On a Relation Between the Integral Image Algorithm and Calculus

This paper is purely theoretical. The main results are: (1) a more rigorous formulation to a parameter at a continuous version of the Integral Image algorithm, and (2) an extension of the algorithm to more general types of domains. The Integral Image algorithm has become an important tool in the image processing community, ever since it was first introduced in the context of objects detection by Viola and Jones in 2001. In this paper, theoretical aspects of this algorithm’s continuous extension from 2007 are being discussed. The paper suggests that a decisive parameter at the formulation of the algorithm should be coherently defined via a novel pointwise operator. This operator examines a function’s instantaneous trend of change, rather than the function’s instantaneous rate of change as examined by the derivative. The paper also depicts a novel integration method over curves in the plane, whose aim is to naturally extend the algorithm to a more general set of domains.