Piecewise linear retractions by reflexion

Polytopal Linear Retractions

We investigate graded retracts of polytopal algebras (essentially the homogeneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are again polytopal algebras and that codimension 1 retractions factor through retractions preserving the semigroup structure. We expect that these results hold in general. This...

Piecewise Constant Controlled Linear Fuzzy Systems

Approximation by sums of piecewise linear polynomials

We present two partitioning algorithms that allow a sum of piecewise linear polynomials over a number of overlaying convex partitions of the unit cube Ω in Rd to approximate a function f ∈ W 3 p (Ω) with the order N−6/(2d+1) in Lp-norm, where N is the total number of cells of all partitions, which makes a marked improvement over the N−2/d order achievable on a single convex partition. The gradi...

Shape Similarity by Piecewise Linear Alignmentz

A key problem when comparing planar shapes is to locate corresponding reference points, such as inflection or high curvature points. High curvature points are to be preferred, since they are psychophysically more important and are shown to be computationally more reliable. Using a scale-space of curves such ‘corners’ are located, and an empirical analysis demonstrates that high curvature points...

Constructing Piecewise Linear

Let P = fp1; : : : ; png and Q = fq1; : : : ; qng be two point sets lying in the interior of rectangles in the plane. We show how to construct a piecewise linear homeomorphism of size O(n2) between the rectangles which maps pi to qi for each i. This bound is optimal in the worst case; i.e., there exist point sets for which any piecewise linear homeomorphism has size (n2). Introduction A homeomo...