The Experts below are selected from a list of 315 Experts worldwide ranked by ideXlab platform
Daniel Cremers - One of the best experts on this subject based on the ideXlab platform.
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Lifting Vectorial Variational Problems: A Natural Formulation Based on Geometric Measure Theory and Discrete Exterior Calculus
2019 IEEE CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2019Co-Authors: Thomas Möllenhoff, Daniel CremersAbstract:Numerous tasks in imaging and vision can be formulated as variational problems over vector-valued maps. We approach the relaxation and convexification of such vectorial variational problems via a lifting to the space of currents. To that end, we recall that functionals with polyconvex Lagrangians can be reparametrized as convex one-homogeneous functionals on the graph of the function. This leads to an equivalent shape optimization problem over oriented surfaces in the product space of domain and codomain. A convex formulation is then obtained by relaxing the search space from oriented surfaces to more general currents. We propose a discretization of the resulting infinite-dimensional optimization problem using Whitney forms, which also generalizes recent "sublabel-accurate" multilabeling approaches.
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CVPR - Lifting Vectorial Variational Problems: A Natural Formulation Based on Geometric Measure Theory and Discrete Exterior Calculus
2019 IEEE CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2019Co-Authors: Thomas Möllenhoff, Daniel CremersAbstract:Numerous tasks in imaging and vision can be formulated as variational problems over vector-valued maps. We approach the relaxation and convexification of such vectorial variational problems via a lifting to the space of currents. To that end, we recall that functionals with polyconvex Lagrangians can be reparametrized as convex one-homogeneous functionals on the graph of the function. This leads to an equivalent shape optimization problem over oriented surfaces in the product space of domain and codomain. A convex formulation is then obtained by relaxing the search space from oriented surfaces to more general currents. We propose a discretization of the resulting infinite-dimensional optimization problem using Whitney forms, which also generalizes recent "sublabel-accurate" multilabeling approaches.
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the natural vectorial total variation which arises from Geometric Measure Theory
Siam Journal on Imaging Sciences, 2012Co-Authors: Bastian Goldluecke, Evgeny Strekalovskiy, Daniel CremersAbstract:Several ways to generalize scalar total variation to vector-valued functions have been proposed in the past. In this paper, we give a detailed analysis of a variant we denote by $\text{TV}_J$, which has not been previously explored as a regularizer. The contributions of the manuscript are twofold: on the theoretical side, we show that $\text{TV}_J$ can be derived from the generalized Jacobians from Geometric Measure Theory. Thus, within the context of this Theory, $\text{TV}_J$ is the most natural form of a vectorial total variation. As an important feature, we derive how $\text{TV}_J$ can be written as the support functional of a convex set in $\mathcal{L}^2$. This property allows us to employ fast and stable minimization algorithms to solve inverse problems. The analysis also shows that in contrast to other total variation regularizers for color images, the proposed one penalizes across a common edge direction for all channels, which is a major theoretical advantage. Our practical contribution consist o...
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An approach to vectorial total variation based on Geometric Measure Theory
2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2010Co-Authors: Bastian Goldluecke, Daniel CremersAbstract:We analyze a previously unexplored generalization of the scalar total variation to vector-valued functions, which is motivated by Geometric Measure Theory. A complete mathematical characterization is given, which proves important invariance properties as well as existence of solutions of the vectorial ROF model. As an important feature, there exists a dual formulation for the proposed vectorial total variation, which leads to a fast and stable minimization algorithm. The main difference to previous approaches with similar properties is that we penalize across a common edge direction for all channels, which is a major theoretical advantage. Experiments show that this leads to a significantly better restoration of color edges in practice.
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CVPR - An approach to vectorial total variation based on Geometric Measure Theory
2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2010Co-Authors: Bastian Goldluecke, Daniel CremersAbstract:We analyze a previously unexplored generalization of the scalar total variation to vector-valued functions, which is motivated by Geometric Measure Theory. A complete mathematical characterization is given, which proves important invariance properties as well as existence of solutions of the vectorial ROF model. As an important feature, there exists a dual formulation for the proposed vectorial total variation, which leads to a fast and stable minimization algorithm. The main difference to previous approaches with similar properties is that we penalize across a common edge direction for all channels, which is a major theoretical advantage. Experiments show that this leads to a significiantly better restoration of color edges in practice.
