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Chapter 4: Algorithm Overview 48of these methods can be exploited as surface shape complexity descriptors. We presentthe implementation issues and the results of analysis in the next chapter and justify theuse of Gauss-Bonnets method of computing curvature.4.2.3 Density EstimationProbability density functions are ubiquitous when it comes to intelligent decisionmaking and modeling. In this section of the document, we survey some of the keydensity estimation techniques. Research in this field of density estimation dates back tothe early 1950’s and was proposed by Fix and Hodges in 1951 [Fix and Hodges, 1951]as a breakthrough of freeing discriminant analysis from rigid distributional assumptions.Since then it has undergone application oriented metamorphosis. Rosenblatt introducesthe concept of non-parametric density estimation as an advanced statistical method[Rosenblatt, 1956]. Parzen follows that up with remarks on a model that aims at nonparametricestimation of a density function [Parzen, 1962].Density estimation is generally approached in two different ways. One of them is theparametric approach that assumes that the data has been drawn from one of theestablished parametric family of distributions such as the Gaussian and Rayleigh witha particular mean, variance and other well defined statistical parameters. The density funderlying the data could then be estimated by finding the estimates of the mean andthe variance from the data and then substituting these values into the formula of theassumed density. The parametric approach to density estimation is bounded by therigid assumption of the shape of the density function independent of the observed data.The non-parametric approach however is less rigid in its assumptions. The data speakfor themselves in determining the estimate of f. Silverman [Silverman, 1986] tracesthe evolution of density estimation techniques for a uni-variate dataset represented as asample of n observations of a data set X ={x 1 , x 2 , x 3 , x 4 ,…., x n }. We briefly survey suchtechniques in the next few paragraphs.The oldest and probably the most widely used non-parametric density estimate is thehistogram. A histogram is constructed by dividing the real line into equally sizedintervals, often called bins. The histogram is then a step function with heights beingthe proportion of the sample contained in that bin divided by the width of the bin. If hdenotes the width of the bins (bin width) and n represents the number of samples inthe dataset then the histogram estimate at a point x is given byfˆ ( x ) =(numberofXiin then hsamebinasx)(4.6)The construction of the histogram depends on the origin and bin width, the choice of thebin width primarily controlling the inherent smoothing of the density estimate.