The operator inpainting_anisoinpainting_anisoInpaintingAnisoinpainting_anisoInpaintingAnisoInpaintingAniso uses the anisotropic diffusion
according to the model of Perona and Malik, to continue image edges
that cross the border of the region RegionRegionRegionRegionRegionregion and to connect
them inside of RegionRegionRegionRegionRegionregion.
With this, the structure of the edges in RegionRegionRegionRegionRegionregion 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
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 ContrastContrastContrastContrastContrastcontrast on the magnitude of
the gray value differences between adjacent
pixels. ContrastContrastContrastContrastContrastcontrast 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 RegionRegionRegionRegionRegionregion. The
standard deviation of this smoothing process is determined by the
parameter RhoRhoRhoRhoRhorho.
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 ModeModeModeModeModemode, the
following functions can be selected:
Choosing the function by setting ModeModeModeModeModemode to
'parabolic'"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
ThetaThetaThetaThetaThetatheta. 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""perona-malik" for ModeModeModeModeModemode, 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 ModeModeModeModeModemode to
'weickert'"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""shock" is possible
for ModeModeModeModeModemode 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
List of values: 'parabolic'"parabolic""parabolic""parabolic""parabolic""parabolic", 'perona-malik'"perona-malik""perona-malik""perona-malik""perona-malik""perona-malik", 'shock'"shock""shock""shock""shock""shock", 'weickert'"weickert""weickert""weickert""weickert""weickert"
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.