Weighted Laplacian Differences Based Multispectral ... - IEEE-GRSS
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: B g : [0, ∞) → (0, ∞) is decreasing, g(0) = 1 B lims→∞ g(s) = 0 with g(s) ≈
√1 s
Examples g1 (s) = exp (−s/K )2
Surya (UC)
g2 (s) = (1 + (s/K )2 )−1
Multispectral
Anisotropic Diffusion
6 / 26
Advantages & caveats
X Use |∇u| to drive the diffusion X 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 = (u01 , · · · , u0N ) : Ω → RN be the noisy input N-D image.
Noisy u0
Denoised u
1
Denoise u0 to find u = (u 1 , · · · , u N )
2
Use information from u0i
3
Detect discontinuities from all u0i
Surya (UC)
Multispectral
Anisotropic Diffusion
9 / 26
Multispectral anisotropic diffusion I Use minimum, median, mean of ∇u:
(Acton & Landis, IJRS ’97)
∂u i = div (g(∇u 1 , ∇u 2 , . . . , ∇u N )∇u i ) ∂t i Minimum g = g(mini ∇u ) i Median g = g(median ∇u ) P Mean g = g( N1 ∇u i ) I 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 X ∂u i ωi ∆u j − ωj ∆u i = div g ∇u i ∇u i + α ∂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 X
j
ωi ∆u − ωj ∆u
i
j=1
(a) (u 1 , u 2 ) Surya (UC)
(b) (∆u 1 , ∆u 2 ) Multispectral
(c) Anisotropic Diffusion
13 / 26
TV based weights The total variation PDE (Rudin, Osher, Fatemi ’92) ! ˜i ˜i ∂u ∇u ˜ i (0) = u0i with u = div i ˜ ∂t ∇u Pre-smooth the gradients ˜i ωi = Gρ ? ∇u
Scheme details X Split Bregman implementation X Fast computation of convolution X 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
Surya (UC)
(b) Denoised
Multispectral
Anisotropic Diffusion
17 / 26
Multispectral image (N = 8) - 3 denoised channels
(a) L-band VV
Surya (UC)
(b) L-band VH
Multispectral
(c) C-band VV
Anisotropic Diffusion
18 / 26
Comparison of different schemes
(a) Acton & Landis (b) Tschumperle´ Deriche
(c) Our scheme Surya (UC)
&
(d) Multi-edges
Multispectral
Anisotropic Diffusion
19 / 26
High noise - San Francisco Bay
(a) Input (N = 4) - (1,2,3)
Surya (UC)
Multispectral
(b) Denoised
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 (4,3,2)
(d) Tschumperle´ Deriche Surya (UC)
&
(e) Original Multispectral
(c) Bresson & Chan
(f) Our scheme Anisotropic Diffusion
26 / 26
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: B g : [0, ∞) → (0, ∞) is decreasing, g(0) = 1 B lims→∞ g(s) = 0 with g(s) ≈
√1 s
Examples g1 (s) = exp (−s/K )2
Surya (UC)
g2 (s) = (1 + (s/K )2 )−1
Multispectral
Anisotropic Diffusion
6 / 26
Advantages & caveats
X Use |∇u| to drive the diffusion X 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 = (u01 , · · · , u0N ) : Ω → RN be the noisy input N-D image.
Noisy u0
Denoised u
1
Denoise u0 to find u = (u 1 , · · · , u N )
2
Use information from u0i
3
Detect discontinuities from all u0i
Surya (UC)
Multispectral
Anisotropic Diffusion
9 / 26
Multispectral anisotropic diffusion I Use minimum, median, mean of ∇u:
(Acton & Landis, IJRS ’97)
∂u i = div (g(∇u 1 , ∇u 2 , . . . , ∇u N )∇u i ) ∂t i Minimum g = g(mini ∇u ) i Median g = g(median ∇u ) P Mean g = g( N1 ∇u i ) I 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 X ∂u i ωi ∆u j − ωj ∆u i = div g ∇u i ∇u i + α ∂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 X
j
ωi ∆u − ωj ∆u
i
j=1
(a) (u 1 , u 2 ) Surya (UC)
(b) (∆u 1 , ∆u 2 ) Multispectral
(c) Anisotropic Diffusion
13 / 26
TV based weights The total variation PDE (Rudin, Osher, Fatemi ’92) ! ˜i ˜i ∂u ∇u ˜ i (0) = u0i with u = div i ˜ ∂t ∇u Pre-smooth the gradients ˜i ωi = Gρ ? ∇u
Scheme details X Split Bregman implementation X Fast computation of convolution X 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
Surya (UC)
(b) Denoised
Multispectral
Anisotropic Diffusion
17 / 26
Multispectral image (N = 8) - 3 denoised channels
(a) L-band VV
Surya (UC)
(b) L-band VH
Multispectral
(c) C-band VV
Anisotropic Diffusion
18 / 26
Comparison of different schemes
(a) Acton & Landis (b) Tschumperle´ Deriche
(c) Our scheme Surya (UC)
&
(d) Multi-edges
Multispectral
Anisotropic Diffusion
19 / 26
High noise - San Francisco Bay
(a) Input (N = 4) - (1,2,3)
Surya (UC)
Multispectral
(b) Denoised
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 (4,3,2)
(d) Tschumperle´ Deriche Surya (UC)
&
(e) Original Multispectral
(c) Bresson & Chan
(f) Our scheme Anisotropic Diffusion
26 / 26
