inpainting_mcfinpainting_mcfInpaintingMcfinpainting_mcfInpaintingMcfInpaintingMcf (Operator)

Name

inpainting_mcfinpainting_mcfInpaintingMcfinpainting_mcfInpaintingMcfInpaintingMcf — Perform an inpainting by smoothing of level lines.

Signature

Description

The operator inpainting_mcfinpainting_mcfInpaintingMcfinpainting_mcfInpaintingMcfInpaintingMcf extends the image edges that adjoin the region RegionRegionRegionRegionRegionregion of the input image ImageImageImageImageImageimage into the interior of RegionRegionRegionRegionRegionregion and connects their ends by smoothing the level lines of the gray value function of ImageImageImageImageImageimage.

This happens through the application of the mean curvature flow or intrinsic heat equation

  u_t = div(grad u/|grad u|) |grad u| = curv(u) |grad u|

on the gray value function u defined in the region RegionRegionRegionRegionRegionregion by the input image ImageImageImageImageImageimage at a time t_0 = 0. The discretized equation is solved in IterationsIterationsIterationsIterationsIterationsiterations time steps of length ThetaThetaThetaThetaThetatheta, so that the output image InpaintedImageInpaintedImageInpaintedImageInpaintedImageInpaintedImageinpaintedImage contains the gray value function at the time IterationsIterationsIterationsIterationsIterationsiterations * ThetaThetaThetaThetaThetatheta .

A stationary state of the mean curvature flow equation, which is also the basis of the operator mean_curvature_flowmean_curvature_flowMeanCurvatureFlowmean_curvature_flowMeanCurvatureFlowMeanCurvatureFlow, has the special property that the level lines of u all have the curvature 0. This means that after sufficiently many iterations there are only straight edges left inside the computation area of the output image InpaintedImageInpaintedImageInpaintedImageInpaintedImageInpaintedImageinpaintedImage. By this, the structure of objects inside of RegionRegionRegionRegionRegionregion can be simplified, while the remaining edges are continuously connected to those of the surrounding image matrix. This allows for a removal of image errors and unwanted objects in the input image, a so called image inpainting, which is only weakly visible to a human beholder since there remain no obvious artefacts or smudges.

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 SigmaSigmaSigmaSigmaSigmasigma 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_diffusionIsotropicDiffusionisotropic_diffusionIsotropicDiffusionIsotropicDiffusion for isotropic image smoothing.

Parallelization

• 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

Alternatives

harmonic_interpolationharmonic_interpolationHarmonicInterpolationharmonic_interpolationHarmonicInterpolationHarmonicInterpolation, inpainting_ctinpainting_ctInpaintingCtinpainting_ctInpaintingCtInpaintingCt, inpainting_anisoinpainting_anisoInpaintingAnisoinpainting_anisoInpaintingAnisoInpaintingAniso, inpainting_cedinpainting_cedInpaintingCedinpainting_cedInpaintingCedInpaintingCed, inpainting_textureinpainting_textureInpaintingTextureinpainting_textureInpaintingTextureInpaintingTexture

References

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.

Module

Foundation