anisotropic_diffusion — Perform an anisotropic diffusion of an image.
The operator anisotropic_diffusion performs an anisotropic diffusion on the input image Image according to the model of Perona and Malik. This procedure is also referred to as nonlinear isotropic diffusion. 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. The equation is iterated Iterations times in time steps of length Theta, so that the output image ImageAniso contains the gray value function at the time t_0 + Iterations * Theta .
The goal of the anisotropic 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 size of the gray value differences between adjacent pixels. Contrast is referred to as the contrast parameter and abbreviated with the letter c.
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 a height 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.
Diffusion coefficient as a function of the edge amplitude.
Default value: 'weickert'
List of values: 'parabolic', 'perona-malik', 'weickert'
Default value: 5.0
Suggested values: 2.0, 5.0, 10.0, 20.0, 50.0, 100.0
Restriction: Contrast > 0
Default value: 1.0
Suggested values: 0.5, 1.0, 3.0
Restriction: Theta > 0
Number of iterations.
Default value: 10
Suggested values: 1, 3, 10, 100, 500
Restriction: Iterations >= 1
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.