If the intensity values of an image are reinterpreted as a distribution of molecular concentration in a liquid or as a distribution of temperature in a solid, then techniques from the field of continuum mechanics can provide certain insights into the shape properties of objects in an image. For example, consider the two-dimensional, isotropic diffusion equation:
The temporal parameter, t, in
equation 
converts the problem into a
dynamic system. The solution of the two-dimensional diffusion
equation is
but equation can also be viewed as
the convolution of a Gaussian function with mean zero and variance
,
Viewed as a convolution process, the time varying problem becomes converted into a purely static problem,
Other forms of the diffusion equation are also possible. For example, if the diffusion constant differs in the x and y directions, the material can be described by the anisotropic diffusion equation,
If the diffusion constant varies depending on location, the material can be described by the amorphous diffusion equation
The significance of the MMA is that it allows for a hierarchical
approach to image understanding through the use of scale parameter
. Using this hierarchical approach, the first step is to
compute medial axes at largest scale of interest. Once the gross
features of an object have been recognized, the smaller features can
be found by reducing the scale parameter depending on the information
obtained at the larger scale. This process is repeated until all the
structures and features of interest have been recognized (see
Figure ). The hierarchical MMA approach is capable of
recognizing large-scale features without sacrificing information
contained at the finer resolutions. There are, however, several
drawbacks to the MMA approach: (1) skeletal shape cannot recover the
original size of the objects, (2) edge and boundary information is
lost, therefore the original shape is not uniquely represented, and
(3) texture and pattern information is lost. Despite these drawbacks,
the MMA can provide a structural road map of objects in an image.
Although the MMA can't capture all the important features of an
object, it can be used to provide some high-level information that
provides the context for low-level image operators.
Figure: The multi-scale medial axis approach performs feature
recognition by starting a the largest image scale of interest and
then repeatedly reduces the scale until finer objects can be
recognized.
