FUNDAMENTAL PROBLEMS IN IMAGE DEBLURRING Dr Slavoljub Mijović University of Montenegro Faculty of Natural Sciences and Mathematics Podgorica; MONTENEGRO smijovic@yahoo.com 1 7/10/2019 SSIP 2019 Timisora , Romania
Content Before all…; Image formation and blurring process; Modeling of Image Formation Process; Inverse Problem and Naïve Solution; General imaging problems; Inverse Ill- posed Problems; Image restoration by regularization; Case study-mammograms; 7/10/2019 CEEPUS Lecture 2
The Bible “ In the beginning God created the heavens and the earth. The earth was formless and void, and darkness was over the surface of the deep, and the Spirit of God was moving over the surface of the waters.... 7/10/2019 3
…Then God said, “Let there be light ”; and there was light. God saw that the light was good; and God separated the light from the darkness… . 7/10/2019 4
Image Formation and Blurring Processes The existence of light-or other forms of electromagnetic radiation is an essential requirement for an image to be created, captured, and perceived. Types of Images: • Reflection Images; • Emission Images; • Absorption Images; The Greek word Optike- ”Theory of vision” 7/10/2019 CEEPUS Lecture 5
IMAGE FORMATION - spatial variation of some physical quantity Contrast Task: • X-ray fluence Sharpness • Optical density of film Find Signal: (the difference noise • Grey-scale value on the between structures of interest monitor and background!) “Structures of interest” “Structures of interest” “Background” “Background” Micro-calcification in the breast glandular tissue Radiograph of a disk-shaped object 7/10/2019 CEEPUS Lecture 6
Image Aquizition, Formation, and digitization An image as a visual two dimensional (2D) representation of an object produced by an imaging system. 7/10/2019 CEEPUS Lecture 7
Modeling of Image Formation process “ Make everything as simple as possible, but not simpler.” Albert Einstein Spherical Cow? How we usually think? A MODEL AP(P)ROXIMATION! 7/10/2019 CEEPUS Lecture 8
Examples of images and blurring processes • There is not an ideal imaging system; • Optical system in the camera is out of focus; • Lens are not perfect; • Motion blur; • Turbulence blur; • Detecting systems is not perfect; 7/10/2019 CEEPUS Lecture 10
Our Task-Image Restoration Do DECONVOLUTION or DEBLURING or Image restoration is based on the attempt INVERSE or just to improve the quality of an image make UNDO! through knowledge of the physical processes which led to its formation ... RECOVERING AS MUCH INFORMATION AS POSSIBLE FROM THE GIVEN IMPERFECT DATA!!! ...i.e. to find object function o, or the IIt is scientific approach to find original image by original information f(x,y) from the image using mathematical model function g(x,y) of the blurring process 7/10/2019 CEEPUS Lecture 11
Remember: We are mainly interested in the characteristics of the object by deriving information from the image! Objective versus subjective information “ You cannot depend on your eyes when your imagination is out of focus- Mark Twain 7/10/2019 CEEPUS Lecture 12
A legitimate question to ask: “ When faced with a practical imaging processing problem, which techniques should I use and in which sequence?” Naturally, there is no universal answer to this question. Most image processing solutions are problem specific and usually involve applica several algorithms – in a meaningful sequenc to achieve the desired goal . My recommendation is at the first step - “Clean” the image from 7/10/2019 13 the influence of the imaging system and any undesired process!!!
