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## Fitting Functions to the Image Brightness Surface

The approach of surface fitting defines a two dimensional function which should be ``flexible'' enough to provide a good approximation to the image brightness function. The best fit of the function to the image is then found at each image position, and the parameters of the fit are used either to estimate image derivatives directly or to find edges using alternative definitions. The problem with the mathematical model approach is that deviations from the surface model cannot (by definition) be accommodated by varying the model's parameters.

In  Hueckel used circular windows within which he fitted a single step edge. Clearly this approach does not perform well at junctions, corners or non-step edges.

The approach of Haralick which was described in the previous section also falls into this category, using a polynomial fit to find zero crossings in the second directional derivative.

In  Nalwa and Binford use the hyberbolic tangent as the function to fit to what are assumed to be short straight sections of edge. Thus no edge curvature is allowed if an ideal fit is to be achieved. It is proposed that other types of edge (than the step edge) can be detected using combinations of the tanh function. The computation necessary in order to achieve the tanh fit is large; first a planar fit and a cubic fit are made, so that the local patch may be classified into ``non-edge'' or ``potential edge'', and so that the edge direction may be found. Next the tanh fit is made, as is a quadratic fit, to allow a further test of the presence of an edge to be made. Finally, the strength and direction of the edge are found from the parameters of the tanh fit, and the strength is thresholded to remove weak edges. As might be expected, edges of high curvature are broken up. A similar fitting method is used in  by Sinha and Schunk. They perform two stage fitting; a planar fit is followed by a bicubic spline fit.   Next: Using Mathematical Morphology Up: Edges Previous: The Second Directional Derivative of Image Brightness

© 1997 Stephen M Smith. LaTeX2HTML conversion by Steve Smith (steve@fmrib.ox.ac.uk)