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Numerical Geometry in Image Processing

Computer Science Department. Technion-Israel Institute of Technology. Numerical Geometry in Image Processing. www.cs.technion.ac.il/~ron. Ron Kimmel. Geometric Image Processing Lab. Heat Equation in Image Analysis. Linear scale space (T. Iijima 59, Witkin 83, Koenderink 84).

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Numerical Geometry in Image Processing

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  1. Computer Science Department • Technion-Israel Institute of Technology Numerical Geometry in Image Processing www.cs.technion.ac.il/~ron Ron Kimmel • Geometric Image Processing Lab

  2. Heat Equation in Image Analysis • Linear scale space (T. Iijima 59, Witkin 83, Koenderink 84)

  3. Geometric Heat Equation in Image Analysis • Geometric scale space, Euclidean (Gage-Hamilton 86, Grayson 89, Osher-Sethian 88, Evans Spruck 91, Alvarez-Guichard-Lions-Morel 93)

  4. Geometric Heat Equation in Image Analysis • Gabor 65 anisotropic reaction-diffusion • Geometric, Special Affine. (Alvarez-Guichard-Lions-Morel 93, Sapiro-Tannenbaum 93)

  5. Geometric Heat Equation in Image Analysis • Multi Channel, Euclidean.(Chambolle 94, Whitaker-Gerig 94, Proesmans-Pauwels-van Gool 94,Sapiro-Ringach 96, Shah 96, Blomgren-Chan 96, Sochen-Kimmel-Malladi 96, Weickert, Romeny, Lopez, and van Enk 97,…) • Geometric, Bending.(Curves: Grayson 89, Kimmel-Sapiro 95 (via Osher-Sethian),Images: Kimmel 97)

  6. Bending Invariant Scale Space • Invariant to surface bending. • Embedding: The gray level sets embedding is preserved. • Existence: The level sets exist for all evolution time, disappear at points or converge into geodesics. • Topology: Image topology is simplified. • Shortening flow:The scale space is a shortening flow of the image level sets. • Implementation: Simple, consistent, and stable numerical implementation.

  7. Curves on Surfaces: The Geodesic Curvature

  8. From Curve to Image Evolution

  9. Geodesic curvature flow

  10. The Beltrami Framework • Brief history of color line element theories. • A simplified color image formation model. • The importance of channel alignment. • Images as surfaces. • Surface area minimization via Beltrami flow. • Applications: Enhancement and scale space. • Beyond the metric, the Gabor connection

  11. Images as Surfaces • Gray level analysis is sometimes misleading… • Is there a `right way’ to link color channels? process texture? enhance volumetric data? • We view images as embedded maps that flow towards minimal surfaces: Gray scale images are surfaces in (x,y, I), and color images are surfaces embedded in (x,y,R,G,B). • Joint with Sochen & Malladi, IEEE T-IP 98, IJCV 2000.

  12. Spatial-Spectral Arclength • Helmholtz 1896: • Schrodinger 1920: • Stiles 1946: • Vos and Walraven 1972: • inductive line elements (above), empirical line elements (MacAdam 1942, CIELAB 1976). Define: the simplest hybrid spatial- color space:

  13. Color Image Formation • F. Guichard 93 Mondrian world: Lambertian surface patches

  14. N l V Image formation Lambetian model

  15. ColorImage Formation • The gradient directions should agree since

  16. Example: Demosaicing • Colorimage reconstruction Solution: • Edges support the colors and the colorssupport the edges

  17. Color Image Formation Lambertian shading model: • R(x,y) = r <N,L> • G(x,y) = r <N,L> • B(x,y) = r <N,L> Thus • Within an object R/G=r/r =constant • We preserve color ratio weighted by an edge indication function. R G B R G

  18. Demosaicing Results Original Bilinear interpolation Weighted interpolation

  19. Demosaicing Results Bilinear interpolation Weighted interpolation

  20. Demosaicing Results Original Bilinear interpolation Weighted interpolation

  21. Demosaicing Results Bilinear interpolation Weighted interpolation

  22. Demosaicing Results Original Bilinear interpolation Weighted interpolation

  23. Demosaicing Results Bilinear interpolation Weighted interpolation

  24. From Arclength to Area Gray level arclength: Color arclength Area

  25. Multi Channel Model

  26. The Beltrami Flow • Gray level:

  27. The Beltrami Flow • Color : where

  28. Matlab Program

  29. Signal processing viewpoint Gaussian Smoothing Beltrami Smoothing Sochen, Kimmel, Bruckstein, JMIV, 2001.

  30. The Beltrami Flow • Texture:

  31. Inverse Diffusion Across the Edge

  32. Inverse Diffusion Across the Edge

  33. Summary: Geometric Framework • From color image formation to the importance of channel alignment. • From color line element theories to the definition of area in color images. • Area minimization as a unified framework for enhancement and scale space. • Inverse heat operator across the edges. • Related applications: Color movies segmentation and demosaicing www.cs.technion.ac.il/~ron

  34. Open Questions • Is there a maximum principle to the Beltrami flow? • Are there simple geometric measures to minimize in color image processing subject to more complicated image formation models? • Can we really invert the geometric heat operator? • Is there a real-time numerical implementation for the Beltrami flow in color? www.cs.technion.ac.il/~ron

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