We present a solver for the 2D high-frequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as @math , where @math is the number of volume unknowns, and @math is the number of processors, as long as @math grows at most like a small fractional power of @math . The solver decomposes the domain into layers, and uses transmission conditions in boundary integral form to explicitly define "polarized traces", i.e., up- and down-going waves sampled at interfaces. Local direct solvers are used in each layer to precompute traces of local Green's functions in an embarrassingly parallel way (the offline part), and incomplete Green's formulas are used to propagate interface data in a sweeping fashion, as a preconditioner inside a GMRES loop (the online part). Adaptive low-rank partitioning of the integral kernels is used to speed up their application to interface data. The method uses second-order finite differences. The complexity scalings are empirical but motivated by an analysis of ranks of off-diagonal blocks of oscillatory integrals. They continue to hold in the context of standard geophysical community models such as BP and Marmousi 2, where convergence occurs in 5 to 10 GMRES iterations.
proposed a large-subdomain sweeping method, based on an approximate Green's function by geometric optics and the butterfly algorithm, which can handle transmitted waves in very favorable complexity @cite_49 ; and a variant of this approach that can handle caustics in the geometric optics ansatz @cite_22 ;
@cite_22 @cite_49
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In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that the Helmholtz equation may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, we propose a new Eulerian computational geometrical-optics method, dubbed the fast Huygens sweeping method, for computing Green functions of Helmholtz equations in inhomogeneous media in the high-frequency regime and in the presence of caustics. The first novelty of the new method is that the Huygens–Kirchhoff secondary source principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics associated with the usual geometrical-optics ansatz can be treated automatically. The second novelty is that a butterfly algorithm is adapted to carry out the matrix–vector products induced by the Huygens–Kirchhoff integration in O(NlogN) operations, where N is the total number of mesh points, and the proportionality constant depends on the desired accuracy and is independent of the frequency parameter. To reduce the storage of the resulting traveltime and amplitude tables, we compress each table into a linear combination of tensor-product based multivariate Chebyshev polynomials so that the information of each table is encoded into a small number of Chebyshev coefficients. The new method enjoys the following desired features: (1) it precomputes a set of local traveltime and amplitude tables; (2) it automatically takes care of caustics; (3) it constructs Green functions of the Helmholtz equation for arbitrary frequencies and for many point sources; (4) for a specified number of points per wavelength it constructs each Green function in nearly optimal complexity in terms of the total number of mesh points, where the prefactor of the complexity only depends on the specified accuracy and is independent of the frequency parameter. Both two-dimensional (2-D) and three-dimensional (3-D) numerical experiments are presented to demonstrate the performance and accuracy of the new method. ABSTRACTMultiarrival Green’s functions are essential in seismic modeling, migration, and inversion. Huygens-Kirchhoff (HK) integrals provide a bridge to integrate locally valid first-arrival Green’s functions into a globally valid multiarrival Green’s function. We have designed robust and accurate finite-difference methods to compute first-arrival traveltimes and amplitudes, so that first-arrival Green’s functions can be constructed rapidly. We adapted a fast butterfly algorithm to evaluate discretized HK integrals. The resulting fast Huygens’ sweeping method has the following unique features: (1) it precomputes a set of local traveltime and amplitude tables, (2) it automatically takes care of caustics, (3) it constructs Green’s functions of the Helmholtz equation for arbitrary frequencies and for many point sources, and (4) for a fixed number of points per wavelength, it constructs each Green’s function in nearly optimal complexity O(N log N) in terms of the total number of mesh points N, where the prefa... Recent attempts to fabricate surfaces with custom reflectance functions boast impressive angular resolution, yet their spatial resolution is limited. In this paper we present a method to construct spatially varying reflectance at a high resolution of up to 220dpi, orders of magnitude greater than previous attempts, albeit with a lower angular resolution. The resolution of previous approaches is limited by the machining, but more fundamentally, by the geometric optics model on which they are built. Beyond a certain scale geometric optics models break down and wave effects must be taken into account. We present an analysis of incoherent reflectance based on wave optics and gain important insights into reflectance design. We further suggest and demonstrate a practical method, which takes into account the limitations of existing micro-fabrication techniques such as photolithography to design and fabricate a range of reflection effects, based on wave interference. We propose a new intrinsic representation of geometric texture over triangle meshes. Our approach extends the conventional height field texture representation by incorporating displacements in the tangential plane in the form of a normal tilt. This texture representation offers a good practical compromise between functionality and simplicity: it can efficiently handle and process geometric texture too complex to be represented as a height field, without having recourse to full blown mesh editing algorithms. The height-and-tilt representation proposed here is fully intrinsic to the mesh, making texture editing and animation (such as bending or waving) intuitively controllable over arbitrary base mesh. We also provide simple methods for texture extraction and transfer using our height-and-field representation. Preface to the Second Edition. Preface to the First Edition. 