Integral Image

Definition

An Integral image is one that is conducive to frequent summation operations. Any rectangular subset of such an image can be evaluated in constant time. Such an image is achieved by converting its pixel values to a SAT (summed area table) by simple interpolation of known pixel values.

An Integral Image is defined as,

I(x,y) = i(x,y) + I(x-1,y) + I(x,y-1) - I(x-1,y-1)

Where,

i(x,y) = Pixel value of base image at (x,y)

I(x,y) = Pixel value of integral image at (x,y)

How this helps

Once the integral image has been calculated, the sum of any rectangular subset can be calculated by just 4 array points, i.e.,

SUM = (Bottom right + top left – top right – bottom left)

Application

Despite being a very simple transform (mathematically), calculating integral images helps perform many complex calculations with ease. Processing an integral image is much simpler than processing a normal pixel table. Hence the prerequisite step of several image processing algorithms is the conversion to an integral image.

Integral Images are useful for calculating HAAR wavelets, gradients, means and other measures. These are also used in image blurring, face recognition and other similar algorithms.

The concept of integral images can be easily extended to continuous domain (using limits) and multidimensional images. Also, the nature of summation operations can be modified to suit the algorithm, such as summation over non rectangular areas.

Method

Integral images are calculated using the following method.