Perform an inpainting by smoothing of level lines.
The operator inpainting_mcf extends the image edges that adjoin the region Region of the input image Image into the interior of Region and connects their ends by smoothing the level lines of the gray value function of Image.
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 Region by the input image Image at a time t_0 = 0. The discretized equation is solved in Iterations time steps of length Theta, so that the output image InpaintedImage contains the gray value function at the time Iterations * Theta .
A stationary state of the mean curvature flow equation, which is also the basis of the operator mean_curvature_flow, 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 InpaintedImage. By this, the structure of objects inside of Region 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 Sigma determines the magnitude of the smoothing by means of the standard deviation of a corresponding Gaussian convolution kernel, as used in the operator isotropic_diffusion for isotropic image smoothing.
|
Image (input_object) |
image(-array) -> object : byte / uint2 / real |
| Input image. | |
|
Region (input_object) |
region -> object |
| Inpainting region. | |
|
InpaintedImage (output_object) |
image(-array) -> object |
| Output image. | |
|
Sigma (input_control) |
real -> real |
| Smoothing for derivative operator. | |
| Default value: 0.5 | |
| Suggested values: 0.0, 0.1, 0.5, 1.0 | |
| Restriction: Sigma >= 0 | |
|
Theta (input_control) |
real -> real |
| Time step. | |
| Default value: 0.5 | |
| Suggested values: 0.1, 0.2, 0.3, 0.4, 0.5 | |
| Restriction: (0 < Theta) <= 0.5 | |
|
Iterations (input_control) |
integer -> integer |
| Number of iterations. | |
| Default value: 10 | |
| Suggested values: 1, 5, 10, 20, 50, 100, 500 | |
| Restriction: Iterations >= 1 | |
inpainting_mcf is reentrant and automatically parallelized (on tuple level).
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.
Foundation
