coherence_enhancing_diffcoherence_enhancing_diffCoherenceEnhancingDiffCoherenceEnhancingDiff (Operator)

Name

coherence_enhancing_diffcoherence_enhancing_diffCoherenceEnhancingDiffCoherenceEnhancingDiff — Perform a coherence enhancing diffusion of an image.

Signature

coherence_enhancing_diff(Image : ImageCED : Sigma, Rho, Theta, Iterations : )

The operator coherence_enhancing_diffcoherence_enhancing_diffCoherenceEnhancingDiffCoherenceEnhancingDiffCoherenceEnhancingDiff performs an anisotropic diffusion process on the input image ImageImageImageImageimage to increase the coherence of the image structures contained in ImageImageImageImageimage. In particular, noncontinuous image edges are connected by diffusion, without being smoothed perpendicular to their dominating direction. For this, coherence_enhancing_diffcoherence_enhancing_diffCoherenceEnhancingDiffCoherenceEnhancingDiffCoherenceEnhancingDiff uses the anisotropic diffusion equation

formulated by Weickert. With a 2x2 coefficient matrix G that depends on the gray values in ImageImageImageImageimage, this is an enhancement of the mean curvature flow or intrinsic heat equation

on the gray value function u defined by the input image ImageImageImageImageimage at a time . The smoothing operator mean_curvature_flowmean_curvature_flowMeanCurvatureFlowMeanCurvatureFlowMeanCurvatureFlow is a direct application of the mean curvature flow equation. The discrete diffusion equation is solved in IterationsIterationsIterationsIterationsiterations time steps of length ThetaThetaThetaThetatheta, so that the output image ImageCEDImageCEDImageCEDImageCEDimageCED contains the gray value function at the time .

To detect the edge direction more robustly, in particular on noisy input data, an additional isotropic smoothing step can precede the computation of the gray value gradients. The parameter SigmaSigmaSigmaSigmasigma determines the magnitude of the smoothing by means of the standard deviation of a corresponding Gaussian convolution kernel, as used in the operator isotropic_diffusionisotropic_diffusionIsotropicDiffusionIsotropicDiffusionIsotropicDiffusion for isotropic image smoothing.

While the matrix G is given by

in the case of the operator mean_curvature_flowmean_curvature_flowMeanCurvatureFlowMeanCurvatureFlowMeanCurvatureFlow, where I denotes the unit matrix, is again smoothed componentwise by a Gaussian filter of standard deviation RhoRhoRhoRhorho for coherence_enhancing_diffcoherence_enhancing_diffCoherenceEnhancingDiffCoherenceEnhancingDiffCoherenceEnhancingDiff. Then, the final coefficient matrix

is constructed from the eigenvalues and eigenvectors of the resulting intermediate matrix, where the functions

were determined empirically and taken from the publication of Weickert.

Hence, the diffusion direction in mean_curvature_flowmean_curvature_flowMeanCurvatureFlowMeanCurvatureFlowMeanCurvatureFlow is only determined by the local direction of the gray value gradient, while considers the macroscopic structure of the image objects on the scale RhoRhoRhoRhorho and the magnitude of the diffusion in coherence_enhancing_diffcoherence_enhancing_diffCoherenceEnhancingDiffCoherenceEnhancingDiffCoherenceEnhancingDiff depends on how well this structure is defined.

Attention

Note that filter operators may return unexpected results if an image with a reduced domain is used as input. Please refer to the chapter Filters.

Execution Information

Multithreading type: reentrant (runs in parallel with non-exclusive operators).
Multithreading scope: global (may be called from any thread).
Automatically parallelized on tuple level.
Automatically parallelized on channel level.

Parameters

ImageImageImageImageimage (input_object) (multichannel-)image(-array) → object (byte / uint2 / real)

Input image.

ImageCEDImageCEDImageCEDImageCEDimageCED (output_object) image(-array) → object (byte / uint2 / real)

Output image.

SigmaSigmaSigmaSigmasigma (input_control) real → (real)

Smoothing for derivative operator.

Default value: 0.5

Suggested values: 0.0, 0.1, 0.5, 1.0

Restriction: Sigma >= 0

RhoRhoRhoRhorho (input_control) real → (real)

Smoothing for diffusion coefficients.

Default value: 3.0

Suggested values: 0.0, 1.0, 3.0, 5.0, 10.0, 30.0

Restriction: Rho >= 0

ThetaThetaThetaThetatheta (input_control) real → (real)

Time step.

Default value: 0.5

Suggested values: 0.1, 0.2, 0.3, 0.4, 0.5

Restriction: 0 < Theta <= 0.5

IterationsIterationsIterationsIterationsiterations (input_control) integer → (integer)

Number of iterations.

Default value: 10

Suggested values: 1, 5, 10, 20, 50, 100, 500

Restriction: Iterations >= 1

References

J. Weickert, V. Hlavac, R. Sara; “Multiscale texture enhancement”; Computer analysis of images and patterns, Lecture Notes in Computer Science, Vol. 970, pp. 230-237; Springer, Berlin; 1995.
J. Weickert, B. ter Haar Romeny, L. Florack, J. Koenderink, M. Viergever; “A review of nonlinear diffusion filtering”; Scale-Space Theory in Computer Vision, Lecture Notes in Comp. Science, Vol. 1252, pp. 3-28; Springer, Berlin; 1997.

Module

Foundation

Operators