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 :
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.