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Learning implicit functions for Dense 3D Shape Correspondence of Generic Objects

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 2024年第3期46卷 1852-1867页

作者： Liu, Feng Liu, Xiaoming Michigan State Univ Dept Comp Sci & Engn E Lansing MI 48824 USA

The objective of this paper is to learn dense 3D shape correspondence for topology-varying generic objects in an unsupervised manner. Conventional implicit functions estimate the occupancy of a 3D point given a shape latent code. Instead, our novel implicit function produces a probabilistic embedding to represent each 3D point in a part embedding space. Assuming the corresponding points are similar in the embedding space, we implement dense correspondence through an inverse function mapping from the part embedding vector to a corresponded 3D point. Both functions are jointly learned with several effective and uncertainty-aware loss functions to realize our assumption, together with the encoder generating the shape latent code. During inference, if a user selects an arbitrary point on the source shape, our algorithm can automatically generate a confidence score indicating whether there is a correspondence on the target shape, as well as the corresponding semantic point if there is one. Such a mechanism inherently benefits man-made objects with different part constitutions. The effectiveness of our approach is demonstrated through unsupervised 3D semantic correspondence and shape segmentation.

关键词： Shape Three-dimensional displays Semantics Uncertainty Strain Probabilistic logic Data models Dense 3D shape correspondence generic objects implicit functions inverse implicit functions topology-varying uncertainty-aware unsupervised learning

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General convex relaxations of implicit functions and inverse functions

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JOURNAL OF GLOBAL OPTIMIZATION 2023年第3期86卷 545-572页

作者： Cao, Huiyi Khan, Kamil A. McMaster Univ Dept Chem Engn 1280 Main St West Hamilton ON L8S 4L8 Canada

Convex relaxations of nonconvex functions provide useful bounding information in applications such as deterministic global optimization and reachability analysis. In some situations, the original nonconvex functions may not be known explicitly, but are instead described implicitly by nonlinear equation systems. In these cases, established convex relaxation methods for closed-form functions are not directly applicable. This article presents a new general strategy to construct convex relaxations for such implicit functions. These relaxations are described as convex parametric programs whose constraints are convex relaxations of the original residual function. This relaxation strategy is straightforward to implement, produces tight relaxations in practice, is particularly efficient to carry out when monotonicity properties can be exploited, and does not assume the existence or uniqueness of an implicit function on the entire intended domain. Unlike all previous approaches to the authors' knowledge, this new approach permits any relaxations of the residual function;it does not require the residual relaxations to be factorable or to be obtained from a McCormick-like traversal of a computational graph. This new convex relaxation strategy is extended to inverse functions, compositions involving implicit functions, feasible-set mappings in constraint satisfaction problems, and solutions of parametric ODEs. Based on a proof-of-concept implementation in Julia, numerical examples are presented to illustrate the convex relaxations produced for various implicit functions and optimal-value functions.

关键词： implicit functions Nonconvex optimization Convex underestimators McCormick relaxations Constraint satisfaction problems

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Critical Points Properties of Ordinary Differential Equations as a Projection of implicit functions Using Spatio-temporal Taylor Expansion 22nd

Critical Points Properties of Ordinary Differential Equation...

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22nd International Conference on Computational Science and its Applications (ICCSA)

作者： Skala, Vaclav Univ West Bohemia Fac Appl Sci Dept Comp Sci & Engn Plzen 30100 Czech Republic

ISBN: (纸本)9783031104503;9783031104497

This contribution describes a new approach to formulation of ODE and PDE critical points using implicit formulation as t-variant scalar function using the Taylor expansion. A general condition for the critical points is derived and specified for t invariant case. It is expected, that the given new formulae lead to more reliable detection of critical points especially for large 3D fluid flow data acquisition, which enable high 3D vector compression and their representation using radial basis functions (RBF). In the case of vector field visualization, e.g. fluid flow, electromagnetic fields, etc., the critical points of ODE are critical for physical phenomena behavior.

