Flow Visualization with Quantified Spatial and Temporal Errors Using Edge Maps

Robust analysis of vector fields has been established as an important tool for deriving insights from the complex systems these fields model. Traditional analysis and visualization techniques rely primarily on computing streamlines through numerical integration. The inherent numerical errors of such approaches are usually ignored, leading to inconsistencies that cause unreliable visualizations and can ultimately prevent in-depth analysis. We propose a new representation for vector fields on surfaces that replaces numerical integration through triangles with maps from the triangle boundaries to themselves. This representation, called edge maps, permits a concise description of flow behaviors and is equivalent to computing all possible streamlines at a user defined error threshold. Independent of this error streamlines computed using edge maps are guaranteed to be consistent up to floating point precision, enabling the stable extraction of features such as the topological skeleton. Furthermore, our representation explicitly stores spatial and temporal errors which we use to produce more informative visualizations. This work describes the construction of edge maps, the error quantification, and a refinement procedure to adhere to a user defined error bound. Finally, we introduce new visualizations using the additional information provided by edge maps to indicate the uncertainty involved in computing streamlines and topological structures.

Optical flow estimation with consistent spatio-temporal coherence models

[EN] In this work we propose a new variational model for the consistent estimation of motion fields. The aim of this work is to develop appropriate spatio-temporal coherence models. In this sense, we propose two main contributions: a nonlinear flow constancy assumption, similar in spirit to the nonlinear brightness constancy assumption, which conveniently relates flow fields at different time instants; and a nonlinear temporal regularization scheme, which complements the spatial regularization and can cope with piecewise continuous motion fields. These contributions pose a congruent variational model since all the energy terms, except the spatial regularization, are based on nonlinear warpings of the flow field. This model is more general than its spatial counterpart, provides more accurate solutions and preserves the continuity of optical flows in time. In the experimental results, we show that the method attains better results and, in particular, it considerably improves the accuracy in the presence of large displacements.

Differential forms on Carnot groups

The main goal of this thesis is to understand and link together some of the early works by Michel Rumin and Pierre Julg. The work is centered around the so-called Rumin complex, which is a construction in subRiemannian geometry. A Carnot manifold is a manifold endowed with a horizontal distribution. If further a metric is given, one gets a subRiemannian manifold. Such data arise in different contexts, such as: - formulation of the second principle of thermodynamics; - optimal control; - propagation of singularities for sums of squares of vector fields; - real hypersurfaces in complex manifolds; - ideal boundaries of rank one symmetric spaces; - asymptotic geometry of nilpotent groups; - modelization of human vision. Differential forms on a Carnot manifold have weights, which produces a filtered complex. In view of applications to nilpotent groups, Rumin has defined a substitute for the de Rham complex, adapted to this filtration. The presence of a filtered complex also suggests the use of the formal machinery of spectral sequences in the study of cohomology. The goal was indeed to understand the link between Rumin's operator and the differentials which appear in the various spectral sequences we have worked with: - the weight spectral sequence; - a special spectral sequence introduced by Julg and called by him Forman's spectral sequence; - Forman's spectral sequence (which turns out to be unrelated to the previous one). We will see that in general Rumin's operator depends on choices. However, in some special cases, it does not because it has an alternative interpretation as a differential in a natural spectral sequence. After defining Carnot groups and analysing their main properties, we will introduce the concept of weights of forms which will produce a splitting on the exterior differential operator d. We shall see how the Rumin complex arises from this splitting and proceed to carry out the complete computations in some key examples. From the third chapter onwards we will focus on Julg's paper, describing his new filtration and its relationship with the weight spectral sequence. We will study the connection between the spectral sequences and Rumin's complex in the n-dimensional Heisenberg group and the 7-dimensional quaternionic Heisenberg group and then generalize the result to Carnot groups using the weight filtration. Finally, we shall explain why Julg required the independence of choices in some special Rumin operators, introducing the Szego map and describing its main properties.

Higher order Sobolev Spaces and polyharmonic boundary value problems in Carnot Groups

The main task of this work is to present a concise survey on the theory of certain function spaces in the contexts of Hörmander vector fields and Carnot Groups, and to discuss briefly an application to some polyharmonic boundary value problems on Carnot Groups of step 2.

