Image Deconvolution: . . . Ideal No-Noise Case Deconvolution in the . . . Blind Image . . . Rotation-Invariance State-of-the-Art . . . Can Further Improve Need for Improvement Rotation-Invariant . . . State-of-the-Art Blind Testing the New . . . Conclusions and . . . Deconvolution Techniques Home Page Title Page Fernando Cervantes 1 , Bryan Usevitch 1 and Vladik Kreinovich 2 ◭◭ ◮◮ ◭ ◮ 1 Department of Electrical and Computer Engineering 2 Department of Computer Science Page 1 of 14 University of Texas at El Paso, El Paso, TX 79968, USA fcervantes@miners.utep.edu, usevitch@utep.edu Go Back vladik@utep.edu Full Screen Close Quit
Image Deconvolution: . . . Ideal No-Noise Case 1. Image Deconvolution: Formulation of the Deconvolution in the . . . Problem Blind Image . . . • The measurement results y k differ from the actual val- State-of-the-Art . . . ues x k dues to additive noise and blurring: Need for Improvement � Rotation-Invariant . . . y k = h i · x k − i + n k . Testing the New . . . i Conclusions and . . . • From the mathematical viewpoint, y is a convolution Home Page of h and x : y = h ⋆ x . Title Page • Similarly, the observed image y ( i, j ) differs from the ◭◭ ◮◮ ideal one x ( i, j ) due to noise and blurring: ◭ ◮ � � h ( i − i ′ , j − j ′ ) · x ( i ′ , j ′ ) + n ( i, j ) . y ( i, j ) = Page 2 of 14 i ′ j ′ Go Back • It is desirable to reconstruct the original signal or im- Full Screen age, i.e., to perform deconvolution . Close Quit
Image Deconvolution: . . . Ideal No-Noise Case 2. Ideal No-Noise Case Deconvolution in the . . . • In the ideal case, when noise n ( i, j ) can be ignored, we Blind Image . . . can find x ( i, j ) by solving a system of linear equations: State-of-the-Art . . . � � Need for Improvement h ( i − i ′ , j − j ′ ) · x ( i ′ , j ′ ) . y ( i, j ) = Rotation-Invariant . . . i ′ j ′ Testing the New . . . • However, already for 256 × 256 images, the matrix h is Conclusions and . . . of size 65,536 × 65,536, with billions entries. Home Page • Direct solution of such systems is not feasible. Title Page ◭◭ ◮◮ • A more efficient idea is to use Fourier transforms, since y = h ⋆ x implies Y ( ω ) = H ( ω ) · X ( ω ); hence: ◭ ◮ – we compute Y ( ω ) = F ( y ); Page 3 of 14 – we compute X ( ω ) = Y ( ω ) Go Back H ( ω ), and Full Screen – finally, we compute x = F − 1 ( X ( ω )). Close Quit
Image Deconvolution: . . . Ideal No-Noise Case 3. Deconvolution in the Presence of Noise with Deconvolution in the . . . Known Characteristics Blind Image . . . • Suppose that signal and noise are independent, and we State-of-the-Art . . . know the power spectral densities Need for Improvement � 1 � � 1 � Rotation-Invariant . . . T · | X T ( ω ) | 2 T · | N T ( ω ) | 2 S I ( ω ) = lim , S N ( ω ) = lim T →∞ E T →∞ E . Testing the New . . . Conclusions and . . . • We minimize the expected mean square difference Home Page �� T/ 2 � 1 x ( t ) − x ( t )) 2 dt def Title Page d = lim T · E ( � . T →∞ − T/ 2 ◭◭ ◮◮ ◭ ◮ • Minimizing d leads to the known Wiener filter formula H ∗ ( ω 1 , ω 2 ) Page 4 of 14 � X ( ω 1 , ω 2 ) = · Y ( ω 1 , ω 2 ) . | H ( ω 1 , ω 2 ) | 2 + S N ( ω 1 , ω 2 ) Go Back S I ( ω 1 , ω 2 ) Full Screen Close Quit
Image Deconvolution: . . . Ideal No-Noise Case 4. Blind Image Deconvolution in the Presence of Deconvolution in the . . . Prior Knowledge Blind Image . . . • Wiener filter techniques assume that we know the blur- State-of-the-Art . . . ring function h . Need for Improvement Rotation-Invariant . . . • In practice, we often only have partial information about h . Testing the New . . . Conclusions and . . . • Such situations are known as blind deconvolution . Home Page • Sometimes, we know a joint probability distribution Title Page p (Ω , x, h, y ) corresponding to some parameters Ω: ◭◭ ◮◮ p (Ω , x, h, y ) = p (Ω) · p ( x | Ω) · p ( h | Ω) · p ( y | x, h, Ω) . ◭ ◮ • In this case, we can find Page 5 of 14 � � � Ω = arg max p (Ω | y ) = p (Ω , x, h, y ) dx dh and Go Back Ω x,h Full Screen x, � x,h p ( x, h | � ( � h ) = arg max Ω , y ) . Close Quit
