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134
Determining Optical Flow
 ARTIFICIAL INTELLIGENCE
, 1981
"... Optical flow cannot be computed locally, since only one independent measurement is available from the image sequence at a point, while the flow velocity has two components. A second constraint is needed. A method for finding the optical flow pattern is presented which assumes that the apparent veloc ..."
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Cited by 1736 (7 self)
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Optical flow cannot be computed locally, since only one independent measurement is available from the image sequence at a point, while the flow velocity has two components. A second constraint is needed. A method for finding the optical flow pattern is presented which assumes that the apparent velocity of the brightness pattern varies smoothly almost everywhere in the image. An iterative implementation is shown which successfully computes the optical flow for a number of synthetic image sequences. The algorithm is robust in that it can handle image sequences that are quantized rather coarsely in space and time. It is also insensitive to quantization of brightness levels and additive noise. Examples are included where the assumption of smoothness is violated at singular points or along lines in the image.
Snakes, Shapes, and Gradient Vector Flow
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 1998
"... Snakes, or active contours, are used extensively in computer vision and image processing applications, particularly to locate object boundaries. Problems associated with initialization and poor convergence to boundary concavities, however, have limited their utility. This paper presents a new extern ..."
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Cited by 491 (16 self)
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Snakes, or active contours, are used extensively in computer vision and image processing applications, particularly to locate object boundaries. Problems associated with initialization and poor convergence to boundary concavities, however, have limited their utility. This paper presents a new external force for active contours, largely solving both problems. This external force, which we call gradient vector flow (GVF), is computed as a diffusion of the gradient vectors of a graylevel or binary edge map derived from the image. It differs fundamentally from traditional snake external forces in that it cannot be written as the negative gradient of a potential function, and the corresponding snake is formulated directly from a force balance condition rather than a variational formulation. Using several twodimensional (2D) examples and one threedimensional (3D) example, we show that GVF has a large capture range and is able to move snakes into boundary concavities.
Numerical Shape from Shading and Occluding Boundaries
 Artifical Intelligence
, 1981
"... An iterative method for computing shape from shading using occluding boundary information is proposed. Some applications of this method are shown. We employ the stereographic plane to express the orientations of surface patches, rather than the more commonly.used gradient space. Use of the stereogra ..."
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Cited by 190 (14 self)
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An iterative method for computing shape from shading using occluding boundary information is proposed. Some applications of this method are shown. We employ the stereographic plane to express the orientations of surface patches, rather than the more commonly.used gradient space. Use of the stereographic plane makes it possible to incorporate occluding boundary information, but forces us to employ a smoothness constraint different from the one previously proposed. The new constraint follows directly from a particular definition of surface smoothness. We solve the set of equations arising from the smoothness constraints and the imageirradiance equation iteratively, using occluding boundary information to supply boundary conditions. Good initial values are found at certain points to help reduce the number of iterations required to reach a reasonable solution. Numerical experiments show that the method is effective and robust. Finally, we analyze scanning electron microscope (SEM) pictures using this method. Other applications are also proposed. 1.
Removing the stiffness from interfacial flows with surface tension
 J. Comput. Phys
, 1994
"... A new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in twodimensional, irrotational, and incompressible fluids. Through the LaplaceYoung condition at the interface, surface tension introduces highorder terms, both nonlinear and nonloca ..."
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Cited by 78 (8 self)
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A new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in twodimensional, irrotational, and incompressible fluids. Through the LaplaceYoung condition at the interface, surface tension introduces highorder terms, both nonlinear and nonlocal, into the dynamics. This leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult. This new formulation has all the nice properties for time integration methods that are associated with having a linear, constant coefficient, highest order term. That is, using this formulation, we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in Fourier space. The approach is based on a boundary integral formulation and applies more generally, even to problems beyond the fluid mechanical context. Here they are applied to computing with high resolution the motion of interfaces in HeleShaw flows and the motion of free surfaces in inviscid flows governed by the Euler equations. One HeleShaw computation shows the behavior of an expanding gas bubble over longtime as the interface undergoes successive tipsplittings and finger competition. A second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification. In Euler flows, the computation of a vortex sheet shows its rollup through the KelvinHelmholtz instability. This motion culminates in the late time selfintersection of the interface, creating trapped bubbles of fluid. This is, we believe, a type of singularity formation previously unobserved for such flows in 2D. Finally, computations of falling plumes in an unstably stratified Boussinesq fluid show a very similar behavior. © 1994 Academic Press, Inc. 1.
Local Parallel Computation of Stochastic Completion Fields
 Neural Computation
, 1997
"... We describe a local parallel method for computing the stochastic completion field introduced in an earlier paper[Williams96]. The stochastic completion field represents the likelihood that a completion joining two contour fragments passes through any given position and orientation in the image plane ..."
