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Tracking and Calibration
Published in Terry M. Peters, Cristian A. Linte, Ziv Yaniv, Jacqueline Williams, Mixed and Augmented Reality in Medicine, 2018
The problem of registration of homologous point-sets is more appropriately known as the Orthogonal Procrustes Problem [16]. It is normally formulated as a minimization of the FRE between two datasets through a similarity transform, consisting of scaling (or weighting) factors, followed by rotation and then translation. If X denotes a mobile measurement point-set with n points, and Y denotes the homologous stationary model points, then the FRE is defined as follows: () FRE2=1N∑N1‖RSxi+t−yi‖2,
Efficacy of quantifying marker-cluster rigidity in a multi-segment foot model: a Monte-Carlo based global sensitivity analysis and regression model
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2022
Po-Hsiang Chan, Julie Stebbins, Amy B. Zavatsky
Rigid body error () originated from solving the orthogonal Procrustes problem (Schönemann, 1966). Proposed by van den Bogert (van den Bogert et al., 1994), the is computed as: where n is the number of markers, [A] is the reference configuration of the marker-cluster, [B] is the deformed configuration of the marker-cluster, and [T] the orthogonal transformation matrix for which the equation is minimised. The minimisation problem is also known as the Procrustes problem, where the algorithm attempts to find the transformation matrix that best describes the movement from to A number of papers have discussed and improved the solution to the orthogonal Procrustes problem (Green, 1951; Schönemann, 1966; Hanson and Norris, 1981; Söderkvist and Wedin, 1994, 1993). One of the solutions that utilises singular value decomposition, proposed by Söderkvist and Wedin (1993), showed more stability and can be easily implemented in programming for various applications, including human motion analysis. In this study, will be determined using the algorithm proposed by Söderkvist and Wedin (1993).
On the equivariance properties of self-adjoint matrices
Published in Dynamical Systems, 2020
Michael Dellnitz, Bennet Gebken, Raphael Gerlach, Stefan Klus
Given two symmetric matrices , the two-sided orthogonal Procrustes problem can be defined as follows: Find an orthogonal matrix such that the cost function is minimized. It is well known – see, for instance, [1,15,17] – that an optimal solution is given by , where and are the eigendecompositions of A and B, respectively. Note that the eigenvalues in and both have to be sorted in non-increasing (or, alternatively, non-decreasing) order. If the cost function is to be maximized, the eigenvalues need to be ordered in opposite order. With the aid of the results from Section 4, we can now characterize all solutions of this form, i.e. since for such P we obtain which is indeed the optimal solution. Here, we used the invariance of the Frobenius norm under unitary transformations and the equivariance properties.