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@ -1100,6 +1100,17 @@ Teillaud"
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,pages = "307--320"
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}
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@article{ cgal:l-lsqp-82,
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title={Least squares quantization in PCM},
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author={Lloyd, Stuart},
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journal={IEEE transactions on information theory},
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volume={28},
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number={2},
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pages={129--137},
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year={1982},
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publisher={IEEE}
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}
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@book{ cgal:l-mrfmi-09,
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author = {Li, Stan Z.},
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title = {Markov Random %Field Modeling in Image Analysis},
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@ -2164,6 +2175,16 @@ location = {Salt Lake City, Utah, USA}
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Geodesy and Photogrammetry)},
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url = {http://www.prs.igp.ethz.ch/research/Source_code_and_datasets.html},
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year = 2014
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@inproceedings{ cgal:wk-srhvs-05,
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title={Structure recovery via hybrid variational surface approximation},
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author={Wu, Jianhua and Kobbelt, Leif},
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booktitle={Computer Graphics Forum},
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volume={24},
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number={3},
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pages={277--284},
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year={2005},
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organization={Wiley Online Library}
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}
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@book{cgal:ww-smgd-02
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@ -16,7 +16,7 @@ The input of the algorithm must be:
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The output is a triangle soup and can be built into a polygon surface mesh.
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Given an input surface triangle mesh, %VSA leverages a discrete clustering algorithm to approximate it by a set of local simple shapes referred to as proxies. Each cluster is represented as a connected set of triangles of the input mesh, and the output mesh is constructed by generating a surface triangle mesh which approximates the clusters.
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The approximation error is one-sided, defined between the clusters and their associated proxies. Two error metrics (\f$ \mathcal{L}^2 \f$, \f$ \mathcal{L}^{2,1} \f$) for planar proxies are provided via the classes `CGAL::VSA::L2_metric_plane_proxy` and `CGAL::VSA::L21_metric_plane_proxy`, and the algorithm design is generic to other user-defined metrics. The current proxies are planes or vectors, however the algorithm design is generic for future extensions to non-planar proxies \cgalCite{cgal:ywly-vmsqs-12}. The default \f$ \mathcal{L}^{2,1} \f$ metric is recommended in terms of computation and visual perception \cgalCite{cgal:cad-vsa-04}. A brief background about <em>%Proxy</em> and <em>%ErrorMetric</em> is provided in Section \ref sma_background.
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The approximation error is one-sided, defined between the clusters and their associated proxies. Two error metrics (\f$ \mathcal{L}^2 \f$, \f$ \mathcal{L}^{2,1} \f$) for planar proxies are provided via the classes `CGAL::VSA::L2_metric_plane_proxy` and `CGAL::VSA::L21_metric_plane_proxy`, and the algorithm design is generic to other user-defined metrics. The current proxies are planes or vectors, however the algorithm design is generic for future extensions to non-planar proxies \cgalCite{cgal:ywly-vmsqs-12}\cgalCite{wk-srhvs-05}. The default \f$ \mathcal{L}^{2,1} \f$ metric is recommended in terms of computation and visual perception \cgalCite{cgal:cad-vsa-04}. A brief background about <em>%Proxy</em> and <em>%ErrorMetric</em> is provided in Section \ref sma_background.
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\cgalFigureBegin{Approximation_teaser, teaser.png}
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Variational shape approximation on two models with the \f$ \mathcal{L}^{2,1} \f$ error metric and planar proxies. From left to right: partition of the input surface triangle mesh, anchor vertices and edges, and output triangle mesh. The partition is optimized via discrete clustering of the input triangles, so as to minimize the approximation error from the clusters to the planar proxies (not shown).
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@ -42,7 +42,7 @@ Workflow of the approximation process in the free function.
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\subsubsection sma_clustering Clustering Iteration
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Figure \cgalFigureRef{iterations} depicts several Lloyd clustering iterations on the plane-sphere model with planar proxies and the \f$ \mathcal{L}^{2,1} \f$ metric. We plot the fitting error against each iteration. After 8 iterations, the error barely changes. Based on this observation, we consider that the clustering converges if the error change between the current and previous iteration is lower than a user-specifed threshold (indicated by two green dash lines).
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Figure \cgalFigureRef{iterations} depicts several Lloyd \cgalCite{cgal:l-lsqp-82} clustering iterations on the plane-sphere model with planar proxies and the \f$ \mathcal{L}^{2,1} \f$ metric. We plot the fitting error against each iteration. After 8 iterations, the error barely changes. Based on this observation, we consider that the clustering converges if the error change between the current and previous iteration is lower than a user-specifed threshold (indicated by two green dash lines).
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\cgalFigureBegin{iterations, iterations.png}
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Discrete Lloyd iterations on the plane-sphere model with planar proxies and the \f$ \mathcal{L}^{2,1} \f$ metric: (left) random seeding of 6 proxies; (center) after one iteration; (right) after 8 iterations, the regions settle. The red lines depict the proxy normals drawn at the seed faces.
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