Textured Image Deconvolution and Decomposition Duy Hoang Thai - PowerPoint PPT Presentation

Textured Image Deconvolution and Decomposition Duy Hoang Thai Joint work with David Banks 1

Inverse Problem in Imaging [1]: Image processing : e.g. segmentation, denoising, deblurring, ... Inverse problem Approximation theory sparsity Optimization Fourier Statistics (PDE) (wavelet) (stochastic process) I The underlying technique in image processing is inverse problem which links to approximation theory . I There are 3 large fields in inverse problem: optimization , harmonic analysis and statistics . I ”Sparsity” is a fundamental concept in approximation theory. 2

Ex: Approximation of a function f ⇡ b + f seg + ✏ + ~ v original image (1) piecewise constant (2) residual (3) bias fi eld texture (4) (5) (6) (7) The purpose of preprocessing is to obtain good feature and approximation theory stays at the heart in this step: I Image segmentation focuses on piecewise constant (2) . I Texture analysis relates to directional texture (4) - (7) . I Image denoising relates to (1) - (7) , except noise (3) . 3

Non-texture v.s. texture: there are three classes of images. (a) Homogeneous (b) Texture (c) Homogeneous & Texture 4

Overview: 1. Motivation 2. Directional mean curvature for image deconvolution and decomposition (DMCDD) 3. Function space & Filter Banks in sampling theory 4. Conclusion 5

1. Motivation for Image Reconstruction Example: Forensic examiner tries to match di ff erent ballistic images (in the same gun) which are captured from a cheap microscope. Feature for matching is texture information . ) a cheap microscope produces noise and blur in the image. Motivation: I Deconvolution is to recover the true underlying signal from a noisy and blurred image . I Decomposition is to extract good feature for post-processing, e.g. pattern recognition. 6

2. Directional mean curvature for image deconvolution and decomposition (DMCDD) Notation: I A bounded domain: n k = [ k 1 , k 2 ] 2 [0 , m � 1] ⇥ [0 , n � 1] ⇢ Z 2 o Ω = I The Euclidean space (space of matrices): X = R | Ω | I A discrete image (size m ⇥ n ): f [ k ] : Ω ! R + , matrix f 2 X Ex: Barbara image (size 512 ⇥ 512) 7

2. DMCDD Go Feature (a) original image f 0 (b) observed image f (c) reconstructed f re , MSE = 0 . 057 f = h ⇤ ( u + v + ⇢ ) + ✏ Model: | {z } = f 0 I h : a blur operator (known) I u : piecewise smooth component I v : texture I ⇢ : small scale objects (residual) I ✏ : noise 8

2. DMCDD I Our strategy is to find value of functions ( u , v , ✏ , ⇢ ) by minimizing an objective function. I The measures for these functions are defined in function space. ) This is a regularization problem in the Banach space (in discrete setting) ! 9

2. The DMCDD model: general idea of the proposed DMCDD model Piecewise smooth u Directional Mean Curvature (high order PDE & reduce stair case e ff ect) T exture v Spatial sparsity T exture v Directional G-norm Bounded condition in curvelet domain Legendre Fenchel trans. Identity condition (Dantzig selector) Note: This model can be explained by the Bayesian framework and L´ evy processes . 10

2. DMCDD: functional spaces [2, 6] To minimize cost func., we have some constraints depending on di ff erent function spaces: Bounded variation dual of Besov space Besov space G-space (space of wavelet coef.) Hilbert space I Dual pairs are ( ˙ 1 , 1 , ˙ B � 1 1 , 1 ) and ( ˙ B 1 BV , G ). BV or ˙ ˙ I Piecewise smooth u is usually measured by B 1 1 , 1 . I Oscillatory components, e.g. texture v or noise ✏ , is measured in L 2 , G , ˙ B � 1 1 , 1 . Example: TV � L 2 model (Rudin-Osher-Fatemi) for f = u + ✏ , ✏ ⇠ N (0 , � 2 ): n o kr u k L 1 + µ 2 k f � u k 2 min L 2 u 2 ˙ BV The ROF model produces “ stair case e ff ect ”. We need higher order approach ! 11

2. DMCDD: space for piecewise smooth u [7] Piecewise smooth u is measured by directional mean curvature (DMC)-norm. Innovation model (UnserT afti2001) Assumption: an original signal is not sparse, but it is sparse under some transform domains. I Multi-direction produces smoother signal rather than 2 direction in MC. I DMC produces higher order PDE than BV and reduces stair case e ff ect . 12

