Piecewise-Linear Classifiers

Suppose a pattern x, actually belonging to subclass m, is presented to all prototypes and subclass I yields the largest scalar product. The weight vector w which should give the maximum scalar product is 

FIGURE 27. Piecewise-linear classification. The two-modal class (+) is represented by two prototypes w and W.  

The weight vector which erronously gave the largest scalar product is corrected by equation (72). 

Factor c is some positive correction increment (Chapter 2.2.3). 

Application of a.multicategory piecewise-linear classifier to the interpretation of mass spectra was of varying success C88, 89, 1173-Recognition of a C=C double bond in a linearly inseparable data set of mass spectra required 3 to 5 weight vectors to obtain 78 to 87 % predictive ability (75 % was obtained with a single weight vector for the same data set). The lack of a satisfactory theory of piecewise-linear classifiers and rather high computational expenses have prevented up to now broader applications of this classification method. 

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Another routine develops a decision tree of binary choices which, taken as a whole, can classify the members of the training set. The decision tree generated implements a piecewise linear discriminant function. The decision tree is developed by splitting the data set into two parts in an optimal way at each node. A node is considered to be terminal when no advantageous split can be made a terminal node is labelled to associate with the pattern class most represented among the patterns present at the node. 

In certain classification problems, a linear separation of two classes by only one decision plane is impossible. Figure 27 shows a two-modal class (+) consisting of two distinct clusters (subclasses). Evidently, this class should be represented by two prototypes (Wg/ w ) and a minimum distance classifier would be successful. In this way, the pattern space is partitioned by several decision planes (piecewise-linear separation). Classification of an unknown pattern requires the calculation of the scalar products with all weight vectors (prototype vectors). The unknown is assigned to the class with the largest scalar product (Chapter 2.1.5.). In the same way, a mu Iticategory classification is possible C89, 3963. 

Piecewise multicategory classifiers have been used for the recognition of C=C double bonds because the data set was not linearly separable C893. Several other pattern recognition methods have been used with more or less success to develop classifiers that recognize molecular structures C9, 12, 134, 142, 144, 173, 184, 197, 2113-... 

Surface (3D) Reconstruction Surface reconstruction from point clouds is fundamental in many applications. Using the raw point clouds or volumetric data acquired from an unknown surface, an approximation of the surface can be constructed and used to compare it with CAD models or for smface-based automated programming. Reconstruction methods can be classified into two types the computational geometry approach focuses on the piecewise-linear interpolation of unorganised points and defines the surface as a carefiilly chosen sub-set of the Delaunay triangulation in a Cartesian coordinate system, and the computer graphics... 

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