Analysis Seminar
Abstract: In his celebrated paper on area distortion under planar quasiconformal mappings (Acta 1994), Astala proved that if E is a compact set of Hausdorff dimension d and f is K-quasiconformal, then fE has Hausdorff dimension at most d' = (2Kd)/(2 (K-1)d), and that this result issharp. He conjectured (Question 4.4) that if the Hausdorff measure H^d(E) = 0, then H^d'(fE) = 0. This conjecture was known to be true if d' = 0 (obvious), d' = 2 (Ahlfors), and d' = 1 (Astala, Clop, Mateu, Orobitg and UT, Duke 2008). The approach in the last mentioned paper does not generalize to other dimensions. UT showed that Astala's conjecture is sharp in the class of all Hausdorff gauge functions (IMRN, 2008). Lacey, Sawyer and UT jointly proved completely Astala's conjecture in all dimensions (Acta, 2009?) The proof uses Astala's 1994 approach, geometric measure theory, and new weighted norm inequalities for Calderon-Zygmund singular integral operators which cannot be deduced from the classical Muckenhoupt A_p theory. These results are intimately related to removability problems for various classes of quasiregular maps. I will particularly mention sharp removability results forbounded K-quasiregular maps (i.e. the quasiconformal analogue of the classical Painleve problem) recently obtained jointly by Tolsa and UT.
Center for Vision Research Seminar Series
Abstract: "There's Always a Way" Michael May was blinded at age three, and lived 42 years of his life without sight. In 1999, at age 45, May was given the possibility to see again through a revolutionary stem-cell transplant surgery. Listen to NPR's Talk of the Nation about the book that was written about May called 'Crashing Through': from Blindness to Sight: http://www.npr.org/templates/story/story.php?storyId=10382528
Pattern Theory and Vision Seminar
Abstract:
Automating the process of measuring human shape characteristics and estimating body postures from images lies at the core of many applications in computer vision, computer graphics, robotics and bio-mechanics. In its most general form, this problem is severely under-constrained. It can be made more tractable by employing simplifying assumptions and relying on domain specific knowledge, or by engineering the environment appropriately.
In this thesis we demonstrate that using a data-driven model of the human body supports the recovery of both human shape and articulated pose from images, and has many benefits over previous body models. Specifically, we represent the body using a recently proposed triangulated mesh model called SCAPE which employs a low-dimensional, but detailed, parametric model of shape and pose-dependent deformations. We show that the parameters of the SCAPE model can be estimated directly from image data in a variety of imaging conditions.
We first consider the case of multiple calibrated and synchronized camera views and assume the subject wears tight-fitting clothing. We define a cost function between image silhouettes and a hypothesized mesh and formulate the problem as an optimization over the body shape and pose parameters. Second, we relax the tight-fitting clothing assumption and develop a robust method that accounts for the fact that observed silhouettes of clothed people do not provide tight bounds on the true 3D shape. We think of these silhouettes as providing only weak constraints, but collect many of them while observing the subject in many poses. These, together with strong constraints from regions detected as skin, can be combined with a prior expectation of typical shapes to infer the most likely shape model under the clothes. Results on a novel database of thousands of images of clothed and "naked" subjects suggest that this method may be accurate enough for biometric shape analysis from images.
[pizza will be provided]
Probability Seminar
Abstract: Despite Ben Bernanke's recent remark to the effect that it is hard to tell if one is in a bubble or not in real time, over the last 10 years there has been much progress in the mathematical modeling, and at least theoretical detection in real time, of financial bubbles. We review the recent progress in the mathematical modeling of financial bubbles.
Scientific Computing Seminar
Abstract: Data assimilation methods have been studied extensively in the recent years and used in many applications, as a means of addressing uncertainty in models. Among many approaches, Ensemble Kalman filter (EnKF) and particle filter (PF) are two important data assimilation methods that have been widely used in practice, primarily due to their ease of implementation. In this talk, we present analysis on the numerical errors of the two filters, e.g., error bounds and convergence rate. We also propose algorithms to enhance the efficiency of the EnKF: one based on deterministic sampling strategy and the other on the general polynomial chaos methodology. Numerical examples are provided to verify the theoretical results and demonstrate the performance of the new proposed filter methods.
PDE Seminar
Pattern Theory and Center for Vision Research Seminar
Abstract: Intelligent tasks such object recognition, auditory scene analysis, or language understanding require the construction of good internal representations of the world. Internal representations (or "features") must be invariant (or robust) to irrelevant variations of the input, but must preserve the information relevant to the task. An important goal of our research, and an important challenge for Machine Learning over the next few years, is to devise methods that can automatically learn good internal representations from labeled and unlabeled data. Theoretical and empirical evidence suggest that the visual world is best represented by a multi-stage hierarchy, in which features in successive stages are increasingly global, invariant, and abstract. The main question is how can one train such deep architectures from unlabeled data and limited amounts of labeled data. We describe a class of methods to train multi-stage systems in which each stage performs a series of convolutions followed by simple non-linearities. The unsupervised learning phase is based on sparse coding methods, but includes a feed-forward predictor that gives a quick approximation of the sparse code. A number of such stages are stacked and trained sequentially in an unsupervised manner. The entire system is then refined in a supervised manner. An application to category-level object recognition with invariance to pose and illumination will be described. By stacking multiple stages of sparse features, and refining the whole system with supervised training, state-the-art accuracy can be achieved on standard datasets with very few labeled samples. A real-time demo will be shown. Another application to vision-based navigation for off-road mobile robots will be shown. After a phase of off-line unsupervised learning, the system autonomously learns to discriminate obstacles from traversable areas at long range using labels produced with stereo vision for nearby areas. This is joint work with Y-Lan Boureau, Karol Gregor, Raia Hadsell, Koray Kavakcuoglu, and Marc'Aurelio Ranzato. http://www.dam.brown.edu/ptg/seminar.html
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