image through the
translucent screen.
The apparent ”wobbling” distortion of scene when viewed through an open fire or
across a hot road is well-known phenomenon. Similar effects is seen in the astronomical
imaging also. These effects are due to gradient in the refractive index in the atmosphere
resulting from temperature variation mostly. The visualization and characterization of
the effect is of considerable interest to atmospheric scientists, earth-based astronomers
and also for long-range surveillance. The deconvolution techniques are required for
proper analysis in the scientific study. The research on various deblurring technique
has been done in the literature to overcome this problem. The thesis presents the
approach for restoring the image degraded due to gaussian effect by an Evolutionary
algorithm. This is desribe in detail in chapter 3.
2.5 Simulated Result
The standard lena image of 256 × 256 is synthetically blurred with different types of
blur. The blurred image are shown as in the figure 2.2. Figure 2.2(a) is Original Lena
image, (b) Motion blurred image with L = 15, θ = 450 , (c) Misfocussed image with
r = 10, (d) Gaussian blurred image with σ = 0.85.
17
(a)
(b)
(c)
(d)
Figure 2.2: Result for different blur effect.
18
Chapter 3
BLIND DECONVOLUTION USING
EVOLUTIONARY ALGORITHM
19
The recent development and the growing popularity of genetic algorithm in the various field, motivated the researchers to utilize the same in the field of image processing
also. The evolutionary algorithm is the generic name for the genetic algorithm. The
evolutionary algorithm (EA), in artificial intelligence, is a subset of evolutionary computation. It is generic population based metaheuristic optimization algorithm. An EA
utilizes the concept of biological evolution. Evolutinary algorithms are search techniques based on the concept of natural selection and survival of the fittest in the natural
world[8]. EA are computer programs used to solve complex problems by imitating the
Darwins theory ”survival of the fittest”[8]. In a EA a number of probable solutions are
generated over the problem space. They then compete each other to find optimal area
of the search space.
EA have number of operators, components and procedures that must be specified
clearly in order to define particular EA[8]. Also,the initialisation and termination
condition must also be well specified.The different components of EA are:• Definition of Individuals
• Fitness Function
• Population
• Recombination Mutation
• Survivor Selection Mechanism
Unlike other traditional optimization techniques, EA involve a search from set of
possible solutions known as ”population”. Each iteration ends with a set of possible
and feasible solutions, discarding the poor solutions based on some ”fitness” criteria.
The solutions with high fitness are then recombined with other solutions by interchanging parts of the solution with one another. These solutions are then again ”mutated”
generating new solution optimal to the given problem. The scheme is shown in Figure 3.1. Several types of evolutionary algorithm search techniques were developed
independently. The history of the technique suggests that there are number of variants
of the EA, but the basic underlying idea is same:-from the given population, only the
fittest candidates are opted for the solution space. Some of the flavors of EA are:• Genetic Programming, which evolve programs
20
Figure 3.1: General scheme of EA
• Evolutionary Programming, which focusses on optimizing continuous functions
without recombinations
• Evolutionary Strategies, which focusses on optimizing continuous functions with
recombinations
• Genetic Algorithm, which focuses on optimizing general combinatorial problem.
3.1 Applications of Evolutionary Algorithm, EA
Evolutionary algorithm is considered as global optimization technique. The most important factor about EA is its robust performance in ”noisy” functions where there
is multiple local optima. EA can find global optimal solutions discarding the local
minima. There exists wide variety of application domains of EA for finding optimization problems such as wire routing, scheduling, travelling salesperson, image
processing, engineering design, parameter fitting, knapsack problem, game playing
and transportation problem. EA are well suited for wide range of combinatorial and
continuous problems [8], but the different variations are tailored depending upon specific applications. The concept of EA has been utilized in the field of image processing.
Different image processing area such as edge detection, segmentation, shape detection,
feature selection, clustering, classification, object recognition use EA to get the optimal
21
solution.
3.2 Multiobjective Optimization using EA
The purpose of simulating the evolutionary process, following the analogy between
evolutionary mechanism and the learning process(optimization), led to the development of evolutionary algorithm. Multi-objective optimization is the fastest growing
technique in the recent years among many other emerging techniques of EA. In multiobjective problem, there is no single objective and possibly no single solution. The
solution is selected from a set by making compromises. A suitable solution is chosen
satisfying the different objectives.
