anisotropic_diffusion — Perform an anisotropic diffusion of an image.
anisotropic_diffusion(Image : ImageAniso : Mode, Contrast, Theta, Iterations : )
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
with the initial value defined by
Image at a time . 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 .
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:
Choosing the function 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.
The choice of 'perona-malik' for Mode, 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 Mode to
'weickert' is an improvement of with
respect to edge sharpening. The transition between smoothing and
sharpening happens very abruptly at x = c^2.
For an explanation of the concept of smoothing filters see the introduction of chapter Filters / Smoothing.
Note that filter operators may return unexpected results if an image with a reduced domain is used as input. Please refer to the chapter Filters.
Image (input_object) (multichannel-)image(-array) → object (byte / uint2 / real)
Input image.
ImageAniso (output_object) image(-array) → object (byte / uint2 / real)
Output image.
Mode (input_control) string → (string)
Diffusion coefficient as a function of the edge amplitude.
Default value: 'weickert'
List of values: 'parabolic', 'perona-malik', 'weickert'
Contrast (input_control) real → (real)
Contrast parameter.
Default value: 5.0
Suggested values: 2.0, 5.0, 10.0, 20.0, 50.0, 100.0
Restriction: Contrast > 0
Theta (input_control) real → (real)
Time step.
Default value: 1.0
Suggested values: 0.5, 1.0, 3.0
Restriction: Theta > 0
Iterations (input_control) integer → (integer)
Number of iterations.
Default value: 10
Suggested values: 1, 3, 10, 100, 500
Restriction: Iterations >= 1
bilateral_filter,
guided_filter
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.
Foundation