Jörg Schumacher - One of the best experts on this subject based on the ideXlab platform.
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Scalar gradient fields by Geometric Measure Theory.
Physical review. E Statistical nonlinear and soft matter physics, 2004Co-Authors: Jörg SchumacherAbstract:Upper bounds of the Hausdorff volume of scalar gradient field graphs are derived by means of Geometric Measure Theory. The approach reproduces that scalar gradient fields along a mean imposed scalar gradient become space filling for sufficiently high values of Schmidt numbers Sc. The bounds are consistent with findings from recent high-resolution numerical experiments for 1< or =Sc< or =64, but too rough when compared with numerical simulations. A Reynolds number dependence of the bounds is found due to the additional scalar gradient stretching term in the equation of motion.
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Scalar gradient fields by Geometric Measure Theory.
Physical Review E, 2004Co-Authors: Jörg SchumacherAbstract:Recent direct numerical simulations (DNS) suggested that the passive scalar mixing in a turbulent flow becomes more isotropic when the Schmidt number, Sc = ν/κ, is increased to values larger than unity, but the Taylor Reynolds number, R�, of the advecting turbulent flow is kept constant [1,2]. Here ν is the kinematic viscosity of the fluid and κ the diffusivity of the passive scalar field θ(x, t). The scalar was driven by a mean scalar gradient, G = Ge�, in both simulations which causes deviations from isotropy of the small-scale statistics. For increasing Sc, scalar filaments can be advected to ever finer scales which steepens up the local gradients, gi = ∂iθ. Large gradients (or fronts) that are aligned with the mean, i.e. g�, occur already for Sc ∼ 1 or even below and are associated with characteristic scalar structures, so-called ramps and cliffs (for further references, see [3]). A return to isotropic mixing is then thought as a growing compensation of those pronounced positive fronts by an increasing number of steep negative gradients with
Bastian Goldluecke - One of the best experts on this subject based on the ideXlab platform.
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the natural vectorial total variation which arises from Geometric Measure Theory
Siam Journal on Imaging Sciences, 2012Co-Authors: Bastian Goldluecke, Evgeny Strekalovskiy, Daniel CremersAbstract:Several ways to generalize scalar total variation to vector-valued functions have been proposed in the past. In this paper, we give a detailed analysis of a variant we denote by $\text{TV}_J$, which has not been previously explored as a regularizer. The contributions of the manuscript are twofold: on the theoretical side, we show that $\text{TV}_J$ can be derived from the generalized Jacobians from Geometric Measure Theory. Thus, within the context of this Theory, $\text{TV}_J$ is the most natural form of a vectorial total variation. As an important feature, we derive how $\text{TV}_J$ can be written as the support functional of a convex set in $\mathcal{L}^2$. This property allows us to employ fast and stable minimization algorithms to solve inverse problems. The analysis also shows that in contrast to other total variation regularizers for color images, the proposed one penalizes across a common edge direction for all channels, which is a major theoretical advantage. Our practical contribution consist o...
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An approach to vectorial total variation based on Geometric Measure Theory
2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2010Co-Authors: Bastian Goldluecke, Daniel CremersAbstract:We analyze a previously unexplored generalization of the scalar total variation to vector-valued functions, which is motivated by Geometric Measure Theory. A complete mathematical characterization is given, which proves important invariance properties as well as existence of solutions of the vectorial ROF model. As an important feature, there exists a dual formulation for the proposed vectorial total variation, which leads to a fast and stable minimization algorithm. The main difference to previous approaches with similar properties is that we penalize across a common edge direction for all channels, which is a major theoretical advantage. Experiments show that this leads to a significantly better restoration of color edges in practice.
