1. A joint work with Nir Sochen & Yehoshua Y. Zeevi.
Computer Vision Seminar, HUJI, December 2002 PDE-based Image Processing and the Triple Well Potential for Image Sharpening
Guy Gilboa
Faculty of Electrical Engineering,
Technion, Haifa, Israel
A joint work with Nir Sochen & Yehoshua Y. Zeevi.

2. Objectives A review on PDE-based methods for image processing.
Show relation to energy minimization.
Present a new well-shaped potential for image sharpening.
Introduce hyper-diffusion for regularization.
Examples and conclusion.

3. Basic Linear and Nonlinear Diffusion
Linear scale-space.
Perona-Malik scheme.
The intuition behind nonlinear-diffusion filtering.
Numerical schemes

4. Uncommitted front end vision “We know nothing, we have no preference whatsoever”
The mathematical formulation for that is:
Linearity (no knowledge, no model)
Spatial shift invariance (no preferred location)
Isotropy (no preferred orientation)
Scale invariance (no preferred size, or scale)

5. Connecting PDE’s to Image processing, introducing: The Linear Scale-Space
Scale Space is represented by the linear diffusion equation:
We add a scale dimension to the original image – using a single scale parameter t.
As shown by Koendrink: The diffusion equation is the unique scheme that incorporates all the above requirements (isotropy, homogeneity, causality).

6. Linear Scale-Space Applying the diffusion equation to the original image – creating a 3rd dimension t
Adopted from
[Romeny ‘96] t

7. Scale-Space
representation
Pyramid
representation

8. Edge and corner detection of images
Adopted from:
[Lindeberg-’94]

9. Application example: edge detection
Edge detection at many scales simultaneously, in the 1D case, by zero-crossing of Laplacian.
Signal at different scales edges
Adopted from:
[Witkin-’83]

10. Linear Diffusion as a LPF
The Gaussian is the Green’s function of the diffusion equation. In the 1D case we get:

11. Diffusion Processes Linear diffusion
Non-linear (inhomogeneous diffusion)

12. Nonlinear diffusion example – Perona Malik:
Smoothing low gradients (mainly noise)
Preserving high gradients (singularities and edges).

13. Linear diffusion example

14. Nonlinear diffusion example

15. Intuition for adaptive denoising
“Do not diffuse the leaves of the tree with the sky at the background” [P-M]
Diffuse (low-pass-filter) only within the same region \ object.
Therefore -> Slow the diffusion near edges.

16. Gaussian averaging along the curve of the signal
The distance is measured not only spatially,
the values of the signal are also considered.
Related to the logic of bilateral filters.
Adopted from [Sochen et al ‘01]

17. Relation to robust statistics
Reducing the effect of outliers – pixels at the other side of an edge are treated as outliers and should not be considered in the estimation (see Black et al `98).

18. Numerics – how do we actually do that ?
Reminder from first year Infi course:
For images we usually take h=1and simply compute the difference between neighboring pixels:
1st order: Forward: Ii+1,j - Ii,j , Backward: Ii,j – Ii_1,j
Central: (Ii+1,j – Ii-1,j )/2
2nd order: Central: Ii+1,j – 2Ii,j + Ii-1,j

19. Explicit Schemes
Computation is done on each pixel separately – using the values of the previous iteration. Example of linear diffusion:

20. Explicit scheme – cont’
Main advantage – simplicity, very easy to implement.
Main disadvantage - time constraints (CFL bound): the scheme is stable only when
(for 2D)
Summary: a very popular scheme (esp. when the process does not need a lot of iterations).

