inpainting_anisoinpainting_anisoInpaintingAnisoInpaintingAnisoinpainting_aniso (Operator)

Name

inpainting_anisoinpainting_anisoInpaintingAnisoInpaintingAnisoinpainting_aniso — Perform an inpainting by anisotropic diffusion.

Signature

Description

The operator inpainting_anisoinpainting_anisoInpaintingAnisoInpaintingAnisoinpainting_aniso uses the anisotropic diffusion according to the model of Perona and Malik, to continue image edges that cross the border of the region RegionRegionRegionregionregion and to connect them inside of RegionRegionRegionregionregion.

With this, the structure of the edges in RegionRegionRegionregionregion will be made consistent with the surrounding image matrix, so that an occlusion of errors or unwanted objects in the input image, a so called inpainting, is less visible to the human beholder, since there remain no obvious artifacts or smudges.

Considering the image as a gray value function u, the algorithm is a discretization of the partial differential equation with the initial value defined by ImageImageImageimageimage at a time . The equation is iterated IterationsIterationsIterationsiterationsiterations times in time steps of length ThetaThetaThetathetatheta, so that the output image InpaintedImageInpaintedImageInpaintedImageinpaintedImageinpainted_image contains the gray value function at the time .

The primary goal of the anisotropic diffusion, which is also referred to as nonlinear isotropic diffusion, is the elimination of image noise in constant image patches while preserving the edges in the image. The distinction between edges and constant patches is achieved using the threshold ContrastContrastContrastcontrastcontrast on the magnitude of the gray value differences between adjacent pixels. ContrastContrastContrastcontrastcontrast is referred to as the contrast parameter and is abbreviated with the letter c. If the edge information is distributed in an environment of the already existing edges by smoothing the edge amplitude matrix, it is furthermore possible to continue edges into the computation area RegionRegionRegionregionregion. The standard deviation of this smoothing process is determined by the parameter RhoRhoRhorhorho.

The algorithm used is basically the same as in the anisotropic diffusion filter anisotropic_diffusionanisotropic_diffusionAnisotropicDiffusionAnisotropicDiffusionanisotropic_diffusion, except that here, border treatment is not done by mirroring the gray values at the border of RegionRegionRegionregionregion. Instead, this procedure is only applicable on regions that keep a distance of at least 3 pixels to the border of the image matrix of ImageImageImageimageimage, since the gray values on this band around RegionRegionRegionregionregion are used to define the boundary conditions for the respective differential equation and thus assure consistency with the neighborhood of RegionRegionRegionregionregion. 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.

The result of the diffusion process depends on the gray values in the computation area of the input image ImageImageImageimageimage. It must be pointed out that already existing image edges are preserved within RegionRegionRegionregionregion. In particular, this holds for gray value jumps at the border of RegionRegionRegionregionregion, which can result for example from a previous inpainting with constant gray value. If the procedure is to be used for inpainting, it is recommended to apply the operator harmonic_interpolationharmonic_interpolationHarmonicInterpolationHarmonicInterpolationharmonic_interpolation first to remove all unwanted edges inside the computation area and to minimize the gray value difference between adjacent pixels, unless the input image already contains information inside RegionRegionRegionregionregion that should be preserved.

The variable diffusion coefficient g can be chosen to follow different monotonically decreasing functions with values between 0 and 1 and determines the response of the diffusion process to an edge. With the parameter ModeModeModemodemode, the following functions can be selected: Choosing the function by setting ModeModeModemodemode to 'parabolic'"parabolic""parabolic""parabolic""parabolic" guarantees that the associated differential equation is parabolic, so that a well-posedness theory exists for the problem and the procedure is stable for an arbitrary step size ThetaThetaThetathetatheta. In this case however, there remains a slight diffusion even across edges of an amplitude larger than c. The choice of 'perona-malik'"perona-malik""perona-malik""perona-malik""perona-malik" for ModeModeModemodemode, as used in the publication of Perona and Malik, does not possess the theoretical properties of , but in practice it has proved to be sufficiently stable and is thus widely used. The theoretical instability results in a slight sharpening of strong edges. The function with the constant C=3.31488, proposed by Weickert, and selectable by setting ModeModeModemodemode to 'weickert'"weickert""weickert""weickert""weickert" is an improvement of with respect to edge sharpening. The transition between smoothing and sharpening happens very abruptly at x = c^2.

Furthermore, the choice of the value 'shock'"shock""shock""shock""shock" is possible for ModeModeModemodemode to select a contrast invariant modification of the anisotropic diffusion. In this variant, the generation of edges is not achieved by variation of the diffusion coefficient g, but the constant coefficient g=1 and thus isotropic diffusion is used. Additionally, a shock filter of type is applied, which, just like a negative diffusion coefficient, causes a sharpening of the edges, but works independent of the absolute value of . In this mode, ContrastContrastContrastcontrastcontrast does not have the meaning of a contrast parameter, but specifies the ratio between the diffusion and the shock filter part applied at each iteration step. Hence, the value 0 would correspond to pure isotropic diffusion, as used in the operator isotropic_diffusionisotropic_diffusionIsotropicDiffusionIsotropicDiffusionisotropic_diffusion. The parameter is scaled in such a way that diffusion and sharpening cancel each other out for ContrastContrastContrastcontrastcontrast=1. A value ContrastContrastContrastcontrastcontrast>1 should not be used, since it would make the algorithm unstable.

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

Example (HDevelop)

read_image (Image, 'fabrik')
gen_rectangle1 (Rectangle, 270, 180, 320, 230)
harmonic_interpolation (Image, Rectangle, InpaintedImage, 0.01)
inpainting_aniso (InpaintedImage, Rectangle, InpaintedImage2, \
                  'perona-malik', 5.0, 100, 50, 0.5)
dev_display(InpaintedImage2)

Alternatives

harmonic_interpolationharmonic_interpolationHarmonicInterpolationHarmonicInterpolationharmonic_interpolation, inpainting_ctinpainting_ctInpaintingCtInpaintingCtinpainting_ct, inpainting_mcfinpainting_mcfInpaintingMcfInpaintingMcfinpainting_mcf, inpainting_textureinpainting_textureInpaintingTextureInpaintingTextureinpainting_texture, inpainting_cedinpainting_cedInpaintingCedInpaintingCedinpainting_ced

References

J. Weickert; “Anisotropic Diffusion in Image Processing”; PhD Thesis; Fachbereich Mathematik, Universität Kaiserslautern; 1996.
P. Perona, J. Malik; “Scale-space and edge detection using anisotropic diffusion”; Transactions on Pattern Analysis and Machine Intelligence 12(7), pp. 629-639; IEEE; 1990.
G. Aubert, P. Kornprobst; “Mathematical Problems in Image Processing”; Applied Mathematical Sciences 147; Springer, New York; 2002.

Module

Foundation