234,153 research outputs found
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Time-varying vector field design on surfaces
Vector field design has a wide variety of applications in computer
graphics, including texture synthesis, non-photorealistic rendering, fluid and crowd simulation, and shape deformation. This paper addresses the problem of the design of time-varying vector fields on surfaces. The additional time dimension poses a number of unique challenges to the design tasks such as the introduction of more complex structural changes. To address these challenges, we present a number of novel techniques to enable efficient design over important characteristics in the vector field such as singularity paths, pathlines, and bifurcations. These vector field features are used to generate a vector field by either blending basis vector fields or performing a constrained optimization process. Unwanted singularities
and bifurcations can lead to visual artifacts, and we address
them through singularity and bifurcation editing. We demonstrate
the capabilities of our system by applying it to the design of two
types of vector fields: orientation field and advection field for the application of texture synthesis and animation
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Vector field editing and periodic orbit extraction using Morse decomposition
Design and control of vector fields is critical for many visualization and graphics tasks such as vector field visualization, fluid simulation, and texture synthesis. The fundamental qualitative structures associated with vector fields are fixed points, periodic orbits, and separatrices. In this paper we provide a new technique that allows for the systematic creation and cancellation of fixed points and periodic orbits. This technique enables vector field design and editing on the plane and surfaces with desired qualitative properties. The technique is based on Conley theory which provides a unified framework that supports the cancellation of fixed points and periodic orbits. We also introduce a novel periodic orbit extraction and visualization algorithm that detects, for the first time, periodic orbits on surfaces. Furthermore, we describe the application of our periodic orbit detection and vector field simplification algorithm to engine simulation data demonstrating the utility of the approach. We apply our design system to vector field visualization by creating datasets containing periodic orbits. This helps us understand the effectiveness of existing visualization techniques. Finally, we propose a new streamline-based technique that allows vector field topology to be easily identified.Keywords: Conley index, Morse decomposition, Vector field visualization, Vector field topology, periodic orbit detection, vector field simplification, Vector field design, connection graph
Vector field processing on triangle meshes
While scalar fields on surfaces have been staples of geometry processing, the use of tangent vector fields has steadily grown in geometry processing over the last two decades: they are crucial to encoding directions and sizing on surfaces as commonly required in tasks such as texture synthesis, non-photorealistic rendering, digital grooming, and meshing. There are, however, a variety of discrete representations of tangent vector fields on triangle meshes, and each approach offers different tradeoffs among simplicity, efficiency, and accuracy depending on the targeted application.
This course reviews the three main families of discretizations used to design computational tools for vector field processing on triangle meshes: face-based, edge-based, and vertex-based representations. In the process of reviewing the computational tools offered by these representations, we go over a large body of recent developments in vector field processing in the area of discrete differential geometry. We also discuss the theoretical and practical limitations of each type of discretization, and cover increasingly-common extensions such as n-direction and n-vector fields.
While the course will focus on explaining the key approaches to practical encoding (including data structures) and manipulation (including discrete operators) of finite-dimensional vector fields, important differential geometric notions will also be covered: as often in Discrete Differential Geometry, the discrete picture will be used to illustrate deep continuous concepts such as covariant derivatives, metric connections, or Bochner Laplacians
Single freeform surface design for prescribed input wavefront and target irradiance
In beam shaping applications, the minimization of the number of necessary
optical elements for the beam shaping process can benefit the compactness of
the optical system and reduce its cost. The single freeform surface design for
input wavefronts, which are neither planar nor spherical, is therefore of
interest. In this work, the design of single freeform surfaces for a given
zero-\'etendue source and complex target irradiances is investigated. Hence,
not only collimated input beams or point sources are assumed. Instead, a
predefined input ray direction vector field and irradiance distribution on a
source plane, which has to be redistributed by a single freeform surface to
give the predefined target irradiance, is considered. To solve this design
problem, a partial differential equation (PDE) or PDE system, respectively, for
the unknown surface and its corresponding ray mapping is derived from energy
conservation and the ray-tracing equations. In contrast to former PDE
formulations of the single freeform design problem, the derived PDE of
Monge-Amp\`ere type is formulated for general zero-\'etendue sources in
cartesian coordinates. The PDE system is discretized with finite differences
and the resulting nonlinear equation system solved by a root-finding algorithm.
The basis of the efficient solution of the PDE system builds the introduction
of an initial iterate constuction approach for a given input direction vector
field, which uses optimal mass transport with a quadratic cost function. After
a detailed description of the numerical algorithm, the efficiency of the design
method is demonstrated by applying it to several design examples. This includes
the redistribution of a collimated input beam beyond the paraxial
approximation, the shaping of point source radiation and the shaping of an
astigmatic input wavefront into a complex target irradiance distribution.Comment: 11 pages, 10 figures version 2: Equation (7) was corrected;
additional minor changes/improvement
Nested invariant tori foliating a vector field and its curl: toward MHD equilibria and steady Euler flows in toroidal domains without continuous Euclidean isometries
This paper studies the problem of finding a three-dimensional solenoidal
vector field such that both the vector field and its curl are tangential to a
given family of toroidal surfaces. We show that this question can be translated
into the problem of determining a periodic solution with periodic derivatives
of a two-dimensional linear elliptic second-order partial differential equation
on each toroidal surface, and prove the existence of smooth solutions. An
example of smooth solution foliated by toroidal surfaces that are not invariant
under continuous Euclidean isometries is also constructed explicitly, and it is
identified as an equilibrium of anisotropic magnetohydrodynamics. The problem
examined here represents a weaker version of a fundamental mathematical problem
that arises in the context of magnetohydrodynamics and fluid mechanics
concerning the existence of regular equilibrium magnetic fields and steady
Euler flows in bounded domains without continuous Euclidean isometries. The
existence of such configurations represents a key theoretical issue for the
design of the confining magnetic field in nuclear fusion reactors known as
stellarators.Comment: 22 pages, 4 figure
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Topological analysis, visualization, and design of vector fields on surfaces
Analysis, visualization, and design of vector fields on surfaces have a wide variety of major applications in both scientific visualization and computer graphics. On the one hand, analysis and visualization of vector fields provide critical insights to the flow data produced from simulation or experiments of various engineering processes. On the other hand, many graphics applications require vector fields as input to drive certain graphical processes. This thesis addresses vector field analysis and design for both visualization and graphics applications.