Image Deconvolution: . . . Ideal No-Noise Case 5. Blind Image Deconvolution in the Absence of Deconvolution in the . . . Prior Knowledge: Sparsity-Based Techniques Blind Image . . . • In many practical situations, we do not have prior State-of-the-Art . . . knowledge about the blurring function h . Need for Improvement Rotation-Invariant . . . • Often, what helps is sparsity assumption: that in the expansion x ( t ) = � Testing the New . . . a i · e i ( t ), most a i are zero. i Conclusions and . . . • In this case, it makes sense to look for a solution with Home Page the smallest value of Title Page def � a � 0 = # { i : a i � = 0 } . ◭◭ ◮◮ ◭ ◮ • The function � a � 0 is not convex and thus, difficult to optimize. Page 6 of 14 Go Back • It is therefore replaced by a close convex objective func- = � def tion � a � 1 | a i | . Full Screen i Close Quit
Image Deconvolution: . . . Ideal No-Noise Case 6. State-of-the-Art Technique for Sparsity-Based Deconvolution in the . . . Blind Deconvolution Blind Image . . . • Sparsity is the main idea behind the algorithm de- State-of-the-Art . . . scribed in (Amizic et al. 2013) that minimizes Need for Improvement Rotation-Invariant . . . β 2 + η 2 ·� y − W a � 2 2 ·� W a − H x � 2 2 + τ ·� a � 1 + α · R 1 ( x )+ γ · R 2 ( h ) . Testing the New . . . • Here, R 1 ( x ) = � 2 1 − o ( d ) � Conclusions and . . . i | ∆ d i ( x ) | p , where ∆ d i ( x ) is Home Page d ∈ D the difference operator, and Title Page • R 2 ( h ) = � C h � 2 , where C is the discrete Laplace oper- ◭◭ ◮◮ ator. ◭ ◮ ( v i ( x ( k ) )) 2 • The ℓ p -sum � | v i ( x ) | p is optimized as � , Page 7 of 14 v 2 − p i i i where v i = v i ( x ( k − 1) ) for x from the previous iteration. Go Back Full Screen • This method results in the best blind image deconvo- lution. Close Quit
Image Deconvolution: . . . Ideal No-Noise Case 7. Need for Improvement Deconvolution in the . . . • The current technique is based on minimizing the sum Blind Image . . . | ∆ x I | p + | ∆ y I | p . State-of-the-Art . . . � � � � p p � � � � Need for Improvement ∂I ∂I � � � � • This is a discrete analog of the term + . � � � � Rotation-Invariant . . . ∂x ∂y Testing the New . . . • For p = 2, this is the square of the length of the gradi- Conclusions and . . . ent vector and is, thus, rotation-invariant. Home Page • However, for p � = 2, the above expression is not Title Page rotation-invariant. ◭◭ ◮◮ • Thus, even if it works for some image, it may not work ◭ ◮ well if we rotate this image. Page 8 of 14 • To improve the quality of image deconvolution, it is Go Back thus desirable to make the method rotation-invariant. Full Screen • We show that this indeed improves the quality of de- convolution. Close Quit
Image Deconvolution: . . . Ideal No-Noise Case 8. Rotation-Invariant Modification: Description Deconvolution in the . . . and Results � � � � Blind Image . . . p p � � � � ∂I ∂I � � � � State-of-the-Art . . . • We want to replace the expression + with � � � � ∂x ∂y Need for Improvement a rotation-invariant function of the gradient. Rotation-Invariant . . . • The only rotation-invariant characteristic of a vector a �� Testing the New . . . a 2 is its length � a � = i . Conclusions and . . . i Home Page • Thus, we replace the above expression with Title Page �� 2 � p/ 2 � � � 2 ◭◭ ◮◮ � � � � ∂I ∂I � � � � + . � � � � ∂x ∂y ◭ ◮ Page 9 of 14 • Its discrete analog is ((∆ x I ) 2 + (∆ y I ) 2 ) p/ 2 . Go Back • This modification leads to a statistically significant im- Full Screen provement in reconstruction accuracy � � x − x � 2 . Close Quit
Image Deconvolution: . . . Ideal No-Noise Case 9. Testing the New Algorithm: Details Deconvolution in the . . . • To test the new method, we compared it with the orig- Blind Image . . . inal methods: State-of-the-Art . . . Need for Improvement – on the same “Cameraman” image use in the origi- Rotation-Invariant . . . nal method, Testing the New . . . – with the same values of the parameters ( α = 1, γ = 5 · 10 5 , τ = 0 . 125, η 1 = 1024); Conclusions and . . . Home Page – we applied the same Gaussian blurring with the variance of 5; Title Page – with the same S/N ratio corr. to σ = 0 . 001. ◭◭ ◮◮ • We used the same criterion � x − � x � 2 to gauge the de- ◭ ◮ convolution quality. Page 10 of 14 • Both methods start with randomly selected initial val- Go Back ues v 1 , 1 d . Full Screen • Because of this, the results differ slightly when we re- Close apply the algorithm to the same image. Quit
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