Scale space implementation

In the areas of computer vision, image analysis and signal processing, the notion of scale space representation is used for processing measurement data at multiple scales, and specifically enhance or suppress image features over different ranges of scale (see the article on scale space). A special type of scale space representation is provided by the Gaussian scale space, where the image data in N dimensions is subjected to smoothing by Gaussian convolution. Most of the theory for Gaussian scale space deals with continuous images, whereas one when implementing this theory will have to face the fact that most measurement data are discrete. Hence, the theoretical problem arises concerning how to discretize the continuous theory while either preserving or well approximating the desirable theoretical properties that lead to the choice of the Gaussian kernel (see the article on scale space axioms). This article describes basic approaches for this that have been developed in the literature.

Since computational efficiency is often important, low order recursive filters are often used for scale space smoothing. For example, Young and van Vliet use a third order recursive filter with one real pole and a pair of complex poles, applied forward and backward to make a sixth order symmetric approximation to the Gaussian with low computational complexity for any smoothing scale. By relaxing a few of the axioms, Lindeberg Tan et al. have analyzed and compared some of these approaches, and have pointed out that the Young and van Vliet filters are a cascade (multiplication) of forward and backward filters, while the Deriche and the Jin et al. filters are sums of forward and backward filters. At fine scales, the recursive filtering approach as well as other separable approaches are not guaranteed to give the best possible approximation to rotational symmetry, so non separable implementations for 2D images may be considered as an alternative. When computing several derivatives in the N jet simultaneously, discrete scale space smoothing with the discrete analogue of the Gaussian kernel, or with a recursive filter approximation, followed by small support difference operators, may be both faster and more accurate than computing recursive approximations of each derivative operator.

Regarding the topic of automatic scale selection based on normalized derivatives, pyramid approximations are frequently used to obtain real time performance. The appropriateness of approximating scale space operations within a pyramid originates from the fact that repeated cascade smoothing with generalized binomial kernels leads to equivalent smoothing kernels that under reasonable conditions approach the Gaussian. Furthermore, the binomial kernels (or more generally the class of generalized binomial kernels) can be shown to constitute the unique class of finite support kernels that guarantee non creation of local extrema or zero crossings with increasing scale (see the article on multi scale approaches for details). Special care may, however, need to be taken to avoid discretization artifacts.

For one dimensional kernels, there is a well developed theory of multi scale approaches, concerning filters that do not create new local extrema or new zero crossings with increasing scales. For continuous signals, filters with real poles in the s plane are within this class, while for discrete signals the above described recursive and FIR filters satisfy these criteria. Combined with the strict requirement of a continuous semi group structure, the continuous Gaussian and the discrete Gaussian constitute the unique choice for continuous and discrete signals. There are many other multi scale signal processing, image processing and data compression techniques, using wavelets and a variety of other kernels, that do not exploit or require the same requirements as scale space descriptions do; that is, they do not depend on a coarser scale not generating a new extremum that was not present at a finer scale (in 1D) or non enhancement of local extrema between adjacent scale levels (in any number of dimensions).