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Hidden Variable Resultant Approach for Classical Computer Vision Problems Thesis Directed by Professor Jonathan Ventura

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Hidden Variable Resultant Approach for Classical Computer Vision Problems Thesis Directed by Professor Jonathan Ventura HIDDEN VARIABLE RESULTANT APPROACH FOR CLASSICAL COMPUTER VISION PROBLEMS by JORDI NONELL PARE B.S., Universitat Politècnica de Catalunya, 2016 M.S., Universitat Politècnica de Catalunya, 2016 A thesis submitted to the Graduate Faculty of the University of Colorado Colorado Springs in partial fulfillment of the requirements for the degree of Master of Science Department of Computer Science 2018 This thesis for the Master of Science degree by Jordi Nonell Paré has been approved for the Department of Computer Science by Terrance Boult, Chair Jonathan Ventura Sudanshu Semwal December 12th, 2018 ii Nonell Paré, Jordi (M.S., Computer Science) Hidden Variable Resultant Approach for Classical Computer Vision Problems Thesis directed by Professor Jonathan Ventura. Abstract Many problems in computer vision may be formulated as minimal problems; problems that require a minimal number of inputs and solving them equals to solve a system of non-linear polynomial equations with a finite number of solutions. Problems like relative and absolute camera pose computation fall into this category. This systems of polynomials usually do not have a straightfor- ward solution, which makes the general algorithms for solving systems of polynomials not performant enough. This raises the need to develop concrete algorithms that solve those particular problems. This thesis will review the state of the art for the current solvers for this problems, and propose a novel method to tackle this prob- lems based on combining the resultant theory with Gröbner basis methods and numerical optimization. iii Acknowledgments I would like to start thanking Dr. Ventura for introducing me to the world of computer vision, first as my professor and later as my advisor, and for all the help, patience and understanding he showed as I was learning about his topic. In addition I would like to express my gratitude to the committee for providing comments and suggestions in a niche topic like this one. I would also make a special mention to the Balsells Foundation, whose fellowship has granted me the incredible opportunity to go through this master program. And finally, I would like to thank my family and all my friends, the ones from Barcelona, and their evenings spent in video calls, and the new ones from Colorado, with whom I have created unbe- lievable memories. iv Contents 1 Introduction 1 2 Contribution of the thesis 3 3 State-of-the-Art 4 3.1 Systems of polynomial equations 4 3.1.1 Numerical methods 5 3.1.2 Algebraic methods 5 3.2 Minimal problems in computer vision 8 4 Solving systems of polynomial equations 10 4.1 Background theory 10 4.2 Gröbner bases 13 4.3 Elimination templates 17 4.4 Hidden variable resultant 18 5 Minimal problems 19 5.1 Preliminaries 20 5.2 Five-point relative pose 21 5.3 Perspective-3-point 25 6 Conclusion 29 v Bibliography 31 vi List of Figures 5.1 Five-point relative pose problem schematic 22 5.2 Residuals comparison for the five point relative pose problem 24 5.3 Time comparison for the five point relative pose prob- lem 25 5.4 Perspective-3-point schematic 26 5.5 Residuals comparison for the five point relative pose problem 28 5.6 Time comparison for the five point relative pose prob- lem 28 vii List of Tables 4.1 Multiplication table in quotient ring 15 viii Chapter 1 Introduction Many problems in computer vision can be formulated as systems of non-linear polynomial equations. Examples of such can be estimate the camera’s absolute pose, which means determine the position and orientation of the camera, and also the intrinsic parameters, de- pending on the problem. Improving the efficiency of this problems has an impact to several applications, from 3D reconstruction to robotics and augmented reality. Solving a system of non-linear polynomial equations is a well- studied problem in algebraic geometry, for which different already developed mathematical methods exist. Because of that, specific solvers for problems regarding computer vision have been devel- oped and improved over and over. Those methods can be classified roughly in two big approaches: numerical methods, which we won’t consider in this thesis, and algebraic methods. Solving a system of non-linear polynomial equations is an old and well-studied problem, whose solutions can roughly be divided into two big groups: numerical methods and algebraic methods. In this thesis we will focus on the algebraic approach. The main problem surrounding the existing mathematical methods is that being so general, usually they are not designed to find the solutions as efficiently and fast as what is needed for computer vision. Among the algebraic methods the problem is usually limited to a minimal