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Weighted Laplacian Differences Based
     Multispectral Anisotropic Diffusion

                  V. B. Surya Prasath

                 Department of Mathematics
                University of Coimbra, Portugal

              Department of Computer Science
             University of Missouri-Columbia, USA




Surya (UC)                Multispectral             Anisotropic Diffusion   1 / 26
Outline


1   Image Denoising
      Basic Problem

2   Proposed Scheme
      Multispectral Diffusion
      Channel Coupling

3   Experimental Results
      Denoising Examples
      Summary




       Surya (UC)               Multispectral   Anisotropic Diffusion   2 / 26
Outline


1   Image Denoising
      Basic Problem

2   Proposed Scheme
      Multispectral Diffusion
      Channel Coupling

3   Experimental Results
      Denoising Examples
      Summary




       Surya (UC)               Multispectral   Anisotropic Diffusion   2 / 26
Outline


1   Image Denoising
      Basic Problem

2   Proposed Scheme
      Multispectral Diffusion
      Channel Coupling

3   Experimental Results
      Denoising Examples
      Summary




       Surya (UC)               Multispectral   Anisotropic Diffusion   2 / 26
Outline


1   Image Denoising
      Basic Problem

2   Proposed Scheme
      Multispectral Diffusion
      Channel Coupling

3   Experimental Results
      Denoising Examples
      Summary




       Surya (UC)               Multispectral   Anisotropic Diffusion   3 / 26
Inverse problem




Imaging Model: u0 = u + n
   Inverse problem
   Ill-posed problem
   n - random noise




     Surya (UC)             Multispectral   Anisotropic Diffusion   4 / 26
Definition of edges in digital images




       | u| determines the edges
       Edge detectors are based on | u|
       Discontinuities of the image function
       Interchannel correlations
       (Color, Multispectral)




     Surya (UC)               Multispectral    Anisotropic Diffusion   5 / 26
Definition of edges in digital images




       | u| determines the edges
       Edge detectors are based on | u|
       Discontinuities of the image function
       Interchannel correlations
       (Color, Multispectral)




     Surya (UC)               Multispectral    Anisotropic Diffusion   5 / 26
Definition of edges in digital images




       | u| determines the edges
       Edge detectors are based on | u|
       Discontinuities of the image function
       Interchannel correlations
       (Color, Multispectral)




     Surya (UC)               Multispectral    Anisotropic Diffusion   5 / 26
Definition of edges in digital images




       | u| determines the edges
       Edge detectors are based on | u|
       Discontinuities of the image function
       Interchannel correlations
       (Color, Multispectral)




     Surya (UC)               Multispectral    Anisotropic Diffusion   5 / 26
Perona-Malik’s idea


Anisotropic diffusion equation
                    ∂u
                       = div (g(| u|) u) with u(x, 0) = u0 (x)
                    ∂t
Required properties:
               g : [0, ∞) → (0, ∞) is decreasing, g(0) = 1
                                                     1
               lims→∞ g(s) = 0 with g(s) ≈           √
                                                       s
Examples

              g1 (s) = exp (−s/K )2        g2 (s) = (1 + (s/K )2 )−1




       Surya (UC)                    Multispectral             Anisotropic Diffusion   6 / 26
Advantages & caveats




   Use | u| to drive the diffusion
   Automatic edge detection & selective smoothing
 × Multichannel correlations
 × Multi-edges alignment




      Surya (UC)               Multispectral        Anisotropic Diffusion   7 / 26
Outline


1   Image Denoising
      Basic Problem

2   Proposed Scheme
      Multispectral Diffusion
      Channel Coupling

3   Experimental Results
      Denoising Examples
      Summary




       Surya (UC)               Multispectral   Anisotropic Diffusion   8 / 26
Multichannel images

Let u0 = (u0 , · · · , u0 ) : Ω → RN be the noisy input N-D image.
           1            N




