Computational geometry

We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.

Triangulating a simple polygon in linear time

Linear mixing models of compositional data have been developed in various branches of the earth sciences (e.g., geochemistry, petrology, mineralogy, sedimentology) for the purpose of summarizing variation among a series of observations in terms of proportional contributions of (theoretical) end members. Methods of parameter estimation range from relatively straightforward normative partitioning by (nonnegative) least squares, to more sophisticated bilinear inversion techniques. Solving the bilinear mixing problem involves the estimation of both mixing proportions and end-member compositions from the data. Normative partitioning, also known as linear unmixing, thus can be regarded as a special situation of bilinear unmixing with (supposedly) known end members. Previous attempts to model linear mixing processes are reviewed briefly, and a new iterative strategy for solving the bilinear problem is developed. This end-member modeling algorithm is more robust and has better convergence properties than previously proposed numerical schemes. The bilinear unmixing solution is intrinsically nonunique, unless additional constraints on the model parameters are introduced. In situations where no a priori knowledge is available, the concept of an “ optimal ” solution may be used. This concept is based on the trade-off between mathematical and geological feasibility, two seemingly contradictory but equally desirable requirements of the unmixing solution.

End-member modeling of compositional data: Numerical-statistical algorithms for solving the explicit mixing problem

Classification Methods For Remotely Sensed Data

Automatic classification is one of the basic tasks required in any pattern recognition and human computer interaction application. In this paper, we discuss training probabilistic classifiers with labeled and unlabeled data. We provide a new analysis that shows under what conditions unlabeled data can be used in learning to improve classification performance. We also show that, if the conditions are violated, using unlabeled data can be detrimental to classification performance. We discuss the implications of this analysis to a specific type of probabilistic classifiers, Bayesian networks, and propose a new structure learning algorithm that can utilize unlabeled data to improve classification. Finally, we show how the resulting algorithms are successfully employed in two applications related to human-computer interaction and pattern recognition: facial expression recognition and face detection.

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Semisupervised learning of classifiers: theory, algorithms, and their application to human-computer interaction

Given a set of n points in R 2 bounded within a rectangular floor F, and a rectangular plate P of specified size, we consider the following two problems: find an isothetic position of P such that it encloses (i) maximum and (ii) minimum number of points, keeping P totally contained within F. For both of these problems, a new algorithm based on interval tree data structure is presented, which runs in O(nlogn) time and consumes O(n) space. If polygonal objects of arbitrary size and shape are distributed in R 2, the proposed algorithm can be tailored for locating the position of the plate to enclose maximum or minimum number of objects with the same time and space complexity. Finally, the algorithm is extended for identifying a cuboid, i.e., a rectangular parallelepiped that encloses maximum number of polyhedral objects in R 3. Thus, the proposed technique serves as a unified paradigm for solving a general class of enclosure problems encountered in computational geometry and pattern recognition.

A unified algorithm for finding maximum and minimum object enclosing rectangles and cuboids

Disclosed is a method and apparatus to provide for fingerprint minutiae matching using scoring techniques. A plurality of minutiae in a fingerprint image is defined. A score associated with each minutia corresponding to the validity of each minutia is estimated. The fingerprint image is then matched against one or more sample fingerprint images utilizing a partial point set pattern matching (PSPM) algorithm.

Fingerprint minutiae matching using scoring techniques

Efficient algorithm for placing a given number of base stations to cover a convex region

Largest empty rectangle among a point set

Given a set of n colored points in IR2 with a total of m (3 ≤ m ≤ n) colors, the problem of identifying the smallest color-spanning object of some predefined shape is studied in this paper. We shall consider two different shapes: (i) corridor and (ii) rectangle of arbitrary orientation. Our proposed algorithm for identifying the smallest color-spanning corridor is simple and runs in O(n2log n) time using O(n) space. A dynamic version of the problem is also studied, where new points may be added, and the narrowest color-spanning corridor at any instance can be reported in O(mn(α(n))2log m) time. Our algorithm for identifying the smallest color-spanning rectangle of arbitrary orientation runs in O(n3log m) time and O(n) space.

Subhas C. Nandy

Papers

A unified algorithm for finding maximum and minimum object enclosing rectangles and cuboids

Fingerprint minutiae matching using scoring techniques

Efficient algorithm for placing a given number of base stations to cover a convex region

Largest empty rectangle among a point set

Smallest color-spanning object revisited