Computing Distance Functions from Generalized Sources on Weighted Polyhedral Surfaces

We present algorithms for computing approximate distance functions and shortest paths from a generalized source (point, segment, polygonal chain or polygonal region) on a weighted non-convex polyhedral surface in which obstacles (represented by polygonal chains or polygons) are allowed. We also describe an algorithm for discretizing, by using graphics hardware capabilities, distance functions. Finally, we present algorithms for computing discrete k-order Voronoi diagrams

Learning and Exploiting Camera Geometry for Computer Vision

This work explores the use of statistical methods in describing and estimating camera poses, as well as the information feedback loop between camera pose and object detection. Surging development in robotics and computer vision has pushed the need for algorithms that infer, understand, and utilize information about the position and orientation of the sensor platforms when observing and/or interacting with their environment.

The first contribution of this thesis is the development of a set of statistical tools for representing and estimating the uncertainty in object poses. A distribution for representing the joint uncertainty over multiple object positions and orientations is described, called the mirrored normal-Bingham distribution. This distribution generalizes both the normal distribution in Euclidean space, and the Bingham distribution on the unit hypersphere. It is shown to inherit many of the convenient properties of these special cases: it is the maximum-entropy distribution with fixed second moment, and there is a generalized Laplace approximation whose result is the mirrored normal-Bingham distribution. This distribution and approximation method are demonstrated by deriving the analytical approximation to the wrapped-normal distribution. Further, it is shown how these tools can be used to represent the uncertainty in the result of a bundle adjustment problem.

Another application of these methods is illustrated as part of a novel camera pose estimation algorithm based on object detections. The autocalibration task is formulated as a bundle adjustment problem using prior distributions over the 3D points to enforce the objects' structure and their relationship with the scene geometry. This framework is very flexible and enables the use of off-the-shelf computational tools to solve specialized autocalibration problems. Its performance is evaluated using a pedestrian detector to provide head and foot location observations, and it proves much faster and potentially more accurate than existing methods.

Finally, the information feedback loop between object detection and camera pose estimation is closed by utilizing camera pose information to improve object detection in scenarios with significant perspective warping. Methods are presented that allow the inverse perspective mapping traditionally applied to images to be applied instead to features computed from those images. For the special case of HOG-like features, which are used by many modern object detection systems, these methods are shown to provide substantial performance benefits over unadapted detectors while achieving real-time frame rates, orders of magnitude faster than comparable image warping methods.

The statistical tools and algorithms presented here are especially promising for mobile cameras, providing the ability to autocalibrate and adapt to the camera pose in real time. In addition, these methods have wide-ranging potential applications in diverse areas of computer vision, robotics, and imaging.

Development of patient-specific computer simulation tools for endoscopic scoliosis surgery

Endoscopic approaches for anterior correction of idiopathic scoliosis are a relatively new surgical technique. This paper describes the development of patient-specific finite element modelling techniques to investigate the biomechanics of single rod anterior scoliosis correction. Spinal geometry is obtained from pre-operative CT scans and material properties for osteo-ligamentous spinal tissues are based on existing literature. The techniques being developed will allow pre-surgical prediction of stresses, forces and deformations in spinal tissues, rods and screws under post-operative physiological loads.

Geometry‐specified troposphere decorrelation for subcentimeter real‐time kinematic solutions over long baselines

Real‐time kinematic (RTK) GPS techniques have been extensively developed for applications including surveying, structural monitoring, and machine automation. Limitations of the existing RTK techniques that hinder their applications for geodynamics purposes are twofold: (1) the achievable RTK accuracy is on the level of a few centimeters and the uncertainty of vertical component is 1.5–2 times worse than those of horizontal components and (2) the RTK position uncertainty grows in proportional to the base‐torover distances. The key limiting factor behind the problems is the significant effect of residual tropospheric errors on the positioning solutions, especially on the highly correlated height component. This paper develops the geometry‐specified troposphere decorrelation strategy to achieve the subcentimeter kinematic positioning accuracy in all three components. The key is to set up a relative zenith tropospheric delay (RZTD) parameter to absorb the residual tropospheric effects and to solve the established model as an ill‐posed problem using the regularization method. In order to compute a reasonable regularization parameter to obtain an optimal regularized solution, the covariance matrix of positional parameters estimated without the RZTD parameter, which is characterized by observation geometry, is used to replace the quadratic matrix of their “true” values. As a result, the regularization parameter is adaptively computed with variation of observation geometry. The experiment results show that new method can efficiently alleviate the model’s ill condition and stabilize the solution from a single data epoch. Compared to the results from the conventional least squares method, the new method can improve the longrange RTK solution precision from several centimeters to the subcentimeter in all components. More significantly, the precision of the height component is even higher. Several geosciences applications that require subcentimeter real‐time solutions can largely benefit from the proposed approach, such as monitoring of earthquakes and large dams in real‐time, high‐precision GPS leveling and refinement of the vertical datum. In addition, the high‐resolution RZTD solutions can contribute to effective recovery of tropospheric slant path delays in order to establish a 4‐D troposphere tomography.