                     Noisy u0                        Denoised u

 1   Denoise u0 to find u = (u 1 , · · · , u N )
 2                         i
     Use information from u0
 3                                    i
     Detect discontinuities from all u0



       Surya (UC)                    Multispectral                Anisotropic Diffusion   9 / 26
Multispectral anisotropic diffusion


    Use minimum, median, mean of                    u:   (Acton & Landis, IJRS ’97)

                    ∂u i
                         = div (g( u 1 , u 2 , . . . , u N ) u i )
                    ∂t
     Minimum g = g(mini           ui )
     Median      g = g(median         ui )
                         1         i)
     Mean        g = g( N        u
    Use vectorial diffusion: (Tschumperle & Deriche, PAMI ’05)
                                        ´

                                 ∂u
                                    = Trace(HD)
                                 ∂t




      Surya (UC)                    Multispectral                        Anisotropic Diffusion   10 / 26
Outline


1   Image Denoising
      Basic Problem

2   Proposed Scheme
      Multispectral Diffusion
      Channel Coupling

3   Experimental Results
      Denoising Examples
      Summary




       Surya (UC)               Multispectral   Anisotropic Diffusion   11 / 26
Proposed scheme



Multispectral Anisotropic Diffusion
                                                 N
           ∂u i
                = div g     ui    ui + α               ωi ∆u j − ωj ∆u i
           ∂t
                                                 j=1

Flexibility:
     Diffusion function g
     Weights ω




        Surya (UC)               Multispectral                  Anisotropic Diffusion   12 / 26
Key idea
Weighted Laplacian Differences
    Laplacian differences (multi-edges)
    Use weights (alignment)
    Keep the intra-channel diffusion

Cross-correlation term (for channel i)
                                       N
                                              ωi ∆u j − ωj ∆u i
                                       j=1




                    (a) (u 1 , u 2 )         (b) (∆u 1 , ∆u 2 )   (c)

       Surya (UC)                               Multispectral           Anisotropic Diffusion   13 / 26
TV based weights

   The total variation PDE (Rudin, Osher, Fatemi ’92)

                    ∂ui
                     ˜                ui
                                      ˜
                        = div                     with u i (0) = u0
                                                       ˜          i
                    ∂t                ui
                                      ˜

   Pre-smooth the gradients

                                 ωi = Gρ            ui
                                                    ˜


Scheme details
   Split Bregman implementation
   Fast computation of convolution
   Additive operator splitting


     Surya (UC)                   Multispectral                 Anisotropic Diffusion   14 / 26
Outline


1   Image Denoising
      Basic Problem

2   Proposed Scheme
      Multispectral Diffusion
      Channel Coupling

3   Experimental Results
      Denoising Examples
      Summary




       Surya (UC)               Multispectral   Anisotropic Diffusion   15 / 26
Color (RGB) image




              (a) Input    (b) Weight       (c) Difference




          (d) Our scheme   (e) Original      (f) Residue

     Surya (UC)             Multispectral          Anisotropic Diffusion   16 / 26
Multispectral image (N = 8) - Mississippi River




                  (a) Input                   (b) Denoised




     Surya (UC)               Multispectral              Anisotropic Diffusion   17 / 26
Multispectral image (N = 8) - 3 denoised channels




           (a) L-band VV   (b) L-band VH     (c) C-band VV



     Surya (UC)              Multispectral         Anisotropic Diffusion   18 / 26
Comparison of different schemes




                                                   ´
                  (a) Acton & Landis (b) Tschumperle         &
                                     Deriche




                   (c) Our scheme          (d) Multi-edges
     Surya (UC)                 Multispectral                    Anisotropic Diffusion   19 / 26
High noise - San Francisco Bay




           (a) Input (N = 4) - (1,2,3)               (b) Denoised




     Surya (UC)                      Multispectral              Anisotropic Diffusion   20 / 26
Outline


1   Image Denoising
      Basic Problem

2   Proposed Scheme
      Multispectral Diffusion
      Channel Coupling

3   Experimental Results
      Denoising Examples
      Summary




       Surya (UC)               Multispectral   Anisotropic Diffusion   21 / 26
Summary




  Selective smoothing & enhancement
  Integrated edge information (multi-edges)
  Fast Split Bregman implementation
  Reliable & efficient
  Extension to Hyperspectral ?




