Differential geometry is deceptively simple. It is surprisingly easyto get the right answer with unclear and informal symbol manipulation.To address this problem we use computer programs to communicate aprecise understanding of the computations in differential geometry.Expressing the methods of differential geometry in a computer languageforces them to be unambiguous and computationally effective. The taskof formulating a method as a computer-executable program and debuggingthat program is a powerful exercise in the learning process. Also,once formalized procedurally, a mathematical idea becomes a tool thatcan be used directly to compute results.

Localized operators, like Gabor wavelets and difference-of-Gaussian filters, are considered to be useful tools for image representation. This is due to their ability to form a ‘sparse code’ that can serve as a basis set for high-fidelity reconstruction of natural images. However, for many visual tasks, the more appropriate criterion of representational efficacy is ‘recognition’, rather than ‘reconstruction’. It is unclear whether simple local features provide the stability necessary to subserve robust recognition of complex objects. In this paper, we search the space of two-lobed differential operators for those that constitute a good representational code under recognition/discrimination criteria. We find that a novel operator, which we call the ‘dissociated dipole’ displays useful properties in this regard. We describe simple computational experiments to assess the merits of such dipoles relative to the more traditional local operators. The results suggest that non-local operators constitute a vocabulary that is stable across a range of image transformations.

Objects can exhibit different dynamics at different scales, a property that isoftenexploited by visual tracking algorithms. A local dynamicmodel is typically used to extract image features that are then used as inputsto a system for tracking the entire object using a global dynamic model.Approximate local dynamicsmay be brittle---point trackers drift due to image noise and adaptivebackground models adapt to foreground objects that becomestationary---but constraints from the global model can make them more robust.We propose a probabilistic framework for incorporating globaldynamics knowledge into the local feature extraction processes.A global tracking algorithm can beformulated as a generative model and used to predict feature values thatinfluence the observation process of thefeature extractor. We combine such models in a multichain graphicalmodel framework.We show the utility of our framework for improving feature tracking and thusshapeand motion estimates in a batch factorization algorithm.We also propose an approximate filtering algorithm appropriate for onlineapplications, and demonstrate its application to problems such as backgroundsubtraction, structure from motion and articulated body tracking.

This paper bridges the gap between variable selection methods (e.g., Pearson coefficients, KS test) and dimensionality reductionalgorithms (e.g., PCA, LDA). Variable selection algorithms encounter difficulties dealing with highly correlated data,since many features are similar in quality. Dimensionality reduction algorithms tend to combine all variables and cannotselect a subset of significant variables.Our approach combines both methodologies by applying variable selection followed by dimensionality reduction. Thiscombination makes sense only when using the same utility function in both stages, which we do. The resulting algorithmbenefits from complex features as variable selection algorithms do, and at the same time enjoys the benefits of dimensionalityreduction.1

In human-human dialogues, face-to-face meetings are often preferred over phone conversations.One explanation is that non-verbal modalities such as gesture provide additionalinformation, making communication more efficient and accurate. If so, computerprocessing of natural language could improve by attending to non-verbal modalitiesas well. We consider the problem of sentence segmentation, using hand-annotatedgesture features to improve recognition. We find that gesture features correlate wellwith sentence boundaries, but that these features improve the overall performance of alanguage-only system only marginally. This finding is in line with previous research onthis topic. We provide a regression analysis, revealing that for sentence boundarydetection, the gestural features are largely redundant with the language model andpause features. This suggests that gestural features can still be useful when speech recognition is inaccurate.

We study properties of algorithms which minimize (or almost minimize) empirical error over a Donsker class of functions. We show that the L2-diameter of the set of almost-minimizers is converging to zero in probability. Therefore, as the number of samples grows, it is becoming unlikely that adding a point (or a number of points) to the training set will result in a large jump (in L2 distance) to a new hypothesis. We also show that under some conditions the expected errors of the almost-minimizers are becoming close with a rate faster than n^{-1/2}.

In this thesis we present LaSLAT, a sensor network algorithm thatsimultaneously localizes sensors, calibrates sensing hardware, andtracks unconstrained moving targets using only range measurementsbetween the sensors and the target. LaSLAT is based on a Bayesian filter, which updates a probabilitydistribution over the quantities of interest as measurementsarrive. The algorithm is distributable, and requires only a constantamount of space with respect to the number of measurementsincorporated. LaSLAT is easy to adapt to new types of hardware and newphysical environments due to its use of intuitive probabilitydistributions: one adaptation demonstrated in this thesis uses amixture measurement model to detect and compensate for bad acousticrange measurements due to echoes.We also present results from a centralized Java implementation ofLaSLAT on both two- and three-dimensional sensor networks in whichranges are obtained using the Cricket ranging system. LaSLAT is ableto localize sensors to within several centimeters of their groundtruth positions while recovering a range measurement bias for eachsensor and the complete trajectory of the mobile.

We present a novel framework to exert a topology control over a level set evolution. Level set methods offer several advantages over parametric active contours, in particular automated topological changes. In some applications, where some a priori knowledge of the target topology is available, topological changes may not be desirable. A method, based on the concept of simple point borrowed from digital topology, was recently proposed to achieve a strict topology preservation during a level set evolution. However, topologically constrained evolutions often generate topological barriers that lead to large geometric inconsistencies. We introduce a topologically controlled level set framework that greatly alleviates this problem. Unlike existing work, our method allows connected components to merge, split or vanish under some specific conditions that ensure that no topological defects are generated. We demonstrate the strength of our method on a wide range of numerical experiments.

When groups of individuals make choices among several alternatives, the most compelling social outcome is the Condorcet winner, namely the alternative beating all others in a pair-wise contest. Obviously the Condorcet winner cannot be overturned if one sub-group proposes another alternative it happens to favor. However, in some cases, and especially with haphazard voting, there will be no clear unique winner, with the outcome consisting of a triple of pair-wise winners that each beat different subsets of the alternatives (i.e. a “top-cycle”.) We explore the sensitivity of Condorcet winners to various perturbations in the voting process that lead to top-cycles. Surprisingly, variations in the number of votes for each alternative is much less important than consistency in a voter’s view of how alternatives are related. As more and more voter’s preference orderings on alternatives depart from a shared model of the domain, then unique Condorcet outcomes become increasingly unlikely.

Research in object detection and recognition in cluttered scenes requires large image collections with ground truth labels. The labels should provide information about the object classes present in each image, as well as their shape and locations, and possibly other attributes such as pose. Such data is useful for testing, as well as for supervised learning. This project provides a web-based annotation tool that makes it easy to annotate images, and to instantly sharesuch annotations with the community. This tool, plus an initial set of 10,000 images (3000 of which have been labeled), can be found at http://www.csail.mit.edu/$\sim$brussell/research/LabelMe/intro.html