关键词： Critical points Vector fields visualization Numerical methods Ordinary differential equations Partial differential equations implicit functions Radial basis functions

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The combinatorics of higher derivatives of implicit functions

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MONATSHEFTE FUR MATHEMATIK 2019年第4期188卷 765-784页

作者： Zemel, Shaul Hebrew Univ Jerusalem Einstein Inst Math Edmund Safra Campus IL-91904 Jerusalem Israel

We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.

关键词： Higher derivatives implicit functions Combinatorial coefficients

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Dependent Object Types with implicit functions 10

Dependent Object Types with Implicit Functions

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10th ACM SIGPLAN International Symposium on Scala (Scala)

作者： Jeffery, Alex Univ Sussex Brighton E Sussex England

ISBN: (纸本)9781450368247

DOT (Dependent Object Types) is an object calculus with path-dependent types and abstract type members, developed to serve as a theoretical foundation for the Scala programming language. As yet, DOT does not model all of Scala's features, but a small subset. We present the calculus DIF (DOT with implicit functions), which extends the set of features modelled by DOT to include implicit functions, a feature of Scala to aid modularity of programs. We show type safety of DIF, and demonstrate that the generic programming focused use cases for implicit functions in Scala are also expressible in DIF.

关键词： implicits implicit parameters implicit functions type classes calculus objects dependent object types DOT Scala Dotty

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Branch-locking AD techniques for nonsmooth composite functions and nonsmooth implicit functions

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OPTIMIZATION METHODS & SOFTWARE 2018年第4-6期33卷 1127-1155页

作者： Khan, Kamil A. Argonne Natl Lab Math & Comp Sci Div Lemont IL 60439 USA McMaster Univ Hamilton ON Canada

A recent nonsmooth vector forward mode of algorithmic differentiation (AD) computes Nesterov's L-derivatives for nonsmooth composite functions;these L-derivatives provide useful sensitivity information to methods for nonsmooth optimization and equation solving. The established reverse AD mode evaluates gradients efficiently for smooth functions, but it does not extend directly to nonsmooth functions. Thus, this article examines branch-locking strategies to harness the benefits of smooth AD techniques even in the nonsmooth case, in order to improve the computational performance of the non smooth vector forward AD mode, In these strategies, each non-smooth elemental function in the original composition is 'locked' into an appropriate linear 'branch'. The original composition is thereby replaced with a smooth variant, which may be subjected to efficient AD techniques for smooth functions such as the reverse AD mode. In order to choose the correct linear branches, we use inexpensive probing steps to ascertain the composite function's local behaviour. A simple implementation in C++11 is described, and the developed techniques are extended to nonsmooth local implicit functions and inverse functions.

关键词： vector forward AD mode nonsmooth functions sensitivity analysis generalized derivatives implicit functions

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Newton step methods for AD of an objective defined using implicit functions

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OPTIMIZATION METHODS & SOFTWARE 2018年第4-6期33卷 907-923页

作者： Bell, Bradley M. Kristensen, Kasper Univ Washington IHME Seattle WA 98195 USA Tech Univ Lyngby DTU Aqua Lyngby Denmark

We consider the problem of computing derivatives of an objective that is defined using implicit functions;i.e., implicit variables are computed by solving equations that are often nonlinear and solved by an iterative process. If one were to apply Algorithmic Differentiation (AD) directly, one would differentiate the iterative process. In this paper we present the Newton step methods for computing derivatives of the objective. These methods make it easy to take advantage of sparsity, forward mode, reverse mode, and other AD techniques. We prove that the partial Newton step method works if the number of steps is equal to the order of the derivatives. The full Newton step method obtains two derivatives order for each step except for the first step. There are alternative methods that avoid differentiating the iterative process;e.g., the method implemented in ADOL-C. An optimal control example demonstrates the advantage of the Newton step methods when computing both gradients and Hessians. We also discuss the Laplace approximation method for nonlinear mixed effects models as an example application.

