This document discusses methods for restoring blurred images, including modeling image degradation using convolution with a point spread function in the spatial and frequency domains. Common point spread functions like Gaussian and motion blur are described. Methods for solving the deconvolution problem to restore blurred images are presented, including inverse filtering, Wiener filtering, regularization filtering, and evaluating the quality of restored images using metrics like PSNR, BSNR, and ISNR.
WEBINAR ON FUNDAMENTALS OF DIGITAL IMAGE PROCESSING DURING COVID LOCK DOWN by by K.Vijay Anand , Associate Professor, Department of Electronics and Instrumentation Engineering , R.M.K Engineering College, Tamil Nadu , India
WEBINAR ON FUNDAMENTALS OF DIGITAL IMAGE PROCESSING DURING COVID LOCK DOWN by by K.Vijay Anand , Associate Professor, Department of Electronics and Instrumentation Engineering , R.M.K Engineering College, Tamil Nadu , India
COM2304: Intensity Transformation and Spatial Filtering – I (Intensity Transf...Hemantha Kulathilake
At the end of this lesson, you should be able to;
describe spatial domain of the digital image.
recognize the image enhancement techniques.
describe and apply the concept of intensity transformation.
express histograms and histogram processing.
describe image noise.
characterize the types of Noise.
describe concept of image restoration.
Image Deblurring using L0 Sparse and Directional FiltersCSCJournals
Blind deconvolution refers to the process of recovering the original image from the blurred image when the blur kernel is unknown. This is an ill-posed problem which requires regularization to solve. The naive MAP approach for solving the blind deconvolution problem was found to favour no-blur solution which in turn led to its failure. It is noted that the success of the further developed successful MAP based deblurring methods is due to the intermediate steps in between, which produces an image containing only salient image structures. This intermediate image is essentially called the unnatural representation of the image. L0 sparse expression can be used as the regularization term to effectively develop an efficient optimization method that generates unnatural representation of an image for kernel estimation. Further, the standard deblurring methods are affected by the presence of image noise. A directional filter incorporated as an initial step to the deblurring process makes the method efficient to be used for blurry as well as noisy images. Directional filtering along with L0 sparse regularization gives a good kernel estimate in spite of the image being noisy. In the final image restoration step, a method to give a better result with lesser artifacts is incorporated. Experimental results show that the proposed method recovers a good quality image from a blurry and noisy image.
COM2304: Intensity Transformation and Spatial Filtering – I (Intensity Transf...Hemantha Kulathilake
At the end of this lesson, you should be able to;
describe spatial domain of the digital image.
recognize the image enhancement techniques.
describe and apply the concept of intensity transformation.
express histograms and histogram processing.
describe image noise.
characterize the types of Noise.
describe concept of image restoration.
Image Deblurring using L0 Sparse and Directional FiltersCSCJournals
Blind deconvolution refers to the process of recovering the original image from the blurred image when the blur kernel is unknown. This is an ill-posed problem which requires regularization to solve. The naive MAP approach for solving the blind deconvolution problem was found to favour no-blur solution which in turn led to its failure. It is noted that the success of the further developed successful MAP based deblurring methods is due to the intermediate steps in between, which produces an image containing only salient image structures. This intermediate image is essentially called the unnatural representation of the image. L0 sparse expression can be used as the regularization term to effectively develop an efficient optimization method that generates unnatural representation of an image for kernel estimation. Further, the standard deblurring methods are affected by the presence of image noise. A directional filter incorporated as an initial step to the deblurring process makes the method efficient to be used for blurry as well as noisy images. Directional filtering along with L0 sparse regularization gives a good kernel estimate in spite of the image being noisy. In the final image restoration step, a method to give a better result with lesser artifacts is incorporated. Experimental results show that the proposed method recovers a good quality image from a blurry and noisy image.
The resolution and performance of an optical microscope can be characterized by a quantity known as the modulation transfer function (MTF), which is a measurement of the microscope's ability to transfer contrast from the specimen to the intermediate image plane at a specific resolution.
GPR systems work by sending a tiny pulse of energy into a material via an antenna. An integrated computer records the strength and time required for the return of any reflected signals. Subsurface variations will create reflections that are picked up by the system and stored on digital media. These reflections are produced by a variety of material such as geological structure differences and man-made objects like pipes and wire.
