A Grassmann-Rayleigh quotient iteration for computing invariant subspaces

The classical Rayleigh quotient iteration (RQI) allows one to compute a one-dimensional invariant subspace of a symmetric matrix A. Here we propose a generalization of the RQI which computes a p-dimensional invariant subspace of A. Cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature.

A Grassman-Rayleigh quotient iteration for computing invariant subspaces

The classical Rayleigh Quotient Iteration (RQI) computes a 1-dimensional invariant subspace of a symmetric matrix A with cubic convergence. We propose a generalization of the RQI which computes a p-dimensional invariant subspace of A. The geometry of the algorithm on the Grassmann manifold Gr(p,n) is developed to show cubic convergence and to draw connections with recently proposed Newton algorithms on Riemannian manifolds.

Efficient topological localization using orientation adjacency coherence histograms

This paper describes an efficient vision-based global topological localization approach that uses a coarse-to-fine strategy. Orientation Adjacency Coherence Histogram (OACH), a novel image feature, is proposed to improve the coarse localization. The coarse localization results are taken as inputs for the fine localization which is carried out by matching Harris-Laplace interest points characterized by the SIFT descriptor. Computation of OACHs and interest points is efficient due to the fact that these features are computed in an integrated process. We have implemented and tested the localization system in real environments. The experimental results demonstrate that our approach is efficient and reliable in both indoor and outdoor environments. © 2006 IEEE.

Combining interest points and edges for content-based image retrieval

This paper presents a novel approach using combined features to retrieve images containing specific objects, scenes or buildings. The content of an image is characterized by two kinds of features: Harris-Laplace interest points described by the SIFT descriptor and edges described by the edge color histogram. Edges and corners contain the maximal amount of information necessary for image retrieval. The feature detection in this work is an integrated process: edges are detected directly based on the Harris function; Harris interest points are detected at several scales and Harris-Laplace interest points are found using the Laplace function. The combination of edges and interest points brings efficient feature detection and high recognition ratio to the image retrieval system. Experimental results show this system has good performance. © 2005 IEEE.

Pothole detection in asphalt pavement images

Pavement condition assessment is essential when developing road network maintenance programs. In practice, the data collection process is to a large extent automated. However, pavement distress detection (cracks, potholes, etc.) is mostly performed manually, which is labor-intensive and time-consuming. Existing methods either rely on complete 3D surface reconstruction, which comes along with high equipment and computation costs, or make use of acceleration data, which can only provide preliminary and rough condition surveys. In this paper we present a method for automated pothole detection in asphalt pavement images. In the proposed method an image is first segmented into defect and non-defect regions using histogram shape-based thresholding. Based on the geometric properties of a defect region the potential pothole shape is approximated utilizing morphological thinning and elliptic regression. Subsequently, the texture inside a potential defect shape is extracted and compared with the texture of the surrounding non-defect pavement in order to determine if the region of interest represents an actual pothole. This methodology has been implemented in a MATLAB prototype, trained and tested on 120 pavement images. The results show that this method can detect potholes in asphalt pavement images with reasonable accuracy.

Defects Detection and Assessment in Concrete Surfaces

Aside from cracks, the impact of other surface defects, such as air pockets and discoloration, can be detrimental to the quality of concrete in terms of strength, appearance and durability. For this reason, local and national codes provide standards for quantifying the quality impact of these concrete surface defects and owners plan for regular visual inspections to monitor surface conditions. However, manual visual inspection of concrete surfaces is a qualitative (and subjective) process with often unreliable results due to its reliance on inspectors’ own criteria and experience. Also, it is labor intensive and time-consuming. This paper presents a novel, automated concrete surface defects detection and assessment approach that addresses these issues by automatically quantifying the extent of surface deterioration. According to this approach, images of the surface shot from a certain angle/distance can be used to automatically detect the number and size of surface air pockets, and the degree of surface discoloration. The proposed method uses histogram equalization and filtering to extract such defects and identify their properties (e.g. size, shape, location). These properties are used to quantify the degree of impact on the concrete surface quality and provide a numerical tool to help inspectors accurately evaluate concrete surfaces. The method has been implemented in C++ and results that validate its performance are presented.

