Finite Differences in Forward and Inverse Imaging Problems--MaxPol Design
For researchers in image processing and numerical computing, this provides a systematic way to design derivative kernels with controlled accuracy and stability.
The paper introduces a comprehensive framework for designing finite impulse response derivative kernels using the maximally flat technique, enabling arbitrary parameter control. The resulting derivative matrices outperform common methods in gradient surface reconstruction and image stitching.
A systematic and comprehensive framework for finite impulse response (FIR) lowpass/fullband derivative kernels is introduced in this paper. Closed form solutions of a number of derivative filters are obtained using the maximally flat technique to regulate the Fourier response of undetermined coefficients. The framework includes arbitrary parameter control methods that afford solutions for numerous differential orders, variable polynomial accuracy, centralized/staggered schemes, and arbitrary side-shift nodes for boundary formulation. Using the proposed framework four different derivative matrix operators are introduced and their numerical stability is analyzed by studying their eigenvalues distribution in the complex plane. Their utility is studied by considering two important image processing problems, namely gradient surface reconstruction and image stitching. Experimentation indicates that the new derivative matrices not only outperform commonly used method but provide useful insights to the numerical issues in these two applications.