21. Perona-Malik example in Matlab
function J=diffusion(J,K,N)
for i=1:N,
% calculate gradient in all directions (N,S,E,W)
In=[J(1,:); J(1:Ny-1,:)]-J;
Is=[J(2:Ny,:); J(Ny,:)]-J;
Ie=[J(:,2:Nx) J(:,Nx)]-J;
Iw=[J(:,1) J(:,1:Nx-1)]-J;
% calculate diffusion coefficients
Cn=exp(-(abs(In)/K).^2);
Cs=exp(-(abs(Is)/K).^2);
Ce=exp(-(abs(Ie)/K).^2);
Cw=exp(-(abs(Iw)/K).^2);
J=J+0.2*(Cn.*In + Cs.*Is + Ce.*Ie + Cw.*Iw);
end; % for i

22. Advanced Schemes
Implicit schemes – need to solve for all the pixels together, no time constraints, produces a very large set of equations. Iterative methods are often used (Jacobi, Gauss-Seidel, Multigrid).
Level-sets – used for curve evolution (like snakes). Representing a curve as a level set of a higher dimensional function. Scheme is stable, non-parametric, able to change topologies.

23. Other PDE-based Processes
Coherence-enhanceing diffusion.
Total-Variation denoising
Beltrami flow color processing
Segmentation – Mumford-Shah functional and active contours (snakes).

24. Anisotropic diffusion
Cottet and Germain, Weickert - coherence enhancing flow: strong diffusion along the edge, weak diffusion across the edge (tensor diffusion coef.) .

25. Total Variation denoising (Osher-Rudin-Fatemi)
Denoising by minimizing the total variation yet staying close to the input image. Reduces the oscillatory part of the signal that contains mostly noise (but also texture and some small details).
Energy to be minimized:
f – original image

26. TV - does not penalize large gradients (edges)
Same energy for any monotone part of the signal, unlike linear diffusion.
For L1 norm all the lines on the right has the same energy, whereas for L2 norm the blue line has the highest energy and the red line has the lowest.

27. TV vs. PM - a sketchy comparison
Top: original+noise(SNR=11.9dB)
Bottom:
left – TV (SNR=17.6dB)
right – PM (SNR=16.9dB) TV PM

28. Color processing by Beltrami flow (Sochen, Kimmel, Malladi)
Representing color image as a 2D surface in a 5D Riemannian manifold. A surface minimizing process that denoises and preserves edges
Evolving each color channel via the Beltrami flow:

29. Beltrami flow – examples

30. Beltrami flow (cont’): denoising JPEG lossy effect – surface rendering of RGB channels.

31. see www.cs.technion.ac.il/~ron
Beltrami flow - movies
CENSORED
see

32. Image Segmentation
Mumford shah functional
Active contours (snakes)

33. Mumford-Shah Functional
A variational approach for image segmentation.
Minimizing the following energy functional:
f – original image, u – piece-wise smooth approx. of f separated by
the contour – C.

34. Active contours (snakes)
Evolving a curve like a rubber-band, with the aim to “close” on the object to be segmented, creating a continuous, smooth curve.
Motivation is drawn from the active contour model of Kass et al (’87) but rely on level set techniques introduced by Osher and Sethian to handle topological changes in a seamless fashion (introduced independently by Caselles et al. and Malladi et al. in ’95-`97, “Geodesic Active Contours”).
Emin (C)  “smooth”+”elastic”+”on edges”

35. Segmentation examples
Adpoted from Chan & Vese, UCLA site

36. Video Segmentation
Adpoted from Julian Jerome ©

37. More processes Texture segmentation Smoothing of vector fields
Image inpainting (filling missing information)
Movies – smoothing, filling frames etc.
Knowledge-based segmentation
Stereo vision
and more..

38. Some References
Black, M., G. Sapiro, D. Marimont, and D. Heeger, Robust anisotropic diffusion, IEEE Transactions on Image Processing Volume 7, PP , 1998.
V. Caselles, R. Kimmel, and G. Sapiro. Geodesic active contours. International Journal of Computer Vision, 22(1):61-79, 1997.
Chan T, Vese L.A., Image Segmentation Using Level Sets and the Mumford-Shah Model , CAM 00-14, April 2000
Chan T, Vese L.A. , Active Contours Without Edges, IEEE Image Proc. Feb 2001.
Chan T, Shen J., Vese L.A., Variational PDE models in image processing, Amer. Math. Soc. Notice, 50,   pp , January 2003.
G.H. Cottet and L. Germain, “Image processing through reaction combined with nonlinear diffusion", Math. Comp., 61 (1993)
M. Kass, A. Witkin and D. Terzopoulos, "Snakes: Active contour models," International Journal of Computer Vision, pp , 1987.
R Kimmel, R Malladi and N Sochen, ``Images as Embedding Maps and Minimal Surfaces: Movies, Color, Texture, and Volumetric Medical Images", Int. J. of Computer Vision, 39(2): , Sept