1 Ray Optics. 2 Wave Optics. 3 Beam Optics. 4 Fourier Optics. 5 Electromagnetic Optics. 6 Ploarization Optics. 7 Photonic-Crystal Optics. 8 Guided-Wave Optics. 9 Fiber Optics. 10 Resonator Optics. 11 Statistical Optics. 12 Photon Optics. 13 Photon and Atoms. 14 Laser Amplifiers. 15 Lasers. 16 Semiconductor Optics. 17 Semiconductor Photon Sources. 18 Semiconductor Photon Detectors. 19 Acousto-Optics. 20 Electro-Optics. 21 Nonlinear Optics. 22 Ultrafast Optics. 23 Optical Interconnects and Switches. 24 Optical Fiber Communications. A Fourier Transform. B Linear Systems. C Modes of Linear Systems. Symbols and Units. Authors. Index. Monotonic surfaces spanning finite regions of Zd arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. We explore how we can sample these surfaces when the distribution is biased to favor higher surfaces. We show that a natural local chain is rapidly mixing with any bias for regions in Z2, and for bias λ > d2 in Zd, when d > 2. Moreover, our bounds on the mixing time are optimal on d-dimensional hyper-cubic regions. The proof uses a geometric distance function and introduces a variant of path coupling in order to handle distances that are exponentially large. The speed of optical flow algorithm is crucial for many video editing tasks such as slow motion synthesis, selection propagation, tone adjustment propagation, and so on. Variational coarse-to-fine optical flow algorithms can generally produce high-quality results but cannot fulfil the speed requirement of many practical applications. Besides, large motions in real-world videos also pose a difficult problem to coarse-to-fine variational approaches. We, in this paper, present a fast optical flow algorithm that can handle large displacement motions. Our algorithm is inspired by recent successes of local methods in visual correspondence searching as well as approximate nearest neighbor field algorithms. The main novelty is a fast randomized edge-preserving approximate nearest neighbor field algorithm, which propagates self-similarity patterns in addition to offsets. Experimental results on public optical flow benchmarks show that our method is significantly faster than state-of-the-art methods without compromising on quality, especially when scenes contain large motions. Finally, we show some demo applications by applying our technique into real-world video editing tasks. Demosaicing is a process of obtaining a full-color image by interpolating the missing colors of an image captured from a single sensor color filter array. This paper provides an effective and low-complexity iterative demosaicing algorithm applying a weighted-edge interpolation to handle green pixels followed by a series of color-difference interpolation to update red, blue, and green pixels. Based on our experiments of images, we enable the algorithm a well-designed stopping condition and pre-determine the proper weights of interpolation. Experimental results show that the proposed method performs much better than three state-of-the-art demosaicing techniques in terms of both computational cost and image quality. In comparison to the algorithm of successive approximation, the algorithm proposed here reduces mean squared error up to 14.5 while requiring computational cost only 22 on average. That is, it takes less time but performs better. Optical flow computation is a key component in many computer vision systems designed for tasks such as action detection or activity recognition. However, despite several major advances over the last decade, handling large displacement in optical flow remains an open problem. Inspired by the large displacement optical flow of Brox and Malik, our approach, termed Deep Flow, blends a matching algorithm with a variational approach for optical flow. We propose a descriptor matching algorithm, tailored to the optical flow problem, that allows to boost performance on fast motions. The matching algorithm builds upon a multi-stage architecture with 6 layers, interleaving convolutions and max-pooling, a construction akin to deep convolutional nets. Using dense sampling, it allows to efficiently retrieve quasi-dense correspondences, and enjoys a built-in smoothing effect on descriptors matches, a valuable asset for integration into an energy minimization framework for optical flow estimation. Deep Flow efficiently handles large displacements occurring in realistic videos, and shows competitive performance on optical flow benchmarks. Furthermore, it sets a new state-of-the-art on the MPI-Sintel dataset. A citing sentence is one that appears in a scientific article and cites previous work. Citing sentences have been studied and used in many applications. For example, they have been used in scientific paper summarization, automatic survey