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Nonparametric Decision Theoretic Classification
Published in Sing-Tze Bow, Pattern Recognition and Image Preprocessing, 2002
A piecewise linear function is a function that is linear over subregions of the feature space. These piecewise linear discriminant functions give piecewise linear boundaries between categories as shown in Figure 3.9a. In Figure 3.9b the boundary surface between class ω1 and class ω2 is nonconvex. However, it can be broken down into piecewise linear boundaries between these two classes ω1 and ω2. The discriminant functions are given by () dk(x)=maxm=1,…,Nk[dkm(x)]k=1,…,M
Identification of Impulse Responses in Heat Transfer: Parameterization, Doses, Partial Time Moments
Published in Heat Transfer Engineering, 2023
The points in a diagram are shown in Figure 1a in a fine grid of points, while the corresponding points are shown in Figure 1b for a coarse grid with points. On both figures, the dosal fractions or correspond to the area below the continuous plot over each time interval. The midpoint of each interval is used to construct a piecewise linear function that can be considered as parsimonious form of or of its discrete version
Toolpath planning for multi-gantry additive manufacturing
Published in IISE Transactions, 2021
Hieu Bui, Harry A. Pierson, Sarah Nurre Pinkley, Kelly M. Sullivan
The trajectory is comprised of paths that gantries follow in the x-direction as a function of time. Each gantry’s trajectory, which is used in the CCR, can be expressed as a piecewise linear function whose line segments (in x–time space) correspond to the raster segments and rapid movements of the printhead. Feasible configurations must prohibit the gantries from occupying the same x-coordinate or passing through each other. With respect to the trajectory plot, this means that the right gantry’s trajectory must not fall below (i.e., its x-coordinate become smaller than) the left gantry’s trajectory. The collision-checking subroutine starts from selecting the earliest segments at time t = 0, one from each gantry. Antonio’s algorithm (Antonio, 1992) is applied to check whether the selected pair of segments intersect. If there is an intersection, the associated segments are recorded, as well as the intersection point. Otherwise, the trajectory segment with the earlier finishing time will be replaced by its next segment. The checking process continues until all pairs of segments are checked, or the first collision is detected.
Global optimization for the three-dimensional open-dimension rectangular packing problem
Published in Engineering Optimization, 2018
(Tsai, Wang, and Lin 2015). To approximatefor, a piecewise linear function, which is constructed by m line segments with m + 1 break points (), can be formulated as the following linear system: where, and.1Notice that the values ofare predetermined.