The main motivation for using EA for solving multi-objective optimizaion problems
is because EAs deal simultaneously with a set of possible solutions (the so-called
population) which allows us to find several members of the Pareto optimal set in a
single run of the algorithm[8], instead of having to perform a series of separate runs as
in the case of the traditional problem solving approach.
3.3 EA for Blind Deconvolution
The development and successful application of EA in different complex problem motivates to utilize the concept in the field of blind image deconvolution. The blind
deconvolution is the practical method of image reconstruction, when it is not possible
to reconstruct the image with all psf. The EA provides large solution space for the
deconvolution and then the optimized solution is selected .
3.4 Underlying Assumptions
The image model as in Equation 1.3 is considered. So, an ideal noiseless scenerio is
assumed. As a matter of convinience, the model is again re-written below:c(x, y) = i(x, y) ∗ d(x, y)
22
(3.1)
The algorithm proceeds based on following main identity:i(x, y) = i(x, y) ∗ δ(x, y),
(3.2)
where δ is finite with all its elements zero except the central one ,given as,

 · · ·

 · · ·


δ =  · · ·


 · · ·


···
···
···
0
0
0
1
0
0
···
···

··· ···


0 · · · 


0 · · · 


0 · · · 


··· ···
where δ is finite square matrix with all its element zero but the central element is
one. The psf is supposed to composed of two matrices given as,
dL (x, y) + dH (x, y) = δ(x, y)
(3.3)
where L and H denotes for low and high, respectively. Hence,( 3.2) can be rewritten as,
i(x, y) = i(x, y) ∗ dL (x, y) + i(x, y) ∗ dH (x, y)
(3.4)
Now, if the observed blurred image, c, in(3.1) is given by
c(x, y) = i(x, y) ∗ dL (x, y)
(3.5)
then an iterative approach is derived based upon the following two equations:
î(x, y) = c(x, y) + ĩ(x, y) ∗ dˆH (x, y)
(3.6)
ĉ(x, y) = î(x, y) ∗ dˆL (x, y)
(3.7)
where ĩ(x, y) and î(x, y) denote the consecutive iterative estimations of the true image
i(x, y) at generation k and k+1, respectively. At every iteration, the psf, dˆL (x, y) is
randomly generated and dˆH (x, y) is obtained by using(3.3).
23
3.5 Underlying Constraint on PSF and Image
The blind deconvolution using EA assume some constraint on the PSF and the Image.
The PSF constraint assume that the blur function should be positive and the mean
value of the image is preserved during degradation. That is,
X
dˆL (x, y) = 1
(3.8)
(x,y)∈S
where S is the finite support of the textitpsf, dˆL (x, y). Also, the blurring function should
be symmetric with zero-phase[8]. Similarly, the algorithm assumes that the pixel value
restored image should be positive. And, the pixel intensity strictly lies between 0 and
255, considering only gray scale image. The image constraint can be formulated as,
if
î(x, y) ≤ 0
î(x, y) = 0
if
î(x, y) ≥ 255
î(x, y) = 255
3.6 Convergence Criteria
In order to control the convergence of the algorithm, the two feature vectors are used,
denoted as ρ1 and ρ2 [8]. The ρ1 is a peak signal-to-noise ratio between c and ĉ. The
optimal estimate of i(x, y) is the one that corresponds to the output î(x, y) with the
highest ρ1 value. Similarly, the ρ2 is defined as a peak signal-to-ratio between î(x, y)
and c. The optimal estimate of i(x, y) corresponds to the output î(x, y) with a ρ2 value
closest to a given minimum. The two conditions are used as a convergence criteria of
the algorithm,collectively denoted as,ρ = (ρ1 , ρ2 ), called as feature vectors[8].
3.7 Realizing BID using EA
The EA for BID can be implemented using by incorporting different steps EA with the
deconvolution steps. The EA provides large solution space thus easying to get optimal
24
Table 3.1: Algorithm for PSF generation
1.
2
3.
4.
5.
6.
Find the psf size,n × n, n should be odd number.
Randomly generate n+1
values in the range (0, 1).
2
Sort the values generated in ascending order.
Produce a vector d1 of n elements,
by flipping the values left to right provided
the middle element is untouched.
Create d = d1 ∗ dT1 .
Finally,mormalize the d.
solution. The basic steps adopted for achieving the objective is shown in the schematic
Figure 3.2. The major steps followed are:• Mutation:- A set of random PSFs is generated in every generation. These PSFs
are then used along with all individual images,obtained from the previous generation. This is done by following Eq. 3.6 and(3.7).
