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.

Bayesian Inference in Large-scale Problems

Many modern applications fall into the category of "large-scale" statistical problems, in which both the number of observations n and the number of features or parameters p may be large. Many existing methods focus on point estimation, despite the continued relevance of uncertainty quantification in the sciences, where the number of parameters to estimate often exceeds the sample size, despite huge increases in the value of n typically seen in many fields. Thus, the tendency in some areas of industry to dispense with traditional statistical analysis on the basis that "n=all" is of little relevance outside of certain narrow applications. The main result of the Big Data revolution in most fields has instead been to make computation much harder without reducing the importance of uncertainty quantification. Bayesian methods excel at uncertainty quantification, but often scale poorly relative to alternatives. This conflict between the statistical advantages of Bayesian procedures and their substantial computational disadvantages is perhaps the greatest challenge facing modern Bayesian statistics, and is the primary motivation for the work presented here.

Two general strategies for scaling Bayesian inference are considered. The first is the development of methods that lend themselves to faster computation, and the second is design and characterization of computational algorithms that scale better in n or p. In the first instance, the focus is on joint inference outside of the standard problem of multivariate continuous data that has been a major focus of previous theoretical work in this area. In the second area, we pursue strategies for improving the speed of Markov chain Monte Carlo algorithms, and characterizing their performance in large-scale settings. Throughout, the focus is on rigorous theoretical evaluation combined with empirical demonstrations of performance and concordance with the theory.

One topic we consider is modeling the joint distribution of multivariate categorical data, often summarized in a contingency table. Contingency table analysis routinely relies on log-linear models, with latent structure analysis providing a common alternative. Latent structure models lead to a reduced rank tensor factorization of the probability mass function for multivariate categorical data, while log-linear models achieve dimensionality reduction through sparsity. Little is known about the relationship between these notions of dimensionality reduction in the two paradigms. In Chapter 2, we derive several results relating the support of a log-linear model to nonnegative ranks of the associated probability tensor. Motivated by these findings, we propose a new collapsed Tucker class of tensor decompositions, which bridge existing PARAFAC and Tucker decompositions, providing a more flexible framework for parsimoniously characterizing multivariate categorical data. Taking a Bayesian approach to inference, we illustrate empirical advantages of the new decompositions.

Latent class models for the joint distribution of multivariate categorical, such as the PARAFAC decomposition, data play an important role in the analysis of population structure. In this context, the number of latent classes is interpreted as the number of genetically distinct subpopulations of an organism, an important factor in the analysis of evolutionary processes and conservation status. Existing methods focus on point estimates of the number of subpopulations, and lack robust uncertainty quantification. Moreover, whether the number of latent classes in these models is even an identified parameter is an open question. In Chapter 3, we show that when the model is properly specified, the correct number of subpopulations can be recovered almost surely. We then propose an alternative method for estimating the number of latent subpopulations that provides good quantification of uncertainty, and provide a simple procedure for verifying that the proposed method is consistent for the number of subpopulations. The performance of the model in estimating the number of subpopulations and other common population structure inference problems is assessed in simulations and a real data application.

In contingency table analysis, sparse data is frequently encountered for even modest numbers of variables, resulting in non-existence of maximum likelihood estimates. A common solution is to obtain regularized estimates of the parameters of a log-linear model. Bayesian methods provide a coherent approach to regularization, but are often computationally intensive. Conjugate priors ease computational demands, but the conjugate Diaconis--Ylvisaker priors for the parameters of log-linear models do not give rise to closed form credible regions, complicating posterior inference. In Chapter 4 we derive the optimal Gaussian approximation to the posterior for log-linear models with Diaconis--Ylvisaker priors, and provide convergence rate and finite-sample bounds for the Kullback-Leibler divergence between the exact posterior and the optimal Gaussian approximation. We demonstrate empirically in simulations and a real data application that the approximation is highly accurate, even in relatively small samples. The proposed approximation provides a computationally scalable and principled approach to regularized estimation and approximate Bayesian inference for log-linear models.

