Finite Differences in Forward and Inverse Imaging Problems--MaxPol Design

TL;DR

This paper introduces a comprehensive framework for designing finite impulse response derivative kernels with adjustable parameters, improving stability and performance in image processing tasks like surface reconstruction and stitching.

Contribution

It presents a systematic method for creating FIR derivative kernels with arbitrary parameters, including boundary handling, and analyzes their stability and effectiveness in practical imaging problems.

Findings

New derivative matrices outperform existing methods in stability and accuracy.

The framework provides flexible solutions for various differential orders and boundary conditions.

Experimental results demonstrate improved performance in gradient surface reconstruction and image stitching.

Abstract

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.