problem, which means that only the minimal number 1 of point correspondences is used to find the solution. Since this approach is susceptible to fail because of outliers, this algorithms are usually implemented inside a RANSAC [5] framework, which also belong to a bigger system. This surrounding conditions force a particular set of requirements to the performance required for computer vision problems, implying that the solvers may need to work under real time situations. The fact that there is a simpler subset of problems that provide value to the computer vision community also means that particular solutions for this subset of concrete problems can be found. In other words, there is no need for an all purpose general solver for systems of polynomials equations, and the concrete developed solver is allowed to take advantage of the particular conditions of the problem. That usually gets translated into precoumputing parts of a general problem so they can be solved faster with concrete coefficients afterwards. In this thesis we revisit some of the current approaches into solving this systems using algebraic methods, and discuss the ad- vantages and drawbacks of them. Also, we explain our general approach to tackle the computer vision problems, and how it can generalize and be used for many different problems. 2 Chapter 2 Contribution of the thesis The focus of this thesis is in how to apply algebraic methods to solve different computer vision problems. It aims to test a new method in two well-known classic minimal problems, to check if it is suitable to improve the current state of the art in some cases. The main contribution of this thesis can be found in the algebraic development on chapter 5 as our method is mathematically proven to be correct, i.e. yield the correct results, by using the tools and theorems provided by the state-of-the-art contributions to algebraic geometry. After that, this thesis checks the performance of this method with synthetic data in two different problems, the five-point relative pose problem and the perspective-3-points problem, so this method can be considered suitable or not for real-world data. 3 Chapter 3 State-of-the-Art In this chapter we summarize the state of the art on how to solve non-linear systems of polynomial equations, and then we describe the state of the art of the problems in computer vision, particularly referring to the so called minimal problems. 3.1 Systems of polynomial equations A system of polynomial equations is a system that takes the follow- ing form f1(x)= ... = fm(x)= 0 (3.1) Cn where x ∈ and fi(x) ∈ K[x1, ..., xn]. Solving systems of polynomial equations, linear and no linear ones, is an ancient problem which has been approached from many different perspectives for centuries. Therefore, we can find many different approaches about how to tackle it. A common way to classify this approaches is to divide them into numerical methods and algebraic methods. This thesis focuses mostly into algebraic methods, but we will talk briefly about the former. 4 3.1.1 Numerical methods This methods can also be divided into two main groups. The first big group of numerical methods to solve a system of polynomials are the iterative methods. This methods take an initial guess of the solution of the system, and successively obtain better approximations to the actual roots of the system. This methods are mostly developed from Newton’s method for one variable poly- nomials, and can be extend to system of multiple equations and several variables, in a diverse number of ways, changing the order of convergence of the method to the actual solution. The biggest problem concerning this methods is that they require a close enough initial guess to each of the solutions so the method can converge, and that guess is not always easy to find. The second numerical approach departs from homotopy meth- ods. This methods try to avoid the problem of the initial guess that iterative methods have to perform, and take a completely different approach. First, the system computes a system with the same struc- ture and number of solutions than the original system, but whose solutions are already known. Next, it transforms that system into the original system, keeping track about how the roots move as the system gets transformed. In practical terms, homotopy methods require a huge amount of computation to approximate one system to the other. 3.1.2 Algebraic methods Algebraic methods are based in partially solving the system so the different variables can be eliminated and the thus the problem is later reduced to finding the roots of an univariate polynomial. This kind of approach relies on obtaining the roots with enough precision, therefore is not suitable for larger systems or high degree polynomials, since the time spent in computation will eventually be excessive. But for the kind of problems in computer vision, those 5 methods are still tractable and produce results with the accuracy desired.
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