Title: Color Image Enhancement by a ForwardandBackward Beltrami Flow
1Color Image Enhancement by a Forward-and-Backward
Beltrami Flow
AFPAC-2000, Kiel, Germany, September 2000
- Faculty of Electrical Engineering
- Technion, Haifa, Israel
By N. Sochen, G. Gilboa, Y.Y. Zeevi
2Presentation outline
- Introduction PDEs in image processing.
- Beltrami flow for color processing.
- The metric as a structure tensor.
- New adaptive structure tensor.
- Results.
- Conclusion.
3Related studies
- 1 N. Sochen, R. Kimmel and R. Malladi , A
general framework for low level vision", IEEE
Trans. on Image Processing, 7, (1998) 310-318. - 2 R. Kimmel, R. Malladi and N. Sochen, Images
as Embedding Maps and Minimal Surfaces Movies,
Color, Texture, and Volumetric Medical Images",
International Journal of Computer Vision,
39(2)111-129, Sept. 2000. - 3 G. Gilboa, Y.Y. Zeevi, N. Sochen Anisotropic
selective inverse diffusion for signal
enhancement in the presence of noise",to appear
in IEEE ICASSP-2000, Istanbul, Turkey, 2000. - 4 J. Weickert, Coherence-enhancing diffusion
of color images, Image and Vision Comp., 17
(1999) 199-210. - 5 I. Pollak, A.S. Willsky, H. Krim, Scale
Space analysis by stabilized inverse diffusion
equations, B. ter Haar Romeney (ed.),
Scale-space theory in computer vision, LNCS, vol.
1252, Springer, Berlin, 200-211, 97.
4Diffusion Processes
- Linear diffusion
- Non-linear (inhomogeneous diffusion)
5Linear Diffusion as a LPF
- The Gaussian is the Greens function of the
diffusion equation. In the 1D case we get
6Adding the scale dimensionApplying the
diffusion equation to the original image
creating a 3rd dimension t
backward
Adopted from B.M. ter Haar Romeney, An
Intorduction to Scale-Space Theory,
VBC-96, Hamburg, Germany.
forward
7Nonlinear diffusion example (Perona and Malik
1990)
- Smoothing low gradients (mainly noise)
- Preserving high gradients (singularities and
edges).
8Denoising by linear vs. nonlinear diffusion
9Color processing by Beltrami Flow
- Representing color image as a 2D surface in a 5D
Riemannian manifold. - Evolving each color channel via the Beltrami
flow
10Beltrami flow example
- An edge-preserving denoising process
Adopted from 2
11Beltrami flow (cont.) denoising JPEG lossy
effect surface rendering of RGB channels.
Adopted from 2
12The metric as a structure tensor
- ?1 corresponds to the eigenvector in the
direction of the gradient. - ?2 corresponds to the eigenvector in the
direction of the level set (I.e. perpendicular to
the gradient). - Previous modifications
- Weickert 4 ?1constgt0 , ?21/?1 gt0.
- Kimmel, Sochen 2 ?1constlt0 , ?21/?1 gt0
13New proposed eigenvalue
- We propose to replace the eigenvalue ?1 by a new
adaptive function that controls the diffusion in
the gradient direction and is proportional to a
gradient measure
14Adaptive Forward-and-Backward (FAB) Process
- Combining two diffusion processes
- A backward process active at medium gradients,
where singularities are expected. - A forward process, used for stabilization and
noise reduction. - Result A new structure tensor, that changes
locally between positive and negative values.
15Adaptive FAB Characteristics
- Sharpening significant edges.
- Avoid explosion by diminishing the value of the
inverse diffusion coefficient at high gradients. - Reduce noise amplification, which after some
pre-smoothing, can be regarded as having mainly
low gradients, by eliminating the inverse
diffusion process at low gradients - Reduce ringing by combining a forward diffusion
process, that smoothes low gradients. -
16FAB new eigenvalue
1
0.5
Lambda_1(s)
0
-0.5
0
5
10
15
20
25
30
35
40
45
50
s
17Linear inverse diffusion A highly unstable
process (ill-posed)Example trying to restore a
blurred step.
18Enhancement by FAB process.
19Mandrill eye Image original (left) and after FAB
process (right)
20Summary
- Conflicting requirements of signal and image
smoothing, and sharpening, are incorporated into
a diffusion-type PDE approach. - A new structure tensor, that varies adaptively as
a function of a gradient measure and assumes
positive and negative values (FAB process), is
used. - Results indicate the potential of the proposed
process in enhancement of noisy images.