Monte Carlo Dose Calculation on Deforming Anatomy

This article presents the implementation and validation of a dose calculation approach for deforming anatomical objects. Deformation is represented by deformation vector fields leading to deformed voxel grids representing the different deformation scenarios. Particle transport in the resulting deformed voxels is handled through the approximation of voxel surfaces by triangles in the geometry implementation of the Swiss Monte Carlo Plan framework. The focus lies on the validation methodology which uses computational phantoms representing the same physical object through regular and irregular voxel grids. These phantoms are chosen such that the new implementation for a deformed voxel grid can be compared directly with an established dose calculation algorithm for regular grids. Furthermore, separate validation of the aspects voxel geometry and the density changes resulting from deformation is achieved through suitable design of the validation phantom. We show that equivalent results are obtained with the proposed method and that no statistically significant errors are introduced through the implementation for irregular voxel geometries. This enables the use of the presented and validated implementation for further investigations of dose calculation on deforming anatomy.

Efficacy of Grazing Stockpiled Perennial Forages for Winter Maintenance of Beef Cows

In a three year study, wintering systems utilizing the grazing of stockpiled perennial hay crop forages or corn crop residues were compared to maintaining cows in a drylot. In the summer of 1992, two cuttings of hay were harvested (June 22 and August 2) from three 10-acre fields containing “Johnstone” endophyte-free tall fescue and “Spreador II” alfalfa, and one cutting of hay was harvested from three 10- acre fields of smooth brome grass. “Arlington” red clover was frost-seeded into the smooth bromegrass fields in 1993 and into tall fescue-alfalfa and smooth bromegrass fields into 1994. Two cuttings of hay were harvested from all fields in subsequent years, and three-year average hay yields for tall fescue-alfalfa and smooth bromegrass-red clover were 4,336 and 3,481 pounds per acre, respectively. Regrowth of the forage following the August hay harvest of each year was accumulated for winter grazing. Following a killing frost in each year, two fields of each stockpiled forage were stocked with cows in midgestation at two acres per cow. Two 10-acre fields of corn crop residues were also stocked at two acres per cow, following the grain harvest. Mean dry matter forage yields at the initiation of grazing were 1,853, 2,173 and 5,797 pounds per acre for fields containing tall fescue-alfalfa, smooth bromegrass-red clover, and cornstalks, respectively. A drylot was stocked with 18 cows in 1992 and 1993 and 10 cows in 1994. All cows were fed hay as necessary to maintain a body condition score of five. During grazing, mean losses of organic matter were -6.4, -7.6, and -10.7 pounds per acre per cow from tall fescue-alfalfa, smooth bromegrass-red clover, and cornstalk fields. Average organic matter loss rates from stockpiled forages due to weathering alone were equal to only 30% of the weathering losses of the corn crop residues. In vitro digestibility of both stockpiled forages and cornstalks decreased at equal rates during grazing each year, with respective annual loss rates of .14, .08, and .06% per day. Cows grazing corn crop residues required an average of 1,321 pounds per cow less hay than cows maintained in the drylot to maintain equivalent body condition during the grazing season. Cows grazing tall fescue-alfalfa or smooth bromegrass-red clover had body weight gains and condition score changes equal to cows maintained in a drylot but required 64% and 62% less harvested hay than cows in the drylot during the grazing season. Over the entire stored forage cows grazing tall fescue-alfalfa and smooth bromegrass-red clover required an average of 2,390 and 2,337 pounds per cow less than those maintained in the drylot. Because less hay was needed to maintain cows grazing stockpiled forages, average annual excesses of 5,629 and 3,868 pounds of hay dry matter per cow remained in the stockpiled tall fescue-alfalfa and smooth bromegrass-red clover systems.

Constraints on the effective fluid theory of stationary branes

We develop further the effective fluid theory of stationary branes. This formalism applies to stationary blackfolds as well as to other equilibrium brane systems at finite temperature. The effective theory is described by a Lagrangian containing the information about the elastic dynamics of the brane embedding as well as the hydrodynamics of the effective fluid living on the brane. The Lagrangian is corrected order-by-order in a derivative expansion, where we take into account the dipole moment of the brane which encompasses finite-thickness corrections, including transverse spin. We describe how to extract the thermodynamics from the Lagrangian and we obtain constraints on the higher-derivative terms with one and two derivatives. These constraints follow by comparing the brane thermodynamics with the conserved currents associated with background Killing vector fields. In particular, we fix uniquely the one- and two-derivative terms describing the coupling of the transverse spin to the background space-time. Finally, we apply our formalism to two blackfold examples, the black tori and charged black rings and compare the latter to a numerically generated solution.