Another challenging and somewhat non-standard joint modeling problem is inference on tail dependence in stochastic processes. In applications where extreme dependence is of interest, data are almost always time-indexed. Existing methods for inference and modeling in this setting often cluster extreme events or choose window sizes with the goal of preserving temporal information. In Chapter 5, we propose an alternative paradigm for inference on tail dependence in stochastic processes with arbitrary temporal dependence structure in the extremes, based on the idea that the information on strength of tail dependence and the temporal structure in this dependence are both encoded in waiting times between exceedances of high thresholds. We construct a class of time-indexed stochastic processes with tail dependence obtained by endowing the support points in de Haan's spectral representation of max-stable processes with velocities and lifetimes. We extend Smith's model to these max-stable velocity processes and obtain the distribution of waiting times between extreme events at multiple locations. Motivated by this result, a new definition of tail dependence is proposed that is a function of the distribution of waiting times between threshold exceedances, and an inferential framework is constructed for estimating the strength of extremal dependence and quantifying uncertainty in this paradigm. The method is applied to climatological, financial, and electrophysiology data.

The remainder of this thesis focuses on posterior computation by Markov chain Monte Carlo. The Markov Chain Monte Carlo method is the dominant paradigm for posterior computation in Bayesian analysis. It has long been common to control computation time by making approximations to the Markov transition kernel. Comparatively little attention has been paid to convergence and estimation error in these approximating Markov Chains. In Chapter 6, we propose a framework for assessing when to use approximations in MCMC algorithms, and how much error in the transition kernel should be tolerated to obtain optimal estimation performance with respect to a specified loss function and computational budget. The results require only ergodicity of the exact kernel and control of the kernel approximation accuracy. The theoretical framework is applied to approximations based on random subsets of data, low-rank approximations of Gaussian processes, and a novel approximating Markov chain for discrete mixture models.

Data augmentation Gibbs samplers are arguably the most popular class of algorithm for approximately sampling from the posterior distribution for the parameters of generalized linear models. The truncated Normal and Polya-Gamma data augmentation samplers are standard examples for probit and logit links, respectively. Motivated by an important problem in quantitative advertising, in Chapter 7 we consider the application of these algorithms to modeling rare events. We show that when the sample size is large but the observed number of successes is small, these data augmentation samplers mix very slowly, with a spectral gap that converges to zero at a rate at least proportional to the reciprocal of the square root of the sample size up to a log factor. In simulation studies, moderate sample sizes result in high autocorrelations and small effective sample sizes. Similar empirical results are observed for related data augmentation samplers for multinomial logit and probit models. When applied to a real quantitative advertising dataset, the data augmentation samplers mix very poorly. Conversely, Hamiltonian Monte Carlo and a type of independence chain Metropolis algorithm show good mixing on the same dataset.

Distribution and Conservation of the Antillean Manatee in Hispaniola

Antillean manatees (Trichechus manatus manatus) were heavily hunted in the past throughout the Wider Caribbean Region (WCR), and are currently listed as endangered on the IUCN Red List of Threatened Species. In most WCR countries, including Haiti and the Dominican Republic, remaining manatee populations are believed to be small and declining, but current information is needed on their status, distribution, and local threats to the species.

To assess the past and current distribution and conservation status of the Antillean manatee in Hispaniola, I conducted a systematic review of documentary archives dating from the pre-Columbian era to 2013. I then surveyed more than 670 artisanal fishers from Haiti and the Dominican Republic in 2013-2014 using a standardized questionnaire. Finally, to identify important areas for manatees in the Dominican Republic, I developed a country-wide ensemble model of manatee distribution, and compared modeled hotspots with those identified by fishers.

Manatees were historically abundant in Hispaniola, but were hunted for their meat and became relatively rare by the end of the 19th century. The use of manatee body parts diversified with time to include their oil, skin, and bones. Traditional uses for folk medicine and handcrafts persist today in coastal communities in the Dominican Republic. Most threats to Antillean manatees in Hispaniola are anthropogenic in nature, and most mortality is caused by fisheries. I estimated a minimum island-wide annual mortality of approximately 20 animals. To understand the impact of this level of mortality, and to provide a baseline for measuring the success of future conservation actions, the Dominican Republic and Haiti should work together to obtain a reliable estimate of the current population size of manatees in Hispaniola.

