Traditional Use of . . . Fuzzy Logic Can Help . . . Case Study Need for Deblurring Fuzzy Techniques Provide In General, Signal and . . . a Theoretical Explanation Tikhonov Regularization Limitations of . . . for the Heuristic Let Us Apply Fuzzy . . . What Next? ℓ p -Regularization Home Page of Signals and Images Title Page ◭◭ ◮◮ Fernando Cervantes 1 , Bryan Usevitch 1 ◭ ◮ Leobardo Valera 2 , Vladik Kreinovich 2 , and Olga Kosheleva 2 Page 1 of 26 1 Department of Electrical and Computer Engineering Go Back 2 Computational Science Program University of Texas at El Paso Full Screen El Paso, TX 79968, USA contact email vladik@utep.edu Close Quit
Traditional Use of . . . Fuzzy Logic Can Help . . . 1. Traditional Use of Fuzzy Logic Case Study • Expert knowledge is often formulated by using impre- Need for Deblurring cise (“fuzzy”) from natural language (like “small”). In General, Signal and . . . Tikhonov Regularization • Fuzzy logic techniques was originally invented to trans- Limitations of . . . late such knowledge into precise terms. Let Us Apply Fuzzy . . . • Such a translation is still the main use of fuzzy tech- What Next? niques. Home Page • Example: we want to control a complex plant for Title Page which: ◭◭ ◮◮ – no good control technique is known, but ◭ ◮ – there are experts how can control this plant reason- Page 2 of 26 ably well. Go Back • So, we elicit rules from the experts. Full Screen • Then we use fuzzy techniques to translate these rules into a control strategy. Close Quit
Traditional Use of . . . Fuzzy Logic Can Help . . . 2. Fuzzy Logic Can Help in Other Cases As Well Case Study • Lately, it turned out that fuzzy techniques can help in Need for Deblurring another class of applied problems: in situations when In General, Signal and . . . Tikhonov Regularization – there are semi-heuristic techniques for solving the Limitations of . . . corresponding problems, i.e., Let Us Apply Fuzzy . . . – techniques for which there is no convincing theo- What Next? retical justification. Home Page • These techniques lack theoretical justification. Title Page • Their previous empirical success does not guarantee ◭◭ ◮◮ that these techniques will work well on new problems. ◭ ◮ • Thus, users are reluctant to use these techniques. Page 3 of 26 Go Back Full Screen Close Quit
Traditional Use of . . . Fuzzy Logic Can Help . . . 3. Additional Problem of Semi-Heuristic Tech- Case Study niques Need for Deblurring • Semi-heuristic techniques are often not perfect. In General, Signal and . . . Tikhonov Regularization • Without an underlying theory, it is not clear how to Limitations of . . . improve their performance. Let Us Apply Fuzzy . . . • For example, linear models can be viewed as first ap- What Next? proximation to Taylor series. Home Page • So, a natural next approximation is to use quadratic Title Page models. ◭◭ ◮◮ • However, e.g., for ℓ p -models: ◭ ◮ – when they do not work well, Page 4 of 26 – it is not immediately clear what is a reasonable next Go Back approximation. Full Screen Close Quit
Traditional Use of . . . Fuzzy Logic Can Help . . . 4. What We Show Case Study • We show that in some situations, the desired theoreti- Need for Deblurring cal justification can be obtained if: In General, Signal and . . . Tikhonov Regularization – in addition to known (crisp) requirements on the Limitations of . . . desired solution, Let Us Apply Fuzzy . . . – we also take into account requirements formulated What Next? by experts in natural-language terms. Home Page • Naturally, we use fuzzy techniques to translate these Title Page imprecise requirements into precise terms. ◭◭ ◮◮ • To make the resulting justification convincing, we need ◭ ◮ to make sure that this justification works: Page 5 of 26 – not only for one specific choice of fuzzy techniques Go Back (membership function, t-norm, etc.), – but for all techniques which are consistent with the Full Screen practical problem. Close Quit
Traditional Use of . . . Fuzzy Logic Can Help . . . 5. Case Study Case Study As an example, we provide the detailed justification of: Need for Deblurring In General, Signal and . . . • ℓ p -regularization techniques in solving inverse problems Tikhonov Regularization – an empirically successful alternative to Tikhonov Limitations of . . . regularization Let Us Apply Fuzzy . . . – which is appropriate for situations when the desired What Next? signal or image is not smooth; Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 26 Go Back Full Screen Close Quit
Traditional Use of . . . Fuzzy Logic Can Help . . . 6. Need for Deblurring Case Study • Cameras and other image-capturing devices are getting Need for Deblurring better and better every day. In General, Signal and . . . Tikhonov Regularization • However, none of them is perfect, there is always some Limitations of . . . blur, that comes from the fact that: Let Us Apply Fuzzy . . . – while we would like to capture the intensity I ( x, y ) What Next? 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 7 of 26 • 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
Traditional Use of . . . Fuzzy Logic Can Help . . . 7. In General, Signal and Image Reconstruction Case Study Are Ill-Posed Problems Need for Deblurring • The image reconstruction problem is ill-posed in the In General, Signal and . . . sense that: Tikhonov Regularization Limitations of . . . – large changes in I ( x, y ) Let Us Apply Fuzzy . . . – can lead to very small changes in s ( x, y ). What Next? • 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 8 of 26 • However, the original images, for large c , may be very Go Back different. Full Screen Close Quit
Traditional Use of . . . Fuzzy Logic Can Help . . . 8. Need for Regularization Case Study • To reconstruct the image reasonably uniquely, we must Need for Deblurring impose additional conditions on the original image. In General, Signal and . . . Tikhonov Regularization • This imposition is known as regularization . Limitations of . . . • Often, a signal or an image is smooth (differentiable). Let Us Apply Fuzzy . . . • Then, a natural idea is to require that the vector What Next? 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 9 of 26 � �� ∂I � 2 � 2 � � ∂I � x ( t )) 2 dt ≤ c or Go Back ( ˙ + dx dy ≤ c. ∂x ∂y Full Screen Close Quit
Traditional Use of . . . Fuzzy Logic Can Help . . . 9. Tikhonov Regularization Case Study • Out of all smooth signals or images, we want to find Need for Deblurring def e 2 the best fit with observation: J = � i → min . In General, Signal and . . . i Tikhonov Regularization • Here, e i is the difference between the actual and the Limitations of . . . reconstructed values. Let Us Apply Fuzzy . . . • Thus, we need to minimize J under the constraint What Next? 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 10 of 26 � �� ∂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
Traditional Use of . . . Fuzzy Logic Can Help . . . 10. From Continuous to Discrete Images Case Study • In practice, we only observe an image with a certain Need for Deblurring spatial resolution. In General, Signal and . . . Tikhonov Regularization • So we can only reconstruct the values I ij = I ( x i , y j ) on Limitations of . . . a certain grid x i = x 0 + i · ∆ x and y j = y 0 + j · ∆ y . Let Us Apply Fuzzy . . . • In this discrete case, instead of the derivatives, we have What Next? 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 11 of 26 def • ∆ y I ij = I ij − I i,j − 1 . Go Back Full Screen Close Quit
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