Topological analysis of vector fields provides the qualitative (or structural) information of the underlying dynamics of the given vector data, which helps the domain experts identify the critical features and behaviors efficiently. In this dissertaion, I introduce a more complete vector field topology called
Entity Connection Graph (ECG) by including periodic orbits,
an essential component in vector field topology. An efficient technique for periodic orbit extraction is introduced and incorporated into the algorithm for ECG construction. The analysis results are visualized using the improved evenly-spaced streamline placement with all separation features being highlighted. This is the first time that periodic orbits have been extracted from surface flows. Through applications to engine simulation datasets, I demonstrate how the extracted topology helps engineers interpret the flow data that contains certain desirable behaviors which indicate the ideal engineering process.
Accuracy is typically of paramount importance for visualization and analysis tasks. However, the trajectory-based vector field topology approaches are sensitive to small perturbations such as error and noise which are contained in the given data and introduced during data acquisition and processing. This makes rigorous interpretation of vector field topology and flow dynamics difficult. To overcome that, I advocate the use of Morse decomposition to define a more reliable vector field topology called Morse Connection Graph (MCG). In particular, I present the pipeline of Morse decomposition of an input vector field. A technique based on the idea of [tau]-map is introduced to produce desirably fine Morse decompositions of vector fields.
To address the issue of slow performance of the global [tau]-map framework, I describe a hierarchical MCG refinement framework. It enables the [tau]-map approach to be conducted
within a Morse set of interest which greatly reduces the computation cost and leads to faster analysis. It is my hope that the work on Morse decomposition will invoke the investigation of other similar data analysis problems such as scalar field and tensor field analysis.
The techniques of time-independent vector field design have been well-studied. However, there is little attention on the systematic design of time-varying vector fields on surfaces. This dissertation addresses this by developing a design system that allows the creation and modification of time-varying vector fields on surfaces. More specifically, I present a number of novel techniques to enable efficient design over important characteristics in the vector field such as singularity paths, pathlines, and bifurcations. These vector field features are used to generate a vector field by either blending basis vector fields or performing a constrained optimization process. Unwanted singularities and bifurcations can lead to visual artifacts, and I address them through singularity and bifurcation editing. I demonstrate the capabilities of the design system by applying it to the design of two types of vector fields: the orientation field and the advection field for the application of texture synthesis and animation
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Interactive tensor field design and visualization on surfaces
Designing tensor fields in the plane and on surfaces
is a necessary task in many graphics applications, such as
painterly rendering, pen-and-ink sketch of smooth surfaces, and
anisotropic remeshing. In this paper, we present an interactive
design system that allows a user to create a wide variety of
surface tensor fields with control over the number and location
of degenerate points. Our system combines basis tensor fields
to make an initial tensor field that satisfies a set of user specifications.
However, such a field often contains unwanted
degenerate points that cannot always be eliminated due to
topological constraints of the underlying surface. To reduce the
artifacts caused by these degenerate points, our system allows the
user to move a degenerate point or to cancel a pair of degenerate
points that have opposite tensor indices.
We observe that a tensor field can be locally converted into
a vector field such that there is a one-to-one correspondence
between the set of degenerate points in the tensor field and the
set of singularities in the vector field. This conversion allows
us to effectively perform degenerate point pair cancellation
and movement by using similar operations for vector fields. In
addition, we adapt the image-based flow visualization technique
to tensor fields, therefore allowing interactive display of tensor
fields on surfaces.
We demonstrate the capabilities of our tensor field design
system with painterly rendering, pen-and-ink sketch of surfaces,
and anisotropic remeshing.Keywords: Tensor field design and visualization, nonphotorealistic rendering, tensor field topology, remeshin
Double freeform illumination design for prescribed wavefronts and irradiances
A mathematical model in terms of partial differential equations (PDE) for the
calculation of double freeform surfaces for irradiance and phase control with
predefined input and output wavefronts is presented. It extends the results of
B\"osel and Gross [J. Opt. Soc. Am. A 34, 1490 (2017)] for the illumination
design of single freeform surfaces for zero-\'etendue light sources to double
freeform lenses and mirrors. The PDE model thereby overcomes the restriction to
paraxiality or the requirement of at least one planar wavefront of the current
design models in the literature. In contrast with the single freeform
illumination design, the PDE system does not reduce to a Monge-Amp\`ere type
equation for the unknown freeform surfaces, if nonplanar input and output
wavefronts are assumed. Additionally, a numerical solving strategy for the PDE
model is presented. To show its efficiency, the algorithm is applied to the
design of a double freeform mirror system and double freeform lens system.Comment: Copyright 2018 Optical Society of America. One print or electronic
copy may be made for personal use only. Systematic reproduction and
distribution, duplication of any material in this paper for a fee or for
commercial purposes, or modifications of the content of this paper are
prohibite
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