• Selection:- The individuals in each generation individually undergo a selection
procedure, as stated previously, called feature vectors. Those individuals which
have ρ2 value greater than the corresponding expectation value for the generation
are excluded.
• Clustering:- There exist only few individuals at each generation. The survivors
are used in the next k + 1th generation. The stopping criterion, |ρ̂k2 − ρ2 | ≤ ε,
is checked at each iteration. And ρ̂k2 is the average estimated value of ρ at kth
generation and εshould be greater than 0 and is experimentally determine.
• Final Image Reconstruction:- There is set of possible estimated image obtained
from each generation at the end. The best image is sort out by adopting fusion
method, pseudowigner distribution, pwd. The detail about pwd is discussed in the
next section.
The steps for generating set of psf generated in every generation is shown in the
Table 3.1.
25
Figure 3.2: General scheme of BID using EA
3.8 Image fusion using pseudo-wigner distribution, PWD
The fusion method used is based on the existence of inut realization of different input
realizations of the same object at the same time. The distance between the pseudo-wigner
distribution, pwd of the images to the pwd of the reference image ,c, on a pixel by pixel is
computed to perform deblurring. The Wigner distribution of 1-D function s(x) is given
by[9]:-
Z
∞
W(x, u) =
s(x +
−∞
α
α ∗
)s (x − ) exp−i(u∗α) dα
2
2
(3.9)
In this algorithm, a discrete Wigner distribution is used, formulated as below,
N
k= 2
X
W(n, m) = 2
m
s(n + k)s∗ (n − k) exp−2i(2π N )k
(3.10)
k=− N2
where, n and, m represent the spatial and space-frequency respectively, and k is the
shifting parameter. Equation(3.10) is interpreted as the discrete Fourier transform
(DFT) of product r(n, k) = s(n + k)s∗ (n − k). Different N-component PWD vector associated with every pixel on the image is computed and the measure is adopted for
determining their greatest distance to a reference image. The more defocused an im26
age is the more the high frequencies get diminish and hence, its pwd is affected. The
observed blurred image is taken as refernce. This helps in establishing at pixel level a
distance measurement for the PWD from each image,
∆i (ξ) = ||Wi (n, m) − Wr (n, m)||n=ξ
(3.11)
where ξ denotes an arbitrary pixel of the image,subindices i and r indicates input and
reference images, respectivel. Wi (n, m) represents the PWD for the reference image.
The norm operator in (3.11) is given as:N
i= 2
X
||ξ|| = (
1
ξ2 (i)) 2
(3.12)
i=− N2
where ξ represents any arbitrary real vector. The largest Euclidean distance corresponds to the pixel belonging to the less locally defocused image[8]. All pixels from
the p input images can be ordered following the same.
3.9 Simulation Results of EA
The algorithm is worked out to reconstruct the image degraded due to atmospheric
turbulence, commonly modelled as the gaussian psf. Three images have been used
for the simulation purpose, the standard lena image of size 64 × 64,128 × 128 and
the lake image of size 128 × 128. The gaussian psf of size 7 × 7 with σ = 0.65 and
σ = 0.85 is convolved with lena image of size 64×64 to artificially generate atmospheric
turbulence degraded image. Similarly, lena image of size 128 × 128 is synthetically
blurred using psf of size 15 × 15 and the σ = 0.85. The lake image is also artificially
blurred using varying psf size with σ values being 0.34,0.65 and 0.85 respectively.
The synthetically generated blurred figures and the corresponding restored are shown
in Figures. Figure 3.3(a) shows Lena image blurred with σ = 0.65, (b) is restored
image. Figure 3.4(a) depicts Lena image blurred with σ = 0.85, (b) is corresponding
restored Image.Similarly, Figure3.5(a) shows Lena image blurred with σ = 0.85and (b)
is restored image. Figure 3.6 shows blurred Lake Image, (a) is blurred with σ = 0.34
and (b) is restored image,(c) shows image blurred with σ = 0.55 and (d) is restored
image of (c),(e) depicts image blurred with σ = 0.85 and (f) is corresponding restored
27
image.
