Started 5/27/2004 by Bob Futrelle.

27 May 2004: There are a limited number of parameters for each edgelet. This affects the number of parameters that can be used to compute a distance, D, between a pair of edgelets for use in clustering. These parameters will also be used to finally construct an overall edge estimate from a small set of edgelets. Here are the parameters for a single edgelet:

• x,y location of the pixel located at the center of the edge operator used, e.g., Prewitt.
• e_x, e_y, the values of the two components of the edgelets output by the edge operator.
• The pixel gray level (probably not used for edge work).

For a pair of edgelets, the parameters that are relevant for computing the distance D involve various combinations of the parameters. There are only six of these that appear relevant: The inner product of the two edgelets, their magnitudes and their relative distance and alignment (not the same as their inner product). I had forgotten about the alignment issue in my discussion with Mingyan on 5/26. It's a tricky concept and we may be able to ignore it, and maybe should ignore it. Alignment has to do with how close an edgelet location is to a line projected along another edgelets direction. Below, I ignore the alignment issue. It might cause problems by creating adjacent parallel edges from a collection of edgelets -- not good.

Here, then, are the relevant parameters for the computation of D, the distance for clustering between two edgelets, along with how D should depend on them.

• The distance d between the two edgelets, between their x,y pixel locations. D(d) is an increasing function of d.
• The magnitude E of each edgelet. D(E,E') should be a decreasing function of both arguments.
• The cosine, c, of the angle between e, e'. D(c) should an increasing function of c.

The details of D need to worked out. We had some useful ideas about this (Jnl. 104, pgs. 66-67, 5/6/04).

A few thoughts on all this: If we ignore relative alignment, edgelets that are transversely aligned will be clustered together, which is a good way to avoid finding adjacent parallel edges. We want to find a single edge. See the figure below. Once we cluster a modest number of edgelets, we'll estimate and edge from them and fit a line model to the edge. Would be nice to find the opposite edge of the line and use the pair. That's a bit tricky. So for now, we could just try a very narrow line model initially and adjust it until a good fit is obtained.