Saurabh Sethia

Hand recognition using geometric classifiers

We discuss the issues and challenges in the design of a hand outline based recognition system. Our system is easier to use, cheaper to build and more accurate than previous systems. Extensive tests on more than 700 images collected from 70 people are reported. Classification, verification and identification of the input images were done using two simple geometric classifiers. We describe a novel minimum enclosing ball classifier which performs well for hand recognition and could be of interest for other applications.

Optimal Covering Tours with Turn Costs

We give the first algorithmic study of a class of “covering tour” problems related to the geometric Traveling Salesman Problem: Find a polygonal tour for a cutter so that it sweeps out a specified region (“pocket”), in order to minimize a cost that depends not only on the length of the tour but also on the number of turns. These problems arise naturally in manufacturing applications of computational geometry to automatic tool path generation and automatic inspection systems, as well as arc routing (“postman”) problems with turn penalties. We prove lower bounds (NP-completeness of minimum-turn milling) and give efficient approximation algorithms for several natural versions of the problem, including a polynomial-time approximation scheme based on a novel adaptation of the m-guillotine method.

When can you fold a map

We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a fiat folding by a sequence of simple folds? There are several models of simple folds; the simplest one-layer simple fold rotates a portion of paper about a crease in the paper by ±180°. We first consider the analogous questions in one dimension lower--bending a segment into a flat object--which lead to interesting problems on strings. We develop efficient algorithms for the recognition of simply foldable 1D crease patterns, and reconstruction of a sequence of simple folds. Indeed, we prove that a 1D crease pattern is flat-foldable by any means precisely if it is by a sequence of one-layer simple folds. n nNext we explore simple foldability in two dimensions, and find a surprising contrast: map folding and variants are polynomial, but slight generalizations are NP-complete. Specifically, we develop a linear-time algorithm for deciding foldability of an orthogonal crease pattern on a rectangular piece of paper, and prove that it is (weakly) NP-complete to decide foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper, (2) a crease pattern of axis-parallel and diagonal (45-degree) creases on a square piece of paper, and (3) crease patterns without a mountain/valley assignment.

On the Reflexivity of Point Sets

We introduce a new measure for planar point sets S that captures a combinatorial distance that S is from being a convex set: The reflexivity rho(S) of S is given by the smallest number of reflex vertices in a simple polygonalization of S. We prove various combinatorial bounds and provide efficient algorithms to compute reflexivity, both exactly (in special cases) and approximately (in general). Our study considers also some closely related quantities, such as the convex cover number kappa_c(S) of a planar point set, which is the smallest number of convex chains that cover S, and the convex partition number kappa_p(S), which is given by the smallest number of convex chains with pairwise-disjoint convex hulls that cover S. We have proved that it is NP-complete to determine the convex cover or the convex partition number and have given logarithmic-approximation algorithms for determining each.

On Computing Enclosing Isosceles Triangles and Related Problems

Given a set of n points in the plane, we show how to compute various enclosing isosceles triangles where different parameters such as area or perimeter are optimized. We then study a 3-dimensional version of the problem where we enclose a point set with a cone of fixed apex angle α.

Information space regionalization using adaptive multiplicatively weighted Voronoi diagrams

In an attempt to regionalize and visualize regions of information space, we offer a method for regionalization based on predetermined area relationships. This problem can be conceptualized as an inverse multiplicatively weighted Voronoi diagram. To compute such a diagram, we offer an adaptive algorithm. Initial testing indicates good results and quick convergence.

Games on triangulations

We analyze several perfect-information combinatorial games played on planar triangulations. We introduce three broad categories of such games--constructing, transforming, and marking triangulations--and several specific games within each category. In various situations of each game, we develop polynomial-time algorithms to determine who wins a given game position under optimal play, and to find a winning strategy. Along the way, we show connections to existing combinatorial games such as Kayles and Nimstring (a variation on Dots-and-Boxes).

Red-blue separability problems in 3D

In this paper we study the problems of separability of two disjoint point sets in 3D by multiple criteria extending some notions on separability of two disjoint point sets in the plane.

When Can You Fold a Map

We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are several models of simple folds; the simplest one-layer simple fold rotates a portion of paper about a crease in the paper by ±180°. We first consider the analogous questions in one dimension lower--bending a segment into a flat object--which lead to interesting problems on strings. We develop efficient algorithms for the recognition of simply foldable 1-D crease patterns, and reconstruction of a sequence of simple folds. Indeed, we prove that a 1-D crease pattern is flat-foldable by any means precisely if it is by a sequence of one-layer simple folds. n nNext we explore simple foldability in two dimensions, and find a surprising contrast: map folding and variants are polynomial, but slight generalizations are NP-complete. Specifically, we develop a linear-time algorithm for deciding foldability of an orthogonal crease pattern on a rectangular piece of paper, and prove that it is (weakly) NP-complete to decide foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper, (2) a crease pattern of axis-parallel and diagonal (45-degree) creases on a square piece of paper, and (3) crease patterns without a mountain/valley assignment.

RED-BLUE SEPARABILITY PROBLEMS IN 3D

In this paper we study the problems of separability of two disjoint point sets in 3D by multiple criteria extending some notions on separability of two disjoint point sets in the plane.