(a)
(b)
Figure 3.3: Gaussian Blurred Lena Image and Restored Lena Image
(a)
(b)
Figure 3.4: More Gaussian Blurred Image Lena Image and Restored Image
28
(a)
(b)
Figure 3.5: Gaussian effected Lena Image
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3.6: Gaussian blurred lake image with varying value of standard deviation.
29
Chapter 4
TWO PASS HEURISTIC APPROACH
FOR MOTION BLUR PARAMETER
IDENTIFICATION
30
Deconvolution is an engineering discipline, which refers to the retrospective improvement of fidelity of the electronic signals such as voice ,music, radar and pictures.
Image processing utilizes the same concept in restoring the true image from degraded
image. The input is the corrupted natural image and one of the many existing deconvolution techniques is used to retrieve the true image. But this restored image is the
estimation of the true image and hence the convergence of deconvolution techniques
should provide the approximate and closest estimate of the true image. The blind
image deconvolution on similar concept estimate the true image but there are almost
no or partial information about the cause of degradation function. The partial information can be in the form of some finite support or nonnegativity of the image, coined
as physical properties of the image. Similarly, this partial information can also be in
the form of any statistcal data such as entropy or probability distribution function of
the signal.The different optimality criteria along with this partial information form the
strong ground in image estimation.
The Visual world around us is always dynamic. The moving objects when captured
with any imaging system leads to ill-effect known as motion blur. The degraded image
can be restored if the motion blur parameters can be identified. Once the blur has been
estimated, the classical restoration can be utilized to restore the degraded image. This
blind technique of restoring the degraded image by first computing the psf and then
utilizing any of the classical restoration technique comes under a priori technique.
4.1 Motivation for Blur Identification
Blind deconvolution techniques have always been a challenging and critical problem.
But, since the techniques are more useful for the practical scenerio compared to classical
ones, the methods can not be ignored. Different blind image deconvolution techniques
assume various parameter to solve the problem. The literature review on different
techniques reveal that some strong underlying concept has to be used as a key to crack
the problem. Though the convergence is not well-defined as well as not sure. This
motivates to search for the parameters responsible for degradation. The parameter
once estimated is used to reverse the ill-effect. This chapter is contribution towards
motion blur identification.
The motion blur is caused whenever there is relative motion between the object
31
and the imaging system. This blur can be estimated if the length of the blur and the
direction can be somehow computed.
4.2 Theoritical Background
The 2-D discrete fourier transform, (DFT) of any f1 representing any MxN image, denoted
by F(u, v), is given by the equation[6] ,
M−1
N−1
XX
F1 (u, v) =
ux
vy
f (x, y) exp− j2π( M + N )
(4.1)
x=0 y=0
foru = 0, 1, 2, . . . , M − 1 and v = 0, 1, 2, . . . , N − 1.The exponential could be expanded
into sines and cosines with the variables v and u determining their frequencies. Since
f1 being an image so it is real, but its fourier transform in general is complex. The
principal method of visually analyzing the transform is to compute its spectrum[i.e.,the
magnitude of F1 (u, v)] and display it as an image. Let Rl (u, v) and Im (u, v) represent the
real and imaginary component of F1 (u, v), the fourier spectrum is defined as,
1
|F1 (u, v)| = [R2l (u, v) + Im2 (u, v)] 2
(4.2)
The phase angle of the transform is defined as,
φ(u, v) = tan−1 [
Im (u, v)
]
Rl (u, v)
(4.3)
The above two equation can be used to represent F1 (u, v) as below,
F1 (u, v) = |F1 (u, v)| exp−jφ(u,v)
(4.4)
The observed degraded image is nothing but the convolution of the true image
with some degradation function. This is true in the spatial domain. In the frequency
domain, the convolution is multiplication of the corresponding fourier transform of the
functions.
C(u, v) = I(u, v)D(u, v)
(4.5)
Substituting the value of I(u, v) and D(u, v) by following the definition of fourier derived
32
in the(4.4) in (4.5), we get,
C(u, v) = |I(u, v)| exp−jφi (u,v) |D(u, v)| exp− jφd (u,v)
(4.6)
Similarly subsequently substituting for C(u, v) in above equation. 4.6, the same fourier
definition,the resulting equation is,
|C(u, v)| exp jφc = |I(u, v)| exp−jφi (u,v) |D(u, v)| exp−jφd (u,v)
(4.7)
Equating the exponential parts of both the sides, omitting u and v for convinience, we
have,
φc = φi + φd
(4.8)
Thus the phase of the observed image is the addition of the phases of the true image
and the degradation function both. Figure4.1 depicts the fourier spectrum plotted for
lena image blurred with angle ∈ 00 , 150 , 300 , 450 respectively.
