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61
Estimating The Tensor Of Curvature Of A Surface From A Polyhedral Approximation
, 1995
"... Estimating principal curvatures and principal directions of a surface from a polyhedral approximation with a large number of small faces, such as those produced by isosurface construction algorithms, has become a basic step in many computer vision algorithms. Particularly in those targeted at medic ..."
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Cited by 184 (5 self)
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Estimating principal curvatures and principal directions of a surface from a polyhedral approximation with a large number of small faces, such as those produced by isosurface construction algorithms, has become a basic step in many computer vision algorithms. Particularly in those targeted at medical applications. In this paper we describe a method to estimate the tensor of curvature of a surface at the vertices of a polyhedral approximation. Principal curvatures and principal directions are obtained by computing in closed form the eigenvalues and eigenvectors of certain # # # symmetric matrices defined by integral formulas, and closely related to the matrix representation of the tensor of curvature. The resulting algorithm is linear, both in time and in space, as a function of the number of vertices and faces of the polyhedral surface. 1
SelfOrganizing Maps on noneuclidean Spaces
 Kohonen Maps
, 1999
"... INTRODUCTION The SelfOrganizing Map, as introduced by Kohonen more than a decade ago, has stimulated an enormous body of work in a broad range of applied and theoretical fields, including pattern recognition, brain theory, biological modeling, mathematics, signal processing, data mining and many m ..."
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Cited by 32 (4 self)
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INTRODUCTION The SelfOrganizing Map, as introduced by Kohonen more than a decade ago, has stimulated an enormous body of work in a broad range of applied and theoretical fields, including pattern recognition, brain theory, biological modeling, mathematics, signal processing, data mining and many more [8]. Much of this impressive success is owed to the combination of elegant simplicity in the SOM's algorithmic formulation, together with a high ability to produce useful answers for a wide variety of applied data processing tasks and even to provide a good model of important aspects of structure formation processes in neural systems. While the applications of the SOM are extremely widespread, the majority of uses still follow the original motivation of the SOM: to create dimensionreduced "feature maps" for various uses, most prominently perhaps for the purpose of data visualization. The suitability of the SOM for this task has been analyzed in great detail and linked to earlier
Nonlinear Partial Least Squares
, 1995
"... We propose a new nonparametric regression method for highdimensional data, nonlinear partial least squares (NLPLS). NLPLS is motivated by projectionbased regression methods, e.g., partial least squares (PLS), projection pursuit (PPR), and feedforward neural networks. The model takes the form of a ..."
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Cited by 16 (0 self)
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We propose a new nonparametric regression method for highdimensional data, nonlinear partial least squares (NLPLS). NLPLS is motivated by projectionbased regression methods, e.g., partial least squares (PLS), projection pursuit (PPR), and feedforward neural networks. The model takes the form of a composition of two functions. The first function in the composition projects the predictor variables onto a lowerdimensional curve or surface yielding scores, and the second predicts the response variable from the scores. We implement NLPLS with feedforward neural networks. NLPLS will often produce a more parsimonious model (fewer score vectors) than projectionbased methods, and the model is well suited for detecting outliers and future covariates requiring extrapolation. The scores are also shown to have useful interpretations. We also extend the model for multiple response variables and discuss situations when multiple response variab...
Introducing Robotic Origami Folding
 IEEE International Conference on Robotics and Automation
, 2004
"... Origami, the human art of paper sculpture, is a fresh challenge for the field of robotic manipulation, and provides a concrete example for many difficult and general manipulation problems. This thesis will present some initial results, including the world’s first origamifolding robot, some new theo ..."
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Cited by 16 (1 self)
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Origami, the human art of paper sculpture, is a fresh challenge for the field of robotic manipulation, and provides a concrete example for many difficult and general manipulation problems. This thesis will present some initial results, including the world’s first origamifolding robot, some new theorems about foldability, definition of a simple class of origami for which I have designed a complete automatic planner, analysis of the kinematics of more complicated folds, and some observations about the configuration spaces of compound spherical closed chains. Acknowledgments Thanks to my family, for everything. And thanks to everyone who has made the development of this thesis and life in Pittsburgh so great. Matt Mason. Wow! Who ever had a better advisor, or friend? Thanks to my colleagues, for all that I’ve learned from them. Jeff Trinkle, “in loco advisoris”. My academic older siblings and cousins, aunts and uncles, for their help and guidance in so many things: Alan Christiansen, Ken Goldberg, Randy
Some Integral Geometry Tools to Estimate the Complexity of 3D Scenes
, 1997
"... Many problems in computer graphics deal with complex 3D scenes where visibility, proximity, collision detection queries have to be answered. Due to the complexity of these queries and the one of the models they are applied to, data structures most often based on hierarchical decompositions have been ..."
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Cited by 15 (4 self)
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Many problems in computer graphics deal with complex 3D scenes where visibility, proximity, collision detection queries have to be answered. Due to the complexity of these queries and the one of the models they are applied to, data structures most often based on hierarchical decompositions have been proposed to solve them. As a result of these involved algorithms/data structures, most of the analysis have been carried out in the worst case and fail to report good average case performances in a vast majority of cases. The goal of this work is therefore to investigate geometric probability tools to characterize average case properties of standard scenes such as architectural scenes, natural models, etc under some standard visibility and proximity requests. In the first part we recall some fundamentals of integral geometry and discuss the classical assumption of measures invariant under the group of motions in the context of non uniform models. In the second one we present simple generali...
