Projective re-normalization for improving the behavior of a homogeneous conic linear system

In this paper we study the homogeneous conic system F : Ax = 0, x ∈ C \ {0}. We choose a point ¯s ∈ intC∗ that serves as a normalizer and consider computational properties of the normalized system F¯s : Ax = 0, ¯sT x = 1, x ∈ C. We show that the computational complexity of solving F via an interior-point method depends only on the complexity value ϑ of the barrier for C and on the symmetry of the origin in the image set H¯s := {Ax : ¯sT x = 1, x ∈ C}, where the symmetry of 0 in H¯s is sym(0,H¯s) := max{α : y ∈ H¯s -->−αy ∈ H¯s} .We show that a solution of F can be computed in O(sqrtϑ ln(ϑ/sym(0,H¯s)) interior-point iterations. In order to improve the theoretical and practical computation of a solution of F, we next present a general theory for projective re-normalization of the feasible region F¯s and the image set H¯s and prove the existence of a normalizer ¯s such that sym(0,H¯s) ≥ 1/m provided that F has an interior solution. We develop a methodology for constructing a normalizer ¯s such that sym(0,H¯s) ≥ 1/m with high probability, based on sampling on a geometric random walk with associated probabilistic complexity analysis. While such a normalizer is not itself computable in strongly-polynomialtime, the normalizer will yield a conic system that is solvable in O(sqrtϑ ln(mϑ)) iterations, which is strongly-polynomialtime. Finally, we implement this methodology on randomly generated homogeneous linear programming feasibility problems, constructed to be poorly behaved. Our computational results indicate that the projective re-normalization methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46% decrease in average IPM iterations for 100 randomly generated poorly-behaved problem instances of dimension 1000 × 5000.Singapore-MIT Allianc

Projective Pre-Conditioners for Improving the Behavior of a Homogeneous Conic Linear System

In this paper we present a general theory for transforming a normalized homogeneous conic system F : Ax = 0, s'x = 1, x in C to an equivalent system via projective transformation induced by the choice of a point w in the set H'(s) = { v : s - A'v in C*}. Such a projective transformation serves to pre-condition the conic system into a system that has both geometric and computational properties with certain guarantees. We characterize both the geometric behavior and the computational behavior of the transformed system as a function of the symmetry of w in H'(s) as well as the complexity parameter of the barrier for C. Under the assumption that F has an interior solution, H'(s) must contain a point w whose symmetry is at least 1/m; if we can find a point whose symmetry is O(1/m) then we can projectively transform the conic system to one whose geometric properties and computational complexity will be strongly-polynomial-time in m and the barrier parameter. We present a method for generating such a point w based on sampling and on a geometric random walk on H'(s) with associated complexity and probabilistic analysis. Finally, we implement this methodology on randomly generated homogeneous linear programming feasibility problems, constructed to be poorly behaved. Our computational results indicate that the projective pre-conditioning methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46% decrease in average IPM iterations for 100 randomly generated poorly-behaved problem instances of dimension 1000 × 5000

