Perona and Malik formulate the anisotropic diffusion filter as a diffusion process that encourages intraregion smoothing while inhibiting interregion smoothing. Mathematically, the process is defined as follows:
In our case, 
is the MR image. 
refers to the image axes (i.e. 
) and 
refers to the
iteration step. 
is called the diffusion
function and is a monotonically decreasing function of the image
gradient magnitude:
It allows for locally adaptive diffusion strengths; edges are
selectively smoothed or enhanced based on the evaluation of the
diffusion function. Although any monotonically decreasing continuous
function of 
would suffice as a diffusion function, two
functions have been suggested [39]:
These functions are plotted in Figure 4.1.
Figure 4.1: Diffusion functions plotted as a
function of image gradient.
is referred to as the diffusion constant or the flow
constant. Obviously, the behavior of the filter depends on . To clarify the effect of , and the diffusion function, on the
diffusion process, it is helpful to define a flow function:
Equation 4.1 can then be rewritten as:
This formulation will also be useful for developing a discrete implementation of the diffusion filter (as will be seen in the following section).
The flow functions, 
and 
, corresponding to the
diffusion functions, 
and 
, are plotted in
Figure 4.2. Notice that flow increases with the
gradient strength to the point where 
, then
decreases to zero. This behavior implies that the diffusion process
maintains homogeneous regions (where 
) since
little flow is generated. Similarly, edges are preserved because the
flow is small in regions where 
.
The greatest flow is produced when the image gradient magnitude is
close to the value of . Therefore, by choosing
to correspond to gradient magnitudes produced by noise, the diffusion
process can be used to reduce noise in images. Assuming an image
contains no discontinuities, object edges can be enhanced by choosing
a value of slightly less than the gradient magnitude of the
edges. These features of nonlinear anisotropic diffusion are
illustrated in Section 4.4.
Figure 4.2: Flow functions plotted as a function of image
gradient.
