Category:Singular integrals

A summed area table (also known as an integral image) is a data structure and algorithm for quickly and efficiently generating the sum of values in a rectangular subset of a grid. It was first introduced to computer graphics in 1984 by Frank Crow for use with mipmaps. In computer vision, it was first prominently used within the Viola–Jones object detection framework in 2002. However, historically, this principle is very well known in the study of multi-dimensional probability distribution functions, namely in computing 2D (or ND) probabilities (area under the probability distribution) from the respective cumulative distribution functions.^[1]

The algorithm

As the name suggests, the value at any point (x, y) in the summed area table is just the sum of all the pixels above and to the left of (x, y), inclusive:^[2][3]

${\displaystyle I(x,y)=\sum _{\begin{array}{c}{x}^{\prime }\le x\\ y^{\prime }\le y\end{array}}i(x^{\prime },y^{\prime })}$

Moreover, the summed area table can be computed efficiently in a single pass over the image, using the fact that the value in the summed area table at (x, y) is just:

${\displaystyle I(x,y)=i(x,y)+I(x-1,y)+I(x,y-1)-I(x-1,y-1)}$

Once the summed area table has been computed, the task of evaluating any rectangle can be accomplished in constant time with just four array references. Specifically, using the notation in the figure at right, having A=(x0, y1), B=(x1, y1), C=(x1, y0) and D=(x0, y0), the sum of ${\displaystyle i(x,y)}$ over the rectangle spanned by A, B,C and D is just

${\displaystyle \sum _{\begin{array}{c}x0\le x\le x1\\ y0\le y\le y1\end{array}}i(x,y)=I(C)+I(A)-I(B)-I(D).}$

Extensions

This method is naturally extended to continuous domains.^[4]

The method can be also extended to high dimensional images.^[5] If the corners of the rectangle are ${\displaystyle x^p}$ with ${\displaystyle p}$ in ${\displaystyle \{0,1{\}}^d}$, then the sum of image values contained in the rectangle are computed with the formula

${\displaystyle \sum _{p\in \{0,1{\}}^d}(-1{)}^{d-\Vert p{\Vert }_1}I(x^p)}$

where ${\displaystyle I(x)}$ is the integral image at ${\displaystyle x}$ and ${\displaystyle d}$ the image dimension. The notation ${\displaystyle x^p}$ correspond in the example to ${\displaystyle d=2}$, ${\displaystyle A=x^{(0,0)}}$, ${\displaystyle B=x^{(1,0)}}$, ${\displaystyle C=x^{(1,1)}}$ and ${\displaystyle D=x^{(0,1)}}$. In neuroimaging, for example, the images have dimension ${\displaystyle d=3}$ or ${\displaystyle d=4}$, when using voxels or voxels with time-stamp.

References

External links

Lecture videos

• An introduction to the theory behind the integral image algorithm
• A demonstration to a continuous version of the integral image algorithm, from the Wolfram Demonstrations Project