Why p -methods in Signal Limitations of . . . and Image - PowerPoint PPT Presentation

Need for Deblurring In General, Signal and . . . Need for Regularization Tikhonov Regularization Why ℓ p -methods in Signal Limitations of . . . and Image Processing: Remaining Problem Let us Apply Fuzzy . . . A Fuzzy-Based Explanation Which Function g ( x ) . . . Main Result Fernando Cervantes 1 , Bryan Usevitch 1 , Home Page and Vladik Kreinovich 2 Title Page 1 Department of Electrical and Computer Engineering ◭◭ ◮◮ 2 Department of Computer Science ◭ ◮ University of Texas at El Paso El Paso, TX 79968, USA Page 1 of 16 fcervantes@miners.utep.edu, usevitch@utep.edu vladik@utep.edu Go Back Full Screen Close Quit

Need for Deblurring In General, Signal and . . . 1. Need for Deblurring Need for Regularization • Cameras and other image-capturing devices are getting Tikhonov Regularization better and better every day. Limitations of . . . Remaining Problem • However, none of them is perfect, there is always some Let us Apply Fuzzy . . . blur, that comes from the fact that: Which Function g ( x ) . . . – while we would like to capture the intensity I ( x, y ) Main Result at each spatial location ( x, y ), Home Page – the signal s ( x, y ) is influenced also by the intensities Title Page I ( x ′ , y ′ ) at nearby locations ( x ′ , y ′ ): ◭◭ ◮◮ � w ( x, y, x ′ , y ′ ) · I ( x ′ , y ′ ) dx ′ dy ′ . s ( x, y ) = ◭ ◮ Page 2 of 16 • When we take a photo of a friend, this blur is barely visible – and does not constitute a serious problem. Go Back • However, when a spaceship takes a photo of a distant Full Screen plant, the blur is very visible – so deblurring is needed. Close Quit

Need for Deblurring In General, Signal and . . . 2. In General, Signal and Image Reconstruction Need for Regularization Are Ill-Posed Problems Tikhonov Regularization • The image reconstruction problem is ill-posed in the Limitations of . . . sense that: Remaining Problem Let us Apply Fuzzy . . . – large changes in I ( x, y ) Which Function g ( x ) . . . – can lead to very small changes in s ( x, y ). Main Result • Indeed, the measured value s ( x, y ) is an average inten- Home Page sity over some small region. Title Page • Averaging eliminates high-frequency components. ◭◭ ◮◮ • Thus, for I ∗ ( x, y ) = I ( x, y ) + c · sin( ω x · x + ω y · y ) , the ◭ ◮ signal is practically the same: s ∗ ( x, y ) ≈ s ( x, y ). Page 3 of 16 • However, the original images, for large c , may be very Go Back different. Full Screen Close Quit

Need for Deblurring In General, Signal and . . . 3. Need for Regularization Need for Regularization • To reconstruct the image reasonably uniquely, we must Tikhonov Regularization impose additional conditions on the original image. Limitations of . . . Remaining Problem • This imposition is known as regularization . Let us Apply Fuzzy . . . • Often, a signal or an image is smooth (differentiable). Which Function g ( x ) . . . • Then, a natural idea is to require that the vector Main Result Home Page d = ( d 1 , d 2 , . . . ) formed by the derivatives is close to 0: Title Page n def � d 2 = C 2 . ρ ( d, 0) ≤ C ⇔ i ≤ c ◭◭ ◮◮ i =1 ◭ ◮ • For continuous signals, sum turns into an integral: Page 4 of 16 � �� ∂I � 2 � 2 � � ∂I � x ( t )) 2 dt ≤ c or Go Back ( ˙ + dx dy ≤ c. ∂x ∂y Full Screen Close Quit

Need for Deblurring In General, Signal and . . . 4. Tikhonov Regularization Need for Regularization • Out of all smooth signals or images, we want to find Tikhonov Regularization def e 2 the best fit with observation: J = � i → min . Limitations of . . . i Remaining Problem • Here, e i is the difference between the actual and the Let us Apply Fuzzy . . . reconstructed values. Which Function g ( x ) . . . • Thus, we need to minimize J under the constraint Main Result Home Page � �� ∂I � 2 � � 2 � ∂I � x ( t )) 2 dt ≤ c and ( ˙ + dx dy ≤ c. Title Page ∂x ∂y ◭◭ ◮◮ • Lagrange multiplier method reduced this constraint ◭ ◮ optimization problem to the unconstrained one: Page 5 of 16 � �� ∂I � 2 � 2 � � ∂I J + λ · + dx dy → min I ( x,y ) . Go Back ∂x ∂y Full Screen • This idea is known as Tikhonov regularization . Close Quit

