The magic sigma

With the explosion in the usage of mobile devices and other smart electronics, embedded devices are becoming ubiquitous. Most such embedded architectures utilize fixed-point rather than floating-point computation to meet power, heat, and speed requirements leading to the need for integer-based processing algorithms. Operations involving Gaussian kernels are common to such algorithms, but the standard methods of constructing such kernels result in approximations and lack a property that enables efficient bitwise shift operations. To overcome these limitations, we present how to precisely combine the power of integer arithmetic and bitwise shifts with intrinsically real valued Gaussian kernels. We prove mathematically that there exist a set of what we call “magic sigmas” for which the integer kernels exactly represent the Gaussian function whose values are all powers-of-two, and we discovered that the maximum sigma that leads to such properties is about 0.85. We also designed a simple and precise algorithm for designing kernels composed exclusively of integers given any arbitrary sigma and show how this can be exploited for Gaussian filter design. Considering the ubiquity of Gaussian filtering and the need for integer computation for increasing numbers of embedded devices, this is an important result for both theoretical and practical purposes.