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Some Blind Deconvolution Techniques in Image Processing Tony Chan Math Dept., UCLA Astronomical Data Analysis Software & Systems Conference Series 2004.

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Presentation on theme: "Some Blind Deconvolution Techniques in Image Processing Tony Chan Math Dept., UCLA Astronomical Data Analysis Software & Systems Conference Series 2004."— Presentation transcript:

1 Some Blind Deconvolution Techniques in Image Processing Tony Chan Math Dept., UCLA Astronomical Data Analysis Software & Systems Conference Series 2004 Pasadena, CA, October 24-27, 2004 Joint work with Frederick Park and Andy M. Yip

2 2 Outline Part I: Total Variation Blind Deconvolution Part II: Simultaneous TV Image Inpainting and Blind Deconvolution Part III: Automatic Parameter Selection for TV Blind Deconvolution Caution: Our work not developed specifically for Astronomical images

3 3 Blind Deconvolution Problem = + Observed image Unknown true image Unknown point spread function Unknown noise Goal: Given u obs, recover both u orig and k

4 4 Typical PSFs PSFs w/ sharp edges: PSFs w/ smooth transitions

5 5 Total Variation Regularization Deconvolution ill-posed: need regularization Total variation Regularization: The characteristic function of D with height h (jump): TV = Length(D) h TV doesn t penalize jumps Co-area Formula: D h

6 6 TV Blind Deconvolution Model Subject to: Objective: (C. and Wong (IEEE TIP, 1998)) 1 determined by signal-to-noise ratio 2 parameterizes a family of solutions, corresponds to different spread of the reconstructed PSF Alternating Minimization Algorithm: Globally convergent with H 1 regularization.

7 7 Blind v.s. non-Blind Deconvolution Observed Image noise-free An out-of-focus blur is recovered automatically Recovered blind deconvolution images almost as good as non-blind Edges well-recovered in image and PSF non-Blind Recovered ImagePSF Blind 1 = , 2 = Clean image True PSF

8 8 Blind v.s. non-Blind Deconvolution: High Noise Observed Image SNR=5 dB non-Blind Clean image True PSF Blind An out-of-focus blur is recovered automatically Even in the presence of high noise level, recovered images from blind deconvolution are almost as good as those recovered with the exact PSF 1 = , 2 =

9 9 Controlling Focal-Length Recovered Images are 1-parameter family w.r.t. 2 Recovered Blurring Functions ( 1 = ) : The parameter 2 controls the focal-length

10 10 Generalizations to Multi-Channel Images Inter-Channel Blur Model –Color image (Katsaggelos et al, SPIE 1994): k 1 : within channel blur k 2 : between channel blur m-channel TV-norm (Color-TV) (C. & Blomgren, IEEE TIP 98)

11 11 Original image Out-of-focus blurred blind non-blind Gaussian blurred blind non-blind Examples of Multi-Channel Blind Deconvolution (C. and Wong (SPIE, 1997)) Blind is as good as non-blind Gaussian blur is harder to recover (zero-crossings in frequency domain)

12 12 TV Blind Deconvolution Patented!

13 13 Outline Part I: Total Variation Blind Deconvolution Part II: Simultaneous TV Image Inpainting and Blind Deconvolution Part III: Automatic Parameter Selection for TV Blind Deconvolution

14 14 TV Inpainting Model (C. & Shen SIAP 2001) Graffiti Removal Scratch Removal

15 15 Images Degraded by Blurring and Missing Regions Blur –Calibration errors of devices –Atmospheric turbulence –Motion of objects/camera Missing regions –Scratches –Occlusion –Defects in films/sensors +

16 16 Problems with Inpaint then Deblur Inpaint first reduce plausible solutions Should pick the solution using more information Original SignalBlurring func. Original SignalBlurring func. Blurred Signal = = Blurred + Occluded =

17 17 Problems with Deblur then Inpaint Different BCs correspond to different image intensities in inpaint region. Most local BCs do not respect global geometric structures OriginalOccludedSupport of PSF DirichletNeumannInpainting

18 18 The Joint Model D o --- the region where the image is observed D i --- the region to be inpainted A natural combination of TV deblur + TV inpaint No BCs needed for inpaint regions 2 parameters (can incorporate automatic parameter selection techniques) Inpainting take place Coupling of inpainting & deblur

19 19 Simulation Results (1) DegradedRestored Zoom-in The vertical strip is completed Not completed Use higher order inpainting methods E.g. Eulers elastica, curvature driven diffusion

20 20 Simulation Results (2) ObservedRestored Deblur then inpaint (many artifacts) Inpaint then deblur (many ringings) Original

21 21 Boundary Conditions for Regular Deblurring Original image domain and artificial boundary outside the scene Dirichlet B.C. Periodic B.C. Neumann B.C. Inpainting B.C.

22 22

23 23 Outline Part I: Total Variation Blind Deconvolution Part II: Simultaneous TV Image Inpainting and Blind Deconvolution Part III: Automatic Parameter Selection for TV Blind Deconvolution (Ongoing Research)

24 24 Automatic Blind Deblurring (ongoing research) Recovered images: 1-parameter family wrt 2 Consider external info like sharpness to choose optimal 2 Problem: Find 2 automatically to recover best u & k SNR = 15 dB Clean image observed image

25 25 Motivation for Sharpness & Support Sharpest image has large gradients Preference for gradients with small support u Support of

26 26 Proposed Sharpness Evaluator F(u) small => sharp image with small support F(u)=0 for piecewise constant images F(u) penalizes smeared edges u Support of

27 27 Planets Example Rel. errors in u (blue) and k (red) v.s. 2 Proposed Objective v.s. 2 Optimal Restored Image Auto-focused Image The minimum of the sharpness function agrees with that of the rel. errors of u and k (minimizer of sharpness func.) (minimizer of rel. error in u) 1 =0.02 (optimal)

28 28 Satellite Example Rel. errors in u (blue) and k (red) v.s. 2 Proposed Objective v.s. 2 Optimal Restored Image Auto-focused Image The minimum of the sharpness function agrees with that of the rel. errors of u and k (minimizer of sharpness func.) (minimizer of rel. error in u) 1 =0.3 (optimal)

29 29 Potential Applications to Astronomical Imaging TV Blind Deconvolution –TV/Sharp edges useful? –Auto-focus: appropriate objective function? –How to incorporate a priori domain knowledge? TV Blind Deconvolution + Inpainting –Other noise models: e.g. salt-and-pepper noise

30 30 References 1.C. and C. K. Wong, Total Variation Blind Deconvolution, IEEE Transactions on Image Processing, 7(3): , C. and C. K. Wong, Multichannel Image Deconvolution by Total Variation Regularization, Proc. to the SPIE Symposium on Advanced Signal Processing: Algorithms, Architectures, and Implementations, vol. 3162, San Diego, CA, July 1997, Ed.: F. Luk. 3.C. and C. K. Wong, Convergence of the Alternating Minimization Algorithm for Blind Deconvolution, UCLA Mathematics Department CAM Report R. H. Chan, C. and C. K. Wong, Cosine Transform Based Preconditioners for Total Variation Deblurring, IEEE Trans. Image Proc., 8 (1999), pp C., A. Yip and F. Park, Simultaneous Total Variation Image Inpainting and Blind Deconvolution, UCLA Mathematics Department CAM Report


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