On the least-squares orthogonalization of an oblique transformation

Gibson, Walker

doi:10.1007/bf02289637

Cited by 29 publications

(13 citation statements)

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“…In the case of G = SO(n), Although the embedding of SO(n) in Euclidean space is not a convex set, the projection onto the matrix manifold is easily achieved by means of the singular value decomposition [21]. Let…”

Section: Regularization Of Maps Onto So(n)mentioning

confidence: 99%

Augmented-Lagrangian regularization of matrix-valued maps

Rosman

Tai

Kimmel

et al. 2014

Methods and Applications of Analysis

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Abstract. We propose a novel framework for fast regularization of matrix-valued images. The resulting algorithms allow a unified treatment for a broad set of matrix groups and manifolds. Using an augmented-Lagrangian technique, we formulate a fast and highly parallel algorithm for matrixvalued image regularization.We demonstrate the applicability of the framework for various problems, such as motion analysis and diffusion tensor image reconstruction, show the formulation of the algorithm in terms of splitBregman iterations and discuss the convergence properties of the proposed algorithms.Key words. Regularization, Lie-groups, total-variation, split-Bregman, matrix-manifolds, diffusion-imaging, rotations, articulated motion.AMS subject classifications. 65K10, 58JXX.1. Introduction. Matrix-manifolds and Matrix-valued images have become an integral part of computer vision and image processing. Matrix-manifolds and groups have been used for tracking [43,56], robotics [38,58,10], motion analysis [45,20], image processing and computer vision [9,40,42,45,60], as well as medical imaging [4,39]. Efficient regularization of matrix-valued images is therefore highly important in the fields of image analysis and computer vision. This includes applications such as direction diffusion [29,53,59] and scene motion analysis [33] in computer vision, as well as diffusion tensor MRI (DT-MRI) regularization [5,15,25,50,54] in medical imaging.We present an augmented Lagrangian method for efficient regularization of matrix-valued images, or maps. We assume the matrix-manifold to have an efficient projection operator onto it from some embedding into a Euclidean space, and that the distortion associated with this mapping is not too large in term of the metric accompanying these spaces.Examples of matrix-manifolds that are of interest include the special-orthogonal and special-Euclidean Lie-groups and the symmetric positive-definite matrices, as well as Stiefel manifolds. We show that the augmented Lagrangian technique allows us to separate the optimization process into a regularization update step of a map onto an embedding-space, and a per-pixel projection step. An efficient regularization step is shown for the total-variation (TV, [48]) regularization, and a second-order regularization penalizing the Hessian norm. Both the regularization step and the projection steps are simple to compute, fast and easily parallelizable using consumer graphic processing units (GPUs), achieving real-time processing rates. The resulting framework unifies algorithms using in several application domains into one framework, since they differ only in the choice of projection operator. While such an optimization problem could have been approached by general saddle-point solvers such as [12], the

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Section: Regularization Of Maps Onto So(n)mentioning

confidence: 99%

Augmented-Lagrangian regularization of matrix-valued maps

Rosman

Tai

Kimmel

et al. 2014

Methods and Applications of Analysis

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“…411-412). Bentler's (1968Bentler's ( , 1971) clustran criterion, based on a proof due to Gibson (1962), yields an orthonormal rotated factor loading matrix, B, fitting b in a least squares sense. If…”

Section: Ah (1)mentioning

confidence: 99%

An Empirical Evaluation of Factor Reliability1

Jackson

Morf

1971

ETS Research Bulletin Series

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The psychometric reliability of a factor, defined as its generalizability across samples drawn from the same population of tests, is considered as a necessary precondition for the scientific meaningfulness of factor analytic results. A solution to the problem of generalizability is illustrated empirically on data from a set of tests designed to measure facets of response styles and of personality dimensions. Parallel sets of measures based on personality scales defining each of seven factors were separately factored. Independent sets of component scores derived from the. orthogonal least squares fit to the oblique factor pattern matrix were computed, and these component scores were intercorrelated between the two sets, yielding factor reliabilities, whose values ranged from .65 to .85 (p < .0001, for each factor). A corresponding analysis based on scores derived from random binary data yielded nonsignificant factor reliabilities ranging from ‐.12 to +.07. It was recommended that such a test of factor generalizability be incorporated routinely into factor analytic investigations, particularly those employing Procrustes‐type rotations.

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“…Its validity for n + 1 is to be demonstrated. Young, 1936, 1939;Gibson, 1962;Johnson, 1963) G l r1 .•• G n+ 1rn+1 = U l i\.U 2 with U l , U 2 orthogonal and i\. diagonal.…”

Section: This Impliesmentioning

confidence: 99%

A theorem on the trace of certain matrix products and some applications

Kristof

1970

Journal of Mathematical Psychology

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On the least-squares orthogonalization of an oblique transformation

Cited by 29 publications

References 4 publications

Augmented-Lagrangian regularization of matrix-valued maps

Augmented-Lagrangian regularization of matrix-valued maps

An Empirical Evaluation of Factor Reliability1

A theorem on the trace of certain matrix products and some applications

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