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CVPR - An approach to vectorial total variation based on Geometric Measure Theory
2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2010Co-Authors: Bastian Goldluecke, Daniel CremersAbstract:We analyze a previously unexplored generalization of the scalar total variation to vector-valued functions, which is motivated by Geometric Measure Theory. A complete mathematical characterization is given, which proves important invariance properties as well as existence of solutions of the vectorial ROF model. As an important feature, there exists a dual formulation for the proposed vectorial total variation, which leads to a fast and stable minimization algorithm. The main difference to previous approaches with similar properties is that we penalize across a common edge direction for all channels, which is a major theoretical advantage. Experiments show that this leads to a significiantly better restoration of color edges in practice.
Frank Morgan - One of the best experts on this subject based on the ideXlab platform.
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Geometric Measure Theory
2009Co-Authors: Frank MorganAbstract:Geometric Measure Theory could be described as differential geometry, generalized through Measure Theory to deal with maps and surfaces that are not necessarily smooth and applied to the calculus of variations. It dates from the 1960 foundational paper of Herbert Federer and Wendell Fleming on Normal and Integral Currents , recognized by the 1986 AMS Steele Prize for a paper of fundamental or lasting importance. This chapter presents an outline of the purpose and basic concepts of Geometric Measure Theory. Along with its successes and advantages, the definition of a surface as a mapping has certain drawbacks: (1) there is an inevitable a priori restriction on the types of singularities that can occur; (2) there is an a priori restriction on the topological complexity; and (3) the natural topology lacks compactness properties. The importance of compactness properties appears in the direct method. The direct method for finding a surface of least area with a given boundary has three steps: (1) take a sequence of surfaces with areas decreasing to the infimum, (2) extract a convergent subsequence, and (3) show that the limit surface is the desired surface of least area. An alternative to surfaces as mappings is provided by rectifiable currents, the m -dimensional, oriented surfaces of Geometric Measure Theory.
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Geometric Measure Theory a beginner s guide
2008Co-Authors: Frank MorganAbstract:Geometric Measure Theory: A Beginner's Guide, Fifth Edition provides the framework readers need to understand the structure of a crystal, a soap bubble cluster, or a universe. The book is essential to any student who wants to learn Geometric Measure Theory, and will appeal to researchers and mathematicians working in the field. Brevity, clarity, and scope make this classic book an excellent introduction to more complex ideas from Geometric Measure Theory and the calculus of variations for beginning graduate students and researchers. Morgan emphasizes geometry over proofs and technicalities, providing a fast and efficient insight into many aspects of the subject, with new coverage to this edition including topical coverage of the Log Convex Density Conjecture, a major new theorem at the center of an area of mathematics that has exploded since its appearance in Perelman's proof of the Poincar conjecture, and new topical coverage of manifolds taking into account all recent research advances in Theory and applications. * Focuses on core geometry rather than proofs, paving the way to fast and efficient insight into an extremely complex topic in Geometric structures* Enables further study of more advanced topics and texts* Demonstrates in the simplest possible way how to relate concepts of Geometric analysis by way of algebraic or topological techniques* Contains full topical coverage of The Log-Convex Density Conjecture* Comprehensively updated throughout
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Geometric Measure Theory (Third Edition) - CHAPTER 4 – Normal and Rectifiable Currents
Geometric Measure Theory, 2000Co-Authors: Frank Morgan, James F. BredtAbstract:The focus of this chapter is on normal and rectifiable currents. To define boundary and establish compactness properties, it is useful to view the rectifiable sets as currents, that is, linear functionals on smooth differential forms (named by analogy with electrical currents). The action of an oriented rectifiable set S on a differential form φ is given by integrating the form φ over the set. Currents thus associated with certain rectifiable sets, with integer multiplicities, will be called rectifiable currents. The larger class of normal currents will allow for real multiplicities and smoothing. The concept of currents is a generalization, by de Rham of distributions. Normal and rectifiable currents are due to Federer and Fleming. Important earlier and contemporaneous work includes the generalized surfaces of L. C. Young the frontiers of E. De Giorgi and the surfaces of E. R. Reifenberg. For hypersurfaces, rectifiable currents are just boundaries of the sets of finite perimeter of Caccioppoli and De Giorgi.
