In artificial vision, the image model underlying many image analysis methods considers
images as the sampling of a smooth world with meaningful discontinuities. Then we have a function , where
both and are discrete and bounded spaces. Usually, is a spatial
domain, so it comes with a given connectivity, while the semantics of the space gives us the image modality:
range, luminance or color images from world scenes; physical parameters from medical images such as MR, CT, DSA, SPECT or PET; etc. Besides, since in this model *L* is a static entity, when we want
to study the dynamic behavior of the objects contained in an image we embed it
in an evolving family of manifolds, ,
being the space where live the parameters which controls the evolution. In practice, of course,
this space is also discrete and bounded. Common evolution parameters are time and scale.
In this setting, artificial vision tries to make a signal to symbol transformation by
computing low-level features
which allow high-level tasks to automatically distinguish between interesting
objects appearing in the image. Most relevant features can be extracted from dissimilarity measures (edgeness), similarity
measures (interiorness) or a summarization of both (medialness). Edgeness and interiorness are dual measures mainly related
to segmentation processes while medialness is more like a 'shape encoder' that gives a degree of how much a point resembles the middle of an object.

When we deal with incommensurable
axes, such as those of grey-level images (luminance, MR, CT, etc), the most common edgeness measure is the gradient magnitude (see notation in appendix A).
In the 2-d case, for example, the idea is that an edge (object boundary) in some direction corresponds to a sharp change of *L* in a direction which is perpendicular to it. Observe that
'sharp change of *L*' can be translated as 'ridge of '. Moreover, as the gradient at each point indicates the direction of maximum change of *L*, the ridge at any point holds perpendicularly to it. Therefore, a scheme for extracting the edges of *L* can consist of computing the ridge-like structures of (without thinking in any particular definition by the moment).

A medialness measure, essentially, assigns to each point inside an object its distance to the boundary of that object, according with a given metric. The points at longest distance are in the middle of the object, therefore, at each point medialness is also accounting for the degree of being on a symmetry axis. Blum [5, 6, 7] was the first proposing this idea. He defined the Symmetry Axis (SA), also called Medial Axis (MA) or skeleton, of a binary object as the loci of centers of maximal disks contained in the object, in a way that the boundary of the union of all maximal disks is the boundary of the object. Then, Blum defined the Symmetric Axis Transform (SAT) of an object, also called Medial Axis Transform (MAT), as its SA together with the radii of the maximal disks at each SA point. In this way, the SAT compresses arbitrary shapes of binary objects. On the other hand, artificial vision daily tasks hardly ever deal with binary objects, at least, in early processing (low-level tasks). Instead, features have to be computed from grey-level images where the boundaries of the objects are not *a priori* known.
Therefore, SAT idea has to be translated to the new data type. Two representative works in this line are:

- The Intensity Axis of Symmetry (IAS) [26] for 2-dimensional grey level images. Conceptually, the IAS is made by stacking the medial axis of each slice having as boundary a level curve of the image graph, level by level. Remember that a level curve is a closed curve such that all its points have the same height. The set of level curves of the terrain can be related hierachically by an inclusion relation, being the border of the terrain (like the sea level in an island) the level curve that includes the rest [47]. In this way, the IAS consists of a set of sheets constituting the skeleton of the image landscape.
- The Multiscale Medial Axis (MMA) [63] for
*d*-dimensional grey level images. The authors computed a medialness measure for a*d*-dimensional image*L*in linear scale-space . To compute the medialness response at each point , they add the values of an edgeness measure computed for each point lying on the '*d*-dimensional circle' of radius*r*centered at , being*r*proportional to . The edgeness measure can (there are other alternatives) consists of the first order directional derivative at in the direction from to . Therefore, it is obtained a medialness measure of*L*defined in space. Then, the authors extracted the ridge-like structures of the measure, calling them Multiscale Medial Axis of*L*. A modification to this approach can be found in [71]. In general, ridge-like structures of a medialness measure are called Cores [18, 23, 24, 4]. A related multiresolution approach comes back to Crowley and Parker [13].

On the other hand, ridge/valley-like structures on grey-level images have been proposed [44, 43, 25, 17] as a reliable approximation to the medial axis provided they are extracted directly from the data instead of from a medialness measure of it. In plain words this can be justifyied because ridge/valley-like structures tend to be in the middle of relatively brighter/darker regions at a given scale, as the brain stem or the skull in MR and CT images, vessels in DSA, roof 'edges' in range images and, a number of phenomena in regular luminance images [34, 97]. Other images having a predominant number of ridge/valley-like structures are fingerprint images (in general, images with oriented textures), hand-written documents, patterns generated by structured light, aerial photographs depicting roads, rivers, cut-fires, etc.

The advantage of using ridge/valley-like structures of the image itself is that the computational complexity is much lower than extracting them from a medialness measure of it. On the other hand, merely extracting ridge/valley-like structures does not give size information of the objects which have them as 'center'. In many applications this is not necessary since we just need the center (e.g. as landmark) but, when it is, the size of the objects can be related to the scale where the ridge/valley-structures are more salient [25, 28, 12, 51]. Another approach is to apply at each point on a ridge/valley-like structure, a filter responding optimally to a simultaneous boundary engagement and whose size (scale) can be related to the size of the object. An example is the Laplacean of the Gaussian (LoG).

As we have seen ridge/valley-like structures are powerful descriptors which can be involved in important tasks: detection of edges when applied to an edgeness measure, detection of medial axes when applied to a medialness measure, and detection of medial structures by themselves. On the other hand, in this section I have spoken about the use
of ridge/valley-like structures in artificial vision without giving their specific mathematical definition. In next section I will review several definitions in
use and their main properties. I have classified them as *creases, separatrices* and *drainage patterns*:

- creases are
*local*descriptors that, basically, looks for*anisotropic shapes*. - separatrices are
*global*descriptors that result from linking critical points of the image by means of*special slopelines*. - drainage patterns are the result of
*simulating*the surface runoff. Taking into account that perturbations do not affect runoff of far away regions, drainage patterns algorithms can be considerated as having a*semi-local*or*regional*character.

Wed Oct 8 17:04:50 MET DST 1997