On a Relation Between the Integral Image Algorithm and Calculus

TL;DR

This paper explores the theoretical foundations of the integral image algorithm, introducing a novel discrete derivative and extending the method to continuous domains for more general applications.

Contribution

It proposes a new discrete derivative operator and develops a continuous domain extension of the integral image algorithm, bridging discrete and continuous analysis.

Findings

Introduces a novel discrete derivative for the integral image algorithm

Extends the algorithm to general continuous domains

Provides a theoretical framework connecting discrete and continuous integration

Abstract

The Integral Image algorithm is often applied in tasks that require efficient integration over images, such as object detection. In this paper we discuss theoretical aspects of the algorithm's continuous version. We suggest to define the coefficients at the formulation of the algorithm by applying a novel kind of discrete derivative. Based on that operator we build a novel integration method over curves in the plane, and apply it in a theorem that extends the algorithm to general continuous domains.