On angles between subspaces of a finite dimensional inner product space

Wedin, Per Åke

doi:10.1007/bfb0062107

Cited by 85 publications

(59 citation statements)

References 9 publications

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“…We adopt a greedy strategy to find the optimal solution, initializing the key frame list with all frames in the original sequence and then incrementally decreasing it by one until the difference between the original PCA subspace and the new PCA subspace spanned by the remaining frames exceeds a user-specified threshold ε. The difference between the two PCA subspaces is measured by principal angle [Wedin 1983], and is experimentally set to 0.4. The details of our analysis algorithm are described in Algorithm 1.…”

Section: Optimizationmentioning

confidence: 99%

Automatic acquisition of high-fidelity facial performances using monocular videos

Shi¹,

et al. 2014

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Figure 1: Our system automatically captures high-fidelity facial performances using Internet videos: (left) input video data; (middle) the captured facial performances; (right) facial editing results: wrinkle removal and facial geometry editing. AbstractThis paper presents a facial performance capture system that automatically captures high-fidelity facial performances using uncontrolled monocular videos (e.g., Internet videos). We start the process by detecting and tracking important facial features such as the nose tip and mouth corners across the entire sequence and then use the detected facial features along with multilinear facial models to reconstruct 3D head poses and large-scale facial deformation of the subject at each frame. We utilize per-pixel shading cues to add finescale surface details such as emerging or disappearing wrinkles and folds into large-scale facial deformation. At a final step, we iterate our reconstruction procedure on large-scale facial geometry and fine-scale facial details to further improve the accuracy of facial reconstruction. We have tested our system on monocular videos downloaded from the Internet, demonstrating its accuracy and robustness under a variety of uncontrolled lighting conditions and overcoming significant shape differences across individuals. We show our system advances the state of the art in facial performance capture by comparing against alternative methods.

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Section: Optimizationmentioning

confidence: 99%

Automatic acquisition of high-fidelity facial performances using monocular videos

Shi¹,

et al. 2014

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“…In this section we give the background information on the angles between two finite dimensional subspaces of a Hilbert space as given in [3,12,17]. Basic results on generalized inverses of (unbounded) operators defined between two Hilbert spaces will be presented as well.…”

Section: The Generalized Inverse and Angle Between The Subspacesmentioning

confidence: 99%

On Eigenvalue and Eigenvector Estimates for Nonnegative Definite Operators

Grubišić¹

2006

SIAM J. Matrix Anal. & Appl.

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Abstract. In this article we further develop a perturbation approach to the RayleighRitz approximations from our earlier work. We both sharpen the estimates and extend the applicability of the theory to nonnegative definite operators . The perturbation argument enables us to solve two problems in one go: We determine which part of the spectrum of the operator is being approximated by the Ritz values and compute the approximation estimates. We also present a Temple-Kato like inequality which -unlike the original Temple-Kato inequality-applies to any test vectors from the quadratic form domain of the operator.

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“…While this notion of angle is less well known, it is nevertheless entirely standard (see, e.g., [33,26,17]). …”

Section: Integration Over Fibresmentioning

confidence: 99%

“…Remark 5 There is a considerable literature on how to compute the angle between two subspaces (see, e.g., [33,26,17]). If we write {∇f 1 , .…”

Section: Integration Over Fibresmentioning

confidence: 99%

Ambient space formulations and statistical mechanics of holonomically constrained Langevin systems

Walter

Hartmann

Maddocks

2011

Eur. Phys. J. Spec. Top.

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The most classic approach to the dynamics of an n-dimensional mechanical system constrained by d independent holonomic constraints is to pick explicitly a new set of (n − d) curvilinear coordinates parametrizing the manifold of configurations satisfying the constraints, and to compute the Lagrangian generating the unconstrained dynamics in these (n−d) configuration coordinates. Starting from this Lagrangian an unconstrained Hamiltonian H(q, p) on 2(n − d) dimensional phase space can then typically be defined in the standard way via a Legendre transform. Furthermore, if the system is in contact with a heat bath, the associated Langevin and Fokker-Planck equations can be introduced. Provided that an appropriate fluctuation-dissipation condition is satisfied, there will be a canonical equilibrium distribution of the Gibbs form exp(−βH) with respect to the flat measure dqdp in these 2(n−d) dimensional curvilinear phase space coordinates. The existence of (n − d) coordinates satisfying the constraints is often guaranteed locally by an implicit function theorem. Nevertheless in many examples these coordinates cannot be constructed in any tractable form, even locally, so that other approaches are of interest. In ambient space formulations the dynamics are defined in the full original n-dimensional configuration space, and associated 2n-dimensional phase space, with some version of Lagrange multipliers introduced so that the 2(n − d) dimensional sub-manifold of phase space implied by the holonomic constraints and their time derivative, is invariant under the dynamics. In this article we review ambient space formulations, and explain that for constrained dynamics there is in fact considerable freedom in how a Hamiltonian form of the dynamics can be constructed. We then discuss and contrast the Langevin and Fokker-Planck equations and their equilibrium distributions for the different forms of ambient space dynamics.

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On angles between subspaces of a finite dimensional inner product space

Cited by 85 publications

References 9 publications

Automatic acquisition of high-fidelity facial performances using monocular videos

Automatic acquisition of high-fidelity facial performances using monocular videos

On Eigenvalue and Eigenvector Estimates for Nonnegative Definite Operators

Ambient space formulations and statistical mechanics of holonomically constrained Langevin systems

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