Bicycle chain shape models

Sommer, Stefan; Tatu, Aditya; Chen, Chen; Jurgensen, Dan R; Bruijne, Marleen de; Loog, Marco; Nielsen, Mads; Lauze, François

doi:10.1109/cvprw.2009.5204053

Cited by 11 publications

(7 citation statements)

References 15 publications

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“…Sommer et.al. introduced the Bicycle chain shape models [11] in which, every landmark on the curve is constrained to be equidistant from its neighbors. Glaunes et al [12] consider a shape as an interior of a planar curve, while the shape space is built by looking at all possible diffeomorphic deformations of the interior of closed curves.…”

Section: Related Workmentioning

confidence: 99%

Tangled Splines

Tatu¹

2016

Preprint

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Extracting shape information from object boundaries is a well studied problem in vision, and has found tremendous use in applications like object recognition. Conversely, studying the space of shapes represented by curves satisfying certain constraints is also intriguing. In this paper, we model and analyze the space of shapes represented by a 3D curve (space curve) formed by connecting n pieces of quarter of a unit circle. Such a space curve is what we call a Tangle, the name coming from a toy built on the same principle.We provide two models for the shape space of n-link open and closed tangles, and we show that tangles are a subset of trigonometric splines of a certain order. We give algorithms for curve approximation using open/closed tangles, computing geodesics on these shape spaces, and to find the deformation that takes one given tangle to another given tangle, i.e., the Log map. The algorithms provided yield tangles upto a small and acceptable tolerance, as shown by the results given in the paper.

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Section: Related Workmentioning

confidence: 99%

Tangled Splines

Tatu¹

2016

Preprint

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“…The best-known example of such an approach is Principal Geodesic Analysis (PGA) Fletcher et al (2003Fletcher et al ( , 2004. PGA and its variants such as Said et al (2007);Huckemann et al (2010); Sommer et al (2010) have been successfully employed for various application, such as analyzing vertebrae outlines Sommer et al (2009) and motion capture data Said et al (2007). PGA can be understood as a generalization of PCA to Riemannian manifolds.…”

Section: Related Workmentioning

confidence: 99%

Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods

Harandi¹,

Salzmann²,

Hartley³

2016

Preprint

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Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices -especially of high-dimensional ones-comes at a high cost that limits the applicability of existing techniques. In this paper, we introduce algorithms able to handle high-dimensional SPD matrices by constructing a lower-dimensional SPD manifold. To this end, we propose to model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. This lets us formulate dimensionality reduction as the problem of finding a projection that yields a low-dimensional manifold either with maximum discriminative power in the supervised scenario, or with maximum variance of the data in the unsupervised one. We show that learning can be expressed as an optimization problem on a Grassmann manifold and discuss fast solutions for special cases. Our evaluation on several classification tasks evidences that our approach leads to a significant accuracy gain over state-of-the-art methods.

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“…A vast body of mathematical literature describes manifolds and Riemannian structures; [5,26] provide excellent introductions to the field. From an applied point of view, the papers [4,22,28,24,35,39] address first-order problems such as computing geodesics and solving the exponential map inverse problem, the logarithm map. Certain second-order problems including computing Jacobi fields on diffeomorphism groups [44,6] have been considered but mainly on limited classes of manifolds.…”

Section: Related Workmentioning

confidence: 99%

“…It has subsequently been used for several applications. To mention a few, the authors in [12,7] study variations of medial atoms, [41] uses a variation of PGA for facial classification, [34] presents examples on motion capture data, and [39] applies PGA to vertebrae outlines. The algorithm presented in [12] for computing PGA with tangent space linearization is most widely used.…”

Section: Related Workmentioning

confidence: 99%

“…The space R m is called the embedding space. When dealing with implicitly defined manifolds, we let m and n denote the dimension of the domain and codomain of F , respectively, so that the dimension η of the manifold equals m − n. Examples of applications using implicit representations include shape and human poses models [39,18], and several shape models use parametric representations [20,25]. 1…”

Section: Manifolds and Their Representationsmentioning

confidence: 99%

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Optimization over geodesics for exact principal geodesic analysis

2013

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In fields ranging from computer vision to signal processing and statistics, increasing computational power allows a move from classical linear models to models that incorporate non-linear phenomena. This shift has created interest in computational aspects of differential geometry, and solving optimization problems that incorporate non-linear geometry constitutes an important computational task. In this paper, we develop methods for numerically solving optimization problems over spaces of geodesics using numerical integration of Jacobi fields and second order derivatives of geodesic families. As an important application of this optimization strategy, we compute exact Principal Geodesic Analysis (PGA), a non-linear version of the PCA dimensionality reduction procedure. By applying the exact PGA algorithm to synthetic data, we exemplify the differences between the linearized and exact algorithms caused by the non-linear geometry. In addition, we use the numerically integrated Jacobi fields to determine sectional curvatures and provide upper bounds for injectivity radii.

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Bicycle chain shape models

Cited by 11 publications

References 15 publications

Tangled Splines

Tangled Splines

Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods

Optimization over geodesics for exact principal geodesic analysis

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