Mobility of Bodies in Contact  I: A New 2nd Order Mobility Index for MultipleFinger Grasps
, 1994
"... Using a configurationspace approach, this paper develops a novel 2 nd order mobility theory for bodies in contact. A major contribution of this paper is the development of a coordinate invariant 2 nd order mobility index for a body, B, in frictionless contact with finger bodies A 1 ; : : : ; A ..."
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Cited by 13 (6 self)
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Using a configurationspace approach, this paper develops a novel 2 nd order mobility theory for bodies in contact. A major contribution of this paper is the development of a coordinate invariant 2 nd order mobility index for a body, B, in frictionless contact with finger bodies A 1 ; : : : ; A k . The index is an integer that captures the inherent mobility of B in an equilibrium grasp due to 2 nd order, or surface curvature, effects. It differentiates between grasps which are deemed equivalent by classical 1 st order theories, but are physically different. We further show that 2 nd order effects can be used to lower the effective mobility of a grasped object, and discuss implications of this result for proving new lower bounds on the number of contacting bodies needed to immobilize an object. Physical interpretation and stability analysis of 2 nd order effects are taken up in the companion paper. 1 Introduction We are concerned with the problem of analyzing the mobility ...
Thickness Of Knots
"... this paper we study physical knots; that is, knots tied (as closed loops) in real pieces of rope, which have diameter. Intuitively, for a given diameter, one needs a certain minimum length of rope in order to tie a (nontrivial) knot, and (more vaguely), the more complicated the knot you want to tie ..."
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Cited by 13 (5 self)
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this paper we study physical knots; that is, knots tied (as closed loops) in real pieces of rope, which have diameter. Intuitively, for a given diameter, one needs a certain minimum length of rope in order to tie a (nontrivial) knot, and (more vaguely), the more complicated the knot you want to tie, the more rope you need. To be specific, we can ask:
A Configuration Space Analysis of Bodies in Contact  Part II: 2nd Order Mobility
, 1994
"... Using a configuration space approach, this paper develops a 2 nd order mobility theory for a body, B, in frictionless quasistatic contact with rigid stationary bodies A 1 ; \Delta \Delta \Delta ; A d . This analysis ultimately leads to a coordinate invariant 2 nd order mobility index, an intege ..."
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Cited by 9 (7 self)
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Using a configuration space approach, this paper develops a 2 nd order mobility theory for a body, B, in frictionless quasistatic contact with rigid stationary bodies A 1 ; \Delta \Delta \Delta ; A d . This analysis ultimately leads to a coordinate invariant 2 nd order mobility index, an integer that captures the inherent mobility of B in an equilibrium grasp. The 2 nd order index differentiates between grasps which are deemed equivalent by 1 st order, or instantaneous, theories, but are physically different. We further show that 2 nd order effects can be used to lower the effective mobility of an equilibrium grasp, and hence can be used to prove new lower bounds on the number of contacting bodies needed to immobilize an object. 1 Introduction Given an object B , we are concerned with determining its mobility when it is in frictionless contact with d rigid and stationary bodies (or "fingers"), A 1 ; \Delta \Delta \Delta ; A d . We introduced in [9] a configuration space (c...
Some Theoretical Results on Nonlinear Principal Components Analysis
 In Proceedings of the American Control Conference
, 1996
"... Nonlinear principal components analysis (NLPCA) neural networks are feedforward autoassociative networks with five layers. The third layer has fewer nodes than the input or output layers. NLPCA has been shown to give better solutions to several feature extraction problems than existing methods, but ..."
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Cited by 9 (0 self)
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Nonlinear principal components analysis (NLPCA) neural networks are feedforward autoassociative networks with five layers. The third layer has fewer nodes than the input or output layers. NLPCA has been shown to give better solutions to several feature extraction problems than existing methods, but very little is know about the theoretical properties of this method or its estimates. This paper studies NLPCA. It proposes a geometric interpretation by showing that NLPCA fits a lowerdimensional curve or surface through the training data. The first three layers project observations onto the curve or surface giving scores. The last three layers define the curve or surface. The first three layers are a continuous function, which I show has several implications: NLPCA "projections" are suboptimal producing larger approximation error, NLPCA is unable to model curves and surfaces that intersect themselves, and NLPCA cannot parameterize curves with parameterizations having discontinuous jumps. ...
THE PROBABILITY THAT A SLIGHTLY PERTURBED NUMERICAL ANALYSIS PROBLEM IS DIFFICULT
, 2008
"... We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of illposed inputs. Several applications to linear and polynomial equation solving ..."
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Cited by 8 (8 self)
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We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of illposed inputs. Several applications to linear and polynomial equation solving show that the estimates obtained in this way are easy to derive and quite accurate. The main theorem is based on a volume estimate of εtubular neighborhoods around a real algebraic subvariety of a sphere, intersected with a spherical disk of radius σ. Besides ε and σ, this bound depends only on the dimension of the sphere and on the degree of the defining equations.