Image denoising with unknown Non-Periodic NoisesSakshiAggarwal85
The experimental study is learnt from application of image restoration techniques. As far as literature review is concerned, the prior knowledge of noise or unwanted signal might have advantage in promising and encouraging results. So, we thought of embedding procedure into our work that completely unacquainted of noise model. In this project, we try to reconstruct original image with incorporation of those restoration models, given the target noisy image. Three known blind deconvolution models; Lucy-Richardson algorithm, Weiner-Hunt algorithm and Total Variation Regularization, are implemented. Results are compared and presented
Image Restitution Using Non-Locally Centralized Sparse Representation ModelIJERA Editor
Sparse representation models uses a linear combination of a few atoms selected from an over-completed
dictionary to code an image patch which have given good results in different image restitution applications. The
reconstruction of the original image is not so accurate using traditional models of sparse representation to solve
degradation problems which are blurring, noisy, and down-sampled. The goal of image restitution is to suppress
the sparse coding noise and to improve the image quality by using the concept of sparse representation. To
obtain a good sparse coding coefficients of the original image we exploit the image non-local self similarity and
then by centralizing the sparse coding coefficients of the observation image to those estimates. This non-locally
centralized sparse representation model outperforms standard sparse representation models in all aspects of
image restitution problems including de-noising, de-blurring, and super-resolution.
digital image processing is Basic concepts, Examples of fields that use Digital Image
Processing, Fundamental steps in Digital Image Processing, Components of
an Image Processing System.
Digital Image Fundamentals: Elements of visual perception, Image sensing
and acquisition, Image sampling and quantization, Some basic relationships
between pixels.
9
2 Image Enhancement in Spatial domain: Some Basic Intensity
Transformation functions, Histogram Processing, Fundamentals of Spatial
Filtering, Smoothing and Sharpening Spatial Filtering.
Self Study:
Image Enhancement In Frequency Domain: Introduction to Fourier
Transform, Smoothing and Sharpening frequency domain filtersBasic concepts, Examples of fields that use Digital Image
Processing, Fundamental steps in Digital Image Processing, Components of
an Image Processing System.
Digital Image Fundamentals: Elements of visual perception, Image sensing
and acquisition, Image sampling and quantization, Some basic relationships
between pixels.
9
2 Image Enhancement in Spatial domain: Some Basic Intensity
Transformation functions, Histogram Processing, Fundamentals of Spatial
Filtering, Smoothing and Sharpening Spatial Filtering.
Self Study:
Image Enhancement In Frequency Domain: Introduction to Fourier
Transform, Smoothing and Sharpening frequency domain filtersBasic concepts, Examples of fields that use Digital Image
Processing, Fundamental steps in Digital Image Processing, Components of
an Image Processing System.
Digital Image Fundamentals: Elements of visual perception, Image sensing
and acquisition, Image sampling and quantization, Some basic relationships
between pixels.
9
2 Image Enhancement in Spatial domain: Some Basic Intensity
Transformation functions, Histogram Processing, Fundamentals of Spatial
Filtering, Smoothing and Sharpening Spatial Filtering.
Self Study:
Image Enhancement In Frequency Domain: Introduction to Fourier
Transform, Smoothing and Sharpening frequency domain filtersBasic concepts, Examples of fields that use Digital Image
Processing, Fundamental steps in Digital Image Processing, Components of
an Image Processing System.
Digital Image Fundamentals: Elements of visual perception, Image sensing
and acquisition, Image sampling and quantization, Some basic relationships
between pixels.
9
2 Image Enhancement in Spatial domain: Some Basic Intensity
Transformation functions, Histogram Processing, Fundamentals of Spatial
Filtering, Smoothing and Sharpening Spatial Filtering.
Self Study:
Image Enhancement In Frequency Domain: Introduction to Fourier
Transform, Smoothing and Sharpening frequency domain filtersBasic concepts, Examples of fields that use Digital Image
Processing, Fundamental steps in Digital Image Processing, Components of
an Image Processing System.
Digital Image Fundamentals: Elements of visual perception, Image sensing
and acquisition, Image sampling and quantization, Some basic relationships
between pixels.
