Fast and Robust QEF Minimization using Probabilistic Quadrics

Error quadrics are a fundamental and powerful building block in many geometry processing algorithms. However, finding the minimizer of a given quadric is in many cases not robust and requires a singular value decomposition or some ad‐hoc regularization. While classical error quadrics measure the squ...

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Published in:	Computer graphics forum Vol. 39; no. 2; pp. 325 - 334
Main Authors:	Trettner, P., Kobbelt, L.
Format:	Journal Article
Language:	English
Published:	Oxford Blackwell Publishing Ltd 01.05.2020
Subjects:	Algorithms CCS Concepts Computing methodologies → Mesh models; Mesh geometry models Error analysis Ground truth Normal distribution Optimization Random noise Regularization Robustness Singular value decomposition Triangles
ISSN:	0167-7055, 1467-8659
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Abstract	Error quadrics are a fundamental and powerful building block in many geometry processing algorithms. However, finding the minimizer of a given quadric is in many cases not robust and requires a singular value decomposition or some ad‐hoc regularization. While classical error quadrics measure the squared deviation from a set of ground truth planes or polygons, we treat the input data as genuinely uncertain information and embed error quadrics in a probabilistic setting (“probabilistic quadrics”) where the optimal point minimizes the expected squared error. We derive closed form solutions for the popular plane and triangle quadrics subject to (spatially varying, anisotropic) Gaussian noise. Probabilistic quadrics can be minimized robustly by solving a simple linear system — 50× faster than SVD. We show that probabilistic quadrics have superior properties in tasks like decimation and isosurface extraction since they favor more uniform triangulations and are more tolerant to noise while still maintaining feature sensitivity. A broad spectrum of applications can directly benefit from our new quadrics as a drop‐in replacement which we demonstrate with mesh smoothing via filtered quadrics and non‐linear subdivision surfaces.
AbstractList	Error quadrics are a fundamental and powerful building block in many geometry processing algorithms. However, finding the minimizer of a given quadric is in many cases not robust and requires a singular value decomposition or some ad‐hoc regularization. While classical error quadrics measure the squared deviation from a set of ground truth planes or polygons, we treat the input data as genuinely uncertain information and embed error quadrics in a probabilistic setting (“probabilistic quadrics”) where the optimal point minimizes the expected squared error. We derive closed form solutions for the popular plane and triangle quadrics subject to (spatially varying, anisotropic) Gaussian noise. Probabilistic quadrics can be minimized robustly by solving a simple linear system — 50× faster than SVD. We show that probabilistic quadrics have superior properties in tasks like decimation and isosurface extraction since they favor more uniform triangulations and are more tolerant to noise while still maintaining feature sensitivity. A broad spectrum of applications can directly benefit from our new quadrics as a drop‐in replacement which we demonstrate with mesh smoothing via filtered quadrics and non‐linear subdivision surfaces. Error quadrics are a fundamental and powerful building block in many geometry processing algorithms. However, finding the minimizer of a given quadric is in many cases not robust and requires a singular value decomposition or some ad‐hoc regularization. While classical error quadrics measure the squared deviation from a set of ground truth planes or polygons, we treat the input data as genuinely uncertain information and embed error quadrics in a probabilistic setting (“probabilistic quadrics”) where the optimal point minimizes the expected squared error. We derive closed form solutions for the popular plane and triangle quadrics subject to (spatially varying, anisotropic) Gaussian noise. Probabilistic quadrics can be minimized robustly by solving a simple linear system — 50× faster than SVD. We show that probabilistic quadrics have superior properties in tasks like decimation and isosurface extraction since they favor more uniform triangulations and are more tolerant to noise while still maintaining feature sensitivity. A broad spectrum of applications can directly benefit from our new quadrics as a drop‐in replacement which we demonstrate with mesh smoothing via filtered quadrics and non‐linear subdivision surfaces.
Author	Kobbelt, L. Trettner, P.
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Copyright	2020 The Author(s) Computer Graphics Forum © 2020 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. 2020 The Eurographics Association and John Wiley & Sons Ltd.
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SubjectTerms	Algorithms CCS Concepts Computing methodologies → Mesh models; Mesh geometry models Error analysis Ground truth Normal distribution Optimization Random noise Regularization Robustness Singular value decomposition Triangles
Title	Fast and Robust QEF Minimization using Probabilistic Quadrics
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