2. Foundations of scale-space"There are many paths to the top of the mountain, but the view is always the same" -Chinese proverb.2.1 Constraints for an uncommitted front-endTo compute any type of representation from the image data, information must be extractedusing certain operators interacting with the data. Basic questions then are: Which operatorsto apply? Where to apply them? How should they look like? How large should they be?Suppose such an operator is the derivative operator. This is a difference operator, comparingtwo neighboring values at a distance close to each other. In mathematics this distance canindeed become infinitesimally small by taking the limit of the separation distance to zero, butin physics this reduces to the sampling distance as the smallest distance possible. Thereforewe may foresee serious problems when we deal with such notions as mathematicaldifferentiation on discrete data (especially for high order), and sub-pixel accuracy.From this moment on we consider the aperture function as an operator: we will search forconstraints to pin down the exact specification of this operator. We will find an importantresult: for an unconstrained front-end there is a unique solution for the operator. This is theGaussian kernel gHx; sL=1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!!!!!!2 p s2 ‰-x2ÅÅÅÅÅÅÅÅÅÅÅÅÅ2 s2, with s the width of the kernel. It is the same bell-shaped kernel we know from probability theory as the probability density function of thenormal distribution, where s is the standard deviation of the distribution.Interestingly, there have been many derivations of the front-end kernel, all leading to theunique Gaussian kernel. This approach was pioneered by Iijima (figure 2.2) in Japan in the sixties [Iijima1962], butwas unnoticed for decades because the work was in Japanese and therefore inaccessible forWestern researchers.Independently Koenderink in the Netherlands developed in the early eighties a rathercomplete multi-scale theory [Koenderink1984a], including the derivation of the Gaussiankernel and the linear diffusion equation.<< FrontEndVision`FEV`;2. Foundations of scale-space 13s = 1; PlotA1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!!!!!!!!!!2 p s2 E-x2ÅÅÅÅÅÅÅÅÅÅÅ2 s2, 8x, -4, 4<, ImageSize -> 200E;-4 -2 2 40.10.20.30.4Figure 2.1 The Gaussian kernel with unit standard deviation in 1D.Koenderink was the first to point out the important relation to the receptive field families inthe visual system, as we will discuss in forthcoming chapters. Koenderink's work turned outto be monumental for the development of scale-space theory. Lindeberg pioneered the fieldwith a tutorial book [Lindeberg1994a]. The papers by Weickert, Ishikawa and Imija (whotogether discovered this Japanese connection) present a very nice review on these earlydevelopments [Weickert1997a, Weickert1999a].Show@Import@"Iijima.gif"D, ImageSize -> 150D;Fig. 2.2 Prof. Taizo Iijima, emeritus prof. of Tokyo Technical University, Japan, was the firstto publish the axiomatic derivation of 'the fundamental equation of figure'.We will select and discuss two fundamentally different example approaches to come to theGaussian kernel in this book:1. An axiomatic approach based on dimensional analysis and the notion of having 'nopreferences' (section 2.2);2. An approach based on the maximization of local entropy in the observation (section 2.5);14 2.1 Constraints for an uncommitted front-end2.2 Axioms of a visual front-endThe line of reasoning presented here is due to Florack et al. [Florack1992a]. Therequirements can be stated as axioms, or postulates for an uncommitted visual front-end. Inessence it is the mathematical formulation for being uncommitted: "we know nothing", or"we have no preference whatsoever". Ë linearity: we do not allow any nonlinearities at this stage, because they involve knowledgeof some kind. So: no knowledge, no model, no memory;Ë spatial shift invariance: no preferred location. Any location should be measured in thesame fashion, with the same aperture function;Ë isotropy: no preferred orientation. Structures with a particular orientation, like verticaltrees or a horizontal horizon, should have no preference, any orientation is just as likely. Thisnecessitates an aperture function with a circular integration area.Ë scale invariance: no preferred size, or scale of the aperture. Any size of structure, object,texture etc. to be measured is at this stage just as likely. We have no reason to look onlythrough the finest of apertures. The visual world consists of structures at any size, and theyshould be measured at any size.In order to use these constraints in a theory that sets up the reasoning to come to the apertureprofile formula, we need to introduce the concept of dimensional analysis.2.2.1 Dimensional analysisEvery physical unit has a physical dimension.It is this that mostly discriminates physics from mathematics. It was Baron Jean-BaptisteFourier who already in 1822 established the concept of dimensional analysis [Fourier1955].This is indeed the same mathematician so famous for his Fourier transformation.Show@Import@"Fourier.jpg"D, ImageSize Ø 140D;Figure 2.3 Jean-Baptiste Fourier, 1792-1842. Fourier described the concept of dimension analysis in his memorable work entitled "Théorieanalytique de la chaleur" [Fourier1955] as follows: "It should be noted that each physicalquantity, known or unknown, possesses a dimension proper to itself and that the terms in anequation cannot be compared one with another unless they possess the same dimensionalexponent".2. Foundations of scale-space 15Fourier described the concept of dimension analysis in his memorable work entitled "Théorieanalytique de la chaleur" [Fourier1955] as follows: "It should be noted that each physicalquantity, known or unknown, possesses a dimension proper to itself and that the terms in anequation cannot be compared one with another unless they possess the same dimensionalexponent".When a physicist inspects a new formula he invariably checks first whether the dimensionsare correct. It is for example impossible