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Yezzi, Anthony

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Publication Search Results

Now showing 1 - 10 of 111
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    Accelerated Optimization in the PDE Framework
    (Georgia Institute of Technology, 2018-10-24) Yezzi, Anthony
    Following the seminal work of Nesterov, accelerated optimization methods (sometimes referred to as momentum methods) have been used to powerfully boost the performance of first-order, gradient-based parameter estimation in scenarios where second-order optimization strategies are either inapplicable or impractical. Not only does accelerated gradient descent converge considerably faster than traditional gradient descent, but it performs a more robust local search of the parameter space by initially overshooting and then oscillating back as it settles into a final configuration, thereby selecting only local minimizers with an attraction basin large enough to accommodate the initial overshoot. This behavior has made accelerated search methods particularly popular within the machine learning community where stochastic variants have been proposed as well. So far, however, accelerated optimization methods have been applied to searches over finite parameter spaces. We show how a variational setting for these finite dimensional methods (recently formulated by Wibisono, Wilson, and Jordan) can be extended to the infinite dimensional setting, both in linear functional spaces as well as to the more complicated manifold of 2D curves and 3D surfaces. Moreover, we also show how extremely simple explicit discretization schemes can be used to efficiently solve the resulting class of high-dimensional optimization problems.
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    A Compact Formula for the Derivative of a 3-D Rotation in Exponential Coordinates
    (Georgia Institute of Technology, 2014-08) Gallego, Guillermo ; Yezzi, Anthony
    We present a compact formula for the derivative of a 3-D rotation matrix with respect to its exponential coordinates. A geometric interpretation of the resulting expression is provided, as well as its agreement with other less-compact but better-known formulas. To the best of our knowledge, this simpler formula does not appear anywhere in the literature. We hope by providing this more compact expression to alleviate the common pressure to reluctantly resort to alternative representations in various computational applications simply as a means to avoid the complexity of differential analysis in exponential coordinates.
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    Space-time Measurements of Oceanic Sea States
    (Georgia Institute of Technology, 2013-10) Fedele, Francesco ; Benetazzo, Alvise ; Gallego, Guillermo ; Shih, Ping-Chang ; Yezzi, Anthony ; Barbariol, Francesco ; Ardhuin,Fabrice
    Stereo video techniques are effective for estimating the space-time wave dynamics over an area of the ocean. Indeed, a stereo camera view allows retrieval of both spatial and temporal data whose statistical content is richer than that of time series data retrieved from point wave probes.We present an application of the Wave Acquisition Stereo System (WASS) for the analysis of offshore video measurements of gravity waves in the Northern Adriatic Sea and near the southern seashore of the Crimean peninsula, in the Black Sea. We use classical epipolartechniques to reconstruct the sea surface from the stereo pairs sequentially in time, viz. asequence of spatial snapshots. We also present a variational approach that exploits the entire data image set providing a global space-time imaging of the sea surface, viz. simultaneous reconstruction of several spatial snapshots of the surface in order to guarantee continuity of the sea surface both in space and time. Analysis of the WASS measurements show that the sea surface can be accurately estimated in space and time together, yielding associated directionalspectra and wave statistics at a point in time that agrees well with probabilistic models. In particular, WASS stereo imaging is able to capture typical features of the wave surface,especially the crest-to-trough asymmetry due to second order nonlinearities, and the observedshape of large waves are fairly described by theoretical models based on the theory of quasi-determinism (Boccotti, 2000). Further, we investigate the space-time extremes of the observed stationary sea states, viz. the largest surface wave heights expected over a given area during thesea state duration. The WASS analysis provides the first experimental proof that a space-time extreme is generally larger than that observed in time via point measurements, in agreement withthe predictions based on stochastic theories for global maxima of Gaussian fields.
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    Variational Stereo Imaging of Oceanic Waves with Statistical Constraints
    (Georgia Institute of Technology, 2013-06) Gallego, Guillermo ; Yezzi, Anthony ; Fedele, Francesco ; Benetazzo, Alvise
    An image processing observational technique for the stereoscopic reconstruction of the wave form of oceanic sea states is developed. The technique incorporates the enforcement of any given statistical wave law modeling the quasi Gaussianity of oceanic waves observed in nature. The problem is posed in a variational optimization framework, where the desired wave form is obtained as the minimizer of a cost functional that combines image observations, smoothness priors and a weak statistical constraint. The minimizer is obtained combining gradient descent and multigrid methods on the necessary optimality equations of the cost functional. Robust photometric error criteria and a spatial intensity compensation model are also developed to improve the performance of the presented image matching strategy. The weak statistical constraint is thoroughly evaluated in combination with other elements presented to reconstruct and enforce constraints on experimental stereo data.
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    Wave statistics and space-time extremes via stereo imaging
    (Georgia Institute of Technology, 2012-06) Fedele, Francesco ; Benetazzo, Alvise ; Gallego, Guillermo ; Shih, Ping-Chang ; Yezzi, Anthony ; Barbariol, Francesco
    We present an analysis of the space-time dynamics of oceanic sea states exploiting stereo imaging techniques. In particular, a novel Wave Acquisition Stereo System (WASS) has been developed and deployed at the oceanographic tower Acqua Alta in the Northern Adriatic Sea, off the Venice coast in Italy. The analysis of WASS video measurements yields accurate estimates of the oceanic sea state dynamics, the associated directional spectra and wave surface statistics that agree well with theoretical models. Finally, we show that a space-time extreme, defined as the expected largest surface wave height over an area, is considerably larger than the maximum crest observed in time at a point, in agreement with theoretical predictions.