    Surya (UC)               Multispectral    Anisotropic Diffusion   22 / 26
References

  G. Aubert and P. Kornprobst.
  Mathematical problems in Image Processing.
  Springer-Verlag, 2006.
  P. Perona and J. Malik.
  Scale space and edge detection using anisotropic diffusion.
  IEEE Trans. on PAMI, 14(8):826–833, 1990.
  S. T. Acton and J. Landis.
  Multi-spectral anisotropic diffusion.
  Int’l J. Remote Sens., 18:2877-2886, 1997.
                 ´
  D. Tschumperle and R. Deriche.
  Vector-valued image regularization with PDEs: A common
  framework for different applications.
  IEEE Trans. on PAMI, 27:1-12, 2005.


    Surya (UC)               Multispectral         Anisotropic Diffusion   23 / 26
References

  G. Aubert and P. Kornprobst.
  Mathematical problems in Image Processing.
  Springer-Verlag, 2006.
  P. Perona and J. Malik.
  Scale space and edge detection using anisotropic diffusion.
  IEEE Trans. on PAMI, 14(8):826–833, 1990.
  S. T. Acton and J. Landis.
  Multi-spectral anisotropic diffusion.
  Int’l J. Remote Sens., 18:2877-2886, 1997.
                 ´
  D. Tschumperle and R. Deriche.
  Vector-valued image regularization with PDEs: A common
  framework for different applications.
  IEEE Trans. on PAMI, 27:1-12, 2005.


    Surya (UC)               Multispectral         Anisotropic Diffusion   23 / 26
Questions?




                 Thank you




    Surya (UC)    Multispectral   Anisotropic Diffusion   24 / 26
Color (RGB) denoising




              (a) Input    (b) Weight       (c) Difference




          (d) Our scheme   (e) Original      (f) Residue

     Surya (UC)             Multispectral          Anisotropic Diffusion   25 / 26
Comparison of different schemes - Paris




      (a) Input (N = 7) - (b) Acton & Landis   (c) Bresson & Chan
      (4,3,2)




                     ´
       (d) Tschumperle   &    (e) Original       (f) Our scheme
       Deriche
     Surya (UC)                Multispectral            Anisotropic Diffusion   26 / 26