39. R. Malladi, J. A. Sethian and B. C. Vemuri
R. Malladi, J. A. Sethian and B.C. Vemuri. Shape modeling with front propagation : A level set approach. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17(2): , February 1995.
D. Mumford and J. Shah, Optimal approximations by piece-wise smooth functions and assosiated variational problems, Comm. Pure and Appl. Math., LII (1989),
P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion", IEEE Trans. PAMI vol. 12,no. 7, pp , 1990.
T. Lindeberg, “Scale-space theory: a basic tool..”, J. App. Statistics, 21(2): , 1994.
Rudin L, Osher S and Fatemi C 1992 Nonlinear total variation based noise removal algorithm, Physica D 60, (1992).
N Sochen, R Kimmel and R Malladi , “`A general framework for low level vision", IEEE Trans. on Image Processing, 7, (1998)
N. Sochen, R. Kimmel, and A.M. Bruckstein. Diffusions and confusions in signal and image processing, Journal of Mathematical Imaging and Vision, 14(3): , 2001.
ter Haar Romeney B.M., “An Intorduction to Scale-Space Theory”, VBC-’96, Hamburg, Germany.

40. J. Weickert,``Coherence-enhancing diffusion of colour images", Image and Vision Comp., 17 (1999)
J. Weickert, A review on nonlinear diffusion filtering, LNCS 1252, Scale-Space Theory in Computer Vision, Springer-Verlag, 1997, 3-28.
A. P. Witkin, ``Scale space filtering", Proc. Int. Joint Conf. On Artificial Intelligence, pp , 1983.
Books
ter Haar Romeny, Geometry-driven diffusion in computer vision.
Weickert, Anisotropic diffusion in image processing .
Sapiro, Geometric partial differential equations and image analysis .
Sites
Ron Kimmel’s course “Numerical Geometry of Images”:
My web site:

41. Nonlinear diffusion as an energy minimizing process
A general nonlinear diffusion process can be viewed as a steepest descent sequence that minimizes the signal’s energy.
The energy functional E is defined as the cumulative potential (energy density) Ψ of the signal in the domain Ω.

42. We define an energy functional E:
where Ψ is a potential which is a function of the gradient magnitude.
The steepest descent process is:

43. we get the nonlinear diffusion equation (“Perona-Malik” style):
Assigning:
we get the nonlinear diffusion equation
(“Perona-Malik” style):
see
- You, Xu, Tannenbaum, Kaveh, IEEE Trans. IP, 5(11), 1996.
- Weickert, LNCS 1252, pp.3-28, 1997.

44. The nature of the diffusion depends on the potential function ψ (or the corresponding diffusion coefficient c)

45. Potentials of several processes

46. Diffusion coefficients of several processes

47. (e.g. linear diffusion, Charbonnier et al., Beltrami)
Convex Potentials
(e.g. linear diffusion, Charbonnier et al., Beltrami)
Reach a global minimum
Well-posed processes
Strong denoising
Edge preservation is weaker

48. Nonconvex Potentials (e.g. Perona-Malik)
In general can have many minima (Hollig, You et al.). produce staircasing.
Need some sort of regularization to be well-posed (Catte et al).
Weaker denoising
Strong edge preservation
Performs well for images-processing

49. Sharpening by going back in time ?
forward
backward

50. Potential requirement for image sharpening
Basic sharpening property: gradients should increase (at least in some range).
High gradients should “cost “ less energy than medium gradients.
Minimum-maximum principle is not kept.
blur
sharpen

51. Inverse diffusion
A classical ill posed sharpening process, attempting to reverse forward diffusion (Gaussian blur).
Drawbacks:
Oscillatory
Amplifies noise exponentially
Causes the explosion of the signal

52. 1D inverse diffusion example: trying to restore a blurred step.