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    Space-time reconstruction of oceanic sea states via variational stereo methods
    (Georgia Institute of Technology, 2012-06) Gallego, Guillermo ; Yezzi, Anthony ; Fedele, Francesco ; Benetazzo, Alvise
    We present a remote sensing observational method for the measurement of the spatio-temporal dynamics of ocean waves. Variational techniques are used to recover a coherent space-time reconstruction of oceanic sea states given stereo video imagery. The stereoscopic reconstruction problem is expressed in a variational optimization framework. There, we design an energy functional whose minimizer is the desired temporal sequence of wave heights. The functional combines photometric observations as well as spatial and temporal regularizers. A nested iterative scheme is devised to numerically solve, via 3-D multigrid methods, the system of partial di erential equations resulting from the optimality condition of the energy functional. The output of our method is the coherent, simultaneous estimation of the wave surface height and radiance at multiple snapshots. We demonstrate our algorithm on real data collected o ffshore. Statistical and spectral analysis are performed. Comparison with respect to an existing sequential method is analyzed.
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    Offshore stereo measurements of gravity waves
    (Georgia Institute of Technology, 2012-06) Benetazzo, Alvise ; Fedele, Francesco ; Gallego, Guillermo ; Shih, Ping-Chang ; Yezzi, Anthony
    Stereo video techniques are effective for estimating the space-time wave dynamics over an area of the ocean. Indeed, a stereo camera view allows retrieval of both spatial and temporal data whose statistical content is richer than that of time series data retrieved from point wave probes. To prove this, we consider an application of the Wave Acquisition Stereo System (WASS) for the analysis of offshore video measurements of gravity waves in the Northern Adriatic Sea. In particular, we deployed WASS at the oceanographic platform Acqua Alta, off the Venice coast, Italy. Three experimental studies were performed, and the overlapping field of view of the acquired stereo images covered an area of approximately 1100 m². Analysis of the WASS measurements show that the sea surface can be accurately estimated in space and time together, yielding associated directional spectra and wave statistics that agree well with theoretical models. From the observed wavenumber-frequency spectrum one can also predict the vertical profile of the current flow underneath the wave surface. Finally, future improvements of WASS and applications are discussed.
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    Optimization of Foraging Multi-Agent System Front: A Flux-Based Curve Evolution Method
    (Georgia Institute of Technology, 2011-12) Haque, Musad A. ; Rahmani, Amir R. ; Egerstedt, Magnus B. ; Yezzi, Anthony
    Numerous social foragers form a foraging front that sweeps through the aggregation of prey. Based on this strategy, and using variational arguments, we develop an algorithm to provide a group-level specification of the shape of the sweeping front for a foraging multi-robot system. The presented flux-based algorithm has the desired property of generating more regular shapes than previously introduced algorithms.
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    A Variational Stereo Method for the Three-Dimensional Reconstruction of Ocean Waves
    (Georgia Institute of Technology, 2011-11) Gallego, Guillermo ; Yezzi, Anthony ; Fedele, Francesco ; Benetazzo, Alvise
    We develop a novel remote sensing technique for the observation of waves on the ocean surface. Our method infers the 3-D waveform and radiance of oceanic sea states via a variational stereo imagery formulation. In this setting, the shape and radiance of the wave surface are given by minimizers of a composite energy functional that combines a photometric matching term along with regularization terms involving the smoothness of the unknowns. The desired ocean surface shape and radiance are the solution of a system of coupled partial differential equations derived from the optimality conditions of the energy functional. The proposed method is naturally extended to study the spatiotemporal dynamics of ocean waves and applied to three sets of stereo video data. Statistical and spectral analysis are carried out. Our results provide evidence that the observed omnidirectional wavenumber spectrum S(k) decays as k-2.5 is in agreement with Zakharov's theory (1999). Furthermore, the 3-D spectrum of the reconstructed wave surface is exploited to estimate wave dispersion and currents.
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    Active Geodesics: Region based Active Contour Segmentation with a Global Edge based Constraint
    (Georgia Institute of Technology, 2011-11) Appia, Vikram ; Yezzi, Anthony
    We present an active geodesic contour model in which we constrain the evolving active contour to be a geodesic with respect to a weighted edge-based energy through its entire evolution rather than just at its final state (as in the traditional geodesic active contour models). Since the contour is always a geodesic throughout the evolution, we automatically get local optimality with respect to an edge fitting criterion. This enables us to construct a purely region-based energy minimization model without having to devise arbitrary weights in the combination of our energy function to balance edge-based terms with the region-based terms. We show that this novel approach of combining edge information as the geodesic constraint in optimizing a purely region-based energy yields a new class of active contours which exhibit both local and global behaviors that are naturally responsive to intuitive types of user interaction. We also show the relationship of this new class of globally constrained active contours with traditional minimal path methods, which seek global minimizers of purely edge-based energies without incorporating region-based criteria. Finally, we present some numerical examples to illustrate the benefits of this approach over traditional active contour models.