Detection of Construction Workers in Video Frames for Automatic Initialization of Vision Trackers

Vision based tracking can provide the spatial location of construction entities such as equipment, workers, and materials in large scale, congested construction sites. It tracks entities in video streams by inferring their locations based on the entities’ visual features and motion histories. To initiate the process, it is necessary to determine the pixel areas corresponding to the construction entities to be tracked in the following consecutive video frames. In order to fully automate the process, an automated way of initialization is needed. This paper presents the method for construction worker detection which can automatically recognize and localize construction workers in video frames. The method first finds the foreground areas of moving objects using a background subtraction method. Within these foreground areas, construction workers are recognized based on the histogram of oriented gradients (HOG) and histogram of the HSV colors. HOG’s have proved to work effectively for detection of people, and the histogram of HSV colors helps differentiate between pedestrians and construction workers wearing safety vests. Preliminary experiments show that the proposed method has the potential to automate the initialization process of vision based tracking.

Construction Worker Detection in Video Frames for Initializing Vision Trackers

Monitoring the location of resources on large scale, congested, outdoor sites can be performed more efficiently with vision tracking, as this approach does not require any pre-tagging of resources. However, the greatest impediment to the use of vision tracking in this case is the lack of detection methods that are needed to automatically mark the resources of interest and initiate the tracking. This paper presents such a novel method for construction worker detection that localizes construction workers in video frames. The proposed method exploits motion, shape, and color cues to narrow down the detection regions to moving objects, people, and finally construction workers, respectively. The three cues are characterized by using background subtraction, the histogram of oriented gradients (HOG), and the HSV color histogram. The method has been tested on videos taken in various environments. The results demonstrate its suitability for automatic initialization of vision trackers.

Low-rank optimization on the cone of positive semidefinite matrices

We propose an algorithm for solving optimization problems defined on a subset of the cone of symmetric positive semidefinite matrices. This algorithm relies on the factorization X = Y Y T , where the number of columns of Y fixes an upper bound on the rank of the positive semidefinite matrix X. It is thus very effective for solving problems that have a low-rank solution. The factorization X = Y Y T leads to a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second-order optimization method with guaranteed quadratic convergence. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. In contrast to existing methods, the proposed algorithm converges monotonically to the sought solution. Its numerical efficiency is evaluated on two applications: the maximal cut of a graph and the problem of sparse principal component analysis. © 2010 Society for Industrial and Applied Mathematics.

Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank

This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is invariant with respect to all transformations that preserve angles (orthogonal transformations, scalings, and pseudoinversion). A meaningful approximation of the associated Riemannian distance is proposed, that can be efficiently numerically computed via a simple algorithm based on SVD. The induced mean preserves the rank, possesses the most desirable characteristics of a geometric mean, and is easy to compute. © 2009 Society for Industrial and Applied Mathematics.

Continuous-time subspace flows related to the symmetric eigenproblem

The classes of continuous-time flows on Rn×p that induce the same flow on the set of p- dimensional subspaces of Rn×p are described. The power flow is briefly reviewed in this framework, and a subspace generalization of the Rayleigh quotient flow [Linear Algebra Appl. 368C, 2003, pp. 343-357] is proposed and analyzed. This new flow displays a property akin to deflation in finite time. © 2008 Yokohama Publishers.

Free energy surface reconstruction from umbrella samples using Gaussian process regression II: Multiple collective variables

We demonstrate how a prior assumption of smoothness can be used to enhance the reconstruction of free energy profiles from multiple umbrella sampling simulations using the Bayesian Gaussian process regression approach. The method we derive allows the concurrent use of histograms and free energy gradients and can easily be extended to include further data. In Part I we review the necessary theory and test the method for one collective variable. We demonstrate improved performance with respect to the weighted histogram analysis method and obtain meaningful error bars without any significant additional computation. In Part II we consider the case of multiple collective variables and compare to a reconstruction using least squares fitting of radial basis functions. We find substantial improvements in the regimes of spatially sparse data or short sampling trajectories. A software implementation is made available on www.libatoms.org.