9
2 Basic
Non-Blind Deblurring Using Partial Differential Equation MethodEditor IJCATR
In this paper, a new idea for two dimensional image deblurring algorithm is introduced which uses basic concepts of PDEs... The various methods to estimate the degradation function (PSF is known in prior called non-blind deblurring) for use in restoration are observation, experimentation and mathematical modeling. Here, PDE based mathematical modeling is proposed to model the degradation and recovery process. Several restoration methods such as Weiner Filtering, Inverse Filtering [1], Constrained Least Squares, and Lucy -Richardson iteration remove the motion blur either using Fourier Transformation in frequency domain or by using optimization techniques. The main difficulty with these methods is to estimate the deviation of the restored image from the original image at individual points that is due to the mechanism of these methods as processing in frequency domain .Another method, the travelling wave de-blurring method is a approach that works in spatial domain.PDE type of observation model describes well several physical mechanisms, such as relative motion between the camera and the subject (motion blur), bad focusing (defocusing blur), or a number of other mechanisms which are well modeled by a convolution. In last PDE method is compared with the existing restoration techniques such as weiner filters, median filters [2] and the results are compared on the basis of calculated PSNR for various noises
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
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Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
3. Image Deblurring: Problem
Statement
Mathematically a variety of image capturing principles can be
described by the Fredholm integral of the first kind
where is a point-spread function (PSF) of an optical
system, is a function of a real object, is
an additive noise, and is an observed signal.
( , )(( , () , ),) f xg x h x y dt yy xy η= +∫
( , )h x y
( , )f x y
( , )g x y
( , )x yη
5. Convolution
• If the PSF h is a linear and shift-invariant
function, then the degraded image is
expressed in the spatial domain by its
convolution with the spatial representation of
the PSF h
5
( , ) ( , ), ) , )( (fg x h x y y yx xy η= ∗ +
6. Frequency Domain Representation
• An equivalent frequency domain
representation is as follows
Where is the Fourier transform of ,
is the Fourier transform of ,
is the Fourier transform of ,
and is the Fourier transform of
6
(( , ) ,( ) ), , )(FG u Nu vH u v u vv= +
( , )G u v
( , )H u v
( , )F u v
( , )N u v
( , )g x y
( , )h x y
( , )f x y
( , )x yη
7. Problem statement: capturing
• Mathematically a variety of capturing principles can be
described by the Fredholm integral of the first kind
• where h(x,y) is a point-spread function (PSF) of a system,
q(x,y) is a function of a real object and g(x,y) is an observed
signal.
2
( , ) ( , ) ( , ) , ,
R
g x y h x y f x y dxdy x y R= ∈∫
Micro
scopy
∈
Photo
7
8. Degradation in Spatial and
Frequency Domain
• In the spatial domain, image degradation (blurring) is
observed as fuzziness, small image details and edges look
unsharpened and many of them become even
undistinguishable
• In the frequency domain, image degradation (blurring)
significantly affects a power spectrum (as a result of
convolution with PSF whose Fourier transform contains
many zeros, some frequencies “disappear” from the Fourier
transform of a blurred image)
• At the same time, symmetric PSFs almost do not affect a
phase spectrum (particularly its low and medium frequency
area is almost not affected, while a high frequency area is
just slightly affected)
8
9. Degradation in Spatial and Frequency Domain
log Z
True Image Horizontal
Motion blur
Gaussian
blur
Images and log of their Power Spectra
10. True Image Rectangular
(2D motion) blur
Optical
Defocus blur
Degradation in Spatial and Frequency Domain
log ZImages and log of their Power Spectra
11. Examples of Different PSFs
The Gaussian PSF:
is a parameter (variance)
The uniform linear motion:
t is a parameter (the length of motion)
The uniform optical defocus:
t is a parameter
(the size of smoothing area)
2 2
2 2
1
( , ) exp
2
x y
h x y
πσ σ
+
= −
2 2
1
, / 2, cos sin ,
( , )
otherwise,0,
x y t x y
h x y t
φ φ
+ < =
=
2
1
, , ,
( , ) 2 2
otherwise,0,
t t
x y
h x y t
< <
=
2
σ
12. Examples of Different PSFs
PSF in spatial domain
PSF in frequency domain
Gaussian Motion Defocus
13. Degradation in Spatial and Frequency Domain
13
True Image Motion blur,
6 pixels
Motion blur,
10 pixels
14. Degradation in Spatial and Frequency Domain
14
True Image Gaussian
blur, var=2
Gaussian
blur, var=5
16. “Ideal” Frequency Domain Model
• Assuming that the effect of noise is negligible
(which, in fact, is never possible for real-world
images), we may use a simpler frequency
domain model
• Respectively, to restore our image, we have to
solve this equation for
16
( , ) ( , ) ( , )G u v H u v F u v=
( , )F u v
( , )ˆ( , )
( , )
G u v
F u v
H u v
=
17. “Ideal” Model in Spatial Domain
• Still assuming that the effect of noise is
negligible, in spatial domain we obtain
• Respectively,
• This is equation is called “deconvolution”.
