Generalized Weiszfeld algorithms for Lq optimization

In many computer vision applications, a desired model of some type is computed by minimizing a cost function based on several measurements. Typically, one may compute the model that minimizes the L₂ cost, that is the sum of squares of measurement errors with respect to the model. However, the Lq solution which minimizes the sum of the qth power of errors usually gives more robust results in the presence of outliers for some values of q, for example, q = 1. The Weiszfeld algorithm is a classic algorithm for finding the geometric L1 mean of a set of points in Euclidean space. It is provably optimal and requires neither differentiation, nor line search. The Weiszfeld algorithm has also been generalized to find the L1 mean of a set of points on a Riemannian manifold of non-negative curvature. This paper shows that the Weiszfeld approach may be extended to a wide variety of problems to find an Lq mean for 1 ≤ q <; 2, while maintaining simplicity and provable convergence. We apply this problem to both single-rotation averaging (under which the algorithm provably finds the global Lq optimum) and multiple rotation averaging (for which no such proof exists). Experimental results of Lq optimization for rotations show the improved reliability and robustness compared to L₂ optimization.This research has been funded by National ICT Australia

Inference of Non-Overlapping Camera Network Topology using Statistical Approaches

This work proposes an unsupervised learning model to infer the topological information of a camera network automatically. This algorithm works on non-overlapped and overlapped cameras field of views (FOVs). The constructed model detects the entry/exit zones of the moving objects across the cameras FOVs using the Data-Spectroscopic method. The probabilistic relationships between each pair of entry/exit zones are learnt to localize the camera network nodes. Increase the certainty of the probabilistic relationships using Computer-Generating to create more Monte Carlo observations of entry/exit points. Our method requires no assumptions, no processors for each camera and no communication among the cameras. The purpose is to figure out the relationship between each pair of linked cameras using the statistical approaches which help to track the moving objects depending on their present location. The Output is shown as a Markov chain model that represents the weighted-unit links between each pair of cameras FOV

Numerical Linear Algebra applications in Archaeology: the seriation and the photometric stereo problems

The aim of this thesis is to explore the application of Numerical Linear Algebra to Archaeology. An ordering problem called the seriation problem, used for dating findings and/or artifacts deposits, is analysed in terms of graph theory. In particular, a Matlab implementation of an algorithm for spectral seriation, based on the use of the Fiedler vector of the Laplacian matrix associated with the problem, is presented. We consider bipartite graphs for describing the seriation problem, since the interrelationship between the units (i.e. archaeological sites) to be reordered, can be described in terms of these graphs. In our archaeological metaphor of seriation, the two disjoint nodes sets into which the vertices of a bipartite graph can be divided, represent the excavation sites and the artifacts found inside them. Since it is a difficult task to determine the closest bipartite network to a given one, we describe how a starting network can be approximated by a bipartite one by solving a sequence of fairly simple optimization problems. Another numerical problem related to Archaeology is the 3D reconstruction of the shape of an object from a set of digital pictures. In particular, the Photometric Stereo (PS) photographic technique is considered

Synchronization Problems in Computer Vision

The goal of \u201csynchronization\u201d is to infer the unknown states of a network of nodes, where only the ratio (or difference) between pairs of states can be measured. Typically, states are represented by elements of a group, such as the Symmetric Group or the Special Euclidean Group. The former can represent local labels of a set of features, which refer to the multi-view matching application, whereas the latter can represent camera reference frames, in which case we are in the context of structure from motion, or local coordinates where 3D points are represented, in which case we are dealing with multiple point-set registration. A related problem is that of \u201cbearing-based network localization\u201d where each node is located at a fixed (unknown) position in 3-space and pairs of nodes can measure the direction of the line joining their locations. In this thesis we are interested in global techniques where all the measures are considered at once, as opposed to incremental approaches that grow a solution by adding pieces iteratively

Nudged Elastic Band in Topological Data Analysis

We use the nudged elastic band method from computational chemistry to analyze high-dimensional data. Our approach is inspired by Morse theory, and as output we produce an increasing sequence of small cell complexes modeling the dense regions of the data. We test the method on data sets arising in social networks and in image processing. Furthermore, we apply the method to identify new topological structure in a data set of optical flow patches

A fractal dimension for measures via persistent homology

We use persistent homology in order to define a family of fractal dimensions, denoted $\mathrm{dim}_{\mathrm{PH}}^i(\mu)$ for each homological dimension $i\ge 0$, assigned to a probability measure $\mu$ on a metric space. The case of $0$-dimensional homology ($i=0$) relates to work by Michael J Steele (1988) studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if $\mu$ is supported on a compact subset of Euclidean space $\mathbb{R}^m$ for $m\ge2$, then Steele's work implies that $\mathrm{dim}_{\mathrm{PH}}^0(\mu)=m$ if the absolutely continuous part of $\mu$ has positive mass, and otherwise $\mathrm{dim}_{\mathrm{PH}}^0(\mu)<m$. Experiments suggest that similar results may be true for higher-dimensional homology $0<i<m$, though this is an open question. Our fractal dimension is defined by considering a limit, as the number of points $n$ goes to infinity, of the total sum of the $i$-dimensional persistent homology interval lengths for $n$ random points selected from $\mu$ in an i.i.d. fashion. To some measures $\mu,$ we are able to assign a finer invariant, a curve measuring the limiting distribution of persistent homology interval lengths as the number of points goes to infinity. We prove this limiting curve exists in the case of $0$-dimensional homology when $\mu$ is the uniform distribution over the unit interval, and conjecture that it exists when $\mu$ is the rescaled probability measure for a compact set in Euclidean space with positive Lebesgue measure