关键词： implicit functions automatic derivatives higher order derivatives Newton step optimal control nonlinear mixed effects

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Semi-Infinite Optimization with implicit functions

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INDUSTRIAL & ENGINEERING CHEMISTRY RESEARCH 2015年第1期54卷 307-317页

作者： Stuber, Matthew D. Barton, Paul I. MIT Dept Chem Engn Proc Syst Engn Lab Cambridge MA 02139 USA

In this work, equality-constrained bilevel optimization problems, arising from engineering design, economics, and operations research problems, are reformulated as an equivalent semi-infinite program (SIP) with implicit functions embedded, which are defined by the original equality constraints that model the system. Using recently developed theoretical tools for bounding implicit functions, a recently developed algorithm for global optimization of implicit functions, and a recently developed algorithm for solving standard SIPs with explicit functions to global optimality, a method for solving SIPs with implicit functions embedded is presented. The method is guaranteed to converge to ?-optimality in finitely many iterations given the existence of a Slater point arbitrarily close to a minimizer. Besides the Slater point assumption, it is assumed only that the functions are continuous and factorable and that the model equations are once continuously differentiable.

关键词： MATHEMATICAL optimization implicit functions ENGINEERING design GLOBAL optimization CONTINUOUS functions

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Numerical integration of implicit functions for the initialization of the VOF function

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COMPUTERS & FLUIDS 2015年 113卷 42-52页

作者： Bna, S. Manservisi, S. Scardovelli, R. Yecko, P. Zaleski, S. Univ Bologna DIN Lab Montecuccolino I-40136 Bologna Italy Montclair Univ Dept Math Montclair NJ USA Univ Paris 06 Inst Jean Rand dAlembert IJLRdA UMR 7190 F-75252 Paris 05 France CNRS F-75252 Paris 05 France

A numerical method of initializing cell volume fraction demarcated by implicitly defined fluid interfaces is presented. Each cell of the computational domain is examined for the presence of the reference phase. When a cell is not full or empty, limits are found that allow volume fraction to be computed by numerical integration. The method enlists a number of algorithms including root finding and minimum search on an oriented segment, a preconditioned conjugate gradient minimum search on a cell face and a double Gauss-Legendre integration with a variable number of nodes, among others. Practical multi-phase fluid examples in two- and three-dimensions are presented to demonstrate the accuracy and robustness of the method. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： implicit functions Numerical integration VOF function

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Surface reconstruction from three-dimensional segmentations using implicit functions 10

Surface reconstruction from three-dimensional segmentations ...

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10th Computing Colombian Conference (10CCC)

作者： Gomez-Mora, Miller Florez-Valencia, Leonardo Univ Dist Francisco Jose de Caldas Bogota Bogota Colombia Pontificia Univ Javeriana Dept Ingn Sistemas Bogota Colombia

ISBN: (纸本)9781467394642

Real-world object modeling is a crucial part of computer graphics, with findings of great importance for many applications in areas of computer assisted design, manufacturing and engineering. However, geometric modeling of such objects proves quite challenging, especially when complex volumetric structures are involved as in the case of biological anatomy. In this paper, we present a method for the 3-D modeling of human organs based on an implicit representation approach. The principal idea consists of sampling the region of interest from three-dimensional labeled images and implicitly reconstructing its surface using an algorithm based on the Poisson surface reconstruction method. From there, either a surface mesh or a 3-D mesh is adjusted to the implicit surface, which is given by the zero level set of a function f(x) = 0. This is done to allow not only the visualization but physically-based simulations. As far as assessment of the new method is concerned, we used the Dice Similarity Coefficient, and the Hausdorff Distance to evaluate accuracy.

关键词： Object modeling surface reconstruction implicit functions mesh generation

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