inpainting_ctinpainting_ctinpainting_ctInpaintingCtInpaintingCt (Operator)

Name

inpainting_ctinpainting_ctinpainting_ctInpaintingCtInpaintingCt — Perform an inpainting by coherence transport.

Signature

Description

The operator inpainting_ctinpainting_ctinpainting_ctInpaintingCtInpaintingCt inpaints a missing region RegionRegionRegionRegionregion of an image ImageImageImageImageimage by transporting image information from the region's boundary along the coherence direction into this region.

Since this operator's basic concept is inpainting by continuing broken contour lines, the image content and inpainting region must be such that this idea makes sense. That is, if a contour line hits the region to inpaint at a pixel p, there should be some opposite point q where this contour line continues so that the continuation of contour lines from two opposite sides can succeed. In cases where there is less geometry in the image, a diffusion-based inpainter, e.g., harmonic_interpolationharmonic_interpolationharmonic_interpolationHarmonicInterpolationHarmonicInterpolation may yield better results. Alternatively, KappaKappaKappaKappakappa can be set to 0. An extreme situation with little global geometries are pure textures. Then the idea behind this operator will fail to produce good results (think of a checkerboard with a big region to inpaint relative to the checker fields). For these kinds of images, a texture-based inpaiting, e.g., inpainting_textureinpainting_textureinpainting_textureInpaintingTextureInpaintingTexture, can be used instead.

The operator uses a so-called upwind scheme to assign gray values to the missing pixels, i.e.,:

• The order of the pixels to process is given by their Euclidean distance to the boundary of the region to inpaint.

• A new value u_i is computed as a weighted average of already known values u_i within a disc of radius EpsilonEpsilonEpsilonEpsilonepsilon around the current pixel. The disc is restricted to already known pixels.

• The size of this scheme's mask depends on EpsilonEpsilonEpsilonEpsilonepsilon.

The initially used image data comes from a stripe of thickness EpsilonEpsilonEpsilonEpsilonepsilon around the region to inpaint. Thus, EpsilonEpsilonEpsilonEpsilonepsilon must be at least 1 for the scheme to work, but should be greater. The maximum value for EpsilonEpsilonEpsilonEpsilonepsilon depends on the gray values that should be transported into the region. Choosing EpsilonEpsilonEpsilonEpsilonepsilon = 5 can be used in many cases.

Since the goal is to close broken contour lines, the direction of the level lines must be estimated and used in the weight. This estimated direction is called the coherence direction, and is computed by means of the structure tensor S.

                T
S = G    * Dv Dv
     rho

and

v = G      * u
     sigma

where * denotes the convolution, u denotes the gray value image, D the derivative and G Gaussian kernels with standard deviation sigma and rho. These standard deviations are defined by the operator's parameters SigmaSigmaSigmaSigmasigma and RhoRhoRhoRhorho. SigmaSigmaSigmaSigmasigma should have the size of the noise or uninportant little objects, which are then not considered in the estimation step by the pre-smoothing. RhoRhoRhoRhorho gives the size of the window around a pixel that will be used for direction estimation. The coherence direction c then is given by the eigendirection of S with respect to the minimal eigenvalue lambda, i.e.

S c = lambda  c , |c|=1

For multichannel or color images, the scheme above is applied to each channel separately, but the weights must be the same for all channels to propagate information in the same direction. Since the weight depends on the coherence direction, the common direction is given by the eigendirection of a composite structure tensor. If u_{1},...,u_{n} denote the n channels of the image, the channel structure tensors S_{1},...,S_{n} are computed and then combined to the composite structure tensor S.

     n
    ----
    \
S = /    a  S
    ----  i  i
    i=1

The coefficients a_{i} are passed in ChannelCoefficientsChannelCoefficientsChannelCoefficientsChannelCoefficientschannelCoefficients, which is a tuple of length n or length 1. If the tuple's length is 1, the arithmetic mean is used, i.e., a_{i} = 1/n. If the length of ChannelCoefficientsChannelCoefficientsChannelCoefficientsChannelCoefficientschannelCoefficients matches the number of channels, the a_{i} are set to

         ChannelCoefficients[i]
a  = -----------------------------
 i     n
      ----
      \
      /    ChannelCoefficients[i]
      ----
      i=1

in order to get a well-defined convex combination. Hence, the ChannelCoefficientsChannelCoefficientsChannelCoefficientsChannelCoefficientschannelCoefficients must be greater than or equal to zero and their sum must be greater than zero. If the tuple's length is neither 1 nor the number of channels or the requirement above is not satisfied, the operator returns an error message.

The purpose of using other ChannelCoefficientsChannelCoefficientsChannelCoefficientsChannelCoefficientschannelCoefficients than the arithmetic mean is to adapt to different color codes. The coherence direction is a geometrical information of the composite image, which is given by high contrasts such as edges. Thus the more contrast a channel has, the more geometrical information it contains, and consequently the greater its coefficient should be chosen (relative to the others). For RGB images, [0.299, 0.587, 0.114] is a good choice.

The weight in the scheme is the product of a directional component and a distance component. If p is the 2D coordinate vector of the current pixel to be inpainted and q the 2D coordinate of a pixel in the neighborhood (the disc restricted to already known pixels), the directional component measures the deviation of the vector p-q from the coherence direction. If the deviation exponentially scaled by beta is large, a low directional component is assigned, whereas if it is small, a large directional component is assigned. beta is controlled by KappaKappaKappaKappakappa (in percent):

beta = 20 * Epsilon * Kappa / 100

KappaKappaKappaKappakappa defines how important it is to propagate information along the coherence direction, so a large KappaKappaKappaKappakappa yields sharp edges, while a low KappaKappaKappaKappakappa allows for more diffusion.

A special case is when KappaKappaKappaKappakappa is zero: In this case the directional component of the weight is constant (one). The direction estimation step is then skipped to save computational costs and the parameters SigmaSigmaSigmaSigmasigma, RhoRhoRhoRhorho, ChannelCoefficientsChannelCoefficientsChannelCoefficientsChannelCoefficientschannelCoefficients become meaningless, i.e, the propagation of information is not based on the structures visible in the image.

The distance component is 1/|p-q|. Consequently, if q is far away from p, a low distance component is assigned, whereas if it is near to p, a high distance component is assigned.

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, 'claudia')
gen_circle (Circle, 333, 164, 35)
inpainting_ct (Image, Circle, InpaintedImage, 15, 25, 1.5, 3,1.0)

Alternatives

harmonic_interpolationharmonic_interpolationharmonic_interpolationHarmonicInterpolationHarmonicInterpolation, inpainting_anisoinpainting_anisoinpainting_anisoInpaintingAnisoInpaintingAniso, inpainting_mcfinpainting_mcfinpainting_mcfInpaintingMcfInpaintingMcf, inpainting_cedinpainting_cedinpainting_cedInpaintingCedInpaintingCed, inpainting_textureinpainting_textureinpainting_textureInpaintingTextureInpaintingTexture

References

Folkmar Bornemann, Tom März: “Fast Image Inpainting Based On Coherence Transport”; Journal of Mathematical Imaging and Vision; vol. 28, no. 3; pp. 259-278; 2007.

Module

Foundation