Precise Hausdorff distance computation between polygonal meshes

Bartoň, Michael; Hanniel, Iddo; Elber, Gershon; Kim, Myung-Soo

doi:10.1016/j.cagd.2010.04.004

Cited by 56 publications

(25 citation statements)

References 14 publications

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“…Other path metrics-e.g., Fréchet with speed limits, direction-based Fréchet [69], or Fréchet with shortcuts for the analysis of noisy data [70]-may offer advantages in carrying out various analyses. The Hausdorff distance can be generalized as well to measure, for instance, distances between surfaces (instead of 1D curves) [71]. The multitudinous permutations that can be selected among the various path metrics, structural similarity metrics, clustering algorithms, etc.…”

Section: Path Metrics For Measuring Transition Path Similaritymentioning

confidence: 99%

Path Similarity Analysis: A Method for Quantifying Macromolecular Pathways

et al. 2015

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Diverse classes of proteins function through large-scale conformational changes and various sophisticated computational algorithms have been proposed to enhance sampling of these macromolecular transition paths. Because such paths are curves in a high-dimensional space, it has been difficult to quantitatively compare multiple paths, a necessary prerequisite to, for instance, assess the quality of different algorithms. We introduce a method named Path Similarity Analysis (PSA) that enables us to quantify the similarity between two arbitrary paths and extract the atomic-scale determinants responsible for their differences. PSA utilizes the full information available in 3N-dimensional configuration space trajectories by employing the Hausdorff or Fréchet metrics (adopted from computational geometry) to quantify the degree of similarity between piecewise-linear curves. It thus completely avoids relying on projections into low dimensional spaces, as used in traditional approaches. To elucidate the principles of PSA, we quantified the effect of path roughness induced by thermal fluctuations using a toy model system. Using, as an example, the closed-to-open transitions of the enzyme adenylate kinase (AdK) in its substrate-free form, we compared a range of protein transition path-generating algorithms. Molecular dynamics-based dynamic importance sampling (DIMS) MD and targeted MD (TMD) and the purely geometric FRODA (Framework Rigidity Optimized Dynamics Algorithm) were tested along with seven other methods publicly available on servers, including several based on the popular elastic network model (ENM). PSA with clustering revealed that paths produced by a given method are more similar to each other than to those from another method and, for instance, that the ENM-based methods produced relatively similar paths. PSA was applied to ensembles of DIMS MD and FRODA trajectories of the conformational transition of diphtheria toxin, a particularly challenging example. For the AdK transition, the new concept of a Hausdorff-pair map enabled us to extract the molecular structural determinants responsible for differences in pathways, namely a set of conserved salt bridges whose charge-charge interactions are fully modelled in DIMS MD but not in FRODA. PSA has the potential to enhance our understanding of transition path sampling methods, validate them, and to provide a new approach to analyzing conformational transitions.

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Section: Path Metrics For Measuring Transition Path Similaritymentioning

confidence: 99%

Path Similarity Analysis: A Method for Quantifying Macromolecular Pathways

et al. 2015

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“…The Hausdorff distance is widely used to measure approximation error in Computer Aided Geometric Design, as a fair measure for the distance between two curves or surfaces; e.g., [1][2][3]10]. We sample points along the curves.…”

Section: A P P R O X I M a T I O N O F G E O D E S I C C I R C L E Smentioning

confidence: 99%

Circle Approximations on Spheroids

2011

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We present an approximation method for geodesic circles on a spheroid. Our approximation curve is the intersection of two spheroids whose axes are parallel, and it interpolates four points of the geodesic circle. Our approximation method has two merits. One is that the approximation curve can be obtained algebraically, and the other is that the approximation error is very small. For example, our approximation of a circle of radius 1000 km on the Earth has error 1·13 cm or less. We analyze the error of our approximation using the Hausdorff distance and confirm it by a geodesic distance computation.K E Y WO R D S 1. geodesic circle.2. spheroid. 3. geodesic distance. 4. geodesic path.

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“…Similar to [17], the Hausdorff distance between a point and a triangle includes three types: Point-Face, Point-Edge, and Point-Point. We describe the main idea in Figure 4 (see also [17]). We denote p to be the projection of p over the plane containing abc.…”

Section: Iterative Adjustment Of Samplingmentioning

confidence: 99%

“…If max D 1 > max D 2 , then p * ∈ P M , q 1 q 2 q 3 ∈ F M and we shall update the locations of q 1 , q 2 , q 3 using the gradient of 2 pf given by (17). Otherwise, we update the locations of p 1 , p 2 , p 3 with the gradient of 2 pf given by …”

Section: Iterative Adjustment Of Samplingmentioning

confidence: 99%

Adaptive Geometry Images for Remeshing

Shi

Kong

Wang

et al. 2017

International Journal of Digital Multimedia Broadcasting

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Geometry images are a kind of completely regular remeshing methods for mesh representation. Traditional geometry images have difficulties in achieving optimal reconstruction errors and preserving manually selected geometric details, due to the limitations of parametrization methods. To solve two issues, we propose two adaptive geometry images for remeshing triangular meshes. The first scheme produces geometry images with the minimum Hausdorff error by finding the optimization direction for sampling points based on the Hausdorff distance between the original mesh and the reconstructed mesh. The second scheme produces geometry images with higher reconstruction precision over the manually selected region-of-interest of the input mesh, by increasing the number of sampling points over the region-of-interest. Experimental results show that both schemes give promising results compared with traditional parametrization-based geometry images.

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Precise Hausdorff distance computation between polygonal meshes

Cited by 56 publications

References 14 publications

Path Similarity Analysis: A Method for Quantifying Macromolecular Pathways

Path Similarity Analysis: A Method for Quantifying Macromolecular Pathways

Circle Approximations on Spheroids

Adaptive Geometry Images for Remeshing

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