Vision-Based Object Recognition and 3-D Pose Estimation Using Conic Features

This thesis deals with monocular vision-based object recognition and 3-D pose estimation based on conic features. Conic features including circles and ellipses are frequently observed in many man-made objects in real word as well as have the merit of robustness potentially in feature extraction in vision-based applications. Although the 3-D pose estimation problem of conic features in 3-D space has been studied well since 1990, the previous work has not provided a unique solution completely for full 3-D pose parameters (i.e., 3-orientations and 3-positions) due to complexity from high nonlinearity of a general conic. This thesis, therefore, renews conic features in a new perspective on geometric invariants in both 3-D space and 2-D projective space, incorporating other geometric features with conics. First, as the most essential step in dealing with conics, this thesis shows that the pose parameters of a circular feature in 3-D space can be derived analytically from incorporating a coplanar point. A procedure of pose parameter recovery is described in detail, and its performance is evaluated and discussed in view of pose estimation errors and sensitivity. Second, it is also revealed that the pose of an elliptic feature can be resolved when two coplanar points are incorporated on the basis of the polarity of two points for a conic in 2-D projective space. This thesis proposes a series of algorithms to determine the 3-D pose parameters uniquely, and evaluates the proposed method through a measure of estimation performance and sensitivity depending on point locations. Third, a pair of two conics is dealt with, which is regarded as an extension of the idea of the incorporation scheme to another conic feature from point features. Under the polarity concept, this thesis proves that the problem involving a pair of two conics can be formulated with the problem of one ellipse with two points so that its solution is derived in the same form as in the ellipse case. In order to treat two or more conic objects as well as to deal with an object recognition problem, the rest of thesis concentrates on the theoretical foundation of multiple object recognition. First, some effective modeling approaches are described. A general object model is specially designed to model multiple objects for object recognition and pose recovery in view of spatial geometry. In particular, this thesis defines a pairwise conic model that can describes the geometrical relation between two conics invariantly in 2-D projective space, which consists of a pairwise conic (PC), a pairwise conic invariant (PCI), and a pairwise conic pole (PCP). Based on the two kinds of models, an object learning and recognition system is proposed as a general framework for multiple object recognition. Considering simplicity and flexibility in object learning stage, this thesis introduces a semi-automatic learning scheme to construct the multiple object model from a model image at once. To utilize geometric relations among multiple objects effectively in object recognition, this thesis specifies some feature functions based on the pairwise conic model, and then describes an object recognition method in a fashion of linear-chain conditional random field (CRF). In particular, as a post refinement step of the recognition, a geometric alignment procedure is also proposed in algorithmic details to improve recognition performance against noisy conditions. Last, the multiple object recognition method is evaluated intensively through two practical applications that deal with a place recognition and an elevator button recognition problem for service robots. A series of experiment results supports the effectiveness of the proposed method, maintaining reliable performance against noisy conditions in the presence of perspective distortion and partial object occlusions.Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Research objective and expected contribution . . . . . . . . . . . . . . . . . . 6 1.4 Organization of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 3-D Pose Estimation of a Circular Feature 10 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.3 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.4 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Preliminaries: an elliptic cone in 3-D space and its homogeneous representation in 2-D projective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.1 Homogeneous representation . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.2 Principal planes of a cone versus diagonalization of a conic matrix Q . 16 2.3 3-D interpretation of a circular feature for 3-D pose estimation . . . . . . . . 19 2.3.1 3-D orientation estimation . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.2 3-D position estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.3 Composition of homogeneous transformation and discrimination for the unique solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Experiment results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.1 A numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.2 Evaluation of pose estimation performance . . . . . . . . . . . . . . . 29 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3 3-D Pose Estimation of an Elliptic Feature 35 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.3 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Interpretation of an elliptic feature with coplanar points in 2-D projective space 38 3.2.1 The minimal number of points for pose estimation . . . . . . . . . . . 39 3.2.2 Analysis of possible constraints for relative positions of two points to an ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.3 Feature selection scheme for stable homography estimation . . . . . . 43 3.3 3-D pose estimation algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3.1 Extraction of triangular features from an elliptic object . . . . . . . . 47 3.3.2 Homography decomposition . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.3 Composition of homogeneous transformation matrix with unique solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4 Experiment results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.2 Evaluation of the proposed method . . . . . . . . . . . . . . . . . . . . 54 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 3-D Pose Estimation of a Pair of Conic Features 61 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 3-D pose estimation of a conic feature incorporated with line features . . . . 61 4.3 3-D pose estimation of a conic feature incorporated with another conic feature 63 4.3.1 Some examples of self-polar triangle and invariants . . . . . . . . . . . 65 4.3.2 3-D pose estimation of a pair of coplanar conics . . . . . . . . . . . . . 67 4.3.3 Examples of 3-D pose estimation of a conic feature incorporated with another conic feature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5 Multiple Object Recognition Based on Pairwise Conic Model 77 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2 Learning of geometric relation of multiple objects . . . . . . . . . . . . . . . . 78 5.3 Pairwise conic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.3.1 De_nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.4 Multiple object recognition based on pairwise conic model and conditional random _elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.4.1 Graphical model for multiple object recognition . . . . . . . . . . . . . 86 5.4.2 Linear-chain conditional random _eld . . . . . . . . . . . . . . . . . . 87 5.4.3 Determination of low-level feature functions for multiple object recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4.4 Range selection trick for e_ciently computing the costs of low-level feature functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4.5 Evaluation of observation sequence . . . . . . . . . . . . . . . . . . . . 93 5.4.6 Object recognition based on hierarchical CRF . . . . . . . . . . . . . . 95 5.5 Geometric alignment algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6 Application to Place Recognition for Service Robots 105 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.1.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.2 Feature extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.2.1 Detection of 2-D geometric shapes . . . . . . . . . . . . . . . . . . . . 107 6.2.2 Examples of shape feature extraction . . . . . . . . . . . . . . . . . . . 109 6.3 Object modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.3.1 A place model that describes multiple landmark objects . . . . . . . . 112 6.3.2 Pairwise conic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.3.3 Incorporation of non-conic features with a pairwise conic model . . . . 114 6.4 Place learning and recognition system . . . . . . . . . . . . . . . . . . . . . . 121 6.4.1 HCRF-based recognition . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.5 Experiment results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.5.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.5.2 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7 Application to Elevator Button Recognition 136 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.1.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.1.3 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.2 Object modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.2.1 Geometric model for multiple button objects . . . . . . . . . . . . . . 140 7.2.2 Pairwise conic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.3 Learning and recognition system . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.3.1 Button object learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.3.2 CRF-based recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.4 Experiment results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.4.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.4.2 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 8 Concluding remarks 159 8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.2 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 References 161 Summary (in Korean) 16

