Weighted Laplacian Differences Based Multispectral Anisotropic Diffusion
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
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6. Inverse problem
Imaging Model: u0 = u + n
Inverse problem
Ill-posed problem
n - random noise
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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)
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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)
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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)
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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)
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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
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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
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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
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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
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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)
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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
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27. Summary
Selective smoothing & enhancement
Integrated edge information (multi-edges)
Fast Split Bregman implementation
Reliable & efficient
Extension to Hyperspectral ?
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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.
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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.
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