mean_curvature_flowmean_curvature_flowMeanCurvatureFlowMeanCurvatureFlow (Operator)


mean_curvature_flowmean_curvature_flowMeanCurvatureFlowMeanCurvatureFlow — Apply the mean curvature flow to an image.


mean_curvature_flow(Image : ImageMCF : Sigma, Theta, Iterations : )

Herror mean_curvature_flow(const Hobject Image, Hobject* ImageMCF, double Sigma, double Theta, const Hlong Iterations)

Herror T_mean_curvature_flow(const Hobject Image, Hobject* ImageMCF, const Htuple Sigma, const Htuple Theta, const Htuple Iterations)

void MeanCurvatureFlow(const HObject& Image, HObject* ImageMCF, const HTuple& Sigma, const HTuple& Theta, const HTuple& Iterations)

HImage HImage::MeanCurvatureFlow(double Sigma, double Theta, Hlong Iterations) const

static void HOperatorSet.MeanCurvatureFlow(HObject image, out HObject imageMCF, HTuple sigma, HTuple theta, HTuple iterations)

HImage HImage.MeanCurvatureFlow(double sigma, double theta, int iterations)


The operator mean_curvature_flowmean_curvature_flowMeanCurvatureFlowMeanCurvatureFlowMeanCurvatureFlow applies the mean curvature flow or intrinsic heat equatio to the gray value function u defined by the input image ImageImageImageImageimage at a time . The discretized equation is solved in IterationsIterationsIterationsIterationsiterations time steps of length ThetaThetaThetaThetatheta, so that the output image contains the gray value function at the time IterationsIterationsIterationsIterationsiterations * ThetaThetaThetaThetatheta .

The mean curvature flow causes a smoothing of ImageImageImageImageimage in the direction of the edges in the image, i.e. along the contour lines of u, while perpendicular to the edge direction no smoothing is performed and hence the boundaries of image objects are not smoothed. To detect the image 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.

The following images show the effect of the parameters SigmaSigmaSigmaSigmasigma, ThetaThetaThetaThetatheta, and IterationsIterationsIterationsIterationsiterations. First, the input image is shown together with the result that is achieved if all parameters are set to their default values.

(1) (2)
(1) Input image. (2) Result when using the default values.

In the following images, the results are shown that are achieved if one parameter is varied while setting the other two parameters to their default values.

SigmaSigmaSigmaSigmasigma controls the amount of smoothing, prior to the computation of the gray value gradient. Be careful with very large values for SigmaSigmaSigmaSigmasigma, because they may lead to undesired effects.

(1) (2) (3)
(1) Sigma = 0.0. (2) Sigma = 1.0. (3) Sigma = 10.0.

ThetaThetaThetaThetatheta controls the step size during the iterative smoothing process. Larger values lead to a stronger smoothing.

(1) (2) (3)
(1) Theta = 0.1. (2) Theta = 0.2. (3) Theta = 0.4.

IterationsIterationsIterationsIterationsiterations controls the number of iterations that are performed. With an increasing number of iterations, the runtime increases, as well. Furthermore, a large number of iterations may lead to a loss of structure in the smoothed image.

(1) (2) (3)
(1) Iterations = 1. (2) Iterations = 50. (3) Iterations = 100.


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


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

Input image.

ImageMCFImageMCFImageMCFImageMCFimageMCF (output_object)  image(-array) objectHImageHImageHobject * (byte / uint2 / real)

Output image.

SigmaSigmaSigmaSigmasigma (input_control)  real HTupleHTupleHtuple (real) (double) (double) (double)

Smoothing parameter for derivative operator.

Default value: 0.5

Suggested values: 0.0, 0.1, 0.5, 1.0

Restriction: Sigma >= 0

ThetaThetaThetaThetatheta (input_control)  real HTupleHTupleHtuple (real) (double) (double) (double)

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 HTupleHTupleHtuple (integer) (int / long) (Hlong) (Hlong)

Number of iterations.

Default value: 10

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

Restriction: Iterations >= 1


M. G. Crandall, P. Lions; “Convergent Difference Schemes for Nonlinear Parabolic Equations and Mean Curvature Motion”; Numer. Math. 75 pp. 17-41; 1996.
G. Aubert, P. Kornprobst; “Mathematical Problems in Image Processing”; Applied Mathematical Sciences 147; Springer, New York; 2002.