Need for Deblurring In General, Signal and . . . 5. From Continuous to Discrete Images Need for Regularization • In practice, we only observe an image with a certain Tikhonov Regularization spatial resolution. Limitations of . . . Remaining Problem • So we can only reconstruct the values I ij = I ( x i , y j ) on Let us Apply Fuzzy . . . a certain grid x i = x 0 + i · ∆ x and y j = y 0 + j · ∆ y . Which Function g ( x ) . . . • In this discrete case, instead of the derivatives, we have Main Result differences: Home Page ((∆ x I ij ) 2 + (∆ y I ij ) 2 ) → min � � J + λ · I ij . Title Page i j ◭◭ ◮◮ • Here: ◭ ◮ def • ∆ x I ij = I ij − I i − 1 ,j , and Page 6 of 16 def • ∆ y I ij = I ij − I i,j − 1 . Go Back Full Screen Close Quit

Need for Deblurring In General, Signal and . . . Limitations of Tikhonov Regularization and ℓ p - 6. Need for Regularization Method Tikhonov Regularization • Tikhonov regularization is based on the assumption Limitations of . . . that the signal or the image is smooth. Remaining Problem Let us Apply Fuzzy . . . • In real life, images are, in general, not smooth. Which Function g ( x ) . . . • For example, many of them exhibit a fractal behavior. Main Result • In such non-smooth situations, Tikhonov regulariza- Home Page tion does not work so well. Title Page • To take into account non-smoothness, researchers have ◭◭ ◮◮ proposed to modify the Tikhonov regularization: ◭ ◮ – instead of the squares of the derivatives, Page 7 of 16 – use the p -th powers for some p � = 2: ( | ∆ x I ij | p + | ∆ y I ij | p ) → min Go Back � � J + λ · I ij . Full Screen i j • This works much better than Tikhonov regularization. Close Quit

Need for Deblurring In General, Signal and . . . 7. Remaining Problem Need for Regularization • Problem: the ℓ p -methods are heuristic. Tikhonov Regularization Limitations of . . . • There is no convincing explanation of why necessarily Remaining Problem we replace the square: Let us Apply Fuzzy . . . – with a p -th power and Which Function g ( x ) . . . – not, for example, with some other function. Main Result Home Page • We show: that a natural formalization of the corre- sponding intuitive ideas indeed leads to ℓ p -methods. Title Page ◭◭ ◮◮ • To formalize the intuitive ideas behind image recon- struction, we use fuzzy techniques . ◭ ◮ • Fuzzy techniques were designed to transform: Page 8 of 16 – imprecise intuitive ideas into Go Back – exact formulas. Full Screen Close Quit

Need for Deblurring In General, Signal and . . . 8. Let us Apply Fuzzy Techniques to Our Problem Need for Regularization • We are trying to formalize the statement that the im- Tikhonov Regularization age is continuous. Limitations of . . . def Remaining Problem • This means that the differences ∆ x k = ∆ x I ij and ∆ y I ij Let us Apply Fuzzy . . . between image intensities at nearby points are small. Which Function g ( x ) . . . • Let µ ( x ) denote the degree to which x is small, and Main Result f & ( a, b ) denote the “and”-operation. Home Page • Then, the degree d to which ∆ x 1 is small and ∆ x 2 is Title Page small, etc., is: ◭◭ ◮◮ d = f & ( µ (∆ x 1 ) , µ (∆ x 2 ) , µ (∆ x 3 ) , . . . ) . ◭ ◮ • Known: each “and”-operation can be approximated, Page 9 of 16 for any ε > 0, by an Archimedean one: Go Back f & ( a, b ) = f − 1 ( f ( a )) · f ( b )) . Full Screen • Thus, without losing generality, we can safely assume Close that the actual “and”-operation is Archimedean. Quit

Need for Deblurring In General, Signal and . . . 9. Analysis of the Problem Need for Regularization • We want to select an image with the largest degree of Tikhonov Regularization satisfying this condition: Limitations of . . . Remaining Problem d = f − 1 ( f ( µ (∆ x 1 )) · f ( µ (∆ x 2 )) · f ( µ (∆ x 3 )) · . . . ) → max . Let us Apply Fuzzy . . . Which Function g ( x ) . . . • Since the function f ( x ) is increasing, maximizing d is equivalent to maximizing Main Result Home Page f ( d ) = f ( µ (∆ x 1 )) · f ( µ (∆ x 2 )) · f ( µ (∆ x 3 )) · . . . Title Page • Maximizing this product is equivalent to minimizing ◭◭ ◮◮ its negative logarithm ◭ ◮ def def � L = − ln( d ) = g (∆ x k ) , where g ( x ) = − ln( f ( µ ( x ))) . Page 10 of 16 k Go Back • In these terms, selecting a membership function is Full Screen equivalent to selecting the related function g ( x ). Close Quit