Summed area table
A summed area table is a data structure and algorithm for quickly and efficiently generating the sum of values in a rectangular subset of a grid. In the image processing domain, it is also known as an integral image. 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 2001. 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.
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:
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, y0), B=(x1, y0), C=(x0, y1) and D=(x1, y1), the sum of over the rectangle spanned by A, B,C and D is just
- This method is naturally extended to continuous domains.
- The method can be also extended to high-dimensional images. If the corners of the rectangle are with in , then the sum of image values contained in the rectangle are computed with the formula
where is the integral image at and the image dimension. The notation correspond in the example to , , , and . In neuroimaging, for example, the images have dimension or , when using voxels or voxels with time-stamp.
- This method has been extended to high-order integral image. In, Phan et al. provided two, three, or four integral images for quickly and efficiently calculating the standard deviation (variance), skewness, and kurtosis of local block in the image.
The variance is given by:
Let and denote the summations of block of and , respectively. and are computed quickly by integral image. Now, we manipulate the variance equation as:
Where and .
- Finkelstein, Amir; neeratsharma (2010). "Double Integrals By Summing Values Of Cumulative Distribution Function". Wolfram Demonstration Project.
- Crow, Franklin (1984). "Summed-area tables for texture mapping". SIGGRAPH '84: Proceedings of the 11th annual conference on Computer graphics and interactive techniques. pp. 207–212.
- Viola, Paul; Jones, Michael (2002). "Robust Real-time Object Detection". International Journal of Computer Vision.
- Tapia, Ernesto (January 2011). "A note on the computation of high-dimensional integral images". Pattern Recognition Letters 32 (2). doi:10.1016/j.patrec.2010.10.007.
- Phan, Thien; Larson, Eric; Sohoni, Sohum; and Chandler, Damon (2012). "Performance-analysis-based acceleration of image quality assessment". IEEE Southwest Symposium on Image Analysis and Interpretation.
- Summed table implementation in object detection
- Integral images to quickly compute local standard deviation (variance), skewness, and kurtosis
- Lecture videos