Computational Methods for Computer Vision : Minimal Solvers and Convex Relaxations

Robust fitting of geometric models is a core problem in computer vision. The most common approach is to use a hypothesize-and-test framework, such as RANSAC. In these frameworks the model is estimated from as few measurements as possible, which minimizes the risk of selecting corrupted measurements. These estimation problems are called minimal problems, and they can often be formulated as systems of polynomial equations. In this thesis we present new methods for building so-called minimal solvers or polynomial solvers, which are specialized code for solving such systems. On several minimal problems we improve on the state-of-the-art both with respect to numerical stability and execution time.In many computer vision problems low rank matrices naturally occur. The rank can serve as a measure of model complexity and typically a low rank is desired. Optimization problems containing rank penalties or constraints are in general difficult. Recently convex relaxations, such as the nuclear norm, have been used to make these problems tractable. In this thesis we present new convex relaxations for rank-based optimization which avoid drawbacks of previous approaches and provide tighter relaxations. We evaluate our methods on a number of real and synthetic datasets and show state-of-the-art results

Implementation and Evaluation of Algorithmic Skeletons: Parallelisation of Computer Algebra Algorithms

This thesis presents design and implementation approaches for the parallel algorithms of computer algebra. We use algorithmic skeletons and also further approaches, like data parallel arithmetic and actors. We have implemented skeletons for divide and conquer algorithms and some special parallel loops, that we call ‘repeated computation with a possibility of premature termination’. We introduce in this thesis a rational data parallel arithmetic. We focus on parallel symbolic computation algorithms, for these algorithms our arithmetic provides a generic parallelisation approach. The implementation is carried out in Eden, a parallel functional programming language based on Haskell. This choice enables us to encode both the skeletons and the programs in the same language. Moreover, it allows us to refrain from using two different languages—one for the implementation and one for the interface—for our implementation of computer algebra algorithms. Further, this thesis presents methods for evaluation and estimation of parallel execution times. We partition the parallel execution time into two components. One of them accounts for the quality of the parallelisation, we call it the ‘parallel penalty’. The other is the sequential execution time. For the estimation, we predict both components separately, using statistical methods. This enables very confident estimations, although using drastically less measurement points than other methods. We have applied both our evaluation and estimation approaches to the parallel programs presented in this thesis. We haven also used existing estimation methods. We developed divide and conquer skeletons for the implementation of fast parallel multiplication. We have implemented the Karatsuba algorithm, Strassen’s matrix multiplication algorithm and the fast Fourier transform. The latter was used to implement polynomial convolution that leads to a further fast multiplication algorithm. Specially for our implementation of Strassen algorithm we have designed and implemented a divide and conquer skeleton basing on actors. We have implemented the parallel fast Fourier transform, and not only did we use new divide and conquer skeletons, but also developed a map-and-transpose skeleton. It enables good parallelisation of the Fourier transform. The parallelisation of Karatsuba multiplication shows a very good performance. We have analysed the parallel penalty of our programs and compared it to the serial fraction—an approach, known from literature. We also performed execution time estimations of our divide and conquer programs. This thesis presents a parallel map+reduce skeleton scheme. It allows us to combine the usual parallel map skeletons, like parMap, farm, workpool, with a premature termination property. We use this to implement the so-called ‘parallel repeated computation’, a special form of a speculative parallel loop. We have implemented two probabilistic primality tests: the Rabin–Miller test and the Jacobi sum test. We parallelised both with our approach. We analysed the task distribution and stated the fitting configurations of the Jacobi sum test. We have shown formally that the Jacobi sum test can be implemented in parallel. Subsequently, we parallelised it, analysed the load balancing issues, and produced an optimisation. The latter enabled a good implementation, as verified using the parallel penalty. We have also estimated the performance of the tests for further input sizes and numbers of processing elements. Parallelisation of the Jacobi sum test and our generic parallelisation scheme for the repeated computation is our original contribution. The data parallel arithmetic was defined not only for integers, which is already known, but also for rationals. We handled the common factors of the numerator or denominator of the fraction with the modulus in a novel manner. This is required to obtain a true multiple-residue arithmetic, a novel result of our research. Using these mathematical advances, we have parallelised the determinant computation using the Gauß elimination. As always, we have performed task distribution analysis and estimation of the parallel execution time of our implementation. A similar computation in Maple emphasised the potential of our approach. Data parallel arithmetic enables parallelisation of entire classes of computer algebra algorithms. Summarising, this thesis presents and thoroughly evaluates new and existing design decisions for high-level parallelisations of computer algebra algorithms

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Précision p-adique

P-Adic numbers are a field in arithmetic analoguous to the real numbers. The advent during the last few decades of arithmetic geometry has yielded many algorithms using those numbers. Such numbers can only by handled with finite precision. We design a method, that we call differential precision, to study the behaviour of the precision in a p-adic context. It reduces the study to a first-order problem. We also study the question of which Gröbner bases can be computed over a p-adic number field.Les nombres p-adiques sont un analogue des nombres réels plus proche de l’arithmétique. L’avènement ces dernières décennies de la géométrie arithmétique a engendré la création de nombreux algorithmes utilisant ces nombres. Ces derniers ne peuvent être de manière générale manipulés qu’à précision finie. Nous proposons une méthode, dite de précision différentielle, pour étudier ces problèmes de précision. Elle permet de se ramener à un problème au premier ordre. Nous nous intéressons aussi à la question de savoir quelles bases de Gröbner peuvent être calculées sur les p-adiques