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suryaiitm.pdf

  • 1. Weighted Laplacian Differences Based Multispectral Anisotropic Diffusion V. B. Surya Prasath Department of Mathematics University of Coimbra, Portugal Department of Computer Science University of Missouri-Columbia, USA Surya (UC) Multispectral Anisotropic Diffusion 1 / 26
  • 2. Outline 1 Image Denoising Basic Problem 2 Proposed Scheme Multispectral Diffusion Channel Coupling 3 Experimental Results Denoising Examples Summary Surya (UC) Multispectral Anisotropic Diffusion 2 / 26
  • 3. Outline 1 Image Denoising Basic Problem 2 Proposed Scheme Multispectral Diffusion Channel Coupling 3 Experimental Results Denoising Examples Summary Surya (UC) Multispectral Anisotropic Diffusion 2 / 26
  • 4. Outline 1 Image Denoising Basic Problem 2 Proposed Scheme Multispectral Diffusion Channel Coupling 3 Experimental Results Denoising Examples Summary Surya (UC) Multispectral Anisotropic Diffusion 2 / 26
  • 5. Outline 1 Image Denoising Basic Problem 2 Proposed Scheme Multispectral Diffusion Channel Coupling 3 Experimental Results Denoising Examples Summary Surya (UC) Multispectral Anisotropic Diffusion 3 / 26
  • 6. Inverse problem Imaging Model: u0 = u + n Inverse problem Ill-posed problem n - random noise Surya (UC) Multispectral Anisotropic Diffusion 4 / 26
  • 7. Definition of edges in digital images | u| determines the edges Edge detectors are based on | u| Discontinuities of the image function Interchannel correlations (Color, Multispectral) Surya (UC) Multispectral Anisotropic Diffusion 5 / 26
  • 8. Definition of edges in digital images | u| determines the edges Edge detectors are based on | u| Discontinuities of the image function Interchannel correlations (Color, Multispectral) Surya (UC) Multispectral Anisotropic Diffusion 5 / 26
  • 9. Definition of edges in digital images | u| determines the edges Edge detectors are based on | u| Discontinuities of the image function Interchannel correlations (Color, Multispectral) Surya (UC) Multispectral Anisotropic Diffusion 5 / 26
  • 10. Definition of edges in digital images | u| determines the edges Edge detectors are based on | u| Discontinuities of the image function Interchannel correlations (Color, Multispectral) Surya (UC) Multispectral Anisotropic Diffusion 5 / 26
  • 11. Perona-Malik’s idea Anisotropic diffusion equation ∂u = div (g(| u|) u) with u(x, 0) = u0 (x) ∂t Required properties: g : [0, ∞) → (0, ∞) is decreasing, g(0) = 1 1 lims→∞ g(s) = 0 with g(s) ≈ √ s Examples g1 (s) = exp (−s/K )2 g2 (s) = (1 + (s/K )2 )−1 Surya (UC) Multispectral Anisotropic Diffusion 6 / 26
  • 12. Advantages & caveats Use | u| to drive the diffusion Automatic edge detection & selective smoothing × Multichannel correlations × Multi-edges alignment Surya (UC) Multispectral Anisotropic Diffusion 7 / 26
  • 13. Outline 1 Image Denoising Basic Problem 2 Proposed Scheme Multispectral Diffusion Channel Coupling 3 Experimental Results Denoising Examples Summary Surya (UC) Multispectral Anisotropic Diffusion 8 / 26
  • 14. Multichannel images Let u0 = (u0 , · · · , u0 ) : Ω → RN be the noisy input N-D image. 1 N Noisy u0 Denoised u 1 Denoise u0 to find u = (u 1 , · · · , u N ) 2 i Use information from u0 3 i Detect discontinuities from all u0 Surya (UC) Multispectral Anisotropic Diffusion 9 / 26
  • 15. Multispectral anisotropic diffusion Use minimum, median, mean of u: (Acton & Landis, IJRS ’97) ∂u i = div (g( u 1 , u 2 , . . . , u N ) u i ) ∂t Minimum g = g(mini ui ) Median g = g(median ui ) 1 i) Mean g = g( N u Use vectorial diffusion: (Tschumperle & Deriche, PAMI ’05) ´ ∂u = Trace(HD) ∂t Surya (UC) Multispectral Anisotropic Diffusion 10 / 26
  • 16. Outline 1 Image Denoising Basic Problem 2 Proposed Scheme Multispectral Diffusion Channel Coupling 3 Experimental Results Denoising Examples Summary Surya (UC) Multispectral Anisotropic Diffusion 11 / 26
