Calculus

Integral

Summed Area Computation

This component calculates the integral image (also known as the summed area table) of the input, producing an output where each pixel represents the cumulative sum of all pixel values above and to the left of its position. This operation is widely used in computer vision and image analysis, providing an efficient way to compute regional sums or averages with constant‐time lookups — fundamental to algorithms like Haar feature detection, adaptive thresholding, and local filtering.

Formally, the integral image is defined as:

${{\rm I}}_{\Sigma }(x,y)=\sum _{i=0}^x\sum _{j=0}^y{\rm I}(i,j)$

In addition to the standard integral, this component also includes an experimental Product mode. Rather than summing pixel values, this mode computes the cumulative product of all pixels above and to the left of each position — effectively generating a product area table. While not commonly applied in practical image processing, this operation serves as a conceptual counterpart to the integral, emphasizing multiplicative relationships instead of additive ones.

Formally, the product image is defined as:

${{\rm I}}_{\Pi }(x,y)=\prod _{i=0}^x\prod _{j=0}^y{\rm I}(i,j)$

Together, these operations extend the notion of cumulative image accumulation into both additive and multiplicative domains, revealing new ways to explore spatial relationships and intensity distributions in images.

Resources:

Download the .tox files
Integral Image on Medium