Digital Step Edges from Zero Crossing of Second Directional Derivatives

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Abstract
We use the facet model to accomplish step edge detection. The essence of the facet model is that any analysis made on the basis of the pixel values in some neighborhood has its final authoritative interpretation relative to the underlying gray tone intensity surface of which the neighborhood pixel values are observed noisy samples. With regard to edge detection, we define an edge to occur in a pixel if and only if there is some point in the pixel's area having a negatively sloped zero crossing of the second directional derivative taken in the direction of a nonzero gradient at the pixel's center. Thus, to determine whether or not a pixel should be marked as a step edge pixel, its underlying gray tone intensity surface must be estimated on the basis of the pixels in its neighborhood. For this, we use a functional form consisting of a linear combination of the tensor products of discrete orthogonal polynomials of up to degree three. The appropriate directional derivatives are easily computed from this kind of a function. Upon comparing the performance of this zero crossing of second directional derivative operator with the Prewitt gradient operator and the Marr-Hildreth zero crossing of the Laplacian operator, we find that it is the best performer; next is the Prewitt gradient operator. The Marr-Hildreth zero crossing of the Laplacian operator performs the worst.

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