Consistent gradient operators

Fine, accurate gradient information is required in many image-processing algorithms and systems including differential geometric methods, orientation analysis, and integrated vision sensors. In this paper, we propose optimal gradient operators based on a newly derived consistency criterion. This criterion is based on an orthogonal decomposition of the difference between a continuous gradient and discrete gradients into the intrinsic smoothing effect and the self-inconsistency involved in the operator. We show that consistency assures the exactness of gradient direction of a locally one-dimensional (1D) pattern in spite of its orientation, spectral composition, and subpixel translation. Stressing that inconsistency reduction is of primary importance, we derive an iterative algorithm which leads to accurate gradient operators of arbitrary size. We compute the optimum 3 x 3, 3 x 4, and 5 x 5 operators, compare them with conventional operators and examine the performance for one synthetic and several real images. The results indicate that the proposed operators are superior with respect to accuracy, bandwidth, and isotropy.