Discrete Scale Spaces via Heat Equation

Scale spaces allow us to organize, compare and analyse differently sized structures of an object. The linear scale space of a monochromatic image is the solution of the heat equation using that image as an initial condition. Alternatively, this linear scale space can also be obtained applying Gaussian filters of increasing variances to the original image. In this work, we compare (by looking at theoretical properties, running time and output differences) five ways of discretizing this Gaussian scale-space: sampling Gaussian distributions; recursively calculating Gaussian approximations; using Splines; approximating by first-order generators; and finally, by a new method we call “Crossed Convolutions”. In particular, we explicitly present a correct way of initializing the recursive method to approximate Gaussian convolutions.