53. Inverse diffusion of a blurred image

54. We would like to find a potential function that has a sharpening ability and yet avoids the inverse diffusion drawbacks.

55. Low gradients should not be enhanced
Restrictions on the potential - 1
Rule 1:
Low gradients should not be enhanced
Avoid amplification of noise
Specifically, the zero gradient should be stable –> have minimum energy.

56. Very high gradients should not be enhanced
Restrictions on the potential - 2
Rule 2:
Very high gradients should not be enhanced
Avoid explosion of the signal
To reduce staricasing – very high gradients should contribute some positive energy.

57. There should be minimal oscilations between low energy states
Restrictions on the potential - 3
Rule 3:
There should be minimal oscilations between low energy states
We assume the original image is with little oscillations.

58. The triple-well potential
Denoising
Sharpening
Sharpening
W(Ix)
Slow smoothing
Slow smoothing Ix
Sharpening potential W(Ix) in one dimension.
Forms the shape of three wells.

59. From inverse diffusion to a Forward-and-Backward (FAB) diffusion
C(|grad(U)|) 1
|grad(U)| -α

60. Stability of smooth regions
Given Mf > Mb then for every x0: |Ix (x0;0)|<rf we satisfy |Ix (x0;t)|<rf for any t>0.

61. The proposed energy functional for sharpening:
where
W is a gradient dependent well-shaped potential.
F is a fidelity term.
R is a high order regularization term.

62. We assign a standard convex fidelity term to the input image I0:
Next:
We discuss the need for a high order regularization term R .

63. Higher order regularization
We search for the “smoothest” energy minimizer with minimal oscillations between the low energy states (similar to the viscosity solutions reasoning).
For that we add a second order term to the energy functional. We use a convex rotationally invariant term:
The steepest descent flow is of hyper-diffusion.

64. Hyper-diffusion Initial and boundary conditions:
n is a unit vector normal to the boundary

65. Hyper-diffusion as a stabilizer of inverse diffusion (from the literature)
In the Cahn-Hilliard and Kuramoto-Sivashinsky equations a hyper diffusion term is used to stabilize linear inverse diffusion. Kuramoto, Dynamics of interacting particles, Springer 1984, Sivashinsky Ann. Rev. Mech. 15, 1983, J.W Cahn, J.E. Hilliard, J. Chem. Phys. 28,2, 1958.
Physical processes modeled by forward-and-backward diffusion and hyper-diffusion are shown to have a unique solution (no proof in our case yet). see Witelski, Studies in Applied Mathematics, 96, pp , 1996.

66. Hyper-diffusion vs. diffusion 1D (Step and noise processed after times 0.1,1,10)
Hyper-diffusion Diffusion

67. Hyper-diffusion 2D (Cameraman processed after times 0.1,1,10)

68. Energy minimization process for selective sharpening
Summary:
Energy minimization process for selective sharpening
Initial and boundary conditions:

69. Processing a 1D step with blur and noise

70. 1D line-edge with blur and noise

71. Processing a 2D step with different blur and noise
Isotropic Gaussian
Anisotropic exponential
5x5 box averaging
Jagginess
+ additive Gaussian and uniform white noise.

72. Enhancement of a toy car by our scheme and a regularized shock filter (Alvarez-Mazorra).
Original Well-potential Regularized shock (ours) (Alvarez Mazorra)

73. Application: super-resolution from a single image
Low
resolution
High
resolution

74. Triple well modification (anisotropic with texture preserving)
Original
Processed

75. Original
Processed

76. Conclusion
PDE-based techniques were shown to be effective in a variety of image-processing applications with concise and well defined formulations.
A well-shaped potential was presented for image sharpening.
The process rewards the increase of gradients in some range, while being able to operate in a noisy environment and avoid oscillations and the explosion of the signal.
Hyper-diffusion was introduced as a means to stabilize inverse-diffusion type processes.