generation, paraphrase identification, and citation function classification. Citing sentences that cite multiple papers are common in scientific writing. This observation should be taken into consideration when using citing sentences in applications. For instance, when a citing sentence is used in a summary of a scientific paper, only the fragments of the sentence that are relevant to the summarized paper should be included in the summary. In this paper, we present and compare three different approaches for identifying the fragments of a citing sentence that are related to a given target reference. Our methods are: word classification, sequence labeling, and segment classification. Our experiments show that segment classification achieves the best results. We consider two of the most commonly cited measures of retrieval performance: average precision and R-precision. It is well known that average precision and R-precision are highly correlated and similarly robust measures of performance, though the reasons for this are not entirely clear. In this paper, we give a geometric argument which shows that under a very reasonable set of assumptions, average precision and R-precision both approximate the area under the precision-recall curve, thus explaining their high correlation. We further demonstrate through the use of TREC data that the similarity or difference between average precision and R-precision is largely governed by the adherence to, or violation of, these reasonable assumptions. Communication systems with all-optical multicasting have better performance than those using optical electrical optical conversion. Multicast protocols assume that all nodes in the network can forward the signal from one input to several outputs. Since fabrication of an optical switch with splitting capability is an expensive technology, there are few switches that are multicast capable. The heuristics designed for all-optical networks have to handle this limitation. We introduce a shortest path based forest algorithm for all-optical networks. We propose a post-processing algorithm to reduce the number of wavelengths needed. The effect of this post-processing algorithm is also examined in a well-known approach. Inner approximation algorithms have had two major roles in the mathematical programming literature. Their first role was in the construction of algorithms for the decomposition of large-scale mathematical programs, such as in the Dantzig-Wolfe decomposition principle. However, recently they have been used in the creation of algorithms that locate Kuhn-Tucker solutions to nonconvex programs. Avriel and Williams' Avriel, M., A. C. Williams. 1970. Complementary geometric programming. SIAM J. Appl. Math.19 125-141. complementary geometric programming algorithm, Duffin and Peterson's Duffin, R. J., E. L. Peterson. 1972. Reversed geometric programs treated by harmonic means. Indiana Univ. Math. J.22 531-550. reversed geometric programming algorithms, Reklaitis and Wilde's Reklaitis, G. V., D. J. Wilde. 1974. Geometric programming via a primal auxiliary problem. AIIE Trans.6 308-317. primal reversed geometric programming algorithm, and Bitran and Novaes' Bitran, G. R., A. G. Novaes. 1973. Linear programming with a fractional objective function. Opns. Res.21 22-29. linear fractional programming algorithm are all examples of this class of inner approximation algorithms. A sequence of approximating convex programs are solved in each of these algorithms. Rosen's Rosen, J. B. 1966. Iterative solution of nonlinear optimal control problems. SIAM J. Control4 223-244. inner approximation algorithm is a special case of the general inner approximation algorithm presented in this note. This paper presents a method for scene flow estimation from a calibrated stereo image sequence. The scene flow contains the 3-D displacement field of scene points, so that the 2-D optical flow can be seen as a projection of the scene flow onto the images. We propose to recover the scene flow by coupling the optical flow estimation in both cameras with dense stereo matching between the images, thus reducing the number of unknowns per image point. The use of a variational framework allows us to properly handle discontinuities in the observed surfaces and in the 3-D displacement field. Moreover our approach handles occlusions both for the optical flow and the stereo. We obtain a partial differential equations system coupling both the optical flow and the stereo, which is numerically solved using an original multi- resolution algorithm. Whereas previous variational methods were estimating the 3-D reconstruction at time t and the scene flow separately, our method jointly estimates both in a single optimization. We present numerical results on synthetic data with ground truth information, and we also compare the accuracy of the scene flow projected in one camera with a state-of-the-art single-camera optical flow computation method. Results are also presented on a real stereo sequence with large motion and stereo discontinuities. Source code and sample data are available for the evaluation of the algorithm. We introduce a novel matching algorithm, called DeepMatching, to compute dense correspondences between images. DeepMatching relies on a hierarchical, multi-layer, correlational architecture designed for matching images and was inspired by deep convolutional approaches. The proposed matching algorithm can handle non-rigid deformations and repetitive textures and efficiently determines dense correspondences in the presence of significant changes between