inpainting_cedinpainting_cedinpainting_cedInpaintingCedInpaintingCed (Operator)

Name

inpainting_cedinpainting_cedinpainting_cedInpaintingCedInpaintingCed — Perform an inpainting by coherence enhancing diffusion.

Signature

Description

The operator inpainting_cedinpainting_cedinpainting_cedInpaintingCedInpaintingCed performs an anisotropic diffusion process on the region RegionRegionRegionRegionregion of the input image ImageImageImageImageimage with the objective of completing discontinuous image edges diffusively by increasing the coherence of the image structures contained in ImageImageImageImageimage and without smoothing these edges perpendicular to their dominating direction. The mechanism is the same as in the operator coherence_enhancing_diffcoherence_enhancing_diffcoherence_enhancing_diffCoherenceEnhancingDiffCoherenceEnhancingDiff, which is based on a discretization of the anisotropic diffusion equation

  u_t = div(G(u) grad u)

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

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

on the gray value function u defined by the input image ImageImageImageImageimage at a time t_0 = 0. The smoothing operator mean_curvature_flowmean_curvature_flowmean_curvature_flowMeanCurvatureFlowMeanCurvatureFlow is a direct application of the mean curvature flow equation. With the operator inpainting_mcfinpainting_mcfinpainting_mcfInpaintingMcfInpaintingMcf, it can also be used for image inpainting. The discrete diffusion equation is solved in IterationsIterationsIterationsIterationsiterations time steps of length ThetaThetaThetaThetatheta, so that the output image InpaintedImageInpaintedImageInpaintedImageInpaintedImageinpaintedImage contains the gray value function at the time IterationsIterationsIterationsIterationsiterations * ThetaThetaThetaThetatheta .

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_diffusionisotropic_diffusionIsotropicDiffusionIsotropicDiffusion for isotropic image smoothing.

Similar to the operator inpainting_mcfinpainting_mcfinpainting_mcfInpaintingMcfInpaintingMcf, the structure of the image data in RegionRegionRegionRegionregion is simplified by smoothing the level lines of ImageImageImageImageimage. By this, image errors and unwanted objects can be removed from the image, while the edges in the neighborhood are extended continuously. This procedure is called image inpainting. The objective is to introduce a minimum amount of artefacts or smoothing effects, so that the image manipulation is least visible to a human beholder.

While the matrix G is given by

  G_MCF(u) = I - (grad u (grad u)^T)/|grad u|^2

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

  G_CED = g_1((l_1-l_2)^2) w_1 (w_1)^T + g_2((l_1-l_2)^2) w_2 (w_2)^T

is constructed from the eigenvalues l_1,l_2 and eigenvectors w_1,w_2 of the resulting intermediate matrix, where the functions

  g_1(p) = 0.001
  g_2(p) = 0.001 + 0.999 exp(-1/p)

were determined empirically and taken from the publication of Weickert.

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

To achieve the highest possible consistency of the newly created edges with the image data from the neighbourhood, the gray values are not mirrored at the border of RegionRegionRegionRegionregion to compute the convolution with the smoothing filter mask of scale RhoRhoRhoRhorho on the pixels close to the border, although this would be the common approach for filter operators. Instead, the existence of gray values on a band of width ceil(3.1*Rho)+2 pixels around RegionRegionRegionRegionregion is presumed and these values are used in the convolution. This means that RegionRegionRegionRegionregion must keep this much distance to the border of the image matrix ImageImageImageImageimage. By involving the gray values and directional information from this extended area, it can be achieved that the continuation of the edges is not only continuous, but also smooth, which means without kinks. Please note that the inpainting progress is restricted to those pixels that are included in the ROI of the input image ImageImageImageImageimage. If the ROI does not include the entire region RegionRegionRegionRegionregion, a band around the intersection of RegionRegionRegionRegionregion and the ROI is used to define the boundary values.

To decrease the number of iterations required for attaining a satisfactory result, it may be useful to initialize the gray value matrix in RegionRegionRegionRegionregion with the harmonic interpolant, a continuous function of minimal curvature, by applying the operator harmonic_interpolationharmonic_interpolationharmonic_interpolationHarmonicInterpolationHarmonicInterpolation to ImageImageImageImageimage before calling inpainting_cedinpainting_cedinpainting_cedInpaintingCedInpaintingCed.

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.

Parameters

Example (HDevelop)

read_image (Image, 'fabrik')
gen_rectangle1 (Rectangle, 270, 180, 320, 230)
harmonic_interpolation (Image, Rectangle, InpaintedImage, 0.01)
inpainting_ced (InpaintedImage, Rectangle, InpaintedImage2, \
                0.5, 3.0, 0.5, 1000)
dev_display(InpaintedImage2)

Alternatives

harmonic_interpolationharmonic_interpolationharmonic_interpolationHarmonicInterpolationHarmonicInterpolation, inpainting_ctinpainting_ctinpainting_ctInpaintingCtInpaintingCt, inpainting_anisoinpainting_anisoinpainting_anisoInpaintingAnisoInpaintingAniso, inpainting_mcfinpainting_mcfinpainting_mcfInpaintingMcfInpaintingMcf, inpainting_textureinpainting_textureinpainting_textureInpaintingTextureInpaintingTexture

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