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The Geometry of Differential Operators with Application to Image Processing
December 1989 (vol. 11 no. 12)
pp. 1252-1264

A family of second-order nonlinear differential operators that are useful in building simple edge detectors is treated. The authors provide a uniform definition for these operators and describe their internal geometry. Based on their geometric analysis, they associate symbolic descriptors with the abstract edges defined by the operators. One of them, Q, is essentially the same operator that J.F. Canny used to define his two-dimensional simple edge detector. The authors give a parse tree for Q, which provides a complete analysis of its local geometry. The analysis shows that Q has a fondness for edges that lift to certain types of asymptotic curves. These include select straight line segments lying on the surface, inflections of mean curvature, and paths with constant rate of ascent. The methods are drawn largely from vector calculus and differential geometry. The authors illustrate their theoretical work with results obtained when Q was applied to image data that were interpolated with a B-spline.

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Index Terms:
picture processing; pattern recognition; interpolation; constant ascent rate; geometry; image processing; second-order nonlinear differential operators; edge detectors; symbolic descriptors; parse tree; asymptotic curves; straight line segments; inflections of mean curvature; vector calculus; differential geometry; B-spline; grammars; pattern recognition; picture processing
W.M. Kruegger, K. Phillips, "The Geometry of Differential Operators with Application to Image Processing," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 11, no. 12, pp. 1252-1264, Dec. 1989, doi:10.1109/34.41364
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