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

Diffeomorphic Transformations for Time Series Analysis: An Efficient Approach to Nonlinear Warping

The proliferation and ubiquity of temporal data across many disciplines has sparked interest for similarity, classification and clustering methods specifically designed to handle time series data. A core issue when dealing with time series is determining their pairwise similarity, i.e., the degree to which a given time series resembles another. Traditional distance measures such as the Euclidean are not well-suited due to the time-dependent nature of the data. Elastic metrics such as dynamic time warping (DTW) offer a promising approach, but are limited by their computational complexity, non-differentiability and sensitivity to noise and outliers. This thesis proposes novel elastic alignment methods that use parametric \& diffeomorphic warping transformations as a means of overcoming the shortcomings of DTW-based metrics. The proposed method is differentiable \& invertible, well-suited for deep learning architectures, robust to noise and outliers, computationally efficient, and is expressive and flexible enough to capture complex patterns. Furthermore, a closed-form solution was developed for the gradient of these diffeomorphic transformations, which allows an efficient search in the parameter space, leading to better solutions at convergence. Leveraging the benefits of these closed-form diffeomorphic transformations, this thesis proposes a suite of advancements that include: (a) an enhanced temporal transformer network for time series alignment and averaging, (b) a deep-learning based time series classification model to simultaneously align and classify signals with high accuracy, (c) an incremental time series clustering algorithm that is warping-invariant, scalable and can operate under limited computational and time resources, and finally, (d) a normalizing flow model that enhances the flexibility of affine transformations in coupling and autoregressive layers.Comment: PhD Thesis, defended at the University of Navarra on July 17, 2023. 277 pages, 8 chapters, 1 appendi

Subspace Clustering: A Possibilistic Approach

Ως συσταδοποίηση υποχώρων ορίζεται το πρόβλημα της μοντελοποίησης δεδομένων που βρίσκονται σε έναν ή και περισσότερους υποχώρους υπό την παρουσία θορύβου και περιέχουν ακραίες παρατηρήσεις και ελλιπή δεδομένα. Εξ όσων γνωρίζουμε, όλοι οι αλγόριθμοι που επιλύουν αυτό το πρόβλημα υποθέτουν ότι μια παρατήρηση ανήκει αυστηρά σε έναν υποχώρο. Η παρούσα διατριβή εξετάζει την περίπτωση όπου ένα σημείο μπορεί ταυτόχρονα και ανεξάρτητα να ανήκει σε παραπάνω από έναν υποχώρο. Ως αποτέλεσμα έχουμε την δημιουργία ενός καινούργιου αλγορίθμου, του sparse adaptive possibilistic K-subspaces (SAP K-subspaces). Ο αλγόριθμος αυτός αποτελεί γενίκευση του αλγορίθμου sparse possibilistic c-means algorithm (SPCM) [2], πράγμα που σημαίνει ότι μπορεί να διαχειριστεί με αξιοπιστία δεδομένα τόσο με θόρυβο και ακραίες τιμές όσο και δεδομένα τα οποία βρίσκονται σε τομές υποχώρων. Επίσης, ο καινούργιος αλγόριθμος αρχικοποιείται με περισσότερες συστάδες από τις πραγματικές, έχοντας την δυνατότητα απαλοιφής των περιττών συστάδων και τελικά την εύρεση αυτών που σχηματίζονται απο τα δεδομένα. Επιπλέον, υιοθετεί μια προσέγγιση εύρεσης γινομένου πινάκων χαμηλής τάξης για την εκτίμηση της διάστασης των υποχώρων [1]. Πειράματα σε συνθετικά και αληθινά δεδομένα επιβεβαιώνουν την αποτελεσματικότητα του αλγορίθμου. [1] Paris V Giampouras, Athanasios A Rontogiannis, and Konstantinos D Koutroumbas. Alternating iteratively reweighted least squares minimization for lowrank matrix factorization. IEEE Transactions on Signal Processing, 67(2):490–503, 2018. [2] Spyridoula D Xenaki, Konstantinos D Koutroumbas, and Athanasios A Rontogiannis. Sparsityaware possibilistic clustering algorithms. IEEE Transactions on Fuzzy Systems, 24(6):1611–1626, 2016.Subspace clustering is the problem of modeling a collection of data points lying in one or more subspaces in the presence of noise, outliers and missing data. To the best of our knowledge, all the algorithms associated to this problem follow a hard clustering philosophy. The study presented in this thesis explores the effectiveness of the possibilistic approach, giving rise to a novel iterative algorithm, called sparse adaptive possibilistic K- subspaces (SAP K-subspaces). SAP K-subspaces algorithm generalizes the sparse possibilistic c-means algorithm (SPCM) [2]. Hence, it inherits the ability to handle reliably data corrupted by noise and containing outliers, as well as data points near the intersections of subspaces. In addition, the new algorithm is suitably initialized with more clusters than those actually exist in the data set and has the ability to gradually eliminate the unnecessary ones in order to conclude with the true clusters, formed by the data. Moreover, it adopts the low-rank approach, introduced in [1], in order to estimate the dimension of the involved subspaces. Experiments on both synthetic and real data illustrate the effectiveness of the proposed method. [1] Paris V Giampouras, Athanasios A Rontogiannis, and Konstantinos D Koutroumbas. Alternating iteratively reweighted least squares minimization for lowrank matrix factorization. IEEE Transactions on Signal Processing, 67(2):490–503, 2018. [2] Spyridoula D Xenaki, Konstantinos D Koutroumbas, and Athanasios A Rontogiannis. Sparsityaware possibilistic clustering algorithms. IEEE Transactions on Fuzzy Systems, 24(6):1611–1626, 2016

A pp-adic RanSaC algorithm for stereo vision using Hensel lifting

A $p$-adic variation of the Ran(dom) Sa(mple) C(onsensus) method for solving the relative pose problem in stereo vision is developped. From two 2-adically encoded images a random sample of five pairs of corresponding points is taken, and the equations for the essential matrix are solved by lifting solutions modulo 2 to the 2-adic integers. A recently devised $p$-adic hierarchical classification algorithm imitating the known LBG quantisation method classifies the solutions for all the samples after having determined the number of clusters using the known intra-inter validity of clusterings. In the successful case, a cluster ranking will determine the cluster containing a 2-adic approximation to the "true" solution of the problem.Comment: 15 pages; typos removed, abstract changed, computation error remove