Iterated-integral signatures in machine learning

The iterated-integral signature, or rough-path signature, of a path has proved useful in several machine learning applications in the last few years. This work is extended in a number of ways. Algorithms for computing the signature and log signature efficiently are investigated and evaluated, which is useful for many applications of signatures when working with large datasets. Online Chinese character recognition using signature features with recurrent neural networks is investigated. A recurrent neural network cell which stores its memory as the signature of a path is suggested and demonstrated on a toy problem. There is an essentially unique element of the signature of a path in space which, under transformations of the space, scales with volume. That element is characterised geometrically. Given two features of curves, you can make a new one by taking the signed area of the 2d curve those two features make as a curve is traced out. A simple algebraic description of those features (which turn out to be signature elements) which can be formed from linear combinations of such combinations of total displacements is conjectured and worked towards. This is know as “areas of areas”

MUSME 2011 4 th International Symposium on Multibody Systems and Mechatronics

El libro de actas recoge las aportaciones de los autores a través de los correspondientes artículos a la Dinámica de Sistemas Multicuerpo y la Mecatrónica (Musme). Estas disciplinas se han convertido en una importante herramienta para diseñar máquinas, analizar prototipos virtuales y realizar análisis CAD sobre complejos sistemas mecánicos articulados multicuerpo. La dinámica de sistemas multicuerpo comprende un gran número de aspectos que incluyen la mecánica, dinámica estructural, matemáticas aplicadas, métodos de control, ciencia de los ordenadores y mecatrónica. Los artículos recogidos en el libro de actas están relacionados con alguno de los siguientes tópicos del congreso: Análisis y síntesis de mecanismos ; Diseño de algoritmos para sistemas mecatrónicos ; Procedimientos de simulación y resultados ; Prototipos y rendimiento ; Robots y micromáquinas ; Validaciones experimentales ; Teoría de simulación mecatrónica ; Sistemas mecatrónicos ; Control de sistemas mecatrónicosUniversitat Politècnica de València (2011). MUSME 2011 4 th International Symposium on Multibody Systems and Mechatronics. Editorial Universitat Politècnica de València. http://hdl.handle.net/10251/13224Archivo delegad

Harnessing the power of GPUs for problems in real algebraic geometry

This thesis presents novel parallel algorithms to leverage the power of GPUs (Graphics Processing Units) for exact computations with polynomials having large integer coefficients. The significance of such computations, especially in real algebraic geometry, is hard to undermine. On massively-parallel architectures such as GPU, the degree of datalevel parallelism exposed by an algorithm is the main performance factor. We attain high efficiency through the use of structured matrix theory to assist the realization of relevant operations on polynomials on the graphics hardware. A detailed complexity analysis, assuming the PRAM model, also confirms that our approach achieves a substantially better parallel complexity in comparison to classical algorithms used for symbolic computations. Aside from the theoretical considerations, a large portion of this work is dedicated to the actual algorithm development and optimization techniques where we pay close attention to the specifics of the graphics hardware. As a byproduct of this work, we have developed high-throughput modular arithmetic which we expect to be useful for other GPU applications, in particular, open-key cryptography. We further discuss the algorithms for the solution of a system of polynomial equations, topology computation of algebraic curves and curve visualization which can profit to the full extent from the GPU acceleration. Extensive benchmarking on a real data demonstrates the superiority of our algorithms over several state-of-the-art approaches available to date. This thesis is written in English.Diese Arbeit beschäftigt sich mit neuen parallelen Algorithmen, die das Leistungspotenzial der Grafik-Prozessoren (GPUs) zur exakten Berechnungen mit ganzzahlige Polynomen nutzen. Solche symbolische Berechnungen sind von großer Bedeutung zur Lösung vieler Probleme aus der reellen algebraischen Geometrie. Für die effziente Implementierung eines Algorithmus auf massiv-parallelen Hardwarearchitekturen, wie z.B. GPU, ist vor allem auf eine hohe Datenparallelität zu achten. Unter Verwendung von Ergebnissen aus der strukturierten Matrix-Theorie konnten wir die entsprechenden Operationen mit Polynomen auf der Grafikkarte leicht übertragen. Außerdem zeigt eine Komplexitätanalyse im PRAM-Rechenmodell, dass die von uns entwickelten Verfahren eine deutlich bessere Komplexität aufweisen als dies für die klassischen Verfahren der Fall ist. Neben dem theoretischen Ergebnis liegt ein weiterer Schwerpunkt dieser Arbeit in der praktischen Implementierung der betrachteten Algorithmen, wobei wir auf der Besonderheiten der Grafikhardware achten. Im Rahmen dieser Arbeit haben wir hocheffiziente modulare Arithmetik entwickelt, von der wir erwarten, dass sie sich für andere GPU Anwendungen, insbesondere der Public-Key-Kryptographie, als nützlich erweisen wird. Darüber hinaus betrachten wir Algorithmen für die Lösung eines Systems von Polynomgleichungen, Topologie Berechnung der algebraischen Kurven und deren Visualisierung welche in vollem Umfang von der GPU-Leistung profitieren können. Zahlreiche Experimente belegen dass wir zur Zeit die beste Verfahren zur Verfügung stellen. Diese Dissertation ist in englischer Sprache verfasst