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
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
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
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
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