Blind Image Deconvolution Need for Theoretical . . . Based on - PowerPoint PPT Presentation

Outline Formulation of the . . . State-of-the-Art . . . Open Problems . . . Blind Image Deconvolution Need for Theoretical . . . Based on Sparsity: Need for Improvement Why Sparsity: . . . Theoretical Justification Why ℓ p -Techniques in . . . Improving the State- . . . and Improvement of Home Page State-of-the-Art Techniques Title Page ◭◭ ◮◮ Fernando Cervantes ◭ ◮ Department of Electrical and Computer Engineering Page 1 of 59 University of Texas at El Paso, El Paso, TX 79968, USA fcervantes@miners.utep.edu Go Back Full Screen Close Quit

Outline Formulation of the . . . 1. Outline State-of-the-Art . . . • Blind image deconvolution: formulation of the general Open Problems . . . problem and description of state-of-the-art techniques Need for Theoretical . . . Need for Improvement • Open problems related to blind image deconvolution: Why Sparsity: . . . – need for theoretical justification and Why ℓ p -Techniques in . . . – need for improvement of the existing techniques Improving the State- . . . Home Page • Theoretical justification of sparsity-based techniques in blind image deconvolution Title Page • Theoretical justification of ℓ p -techniques in blind image ◭◭ ◮◮ deconvolution ◭ ◮ • The idea of rotation invariance enables us to improve Page 2 of 59 the state-of-the-art blind deconvolution technique Go Back • Conclusions and future work Full Screen Close Quit

Outline Formulation of the . . . Part I State-of-the-Art . . . Blind Image Deconvolution: Open Problems . . . Formulation of the General Need for Theoretical . . . Problem and Description of Need for Improvement Why Sparsity: . . . State-of-the-Art Techniques Why ℓ p -Techniques in . . . Improving the State- . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 59 Go Back Full Screen Close Quit

Outline Formulation of the . . . 2. Blind Image Deconvolution: Formulation of the State-of-the-Art . . . Problem Open Problems . . . • The measurement results y k differ from the actual val- Need for Theoretical . . . ues x k dues to additive noise and blurring: Need for Improvement � Why Sparsity: . . . y k = h i · x k − i + n k . Why ℓ p -Techniques in . . . i Improving the State- . . . • 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 4 of 59 i ′ j ′ Go Back • It is desirable to reconstruct the original signal or im- Full Screen age, i.e., to perform deconvolution . Close Quit

Outline Formulation of the . . . 3. Ideal No-Noise Case State-of-the-Art . . . • In the ideal case, when noise n ( i, j ) can be ignored, we Open Problems . . . can find x ( i, j ) by solving a system of linear equations: Need for Theoretical . . . � � Need for Improvement h ( i − i ′ , j − j ′ ) · x ( i ′ , j ′ ) . y ( i, j ) = Why Sparsity: . . . i ′ j ′ Why ℓ p -Techniques in . . . • However, already for 256 × 256 images, the matrix h is Improving the State- . . . 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 5 of 59 – we compute X ( ω ) = Y ( ω ) Go Back H ( ω ), and Full Screen – finally, we compute x = F − 1 ( X ( ω )). Close Quit

Outline Formulation of the . . . 4. Deconvolution in the Presence of Noise with State-of-the-Art . . . Known Characteristics Open Problems . . . • Suppose that signal and noise are independent, and we Need for Theoretical . . . know the power spectral densities Need for Improvement � 1 � � 1 � Why Sparsity: . . . T · | X T ( ω ) | 2 T · | N T ( ω ) | 2 S I ( ω ) = lim T →∞ E , S N ( ω ) = lim T →∞ E . Why ℓ p -Techniques in . . . Improving the State- . . . • 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 6 of 59 � 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

Outline Formulation of the . . . 5. Blind Image Deconvolution in the Presence of State-of-the-Art . . . Prior Knowledge Open Problems . . . • Wiener filter techniques assume that we know the blur- Need for Theoretical . . . ring function h . Need for Improvement Why Sparsity: . . . • In practice, we often only have partial information Why ℓ p -Techniques in . . . about h . Improving the State- . . . • 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 7 of 59 � � � 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

Outline Formulation of the . . . 6. Blind Image Deconvolution in the Absence of State-of-the-Art . . . Prior Knowledge: Sparsity-Based Techniques Open Problems . . . • In many practical situations, we do not have prior Need for Theoretical . . . knowledge about the blurring function h . Need for Improvement Why Sparsity: . . . • Often, what helps is sparsity assumption: that in the expansion x ( t ) = � Why ℓ p -Techniques in . . . a i · e i ( t ), most a i are zero. i Improving the State- . . . • 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 8 of 59 Go Back • It is therefore replaced by a close convex objective func- = � def tion � a � 1 | a i | . Full Screen i Close Quit

Outline Formulation of the . . . 7. State-of-the-Art Technique for Sparsity-Based State-of-the-Art . . . Blind Deconvolution Open Problems . . . • Sparsity is the main idea behind the algorithm de- Need for Theoretical . . . scribed in (Amizic et al. 2013) that minimizes Need for Improvement Why Sparsity: . . . 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 ) . Why ℓ p -Techniques . . . • Here, R 1 ( x ) = � 2 1 − o ( d ) � Improving the State- . . . 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 9 of 59 v 2 − p i i i Go Back where v i = v i ( x ( k − 1) ) for x from the previous iteration. Full Screen • This method results in the best blind image deconvo- Close lution. Quit

Outline Formulation of the . . . Part II State-of-the-Art . . . Open Problems Related to Blind Open Problems . . . Image Deconvolution Need for Theoretical . . . Need for Improvement Why Sparsity: . . . Why ℓ p -Techniques in . . . Improving the State- . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 59 Go Back Full Screen Close Quit

Outline Formulation of the . . . 8. First Problem Related to Blind Image Decom- State-of-the-Art . . . position: Need for Theoretical Justification Open Problems . . . • The state-of-the-art technique works well on several Need for Theoretical . . . examples. Need for Improvement Why Sparsity: . . . • However, many details of this technique are purely em- Why ℓ p -Techniques in . . . pirical, with no theoretical justification. Improving the State- . . . • Thus, there is no guarantee that this method will work Home Page well on other examples. Title Page • As a result, practitioners are somewhat reluctant to ◭◭ ◮◮ use this technique. ◭ ◮ • Specifically, it is not clear: Page 11 of 59 – why sparsity-based method are efficient, and – why ℓ p -methods are efficient. Go Back Full Screen • In this dissertation, we provide a theoretical answer to both questions Close Quit