Geometry vs appearance for discriminating between posed and spontaneous emotions

Spontaneous facial expressions differ from posed ones in appearance, timing and accompanying head movements. Still images cannot provide timing or head movement information directly. However, indirectly the distances between key points on a face extracted from a still image using active shape models can capture some movement and pose changes. This information is superposed on information about non-rigid facial movement that is also part of the expression. Does geometric information improve the discrimination between spontaneous and posed facial expressions arising from discrete emotions? We investigate the performance of a machine vision system for discrimination between posed and spontaneous versions of six basic emotions that uses SIFT appearance based features and FAP geometric features. Experimental results on the NVIE database demonstrate that fusion of geometric information leads only to marginal improvement over appearance features. Using fusion features, surprise is the easiest emotion (83.4% accuracy) to be distinguished, while disgust is the most difficult (76.1%). Our results find different important facial regions between discriminating posed versus spontaneous version of one emotion and classifying the same emotion versus other emotions. The distribution of the selected SIFT features shows that mouth is more important for sadness, while nose is more important for surprise, however, both the nose and mouth are important for disgust, fear, and happiness. Eyebrows, eyes, nose and mouth are important for anger.

Practical improvements to simultaneous computation of multi-view geometry and radial lens distortion

This paper discusses practical issues related to the use of the division model for lens distortion in multi-view geometry computation. A data normalisation strategy is presented, which has been absent from previous discussions on the topic. The convergence properties of the Rectangular Quadric Eigenvalue Problem solution for computing division model distortion are examined. It is shown that the existing method can require more than 1000 iterations when dealing with severe distortion. A method is presented for accelerating convergence to less than 10 iterations for any amount of distortion. The new method is shown to produce equivalent or better results than the existing method with up to two orders of magnitude reduction in iterations. Through detailed simulation it is found that the number of data points used to compute geometry and lens distortion has a strong influence on convergence speed and solution accuracy. It is recommended that more than the minimal number of data points be used when computing geometry using a robust estimator such as RANSAC. Adding two to four extra samples improves the convergence rate and accuracy sufficiently to compensate for the increased number of samples required by the RANSAC process.

Facilitating growth in prospective teachers’ knowledge : teaching geometry in primary schools

This paper reports on a study that focused on growth of understanding about teaching geometry by a group of prospective teachers engaged in lesson plan study within a computer-supported collaborative learning (CSCL) environment. Participation in the activity was found to facilitate considerable growth in the participants’ pedagogical-content knowledge (PCK). Factors that influenced growth in PCK included the nature of the lesson planning task, the cognitive scaffolds inserted into the CSCL virtual space, the meta-language scaffolds provided to the participants, and the provision of both private and public discourse spaces. The paper concludes with recommendations for enhancing effective knowledge-building discourse about mathematics PCK within prospective teacher education CSCL environments.

A multi-layered approach for site detection in UAS emergency landing scenarios using geometry-based image segmentation

This paper presents an alternative approach to image segmentation by using the spatial distribution of edge pixels as opposed to pixel intensities. The segmentation is achieved by a multi-layered approach and is intended to find suitable landing areas for an aircraft emergency landing. We combine standard techniques (edge detectors) with novel developed algorithms (line expansion and geometry test) to design an original segmentation algorithm. Our approach removes the dependency on environmental factors that traditionally influence lighting conditions, which in turn have negative impact on pixel-based segmentation techniques. We present test outcomes on realistic visual data collected from an aircraft, reporting on preliminary feedback about the performance of the detection. We demonstrate consistent performances over 97% detection rate.

Evaluation of two-view geometry methods with automatic ground-truth generation

A large number of methods have been published that aim to evaluate various components of multi-view geometry systems. Most of these have focused on the feature extraction, description and matching stages (the visual front end), since geometry computation can be evaluated through simulation. Many data sets are constrained to small scale scenes or planar scenes that are not challenging to new algorithms, or require special equipment. This paper presents a method for automatically generating geometry ground truth and challenging test cases from high spatio-temporal resolution video. The objective of the system is to enable data collection at any physical scale, in any location and in various parts of the electromagnetic spectrum. The data generation process consists of collecting high resolution video, computing accurate sparse 3D reconstruction, video frame culling and down sampling, and test case selection. The evaluation process consists of applying a test 2-view geometry method to every test case and comparing the results to the ground truth. This system facilitates the evaluation of the whole geometry computation process or any part thereof against data compatible with a realistic application. A collection of example data sets and evaluations is included to demonstrate the range of applications of the proposed system.