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Geometric Measure Theory (Third Edition) - CHAPTER 16 – Immiscible Fluids and Crystals
Geometric Measure Theory, 2000Co-Authors: Frank Morgan, James F. BredtAbstract:This chapter focuses on immiscible fluids and crystals. Clusters of immiscible fluids F 1 ,…., F m (with ambient Fo) such as oil, water, and mercury in air tend to minimize an energy proportional to surface area, where now the constant of proportionality a ij > 0 depends on which fluids the surface separates. It is assumed that triangle inequalities a ik ≤ a ij + a jk , since otherwise an interface between F i and F k could profitably be replaced by a thin layer of F j . The existence of least-energy clusters of immiscible fluids in R n follows as for soap bubble clusters. Technically it is very convenient to use flat chains with “fluid” rather than integer coefficients, so that, for example, the superposition of an oil-water interface and a water-mercury interface is automatically an oil-mercury interface. The structure of the singularities is not as well understood as for soap bubble clusters. In planar singularities, many circular arcs can meet at an isolated point. A conjecture on the classification of energy-minimizing cones in R 2 was proved for up to five fluids and disproved for six.
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Geometric Measure Theory (Third Edition) - CHAPTER 17 – Isoperimetric Theorems in General Codimension
Geometric Measure Theory, 2000Co-Authors: Frank Morgan, James F. BredtAbstract:This chapter discusses isoperimetric theorems in general codimension. The classical isoperimetric inequality for the volume of a region in R n in terms of its perimeter, maximized by the round ball, has important generalizations to higher codimension and to other ambients. It was not until 1986 that the classical isoperimetric inequality was extended to general codimension by Fred Almgren. While in codimension 0 there is a unique region with given boundary, in higher codimension there are many surfaces (of unbounded area) with given boundary and the isoperimetric inequality applies only to the one of least area. An m-dimensional area-minimizing integral current in R n (2 ≤ m ≤ n) has no more area than a round disc of the same boundary area, with equality only for the round disc itself. Previously for m n , may by decomposition be assumed to be a closed curve through the origin. The cone over the curve can be developed in the plane and thus shown to have no more area than a round planar disc.
Thomas Möllenhoff - One of the best experts on this subject based on the ideXlab platform.
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Lifting Vectorial Variational Problems: A Natural Formulation Based on Geometric Measure Theory and Discrete Exterior Calculus
2019 IEEE CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2019Co-Authors: Thomas Möllenhoff, Daniel CremersAbstract:Numerous tasks in imaging and vision can be formulated as variational problems over vector-valued maps. We approach the relaxation and convexification of such vectorial variational problems via a lifting to the space of currents. To that end, we recall that functionals with polyconvex Lagrangians can be reparametrized as convex one-homogeneous functionals on the graph of the function. This leads to an equivalent shape optimization problem over oriented surfaces in the product space of domain and codomain. A convex formulation is then obtained by relaxing the search space from oriented surfaces to more general currents. We propose a discretization of the resulting infinite-dimensional optimization problem using Whitney forms, which also generalizes recent "sublabel-accurate" multilabeling approaches.
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CVPR - Lifting Vectorial Variational Problems: A Natural Formulation Based on Geometric Measure Theory and Discrete Exterior Calculus
2019 IEEE CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2019Co-Authors: Thomas Möllenhoff, Daniel CremersAbstract:Numerous tasks in imaging and vision can be formulated as variational problems over vector-valued maps. We approach the relaxation and convexification of such vectorial variational problems via a lifting to the space of currents. To that end, we recall that functionals with polyconvex Lagrangians can be reparametrized as convex one-homogeneous functionals on the graph of the function. This leads to an equivalent shape optimization problem over oriented surfaces in the product space of domain and codomain. A convex formulation is then obtained by relaxing the search space from oriented surfaces to more general currents. We propose a discretization of the resulting infinite-dimensional optimization problem using Whitney forms, which also generalizes recent "sublabel-accurate" multilabeling approaches.