Body-and-cad geometric constraint systems

Haller, Kirk; John, Audrey Lee-St.; Sitharam, Meera; Streinu, Ileana; White, Neil

doi:10.1145/1529282.1529530

Cited by 13 publications

(29 citation statements)

References 31 publications

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“…Previous works on related types of geometric constraint frameworks include pin-collinear body-pin frameworks [9], direction networks [20], slider-pinning rigidity [16], body-cad constraint system [8], k-frames [18,19], and affine rigid- ity [6]. However, we are not aware of any previous results on systems that are similar to pinned subspace-incidence systems.…”

Section: Contributionsmentioning

confidence: 98%

Combinatorial Rigidity and Independence of Generalized Pinned Subspace-Incidence Constraint Systems

Wang

Sitharam

2015

Automated Deduction in Geometry

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Given a hypergraph H with m hyperedges and a set X of m pins, i.e. globally fixed subspaces in Euclidean space R d , a pinned subspaceincidence system is the pair (H, X), with the constraint that each pin in X lies on the subspace spanned by the point realizations in R d of vertices of the corresponding hyperedge of H. We are interested in combinatorial characterization of pinned subspace-incidence systems that are minimally rigid, i.e. those systems that are guaranteed to generically yield a locally unique realization. As is customary, this is accompanied by a characterization of generic independence as well as rigidity.In a previous paper [13], we used pinned subspace-incidence systems towards solving the fitted dictionary learning problem, i.e. dictionary learning with specified underlying hypergraph, and gave a combinatorial characterization of minimal rigidity for a more restricted version of pinned subspace-incidence system, with H being a uniform hypergraph and pins in X being 1-dimension subspaces. Moreover in a recent paper [2], the special case of pinned line incidence systems was used to model biomaterials such as cellulose and collagen fibrils in cell walls. In this paper, we extend the combinatorial characterization to general pinned subspace-incidence systems, with H being a non-uniform hypergraph and pins in X being subspaces with arbitrary dimension. As there are generally many data points per subspace in a dictionary learning problem, which can only be modeled with pins of dimension larger than 1, such an extension enables application to a much larger class of fitted dictionary learning problems.

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Section: Contributionsmentioning

confidence: 98%

Combinatorial Rigidity and Independence of Generalized Pinned Subspace-Incidence Constraint Systems

Wang

Sitharam

2015

Automated Deduction in Geometry

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“…In the setting of body-bar frameworks, including the specific setting of Body CAD frameworks, there are preliminary results [7,12] which include (nongeneric) pinned and slider-joints, and point-line, as well as point-plane, distance constraints. These results and the results in this paper, can be refined and extended to explore body-bar-point frameworks -which do occur in built linkages in 2D and 3D.…”

Section: Further Remarks and Open Problemsmentioning

confidence: 99%

Point-hyperplane frameworks, slider joints, and rigidity preserving transformations

Eftekhari

Jackson

Nixon

et al. 2019

Journal of Combinatorial Theory, Series B

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A one-to-one correspondence between the infinitesimal motions of bar-joint frameworks in R d and those in S d is a classical observation by Pogorelov, and further connections among different rigidity models in various different spaces have been extensively studied. In this paper, we shall extend this line of research to include the infinitesimal rigidity of frameworks consisting of points and hyperplanes. This enables us to understand correspondences between point-hyperplane rigidity, classical bar-joint rigidity, and scene analysis.Among other results, we derive a combinatorial characterization of graphs that can be realized as infinitesimally rigid frameworks in the plane with a given set of points collinear. This extends a result by Jackson and Jordán, which deals with the case when three points are collinear.

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“…To our best knowledge, the only known results with a similar flavor are [17,30] which characterize the rigidity of Body-and-Cad frameworks. However, these results are dedicated to specific frameworks in 3D instead of arbitrary dimension subspace arrangements and hypergraphs, and their formulation process start directly with the linearized Jacobian (omitting the first bullet alone).…”

Section: Contributionsmentioning

confidence: 99%

Combinatorial rigidity of incidence systems and application to dictionary learning

Sitharam¹,

Tarifi²,

Wang³

2018

Journal of Symbolic Computation

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Given a hypergraph H with m hyperedges and a set Q of m pinning subspaces, i.e. globally fixed subspaces in Euclidean space R d , a pinned subspace-incidence system is the pair (H, Q), with the constraint that each pinning subspace in Q is contained in the subspace spanned by the point realizations in R d of vertices of the corresponding hyperedge of H. This paper provides a combinatorial characterization of pinned subspace-incidence systems that are minimally rigid, i.e. those systems that are guaranteed to generically yield a locally unique realization.Pinned subspace-incidence systems have applications in the Dictionary Learning (aka sparse coding) problem, i.e. the problem of obtaining a sparse representation of a given set of data vectors by learning dictionary vectors upon which the data vectors can be written as sparse linear combinations. Viewing the dictionary vectors from a geometry perspective as the spanning set of a subspace arrangement, the result gives a tight bound on the number of dictionary vectors for sufficiently randomly chosen data vectors, and gives a way of constructing a dictionary that meets the bound. For less stringent restrictions on data, but a natural modification of the dictionary learning problem, a further dictionary learning algorithm is provided. Although there are recent rigidity based approaches for low rank matrix completion, we are unaware of prior application of combinatorial rigidity techniques in the setting of Dictionary Learning. We also provide a systematic classification of problems related to dictionary learning together with various algorithms, their assumptions and performance. *

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Body-and-cad geometric constraint systems

Cited by 13 publications

References 31 publications

Combinatorial Rigidity and Independence of Generalized Pinned Subspace-Incidence Constraint Systems

Combinatorial Rigidity and Independence of Generalized Pinned Subspace-Incidence Constraint Systems

Point-hyperplane frameworks, slider joints, and rigidity preserving transformations

Combinatorial rigidity of incidence systems and application to dictionary learning

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