  • 17. Proposed scheme Multispectral Anisotropic Diffusion N ∂u i = div g ui ui + α ωi ∆u j − ωj ∆u i ∂t j=1 Flexibility: Diffusion function g Weights ω Surya (UC) Multispectral Anisotropic Diffusion 12 / 26
  • 18. Key idea Weighted Laplacian Differences Laplacian differences (multi-edges) Use weights (alignment) Keep the intra-channel diffusion Cross-correlation term (for channel i) N ωi ∆u j − ωj ∆u i j=1 (a) (u 1 , u 2 ) (b) (∆u 1 , ∆u 2 ) (c) Surya (UC) Multispectral Anisotropic Diffusion 13 / 26
  • 19. TV based weights The total variation PDE (Rudin, Osher, Fatemi ’92) ∂ui ˜ ui ˜ = div with u i (0) = u0 ˜ i ∂t ui ˜ Pre-smooth the gradients ωi = Gρ ui ˜ Scheme details Split Bregman implementation Fast computation of convolution Additive operator splitting Surya (UC) Multispectral Anisotropic Diffusion 14 / 26
  • 20. Outline 1 Image Denoising Basic Problem 2 Proposed Scheme Multispectral Diffusion Channel Coupling 3 Experimental Results Denoising Examples Summary Surya (UC) Multispectral Anisotropic Diffusion 15 / 26
  • 21. Color (RGB) image (a) Input (b) Weight (c) Difference (d) Our scheme (e) Original (f) Residue Surya (UC) Multispectral Anisotropic Diffusion 16 / 26
  • 22. Multispectral image (N = 8) - Mississippi River (a) Input (b) Denoised Surya (UC) Multispectral Anisotropic Diffusion 17 / 26
  • 23. Multispectral image (N = 8) - 3 denoised channels (a) L-band VV (b) L-band VH (c) C-band VV Surya (UC) Multispectral Anisotropic Diffusion 18 / 26
  • 24. Comparison of different schemes ´ (a) Acton & Landis (b) Tschumperle & Deriche (c) Our scheme (d) Multi-edges Surya (UC) Multispectral Anisotropic Diffusion 19 / 26
  • 25. High noise - San Francisco Bay (a) Input (N = 4) - (1,2,3) (b) Denoised Surya (UC) Multispectral Anisotropic Diffusion 20 / 26
  • 26. Outline 1 Image Denoising Basic Problem 2 Proposed Scheme Multispectral Diffusion Channel Coupling 3 Experimental Results Denoising Examples Summary Surya (UC) Multispectral Anisotropic Diffusion 21 / 26
  • 27. Summary Selective smoothing & enhancement Integrated edge information (multi-edges) Fast Split Bregman implementation Reliable & efficient Extension to Hyperspectral ? Surya (UC) Multispectral Anisotropic Diffusion 22 / 26
  • 28. References G. Aubert and P. Kornprobst. Mathematical problems in Image Processing. Springer-Verlag, 2006. P. Perona and J. Malik. Scale space and edge detection using anisotropic diffusion. IEEE Trans. on PAMI, 14(8):826–833, 1990. S. T. Acton and J. Landis. Multi-spectral anisotropic diffusion. Int’l J. Remote Sens., 18:2877-2886, 1997. ´ D. Tschumperle and R. Deriche. Vector-valued image regularization with PDEs: A common framework for different applications. IEEE Trans. on PAMI, 27:1-12, 2005. Surya (UC) Multispectral Anisotropic Diffusion 23 / 26
  • 29. References G. Aubert and P. Kornprobst. Mathematical problems in Image Processing. Springer-Verlag, 2006. P. Perona and J. Malik. Scale space and edge detection using anisotropic diffusion. IEEE Trans. on PAMI, 14(8):826–833, 1990. S. T. Acton and J. Landis. Multi-spectral anisotropic diffusion. Int’l J. Remote Sens., 18:2877-2886, 1997. ´ D. Tschumperle and R. Deriche. Vector-valued image regularization with PDEs: A common framework for different applications. IEEE Trans. on PAMI, 27:1-12, 2005. Surya (UC) Multispectral Anisotropic Diffusion 23 / 26
  • 30. Questions? Thank you Surya (UC) Multispectral Anisotropic Diffusion 24 / 26
  • 31. Color (RGB) denoising (a) Input (b) Weight (c) Difference (d) Our scheme (e) Original (f) Residue Surya (UC) Multispectral Anisotropic Diffusion 25 / 26
  • 32. Comparison of different schemes - Paris (a) Input (N = 7) - (b) Acton & Landis (c) Bresson & Chan (4,3,2) ´ (d) Tschumperle & (e) Original (f) Our scheme Deriche Surya (UC) Multispectral Anisotropic Diffusion 26 / 26