Analysis of terrain geometry representations for traversability of a Mars rover

For a planetary rover to successfully traverse across unstructured terrain autonomously, one of the major challenges is to assess its local traversability such that it can plan a trajectory to achieve its mission goals efficiently while minimising risk to the vehicle itself. This paper aims to provide a comparative study on different approaches for representing the geometry of Martian terrain for the purpose of evaluating terrain traversability. An accurate representation of the geometric properties of the terrain is essential as it can directly affect the determination of traversability for a ground vehicle. We explore current state-of-the-art techniques for terrain estimation, in particular Gaussian Processes (GP) in various forms, and discuss the suitability of each technique in the context of an unstructured Martian terrain. Furthermore, we present the limitations of regression techniques in terms of spatial correlation and continuity assumptions, and the impact on traversability analysis of a planetary rover across unstructured terrain. The analysis was performed on datasets of the Mars Yard at the Powerhouse Museum in Sydney, obtained using the onboard RGB-D camera.

Object tracking via non-Euclidean geometry : a Grassmann approach

A robust visual tracking system requires an object appearance model that is able to handle occlusion, pose, and illumination variations in the video stream. This can be difficult to accomplish when the model is trained using only a single image. In this paper, we first propose a tracking approach based on affine subspaces (constructed from several images) which are able to accommodate the abovementioned variations. We use affine subspaces not only to represent the object, but also the candidate areas that the object may occupy. We furthermore propose a novel approach to measure affine subspace-to-subspace distance via the use of non-Euclidean geometry of Grassmann manifolds. The tracking problem is then considered as an inference task in a Markov Chain Monte Carlo framework via particle filtering. Quantitative evaluation on challenging video sequences indicates that the proposed approach obtains considerably better performance than several recent state-of-the-art methods such as Tracking-Learning-Detection and MILtrack.

Automated fit quantification of tibial nail designs during the insertion using computer three-dimensional modelling

Intramedullary nailing is the standard fixation method for displaced diaphyseal fractures of the tibia. An optimal nail design should both facilitate insertion and anatomically fit the bone geometry at its final position in order to reduce the risk of stress fractures and malalignments. Due to the nonexistence of suitable commercial software, we developed a software tool for the automated fit assessment of nail designs. Furthermore, we demonstrated that an optimised nail, which fits better at the final position, is also easier to insert. Three-dimensional models of two nail designs and 20 tibiae were used. The fitting was quantified in terms of surface area, maximum distance, sum of surface areas and sum of maximum distances by which the nail was protruding into the cortex. The software was programmed to insert the nail into the bone model and to quantify the fit at defined increment levels. On average, the misfit during the insertion in terms of the four fitting parameters was smaller for the Expert Tibial Nail Proximal bend (476.3 mm2, 1.5 mm, 2029.8 mm2, 6.5 mm) than the Expert Tibial Nail (736.7 mm2, 2.2 mm, 2491.4 mm2, 8.0 mm). The differences were statistically significant (p ≤ 0.05). The software could be used by nail implant manufacturers for the purpose of implant design validation.

Geometry-free stochastic analysis of BDS triple frequency signals

The paper presents a geometry-free approach to assess the variation of covariance matrices of undifferenced triple frequency GNSS measurements and its impact on positioning solutions. Four independent geometryfree/ ionosphere-free (GFIF) models formed from original triple-frequency code and phase signals allow for effective computation of variance-covariance matrices using real data. Variance Component Estimation (VCE) algorithms are implemented to obtain the covariance matrices for three pseudorange and three carrier-phase signals epoch-by-epoch. Covariance results from the triple frequency Beidou System (BDS) and GPS data sets demonstrate that the estimated standard deviation varies in consistence with the amplitude of actual GFIF error time series. The single point positioning (SPP) results from BDS ionosphere-free measurements at four MGEX stations demonstrate an improvement of up to about 50% in Up direction relative to the results based on a mean square statistics. Additionally, a more extensive SPP analysis at 95 global MGEX stations based on GPS ionosphere-free measurements shows an average improvement of about 10% relative to the traditional results. This finding provides a preliminary confirmation that adequate consideration of the variation of covariance leads to the improvement of GNSS state solutions.