Neither in spatial, nor in frequency domain it
can be solved directly in the straightforward
way due to zeros contained in h and H.
17
( , ) ( , ) ( , )g x y h x y f x y= ∗
( )
1
( , ) ( , ) ( , )f x y g x y h x y
−
= ∗
18. Methods of Solving Deconvolution
Problem
• A necessary condition to restore a blurred
image is to know the exact PSF model – to
know function h(x,y) and its spectrum H(u,v).
• The alternative is to apply a blind
deconvolution technique, which is resulted in
the long iterative process whose results can be
good or not good. So it is “50/50”. Thus, it is
preferable to identify PSF
18
19. Methods of Solving Deconvolution
Problem – Inverse Filtering
• Inverse filtering – deconvolution equation is
going to be solved in the frequency domain.
Those frequencies where H(u,v)=0, are not
involved in the operation
• Disadvantage: just a light blur can be removed
in this way
19
( , )ˆ( , )
( , )
G u v
F u v
H u v
=
20. Methods for PSF identification
• To identify PSF, if an image is even slightly
corrupted by noise, it is necessary to use machine
learning tools (a neural network, a support vector
machine, etc.) and to analyze a power spectrum
• A problem is that when an image is noisy, there
are no actual zeros in its power spectrum,
because the Fourier transform of an image
contains the one of noise as an additive
component
20
21. Methods of Solving Deconvolution
Problem – Wiener Filtering
• Wiener filtering is a minimum Mean Square
Error Filtering. The error is
• The filter in frequency domain is determined
as follows
• Disadvantages: and are
unknown and can just be estimated
21
{ }2 2ˆ( )e E f f= −
2
2
2
2
( , )1ˆ( , ) ( , )
( , ) ( , )
( , )
( , )
H u v
F u v G u v
H u v N u v
H u v
F u v
=
+
( , )N u v ( , )F u v
22. Wiener Filtering for
White Noise Case
• Wiener filtering for white noise case is as
follows
• Where K is a power spectrum of noise, which
is a constant for white noise
• Disadvantage: noise is not always white
22
2
2
( , )1ˆ( , ) ( , )
( , ) ( , )
H u v
F u v G u v
H u v H u v K
=
+
24. Methods of Solving Deconvolution
Problem – Regularization Filtering
• Regularization filtering is reduced to solving
the optimization problem
in the frequency domain
• The filter in frequency domain is determined
as follows
where * stands for complex conjugation and α
is the regularization parameter
24
2 2
ˆg Hf η− =
*
2
( , )ˆ( , ) ( , )
( , )
H u v
F u v G u v
H u v α
=
+
25. Restoration of an image after
deconvolution is done
• After restoring an image Fourier transform
by solving a deconvolution problem, an image
can be restored by finding its inverse Fourier
transform
25
ˆ( , )F u v
( ) ( )1ˆ ˆ, ( , )f x y F u v−
= Φ
26. Evaluation of the quality of a
restored image
• The quality of restoration can be evaluated as
usually in terms of PSNR
• In image deblurring there are also two other
measures, which are preferable:
the blurred signal-to-noise ratio (BSNR) and
the improved signal-to-noise-ratio (ISNR)
26
27. BSNR and ISNR
• BSNR
where is a blurred image without noise, is a
restored image, M is the number of pixels, and is
the noise variance
• ISNR
where is an original image, is a degraded
image, and is a restored image
27
( )
2
10 2
ˆ
10 log
g f
BSNR
Mσ
−
= ⋅
g ˆf
2
σ
( )
( )
2
10 2
10 log
ˆ
f g
ISNR
f f
−
= ⋅
−
f g
ˆf