Mathematical derivation for Vora-Value based filter design method: Gradient and Hessian

TL;DR

This paper provides a detailed mathematical derivation of the gradient and Hessian matrices for Vora-Value based filter optimization, clarifying the differentiation process and the positive-definiteness of the Hessian.

Contribution

It offers a comprehensive derivation of the gradient and Hessian for Vora-Value filter optimization, enhancing understanding of the mathematical foundations.

Findings

Hessian matrix is positive-definite with regularization

Detailed derivation steps are provided

Supports the theoretical framework for filter design

Abstract

In this paper, we present the detailed mathematical derivation of the gradient and Hessian matrix for the Vora-Value based colorimetric filter optimization. We make a full recapitulation of the steps involved in differentiating the objective function and reveal the positive-definite Hessian matrix when a positive regularizer is applied. This paper serves as a supplementary material for our paper in the colorimetric filter design theory.