In Haiti, the survey of fishers showed a wider distribution range of the species than suggested by the documentary archive review: fishers reported recent manatee sightings in seven of nine coastal departments, and three manatee hotspot areas were identified in the north, central, and south coasts. Thus, the contracted manatee distribution range suggested by the documentary archive review likely reflects a lack of research in Haiti. Both the review and the interviews agreed that manatees no longer occupy freshwater habitats in the country. In general, more dedicated manatee studies are needed in Haiti, employing aerial, land, or boat surveys.

In the Dominican Republic, the documentary archive review and the survey of fishers showed that manatees still occur throughout the country, and occasionally occupy freshwater habitats. Monte Cristi province in the north coast, and Barahona province in the south coast, were identified as focal areas. Sighting reports of manatees decreased from Monte Cristi eastwards to the adjacent province in the Dominican Republic, and westwards into Haiti. Along the north coast of Haiti, the number of manatee sighting and capture reports decreased with increasing distance to Monte Cristi province. There was good agreement among the modeled manatee hotspots, hotspots identified by fishers, and hotspots identified during previous dedicated manatee studies. The concordance of these results suggests that the distribution and patterns of habitat use of manatees in the Dominican Republic have not changed dramatically in over 30 years, and that the remaining manatees exhibit some degree of site fidelity. The ensemble modeling approach used in the present study produced accurate and detailed maps of manatee distribution with minimum data requirements. This modeling strategy is replicable and readily transferable to other countries in the Caribbean or elsewhere with limited data on a species of interest.

The intrinsic value of manatees was stronger for artisanal fishers in the Dominican Republic than in Haiti, and most Dominican fishers showed a positive attitude towards manatee conservation. The Dominican Republic is an upper middle income country with a high Human Development Index. It possesses a legal framework that specifically protects manatees, and has a greater number of marine protected areas, more dedicated manatee studies, and more manatee education and awareness campaigns than Haiti. The constant presence of manatees in specific coastal segments of the Dominican Republic, the perceived decline in the number of manatee captures, and a more conservation-minded public, offer hope for manatee conservation, as non-consumptive uses of manatees become more popular. I recommend a series of conservation actions in the Dominican Republic, including: reducing risks to manatees from harmful fishing gear and watercraft at confirmed manatee hotspots; providing alternative economic alternatives for displaced fishers, and developing responsible ecotourism ventures for manatee watching; improving law enforcement to reduce fisheries-related manatee deaths, stop the illegal trade in manatee body parts, and better protect manatee habitat; and continuing education and awareness campaigns for coastal communities near manatee hotspots.

In contrast, most fishers in Haiti continue to value manatees as a source of food and income, and showed a generally negative attitude towards manatee conservation. Haiti is a low income country with a low Human Development Index. Only a single dedicated manatee study has been conducted in Haiti, and manatees are not officially protected. Positive initiatives for manatees in Haiti include: protected areas declared in 2013 and 2014 that enclose two of the manatee hotspots identified in the present study; and local organizations that are currently working on coastal and marine environmental issues, including research and education on marine mammals. Future conservation efforts for manatees in Haiti should focus on addressing poverty and providing viable economic alternatives for coastal communities. I recommend a community partnership approach for manatee conservation, paired with education and awareness campaigns to inform coastal communities about the conservation situation of manatees in Haiti, and to help change their perceived value. Haiti should also provide legal protection for manatees and their habitat.

Does the degree of coarctation of the aorta influence wall shear stress focal heterogeneity?

The development of atherosclerosis in the aorta is associated with low and oscillatory wall shear stress for normal patients. Moreover, localized differences in wall shear stress heterogeneity have been correlated with the presence of complex plaques in the descending aorta. While it is known that coarctation of the aorta can influence indices of wall shear stress, it is unclear how the degree of narrowing influences resulting patterns. We hypothesized that the degree of coarctation would have a strong influence on focal heterogeneity of wall shear stress. To test this hypothesis, we modeled the fluid dynamics in a patient-specific aorta with varied degrees of coarctation. We first validated a massively parallel computational model against experimental results for the patient geometry and then evaluated local shear stress patterns for a range of degrees of coarctation. Wall shear stress patterns at two cross sectional slices prone to develop atherosclerotic plaques were evaluated. Levels at different focal regions were compared to the conventional measure of average circumferential shear stress to enable localized quantification of coarctation-induced shear stress alteration. We find that the coarctation degree causes highly heterogeneous changes in wall shear stress.