Color Image Enhancement by a ForwardandBackward Beltrami Flow

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Color 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
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Presentation outline

• Introduction PDEs in image processing.
• Beltrami flow for color processing.
• The metric as a structure tensor.
• New adaptive structure tensor.
• Results.
• Conclusion.

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Related 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.

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Diffusion Processes

• Linear diffusion
• Non-linear (inhomogeneous diffusion)

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Linear Diffusion as a LPF

• The Gaussian is the Greens function of the
  diffusion equation. In the 1D case we get

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Adding 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
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Nonlinear diffusion example (Perona and Malik
1990)

• Smoothing low gradients (mainly noise)
• Preserving high gradients (singularities and
  edges).

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Denoising by linear vs. nonlinear diffusion
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Color processing by Beltrami Flow

• Representing color image as a 2D surface in a 5D
  Riemannian manifold.
• Evolving each color channel via the Beltrami
  flow

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Beltrami flow example

• An edge-preserving denoising process

Adopted from 2
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Beltrami flow (cont.) denoising JPEG lossy
effect surface rendering of RGB channels.
Adopted from 2
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The 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

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New 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

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Adaptive 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.

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Adaptive 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.

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FAB new eigenvalue

• New adaptive eigenvalue

1
0.5
Lambda_1(s)
0
-0.5
0
5
10
15
20
25
30
35
40
45
50
s
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Linear inverse diffusion A highly unstable
process (ill-posed)Example trying to restore a
blurred step.

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Enhancement by FAB process.
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Mandrill eye Image original (left) and after FAB
process (right)
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Summary

• 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.