Convexity preserving interpolatory subdivision with conic precision

The paper is concerned with the problem of shape preserving interpolatory subdivision. For arbitrarily spaced, planar input data an efficient non-linear subdivision algorithm is presented that results in $G^1$ limit curves, reproduces conic sections and respects the convexity properties of the initial data. Significant numerical examples illustrate the effectiveness of the proposed method

Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian

We prove the existence of various families of irreducible homaloidal hypersurfaces in projective space $\mathbb P^ r$, for all $r\geq 3$. Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared to the dimension of the ambient projective space. The existence of such a family solves a question that has naturally arisen from the consideration of the classes of homaloidal hypersurfaces known so far. The result relies on a fine analysis of dual hypersurfaces to certain scroll surfaces. We also introduce an infinite family of determinantal homaloidal hypersurfaces based on a certain degeneration of a generic Hankel matrix. These examples fit non--classical versions of de Jonqui\`eres transformations. As a natural counterpoint, we broaden up aspects of the theory of Gordan--Noether hypersurfaces with vanishing Hessian determinant, bringing over some more precision to the present knowledge.Comment: 56 pages. Some material added in section 1; minor changes. Final version to appear in Advances in Mathematic

Zoll Manifolds and Complex Surfaces

We classify compact surfaces with torsion-free affine connections for which every geodesic is a simple closed curve. In the process, we obtain completely new proofs of all the major results concerning the Riemannian case. In contrast to previous work, our approach is twistor-theoretic, and depends fundamentally on the fact that, up to biholomorphism, there is only one complex structure on CP2

Studies integrating geometry, probability, and optimization under convexity

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2006.Includes bibliographical references (p. 197-202).Convexity has played a major role in a variety of fields over the past decades. Nevertheless, the convexity assumption continues to reveal new theoretical paradigms and applications. This dissertation explores convexity in the intersection of three fields, namely, geometry, probability, and optimization. We study in depth a variety of geometric quantities. These quantities are used to describe the behavior of different algorithms. In addition, we investigate how to algorithmically manipulate these geometric quantities. This leads to algorithms capable of transforming ill-behaved instances into well-behaved ones. In particular, we provide probabilistic methods that carry out such task efficiently by exploiting the geometry of the problem. More specific contributions of this dissertation are as follows. (i) We conduct a broad exploration of the symmetry function of convex sets and propose efficient methods for its computation in the polyhedral case. (ii) We also relate the symmetry function with the computational complexity of an interior-point method to solve a homogeneous conic system. (iii) Moreover, we develop a family of pre-conditioners based on the symmetry function and projective transformations for such interior-point method.(cont.) The implementation of the pre-conditioners relies on geometric random walks. (iv) We developed the analysis of the re-scaled perceptron algorithm for a linear conic system. In this method a sequence of linear transformations is used to increase a condition measure associated with the problem. (v) Finally, we establish properties relating a probability density induced by an arbitrary norm and the geometry of its support. This is used to construct an efficient simulating annealing algorithm to test whether a convex set is bounded, where the set is represented only by a membership oracle.by Alexandre Belloni Nogueira.Ph.D