Orthogonal Procrustes Analysis

Crosilla, Fabio; Beinat, Alberto; Fusiello, Andrea; Maset, Eleonora; Visintini, D.

doi:10.1007/978-3-030-11760-3_2

Cited by 5 publications

(5 citation statements)

References 43 publications

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“…F is the Frobenius Norm, and I 3 is the identity matrix in R 3 . In the case of triaxial accelerometer data, 3 , with m rows corresponding to m three spatial coordinates in time. One can decompose each object into a matrix of the true signal and a noise matrix.…”

Section: Appendix: Proof Of Trivariate Normally Random Residuals For ...mentioning

confidence: 99%

“…Alternatively, if the victim was longer than the bed, Procrustes would cut off the legs to make the body fit the bed's length. In either event, the victim died [2,3]. In a similar, but less gruesome manner, Procrustes Analysis or Procrustes methods refer to a set of least squares mathematical techniques used to perform transformations among corresponding points belonging to a generic m-dimensional space, to satisfy some maximum agreement [3].…”

Section: Introductionmentioning

confidence: 99%

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Robust and Optimal Alignment of High-Dimensional Data Using Maximum Likelihood Estimation through a Random Sample Consensus Framework

Watts,

Harvey

2023

Preprint

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Correcting spatial orientations of groups of high-dimensional data sets such that they are all in a consistent coordinate system is often a time consuming and error-prone process. Automation of this process can be accomplished by using Generalized Procrustes Analysis to estimate the relative orientations among a population of high-dimensional data sets. A least squares Procrustes solution is applied through a maximum likelihood estimation and random sample consensus framework for robustness. The likelihood model is comprised of a mixture distribution where inliers are assumed to originate from a Student's t-distribution and outliers from a uniform distribution. Applications will focus on a synthetic data set that emulates tri-axial acceleration data from a population of accelerometers. Outliers represent either non-rigid body responses, data acquisition, and/or sensor issues. The intended application for the methodology is to robustly automate the rotation of populations of experimentally collected tri-axial accelerometer data sets to a single global coordinate system.

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Section: Appendix: Proof Of Trivariate Normally Random Residuals For ...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Robust and Optimal Alignment of High-Dimensional Data Using Maximum Likelihood Estimation through a Random Sample Consensus Framework

Watts,

Harvey

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Matrix transformation of vector spaces has been accomplished using Generalized Procrustes Analysis (GPA), ever since a computationally viable solution was devised (Gower, 1975). In particular, GPA has been used to great effect in geo-spatial shape manipulation (Duta, 2015;Crosilla et al, 2019) and qualitative data analysis (Maurício et al, 2016).…”

Section: Procrustes Analysismentioning

confidence: 99%

“…Thus, distance measurement among image vectors can use L p norm or trigonometric calculations such as cosine distance. One specific kind of euclidean distance measurement is called Procrustes Distance and is the basis of Procrustes Analysis (Crosilla et al, 2019).…”

Section: Image Data and Language Datamentioning

confidence: 99%

Language Model Metrics and Procrustes Analysis for Improved Vector Transformation of NLP Embeddings

Conley,

Kalita

2021

Preprint

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Artificial Neural networks are mathematical models at their core. This truism presents some fundamental difficulty when networks are tasked with Natural Language Processing.A key problem lies in measuring the similarity or distance among vectors in NLP embedding space, since the mathematical concept of distance does not always agree with the linguistic concept. We suggest that the best way to measure linguistic distance among vectors is by employing the Language Model (LM) that created them. We introduce Language Model Distance (LMD) for measuring accuracy of vector transformations based on the Distributional Hypothesis (LMD Accuracy). We show the efficacy of this metric by applying it to a simple neural network learning the Procrustes algorithm for bilingual word mapping.

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“…However, only ground control points (GCPs) that meet certain criteria can be used [ 10 ], which poses a limitation. To estimate the camera’s EOPs, SPR algorithms based on the law of cosines [ 11 ] and the Procrustes algorithm [ 12 , 13 , 14 ] were proposed. The PnP algorithm, proposed by Fischler and Bolles, is a method of estimating the position and orientation of a camera using a point corresponding to a 3D object and a 2D image [ 15 ].…”

Section: Introductionmentioning

confidence: 99%

Absolute IOP/EOP Estimation Models without Initial Information of Various Smart City Sensors

Kim

Baek

Kim

2023

Sensors

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In smart cities, a large amount of optical camera equipment is deployed and used. Closed-circuit television (CCTV), unmanned aerial vehicles (UAVs), and smartphones are some examples of such equipment. However, additional information about these devices, such as 3D position, orientation information, and principal distance, is not provided. To solve this problem, the structured mobile mapping system point cloud was used in this study to investigate methods of estimating the principal point, position, and orientation of optical sensors without initial given values. The principal distance was calculated using two direct linear transformation (DLT) models and a perspective projection model. Methods for estimating position and orientation were discussed, and their stability was tested using real-world sensors. When the perspective projection model was used, the camera position and orientation were best estimated. The original DLT model had a significant error in the orientation estimation. The correlation between the DLT model parameters was thought to have influenced the estimation result. When the perspective projection model was used, the position and orientation errors were 0.80 m and 2.55°, respectively. However, when using a fixed-wing UAV, the estimated result was not properly produced owing to ground control point placement problems.

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Orthogonal Procrustes Analysis

Cited by 5 publications

References 43 publications

Robust and Optimal Alignment of High-Dimensional Data Using Maximum Likelihood Estimation through a Random Sample Consensus Framework

Robust and Optimal Alignment of High-Dimensional Data Using Maximum Likelihood Estimation through a Random Sample Consensus Framework

Language Model Metrics and Procrustes Analysis for Improved Vector Transformation of NLP Embeddings

Absolute IOP/EOP Estimation Models without Initial Information of Various Smart City Sensors

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