Differentiation of Discrete Multi-Dimensional Signals

H. Farid and E.P. Simoncelli

IEEE Transactions on Image Processing, 13(4):496-508, 2004

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We describe the design of finite-size linear-phase separable kernels for differentiation of discrete multi-dimensional signals. The problem is formulated as an optimization of the rotation-invariance of the gradient operator, which results in a simultaneous constraint on a set of one-dimensional lowpass prefilter and differentiator filters up to the desired order. We also develop extensions of this formulation to both higher dimensions and higher-order directional derivatives. We develop a numerical procedure for optimizing the constraint, and demonstrate its use in constructing a set of example filters. The resulting filters are significantly more accurate than those commonly used in the image and multi-dimensional signal processing literature.