The degradation process can thus be understood as a filtering in the spatial domain
consisting of the true image i(x, y) with a filter mask, d(x, y). According to the convolution theorem, we can obtain the same result in the frequency domain by multiplying
I(u, v), the fourier transform of the true image by D(u, v), the fourier transform of the
filter. The D(u, v) is commonly referred as the filter transfer function. This D(u, v) is
the function that modifies F(u, v) in a specified manner. Hence, when formulated as
motion blur function, the filter operates on the frequency spectrum of the true image
inturn. The motion blur used is linear shift-invariant in nature and is characterized by
the length of the blur and the angle. Thus, when the filter operates on the true image,
the angular information of the mask gets embedded in the image along with other
parameters. This can be easily visualize in the fourier spectrum of the degraded image.
The angular information is investigated and the length is then computed by estimating the approximate of the true image. The quality of the estimated image is measured
using the Shannon Entropy. It is well-known that the digital image is a realization from
a source S generating symbol, si , identified as the gray values of the image. Associated
with a symbol source S, there is a number,H(S), called the entropy of the source, which
measure the amount of information in the source. The entropy of a source S is defined
33
by,
q
X
H(S) = −
P(si ) log2 P(si )
(4.9)
i=1
where P(si )= number of times si occurs in the image M × imesN. M × N is the size of the
image and q the number of gray levels in the source, i.e., 256 for a 8bits/pixel image. The
estimated image with the values closer to the highest entropy values are again checked
using othe metric , RMSE, with reference to the observed blurred image. The RMSE
between two images say, F1 and F2 is given as,
s
RMSE(F1 , F2 ) =
PN PM
x=1
y=1 [F1 (x,
y) − F2 (x, y)]2
N∗M
(4.10)
4.3 Implementing the Algorithm
The approach adopted here starts with closely visualizing the Fourier spectrum of
the blurred image. Whenever the image suffers from motion blur,the direction of the
motion is well preserved in the fourier domain of the blurred image.This can be clearly
analyze from the figure.4.1.
The angular direction gets captured in the fourier spectrum. The periodic blackwhite band subtend the angle with the verical axis from the origin. Thus, the fourier
spectrum of the blurred image is utilized to get an approximate value of the motion
direction. This angle is roughly computed to get an closer value of the actual blur
angle.
The bright band passing through the origin is used to get a close approximation of
the angular direction. A pixel with a maximum intensity can be traced out at some
radial distance from the origin. The radial distance is set and the pixel is searched
for each degree of rotation. Here, the search has been done for the angles ranging
from 0 to 90 degree. The angle which gives the maximum intensity pixel at radial axis
being set before is the required rough estimated direction. The approximate angle so
obtained is then ranged to get the more refined angle. For each value of the angle in
the newly approximated domain,the length is estimated. For each value of length the
psf is computed and the image estimate is obtained. The image estimate is then used to
calculate the entropy of the corresponding image. The image estimates with the entropy
values closer to the highest entropy of the all estimated image are sorted out. These
34
image estimate are further tested for closed approximation of the true image. This is
done by computing the RMSE,root mean square error with referenced to the observed
image. This steps helps in filtering out the image estimates that have entropy closer to
the high value but contains no information.
35
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 4.1: Motion Blurred Lena Image with varying angle and its corresponding
Fourier Spectrum
36
The computed RMSE values are then arranged in the ascending order. The length
and the angle corresponding to the minimum RMSE is the required parameter. The
different algorithm steps are tabulated as below:
Table 4.1: Algorithm
1.
Find FFT of the given blurred image c,
C1 = F (c).
2.
Shift C1 to origin.
3.
C2 = log(1 + |C1|)
4.
Find the approximate angle subtended by the spectrum C2
to the origin from vertical axis, Ainit .
5.
Variate the approximated angle from −α to α, A = {−α + Ainit : α + Ainit }.
6.
For each value of A and length, L = {3 :
Obtain Image estimate for each psf.
7.
Compute Entropy for each estimated image.
8.
Obtain angle and length of the estimated image giving values
closer to the maximum entropy.
9.
Calculate RMSE using the filtered image at step8 with respect to blurred image.
sizeo f (c)
},
2
calculate the psf.