images. We evaluate the performance of DeepMatching, in comparison with state-of-the-art matching algorithms, on the Mikolajczyk ( A comparison of affine region detectors, 2005), the MPI-Sintel ( A naturalistic open source movie for optical flow evaluation, 2012) and the Kitti ( Vision meets robotics: The KITTI dataset, 2013) datasets. DeepMatching outperforms the state-of-the-art algorithms and shows excellent results in particular for repetitive textures. We also apply DeepMatching to the computation of optical flow, called DeepFlow, by integrating it in the large displacement optical flow (LDOF) approach of Brox and Malik (Large displacement optical flow: descriptor matching in variational motion estimation, 2011). Additional robustness to large displacements and complex motion is obtained thanks to our matching approach. DeepFlow obtains competitive performance on public benchmarks for optical flow estimation. Fast scramblers process information in characteristic times scaling logarithmically with the entropy, a behavior which has been conjectured for black hole horizons. In this note we use the AdS CFT fold to argue that causality bounds on information flow only depend on the properties of a single thermal cell, and admit a geometrical interpretation in terms of the optical depth, i.e. the thickness of the Rindler region in the so-called optical metric. The spatial sections of the optical metric are well approximated by constant-curvature hyperboloids. We use this fact to propose an effective kinetic model of scrambling which can be assimilated to a compact hyperbolic billiard, furnishing a classic example of hard chaos. It is suggested that classical chaos at large N is a crucial ingredient in reconciling the notion of fast scrambling with the required saturation of causality. Several state-of-the-art video deblurring methods are based on a strong assumption that the captured scenes are static. These methods fail to deblur blurry videos in dynamic scenes. We propose a video deblurring method to deal with general blurs inherent in dynamic scenes, contrary to other methods. To handle locally varying and general blurs caused by various sources, such as camera shake, moving objects, and depth variation in a scene, we approximate pixel-wise kernel with bidirectional optical flows. Therefore, we propose a single energy model that simultaneously estimates optical flows and latent frames to solve our deblurring problem. We also provide a framework and efficient solvers to optimize the energy model. By minimizing the proposed energy function, we achieve significant improvements in removing blurs and estimating accurate optical flows in blurry frames. Extensive experimental results demonstrate the superiority of the proposed method in real and challenging videos that state-of-the-art methods fail in either deblurring or optical flow estimation. Citation relationship between scientific publications has been successfully used for scholarly bibliometrics, information retrieval and data mining tasks, and citation-based recommendation algorithms are well documented. While previous studies investigated citation relations from various viewpoints, most of them share the same assumption that, if paper1 cites paper2 (or author1 cites author2), they are connected, regardless of citation importance, sentiment, reason, topic, or motivation. However, this assumption is oversimplified. In this study, we employ an innovative "context-rich heterogeneous network" approach, which paves a new way for citation recommendation task. In the network, we characterize 1) the importance of citation relationships between citing and cited papers, and 2) the topical citation motivation. Unlike earlier studies, the citation information, in this paper, is characterized by citation textual contexts extracted from the full-text citing paper. We also propose algorithm to cope with the situation when large portion of full-text missing information exists in the bibliographic repository. Evaluation results show that, context-rich heterogeneous network can significantly enhance the citation recommendation performance. We investigate the reconstruction of plenoptic camera images in a scalar wave-optical framework. Previous publications relating to this topic numerically simulate light propagation on the basis of ray tracing. However, due to continuing miniaturization of hardware components it can be assumed that in combination with low-aperture optical systems this technique may not be generally valid. Therefore, we study the differences between ray- and wave-optical object reconstructions of true plenoptic camera images. For this purpose we present a wave-optical reconstruction algorithm, which can be run on a regular computer. Our findings show that a wave-optical treatment is capable of increasing the detail resolution of reconstructed objects. We propose a sweep-based approach to the freeform deformation of three-dimensional objects. Instead of using a volume enclosing the whole object, we approximate only its deformable parts using sweep surfaces. The vertices on the object boundary are bound to the sweep surfaces and follow their deformation. Several sweep surfaces can be organized into a hierarchy so that they interact with each other in a controlled manner. Thus we can support intuitively plausible shape deformation of objects of arbitrary topology with multiple control handles. A sweep-based approach also provides important advantages such as volume preservation. We demonstrate the effectiveness of our technique in several examples.