2. DMCDD: space for texture v Texture isn’t well presented in the Fourier domain. In function space, we use the discrete directional G S -norm (approximation version) [2] S � 1 n s =0 2 X S o X ⇥ ⇤ S � 1 k g s k ` 1 , v = div � k v k G S = inf k ~ g k ` 1 = S ~ g , ~ g = g s s =0 to minimize the cost function. 13

2. DMCDD: space for residual ⇢ & noise ✏ To handle residual and noise, we need to bound ` 1 -norm in curvelet space by a constant ⌫ : � � � C{ ✏ } ` 1  ⌫ , ✏ 2 X � This is because the oscillatory components do not have small norms in L 2 ( Ω ) or L 1 ( Ω ). � � Constraint � C{·} ` 1  ⌫ is similar to Dantzig selector [5]. � (d) wavelet ( ˙ B 1 1 , 1 , ˙ B � 1 (e) curv � � � � 1 , 1 ) � C{·} � C{·} ` 1 , � � ` 1 Signal representation in curvelet is sparser than wavelet , i.e. signal is smoother in spatial domain, because the subband is more refine. 14

2. DMCDD: component in directional G-norm texture observed image blur piecewise smooth h original noise residual bounded condition in curvelet space To estimate functions ( u , v , ⇢ , ✏ ), we solve the DMCDD model ⇢� � �  d � min u + µ 1 k v k G S + µ 2 k v k ` 1 � � L ( u , v , ⇢ , ✏ ) 2 X 4 � ` 1 � � � � � s.t. f = h ⇤ ( u + v + ⇢ ) + ✏ , � C{ ⇢ } ` 1  ⌫ ⇢ , � C{ ✏ } ` 1  ⌫ ✏ � � by Augmented Lagrangian method , alternating directional method of multipliers and Iterative Shrinkage/Thresholding Algorithms [4]. Go ALM Go ADMM 15

2. Numerical result: Texture is well recovered on the Barbara image. (a) Original image f 0 (b) Blur image f = f 0 ⇤ h (c) f re = u + v + ⇢ , MSE = 0 . 016 (d) u (e) v ( very sparse ) (f) ⇢ (g) ✏ = 0 Back 16

2. Numerical result: Barbara image The previous slide shows reconstructed image from the original Barbara image. Note: I Since there is no noise in the observed image (b), the noise in (g) is set to 0 because of the bounded norm in curvelet space : � � � C{ ✏ } ` 1  ⌫ ✏ � which is di ff erent from regularization in ` 2 -norm. I Texture is very sparse with 29.59% of nonzero coef. . 17

3. Approximation scheme: from sampling theory to function space N -th System 2 Lowpass 2 Highpass (a) function space (b) wavelet Approximation by function space: I System contains shrinkage operator and ” frame functions ” H ( z ) , Φ ( z ) , ˜ Ψ l ( z ) , Ψ l ( z ) , Ξ ( z ) , ˜ Θ l ( z ) , Θ l ( z ) which have definition in the Fourier domain and satisfy the unity condition . Go I These ”frame functions” are combined in this system as sampling theory in the Banach space (with ` 1 -norm) . I These ”frame functions” are similar to wavelet-like operator [3]. 18

3. Filter banks in the DMCDD model: L � 1 Figure: Ψ L , c ( z ) = Ψ L , c ( z ) Ψ L X ˜ l ( z ). l 19 l =0

3. Comparison between multi-directional mean curvature and MC Go 20

3. Comparison between multi-directional mean curvature and MC Directional mean curvature is better because I When increasing # direction, the bandwidth of lowpass is shrinked . It ensures there is no texture or residual in piecewise smooth component u . I The shrinkage operator removes small coe ffi cients in highpass by a constant depending on Lagrange multiplier, i.e. reconstructed image is smoother . Solution of piecewise smooth component u (at iteration ⌧ ): piecewise smooth lowpass (interpolant in sampling theory) shrinkage operator (~ sampling theory with L1 norm) frame fucn. & its dual constant depends on Lagrange multiplier (~ biorthogonal wavelet) due to high order approach 21

4. Conclusion 1. This research proposes a new method for image recovery. The method uses constrained minimization in some appropriate spaces to archive: I good texture recovery I small Mean square error I avoid stair case e ff ect by extending two directional mean curvature to multidirectional one (high order PDE approach). 2. We also set the link between sampling theory and function space which is similar to wavelet-like operator . This research is based on two projects at SAMSI: I D.H. Thai and D. Banks, Directional Mean Curvature for Textured Image Deconvolution and Decomposition (ongoing) I D.H. Thai and L. Mentch, Multiphase Segmentation For Simultaneously Homogeneous and Textural Images (in submission) 22

Why do we need some complicated models? because we’d like to solve some complicated problems: extract fingerprint pattern from noisy background. 23