inpainting_aniso — Perform an inpainting by anisotropic diffusion.
The operator inpainting_aniso uses the anisotropic diffusion according to the model of Perona and Malik, to continue image edges that cross the border of the region Region and to connect them inside of Region.
With this, the structure of the edges in Region 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 artefacts or smudges.
Considering the image as a gray value function u, the algorithm is a discretization of the partial differential equation
u_t = div(g(|grad u|^2, c) grad u)
with the initial value u = u_0 defined by Image at a time t_0 = 0. The equation is iterated Iterations times in time steps of length Theta, so that the output image InpaintedImage contains the gray value function at the time Iterations * Theta .
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 Contrast on the magnitude of the gray value differences between adjacent pixels. Contrast 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 Region. The standard deviation of this smoothing process is determined by the parameter Rho.
The algorithm used is basically the same as in the anisotropic diffusion filter anisotropic_diffusion, except that here, border treatment is not done by mirroring the gray values at the border of Region. 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 Image, since the gray values on this band around Region are used to define the boundary conditions for the respective differential equation and thus assure consistency with the neighborhood of Region. Please note that the inpainting progress is restricted to those pixels that are included in the ROI of the input image Image. If the ROI does not include the entire region Region, a band around the intersection of Region 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 Image. It must be pointed out that already exisiting image edges are preserved within Region. In particular, this holds for gray value jumps at the border of Region, 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_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 Region 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 Mode, the following functions can be selected:
g_1(x,c) = 1/sqrt( 1 + 2*x/c^2 )
Choosing the function g_1 by setting Mode to '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 Theta. In this case however, there remains a slight diffusion even across edges of an amplitude larger than c.
g_2(x,c) = 1/( 1 + (x/c^2) )
The choice of 'perona-malik' for Mode, as used in the publication of Perona and Malik, does not possess the theoretical properties of g_1, 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.
g_3(x,c) = 1-exp(-C*c^8/x^4)
The function g_3 with the constant C=3.31488, proposed by Weickert, and selectable by setting Mode to 'weickert' is an improvement of g_2 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' is possible for Mode 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
u_t = -sgn(grad |grad u|) |grad u|
is applied, which, just like a negative diffusion coefficient, causes a sharpening of the edges, but works independent of the absolute value of |grad u|. In this mode, Contrast 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_diffusion. The parameter is scaled in such a way that diffusion and sharpening cancel each other out for Contrast=1. A value Contrast>1 should not be used, since it would make the algorithm unstable.
Type of edge sharpening algorithm.
Default value: 'weickert'
List of values: 'parabolic', 'perona-malik', 'shock', 'weickert'
Default value: 5.0
Suggested values: 0.5, 2.0, 5.0, 10.0, 20.0, 50.0, 100.0
Restriction: Contrast > 0
Default value: 0.5
Suggested values: 0.5, 1.0, 5.0, 10.0, 30.0, 100.0
Restriction: Theta > 0
Number of iterations.
Default value: 10
Suggested values: 1, 3, 10, 100, 500
Restriction: Iterations >= 1
Smoothing coefficient for edge information.
Default value: 3.0
Suggested values: 0.0, 0.1, 0.5, 1.0, 3.0, 10.0
Restriction: Rho >= 0
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)
harmonic_interpolation, inpainting_ct, inpainting_mcf, inpainting_texture, inpainting_ced
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.