How is mathematically described an image formation ? g h f n • PSF -a characteristic of the imaging device and is a deterministic function; • Object function -describes object surface or its internal structure; • Noise- a stochastic function which is a consequence of all the unwanted external disturbances • - Convolution operator which ‘smears’ (convolves) one function with another 7/10/2019 CEEPUS Lecture 14
Linear Imaging System 7/10/2019 CEEPUS Lecture 15
Linear imaging systems- cont’d ' ' ' ' ' ' g x , y h x , y ; x , y f x , y dx dy Linear superposition integral 7/10/2019 CEEPUS Lecture 16
The Point-Spread Function The Point Spread Function ( PSF ) describes the response of an imaging system to a point source or point object ' ' ' ' f x , y x x , y y 0 0 ' ' ' ' ' ' g x , y x x , y y h x , y ; x , y dx dy 0 0 g x , y h x , y ; x , y 0 0 7/10/2019 CEEPUS Lecture 17
Linear Shift-Invariant (LSI) systems and the convolution integral ' ' ' ' h x , y ; x y h x x , y y ' ' ' ' ' ' g x , y h x x , y y f x , y dx dy g x , y f x , y h x , y (2 - D) A very large number of image formation process are well described by the process of convolution. If a system is Linear Shift-Invariant then the image formation is necessarily described by convolution 7/10/2019 CEEPUS Lecture 18
An example of the LSI system and the convolution integral The scan was aquired with uniform speed over the patient. The derived signal is proportional To the gamma activity emanating from that region of the body beneath the aperture.
Inverse problems and Naïve Solution • Complex links among measured image quantities and object parameters: • the cause-effect connection of investigated phenomenon is inverse; Data → Model parameters • a characteristic of the object plays a role of “cause”, and the observed data of the image, such as brightness - “effect” f x , y • Our Task: Find • from integral equation q ( x , y ) h ( x , y ) f ( x , y ) 2 - D Hf • or in matrix formulation g
Frequency space and Fourier transforms “a big picture” • The Fourier representation is a complete alternative ; • The space domain and Fourier domain are reciprocal The harmonic content of signals: The fundamental idea of Fourier analysis is that any signal, be it a function of time, space or any other variables, may be expressed as a weighted linear combination of harmonic (i.e. sine and cosine) functions having different periods or frequencies. 7/10/2019 CEEPUS Lecture 21
Image Transformation Analysis &Processing 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 20 40 60 80 100 120 140 160 180 200 Inten Spatial period Spatial 1 frequency=1/spatial 0.9 0.8 period 0.7 0.6 0.5 0.4 0.3 0.2 Distance 0.1 0 0 20 40 60 80 100 120 140 160 180 200 FOURIER TRANSFORM Quite generally, we can transform the information with any scan line signal into a series of sinusoidal functions of the appropriate amplitude and spatial frequency (the spatial frequency spectrum) and vice-versa, we can synthesize any spatial signal by 7/10/2019 CEEPUS Lecture 22 summing its harmonic components
Fourier Transform examples con’d Superposition of two waves-a beat pattern 2 3 0.5 1.5 0.6 2 1 0.7 0.8 0.5 1 magnitude 0.9 0 1 0 1.1 -0.5 1.2 -1 -1 1.3 1.4 -2 -1.5 1.5 20 40 60 80 100 120 140 160 180 200 -2 -3 0 100 200 300 400 500 600 700 800 900 1000 0 20 40 60 80 100 120 140 160 180 200 x Image Rearranged Fourier transform Fourier transform of the beat pattern 10 10 500 20 20 400 amlitudes rel.units 30 30 300 40 40 200 50 50 100 60 60 20 40 60 20 40 60 0 65 70 75 80 85 90 spatial frequency Different signals and its Fourier transform pairs 7/10/2019 CEEPUS Lecture 23
Image Rearranged Fourier transform Filtering 10 10 Original image and its Fourier 20 20 transform 30 30 40 40 50 50 60 60 20 40 60 20 40 60 High pass filtered transform High pass filtered image Low-pass filtered image Low-pass filtered Fourier transform 10 10 10 10 20 20 20 20 30 30 30 30 40 40 40 40 50 50 50 50 60 60 10 20 30 40 50 60 60 60 10 20 30 40 50 60 20 40 60 20 40 60 Low-pass High-pass filtered image filtered image 7/10/2019 CEEPUS Lecture 24
Linear systems and Fourier transforms The Convolutio n theorem An imaging system operates on the constituent input harmonics and its quality can be assessed by its ability to transmit the input harmonics to the output The convolution theorem { f x , y h x , y } F k , k H k , k x y x y The Fourier transform of the convolution of the two functions is equal to the product of the individual transforms 7/10/2019 CEEPUS Lecture 25
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