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Cited by 35 (5 self)
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We describe a local parallel method for computing the stochastic completion field introduced in an earlier paper[Williams96]. The stochastic completion field represents the likelihood that a completion joining two contour fragments passes through any given position and orientation in the image plane. It is based upon the assumption that the prior probability distribution of completion shape can be modeled as a random walk in a lattice of discrete positions and orientations. The local parallel method can be interpreted as a stable finite difference scheme for solving the underlying FokkerPlanck equation identified by Mumford[Mumford94]. The resulting algorithm is significantly faster than the previously employed method which relied on convolution with largekernel filters computed by Monte Carlo simulation. The complexity of the new method is O(n 3 m) while that of the previous algorithm was O(n 4 m 2 ) (for an n \Theta n image with m discrete orientations). Perhaps most significa...
Loop quantum cosmology
, 2006
"... Quantum gravity is expected to be necessary in order to understand situations where classical general relativity breaks down. In particular in cosmology one has to deal with initial singularities, i.e. the fact that the backward evolution of a classical spacetime inevitably comes to an end after a ..."
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Cited by 28 (9 self)
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Quantum gravity is expected to be necessary in order to understand situations where classical general relativity breaks down. In particular in cosmology one has to deal with initial singularities, i.e. the fact that the backward evolution of a classical spacetime inevitably comes to an end after a finite amount of proper time. This presents a breakdown of the classical picture and requires an extended theory for a meaningful description. Since small length scales and high curvatures are involved, quantum effects must play a role. Not only the singularity itself but also the surrounding spacetime is then modified. One particular realization is loop quantum cosmology, an application of loop quantum gravity to homogeneous systems, which removes classical singularities. Its implications can be studied at different levels. Main effects are introduced into effective classical equations which allow to avoid interpretational problems of quantum theory. They give rise to new kinds of early universe phenomenology with applications to inflation and cyclic models. To resolve classical singularities and to understand the structure of geometry around them, the quantum description is necessary. Classical evolution is then replaced by a difference equation for a wave function which allows to extend spacetime beyond classical singularities. One main question is how these homogeneous scenarios are related to full loop quantum gravity, which can be dealt with at the level of distributional symmetric states. Finally, the new structure of spacetime arising in loop quantum gravity and its application to cosmology sheds new light on more general issues such as time.
ModelBased Image Reconstruction From TimeResolved Diffusion Data
"... This paper addresses the issue of reconstructing the unknown field of absorption and scattering coefficients from timeresolved measurements of diffused light in a computationally efficient manner. The intended application is optical tomography, which has generated considerable interest in recent ti ..."
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Cited by 21 (7 self)
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This paper addresses the issue of reconstructing the unknown field of absorption and scattering coefficients from timeresolved measurements of diffused light in a computationally efficient manner. The intended application is optical tomography, which has generated considerable interest in recent times. The inverse problem is posed in the Bayesian framework. The maximum aposteriori (MAP) estimate is used to compute the reconstruction. We use an edgepreserving generalized Gaussian Markov random field to model the unknown image. The diffusion model used for the measurements is solved forward in time using a finitedifference approach known as the alternatingdirections implicit method. This method requires the inversion of a tridiagonal matrix at each time step and is therefore of O(N) complexity, where N is the dimensionality of the image. Adjoint differentiation is used to compute the sensitivity of the measurements with respect to the unknown image. The novelty of our method lies in the computation of the sensitivity since we can achieve it in O(N) time as opposed to O(N²) time required by the perturbation approach. We present results using simulated data to show that the proposed method yields superior quality reconstructions with substantial savings in computation.
Loop Quantum Cosmology IV: Discrete Time Evolution
, 2000
"... Using general features of recent quantizations of the Hamiltonian constraint in loop quantum gravity and loop quantum cosmology, a dynamical interpretation of the constraint equation as evolution equation is presented. This involves a transformation from the connection to a dreibein representation a ..."
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Cited by 18 (11 self)
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Using general features of recent quantizations of the Hamiltonian constraint in loop quantum gravity and loop quantum cosmology, a dynamical interpretation of the constraint equation as evolution equation is presented. This involves a transformation from the connection to a dreibein representation and the selection of an internal time variable. Due to the discrete nature of geometrical quantities in loop quantum gravity also time turns out to be discrete leading to a difference rather than differential evolution equation. Furthermore, evolving observables are discussed in this framework which enables an investigation of physical spectra of geometrical quantities. In particular, the physical volume spectrum is proven to equal the discrete kinematical volume spectrum in loop quantum cosmology.
A Cost Analysis for a Higherorder Parallel Programming Model
, 1996
"... Programming parallel computers remains a difficult task. An ideal programming environment should enable the user to concentrate on the problem solving activity at a convenient level of abstraction, while managing the intricate lowlevel details without sacrificing performance. This thesis investiga ..."
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Cited by 17 (1 self)
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Programming parallel computers remains a difficult task. An ideal programming environment should enable the user to concentrate on the problem solving activity at a convenient level of abstraction, while managing the intricate lowlevel details without sacrificing performance. This thesis investigates a model of parallel programming based on the BirdMeertens Formalism (BMF). This is a set of higherorder functions, many of which are implicitly parallel. Programs are expressed in terms of functions borrowed from BMF. A parallel implementation is defined for each of these functions for a particular topology, and the associated execution costs are derived. The topologies which have been considered include the hypercube, 2D torus, tree and the linear array. An analyser estimates the costs associated with different implementations of a given program and selects a costeffective one for a given topology. All the analysis is performed at compiletime which has the advantage of reducing run...