Uniqueness of black holes with bubbles in minimal supergravity

We generalize uniqueness theorems for non-extremal black holes with three mutually independent Killing vector fields in five-dimensional minimal supergravity in order to account for the existence of non-trivial two-cycles in the domain of outer communication. The black hole space-times we consider may contain multiple disconnected horizons and be asymptotically flat or asymptotically Kaluza–Klein. We show that in order to uniquely specify the black hole space-time, besides providing its domain structure and a set of asymptotic and local charges, it is necessary to measure the magnetic fluxes that support the two-cycles as well as fluxes in the two semi-infinite rotation planes of the domain diagram.

Coded Aperture Flow

Real cameras have a limited depth of field. The resulting defocus blur is a valuable cue for estimating the depth structure of a scene. Using coded apertures, depth can be estimated from a single frame. For optical flow estimation between frames, however, the depth dependent degradation can introduce errors. These errors are most prominent when objects move relative to the focal plane of the camera. We incorporate coded aperture defocus blur into optical flow estimation and allow for piecewise smooth 3D motion of objects. With coded aperture flow, we can establish dense correspondences between pixels in succeeding coded aperture frames. We compare several approaches to compute accurate correspondences for coded aperture images showing objects with arbitrary 3D motion.

Multigradient field-active contour model for multilayer boundary detection of ultrasound rectal wall image

Extraction and reconstruction of rectal wall structures from an ultrasound image is helpful for surgeons in rectal clinical diagnosis and 3-D reconstruction of rectal structures from ultrasound images. The primary task is to extract the boundary of the muscular layers on the rectal wall. However, due to the low SNR from ultrasound imaging and the thin muscular layer structure of the rectum, this boundary detection task remains a challenge. An active contour model is an effective high-level model, which has been used successfully to aid the tasks of object representation and recognition in many image-processing applications. We present a novel multigradient field active contour algorithm with an extended ability for multiple-object detection, which overcomes some limitations of ordinary active contour models—"snakes." The core part in the algorithm is the proposal of multigradient vector fields, which are used to replace image forces in kinetic function for alternative constraints on the deformation of active contour, thereby partially solving the initialization limitation of active contour for rectal wall boundary detection. An adaptive expanding force is also added to the model to help the active contour go through the homogenous region in the image. The efficacy of the model is explained and tested on the boundary detection of a ring-shaped image, a synthetic image, and an ultrasound image. The experimental results show that the proposed multigradient field-active contour is feasible for multilayer boundary detection of rectal wall

Sigmoidal cosine series on the interval

We construct a set of functions, say, psi([r])(n) composed of a cosine function and a sigmoidal transformation gamma(r) of order r > 0. The present functions are orthonormal with respect to a proper weight function on the interval [-1, 1]. It is proven that if a function f is continuous and piecewise smooth on [-1, 1] then its series expansion based on psi([r])(n) converges uniformly to f so long as the order of the sigmoidal transformation employed is 0 < r

Special compositions in affinely connected spaces without a torsion

2000 Mathematics Subject Classification: 53B05, 53B99.

Thermalized polarization dynamics of a discrete optical-waveguide system with four-wave mixing

Statistical mechanics of two coupled vector fields is studied in the tight-binding model that describes propagation of polarized light in discrete waveguides in the presence of the four-wave mixing. The energy and power conservation laws enable the formulation of the equilibrium properties of the polarization state in terms of the Gibbs measure with positive temperature. The transition line T=∞ is established beyond which the discrete vector solitons are created. Also in the limit of the large nonlinearity an analytical expression for the distribution of Stokes parameters is obtained, which is found to be dependent only on the statistical properties of the initial polarization state and not on the strength of nonlinearity. The evolution of the system to the final equilibrium state is shown to pass through the intermediate stage when the energy exchange between the waveguides is still negligible. The distribution of the Stokes parameters in this regime has a complex multimodal structure strongly dependent on the nonlinear coupling coefficients and the initial conditions.

Rank and k-nullity of contact manifolds

We prove that the dimension of the 1-nullity distribution N(1) on a closed Sasakian manifold M of rankl is at least equal to 2l−1 provided that M has an isolated closed characteristic. The result is then used to provide some examples of k-contact manifolds which are not Sasakian. On a closed, 2n+1-dimensional Sasakian manifold of positive bisectional curvature, we show that either the dimension of N(1) is less than or equal to n+1 or N(1) is the entire tangent bundle TM. In the latter case, the Sasakian manifold Mis isometric to a quotient of the Euclidean sphere under a finite group of isometries. We also point out some interactions between k-nullity, Weinstein conjecture, and minimal unit vector fields.