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Some Blind Deconvolution Techniques in Image Processing

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

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Some Blind Deconvolution Techniques in Image Processing

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

  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. Blind Deconvolution Problem  = + Observed image Unknown true image Unknown point spread function Unknown noise Goal: Given uobs, recover both uorig and k

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

  5. h D 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:

  6. TV Blind Deconvolution Model (C. and Wong (IEEE TIP, 1998)) Objective: Subject to: • 1determined 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 H1 regularization.

  7. Clean image Recovered Image PSF Blind 1= 2106, 2 = 1.5105 Blind v.s. non-Blind Deconvolution Observed Image noise-free non-Blind True PSF • 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

  8. Clean image Blind v.s. non-Blind Deconvolution: High Noise Blind Observed Image SNR=5 dB non-Blind True PSF 1= 2105, 2 = 1.5105 • 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

  9. 1107 1105 1104 Controlling Focal-Length Recovered Images are 1-parameter family w.r.t. 2 Recovered Blurring Functions (1 = 2106) 2: 0 The parameter 2 controls the focal-length

  10. Generalizations to Multi-Channel Images • Inter-Channel Blur Model • Color image (Katsaggelos et al, SPIE 1994): k1: within channel blur k2: between channel blur m-channel TV-norm (Color-TV) (C. & Blomgren, IEEE TIP ‘98)

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

  12. TV Blind Deconvolution Patented!

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

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

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

  17. Problems with Deblur then Inpaint Original Occluded Support of PSF • Different BC’s correspond to different image intensities in inpaint region. • Most local BC’s do not respect global geometric structures Dirichlet Neumann Inpainting

  18. Coupling of inpainting & deblur Inpainting take place The Joint Model • Do --- the region where the image is observed • Di --- the region to be inpainted • A natural combination of TV deblur + TV inpaint • No BC’s needed for inpaint regions • 2 parameters (can incorporate automatic parameter selection techniques)

  19. Degraded Restored Zoom-in Simulation Results (1) • The vertical strip is completed • Not completed • Use higher order inpainting methods • E.g. Euler’s elastica, curvature driven diffusion

  20. Simulation Results (2) Original Observed Restored Inpaint then deblur (many ringings) Deblur then inpaint (many artifacts)

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

  22. 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)

  23. Automatic Blind Deblurring (ongoing research) observed image Clean image SNR = 15 dB • 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

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

  25. 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

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

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

  28. 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

  29. References • C. and C. K. Wong, Total Variation Blind Deconvolution, IEEE Transactions on Image Processing, 7(3):370-375, 1998. • 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. • C. and C. K. Wong, Convergence of the Alternating Minimization Algorithm for Blind Deconvolution, UCLA Mathematics Department CAM Report 99-19. • R. H. Chan, C. and C. K. Wong,Cosine Transform Based Preconditioners for Total Variation Deblurring, IEEE Trans. Image Proc., 8 (1999), pp. 1472-1478 • C., A. Yip and F. Park, Simultaneous Total Variation Image Inpainting and Blind Deconvolution, UCLA Mathematics Department CAM Report 04-45.

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