10. The parameters giving minimum RMSE is the required angle and length.
4.4 Simulation Result
The proposed algorithm is simulated for standard grayscale Lena image of size 64 × 64.
The true ground image is synthetically blurred with the psf of varying length and angle.
The psf length is varied from 5 to 30. The motion blur with angle ranging from 00 to
450 . The approximated angle range is increased using α = 5. The entropy is iteratively
caluted by using the estimated angle but with varying blur length. This provide with a
solution space consisting of high entropy value. But, the true image estimate entropy
lies closer to the highest entropy among all. This can be observed from the Figure 4.3.
37
Entropy Plot of Image at different Length and Angle
Peak Entropy
Values
Entropy
6
5
4
3
20
2
15
1
10
Angle
5
0
0
5
10
15
20
25
Blur Length
0
30
Figure 4.2: Plotted Entropy of images.
Figure 4.2 depicts the entropy plot of the images estimated for blurred image with
Length=11 and θ = 150 which shows prominent peaks at various places. Similarly,
Figure 4.3 is entropy plot of the images estimated for image blurred with Length=17
and θ = 170 which shows peak at different places including at desired place. The
prominent peaks at some value of angle and length actually correspond to informative
image estimate but not exactly the true image. Some of these estimate correspond to
the obsolete result. To get rid of these estimates, the estimated image are again chosen
based on the RMSE with reference to the observed blurred image. Then the minimum
RMSE correspond to the desired length and the angle. The image is then restored using
the classical inverse filter. The following figure shows the restoration. The 64 × 64 lena
image is blurred with the psf of length=11 and θ = 230 . The result is shown in the
figure 4.4. The ground image is blurred with different values of the psf. The algorithm
shows the good result till length=25. The result is tabulated in the Table 4.4, showing
succesful estimation in many cases tested. The deviations are obtained mostly when
the image is blurred with length greater than half of its size. But the result includes
38
Entropy plot of image at varying Length and Angle
6
5
Peak Entropy
values
Entropy
4
3
2
1
0
15
10
Angle
5
0
0
15
10
Blur Length
5
20
25
30
Figure 4.3: Entropy plot of images.
more number of favourable solution as compare to the obsolete ones.
(a)
(b)
Figure 4.4: (a)Lena image blurred with psf length=11 and angle=23. (b) Restored
image.
39
Table 4.2: Results of Observed Blur Parameter
Actual
Observed
Length & Angle
7,0
7,45
11,45
13,19
15,25
9,19
7,30
15,19
23,45
7,19
27,23
Length & Angle
7,0
7,45
11,45
13,20
15,25
9,19
7,32
15,19
23,45
7,24
27,23
40
Chapter 5
CONCLUSIONS AND FUTURE
WORKS
Optically stabilised lenses are often used in video cameras and more expensive still
cameras to reduce the effects of small amounts of camera shake. These use a system of
gyroscopes and inertial sensors to keep the optical systems of the camera steady during
image capture. This is only really effective for removing a small amount of camera
shake at relatively short exposures (less than 1/15th second). Due to this hardware
incapability, the image blurring is unavoidable in practical scenarios, and the required
information is lost. Human visual system is primary tool for the information extraction
from the blurred image. But this is possible only if the information is lost up to certain
extent due to blurring. Hence, the image deblurring techniques has to be applied.
For practical applications, the exact modelling of the system is always not possible,
thus, we go for blind technique for image deblurring. The focus in this thesis is the
motion blur and the gaussian blur. To overcome the gaussian blur, the evolutionary
algorithm is used. Even for maximum standard deviation, this simulation shows
satisfactory result. A novel heuristic approach is introduced to estimate motion blur
parameters. This algorithm is tested for varying motion direction and blur length.
The field of blind image deconvolution is critical as well as challenging problem.
The thesis has been worked out considering only spatial-invariant type of blur to
reduce the problem complexity. But spatial-invariant blur fails to model the blur in
most of the practical case[24]. The noise effect is considered zero which is normally
impractical. The irreducible demand of psf for unambiguous deconvolution is another
limitation. The ground truth image used is grayscale and is synthetically blurred. The
41
blind image deconvolution approach adopted requires a well classification of the type
of the degradation that the image has undergone, and then a particular method could
be applied.
The thesis can be more useful for practical application if the spatial-variant degradation and noise parameters are considered. This opens the future research of the
current work leading to robustness of the algorithms. The work can also be extended
for the color images.
42
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