Operators |
inpainting_ct — Perform an inpainting by coherence transport.
inpainting_ct(Image, Region : InpaintedImage : Epsilon, Kappa, Sigma, Rho, ChannelCoefficients : )
The operator inpainting_ct inpaints a missing region Region of an image Image 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_interpolation may yield better results. Alternatively, Kappa 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_texture, 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 Epsilon around the current pixel. The disc is restricted to already known pixels.
The size of this scheme's mask depends on Epsilon.
The initially used image data comes from a stripe of thickness Epsilon around the region to inpaint. Thus, Epsilon must be at least 1 for the scheme to work, but should be greater. The maximum value for Epsilon depends on the gray values that should be transported into the region. Choosing Epsilon = 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 Sigma and Rho. Sigma should have the size of the noise or uninportant little objects, which are then not considered in the estimation step by the pre-smoothing. Rho 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 ChannelCoefficients, 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 ChannelCoefficients 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 ChannelCoefficients 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 ChannelCoefficients 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 Kappa (in percent):
beta = 20 * Epsilon * Kappa / 100
Kappa defines how important it is to propagate information along the coherence direction, so a large Kappa yields sharp edges, while a low Kappa allows for more diffusion.
A special case is when Kappa 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 Sigma, Rho, ChannelCoefficients 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.
Input image.
Inpainting region.
Output image.
Radius of the pixel neighborhood.
Default value: 5.0
Typical range of values: 1.0 ≤ Epsilon ≤ 20.0
Minimum increment: 1.0
Recommended increment: 1.0
Sharpness parameter in percent.
Default value: 25.0
Typical range of values: 0.0 ≤ Kappa ≤ 100.0
Minimum increment: 1.0
Recommended increment: 1.0
Pre-smoothing parameter.
Default value: 1.41
Typical range of values: 0.0 ≤ Sigma ≤ 20.0
Minimum increment: 0.001
Recommended increment: 0.01
Smoothing parameter for the direction estimation.
Default value: 4.0
Typical range of values: 0.001 ≤ Rho ≤ 20.0
Minimum increment: 0.001
Recommended increment: 0.01
read_image (Image, 'claudia') gen_circle (Circle, 333, 164, 35) inpainting_ct (Image, Circle, InpaintedImage, 15, 25, 1.5, 3,1.0)
harmonic_interpolation, inpainting_aniso, inpainting_mcf, inpainting_ced, inpainting_texture
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.
Foundation
Operators |