38-Issue 5

Permanent URI for this collection

https://diglib7.eg.org/handle/10.2312/2632795
Geometry Processing 2019 - Symposium Proceedings
8-10 July in Milan, Italy
(for Posters see SGP 2019 - Posters)
Issue Information
Shape Correspondences
Divergence-Free Shape Correspondence by Deformation
[full paper Image] [meta data Image]
Marvin Eisenberger, Zorah Lähner, and Daniel Cremers
Consistent Shape Matching via Coupled Optimization
[full paper Image] [meta data Image]
Omri Azencot, Anastasia Dubrovina, and Leonidas Guibas
Dense Point-to-Point Correspondences Between Genus-Zero Shapes
[full paper Image] [meta data Image]
Sing Chun Lee and Misha Kazhdan
Functional Maps
Structured Regularization of Functional Map Computations
[full paper Image] [meta data Image]
Jing Ren, Mikhail Panine, Peter Wonka, and Maks Ovsjanikov
Hierarchical Functional Maps between Subdivision Surfaces
[full paper Image] [meta data Image]
‪Meged Shoham‬‏, Amir Vaxman, and Mirela Ben-Chen
Modeling and Deformation
A Family of Barycentric Coordinates for Co-Dimension 1 Manifolds with Simplicial Facets
[full paper Image] [meta data Image]
Zhipei Yan and Scott Schaefer
Structural Design Using Laplacian Shells
[full paper Image] [meta data Image]
Erva Ulu, Jim McCann, and Levent Burak Kara
Synthesis and Learning
A Convolutional Decoder for Point Clouds using Adaptive Instance Normalization
[full paper Image] [meta data Image]
Isaak Lim, Moritz Ibing, and Leif Kobbelt
Point Pattern Synthesis via Irregular Convolution
[full paper Image] [meta data Image]
Peihan Tu, Dani Lischinski, and Hui Huang
Unsupervised Cycle-consistent Deformation for Shape Matching
[full paper Image] [meta data Image]
Thibault Groueix, Matthew Fisher, Vladimir G. Kim, Bryan C. Russel, and Mathieu Aubry
Shape Representations
Feature Preserving Octree-Based Hexahedral Meshing
[full paper Image] [meta data Image]
Xifeng Gao, Hanxiao Shen, and Daniele Panozzo
High Quality Refinable G-splines for Locally Quad-dominant Meshes With T-gons
[full paper Image] [meta data Image]
Kestutis Karciauskas and Jorg Peters
2D and 3D Reconstruction
Parallel Globally Consistent Normal Orientation of Raw Unorganized Point Clouds
[full paper Image] [meta data Image]
Johannes Jakob, Christoph Buchenau, and Michael Guthe
On Evaluating Consensus in RANSAC Surface Registration
[full paper Image] [meta data Image]
Lukáš Hruda, Jan Dvořák, and Libor Váša
Shape Collections and Analysis
Limit Shapes - A Tool for Understanding Shape Differences and Variability in 3D Model Collections
[full paper Image] [meta data Image]
Ruqi Huang, Panos Achlioptas, Leonidas Guibas, and Maks Ovsjanikov

BibTeX (38-Issue 5)
                
@article{
10.1111:cgf.13785,

journal = {Computer Graphics Forum},
title = {{
Divergence-Free Shape Correspondence by Deformation}},

author = {
Eisenberger, Marvin
 and 
Lähner, Zorah
 and 
Cremers, Daniel
},
year = {
2019},

publisher = {
The Eurographics Association and John Wiley & Sons Ltd.},


ISSN = {1467-8659},

DOI = {
10.1111/cgf.13785}

}
                
@article{
10.1111:cgf.13786,

journal = {Computer Graphics Forum},
title = {{
Consistent Shape Matching via Coupled Optimization}},

author = {
Azencot, Omri
 and 
Dubrovina, Anastasia
 and 
Guibas, Leonidas
},
year = {
2019},

publisher = {
The Eurographics Association and John Wiley & Sons Ltd.},


ISSN = {1467-8659},

DOI = {
10.1111/cgf.13786}

}
                
@article{
10.1111:cgf.13787,

journal = {Computer Graphics Forum},
title = {{
Dense Point-to-Point Correspondences Between Genus-Zero Shapes}},

author = {
Lee, Sing Chun
 and 
Kazhdan, Misha
},
year = {
2019},

publisher = {
The Eurographics Association and John Wiley & Sons Ltd.},


ISSN = {1467-8659},

DOI = {
10.1111/cgf.13787}

}
                
@article{
10.1111:cgf.13788,

journal = {Computer Graphics Forum},
title = {{
Structured Regularization of Functional Map Computations}},

author = {
Ren, Jing
 and 
Panine, Mikhail
 and 
Wonka, Peter
 and 
Ovsjanikov, Maks
},
year = {
2019},

publisher = {
The Eurographics Association and John Wiley & Sons Ltd.},


ISSN = {1467-8659},

DOI = {
10.1111/cgf.13788}

}
                
@article{
10.1111:cgf.13789,

journal = {Computer Graphics Forum},
title = {{
Hierarchical Functional Maps between Subdivision Surfaces}},

author = {
Shoham‬‏, ‪Meged
 and 
Vaxman, Amir
 and 
Ben-Chen, Mirela
},
year = {
2019},

publisher = {
The Eurographics Association and John Wiley & Sons Ltd.},


ISSN = {1467-8659},

DOI = {
10.1111/cgf.13789}

}
                
@article{
10.1111:cgf.13790,

journal = {Computer Graphics Forum},
title = {{
A Family of Barycentric Coordinates for Co-Dimension 1 Manifolds with Simplicial Facets}},

author = {
Yan, Zhipei
 and 
Schaefer, Scott
},
year = {
2019},

publisher = {
The Eurographics Association and John Wiley & Sons Ltd.},


ISSN = {1467-8659},

DOI = {
10.1111/cgf.13790}

}
                
@article{
10.1111:cgf.13791,

journal = {Computer Graphics Forum},
title = {{
Structural Design Using Laplacian Shells}},

author = {
Ulu, Erva
 and 
McCann, Jim
 and 
Kara, Levent Burak
},
year = {
2019},

publisher = {
The Eurographics Association and John Wiley & Sons Ltd.},


ISSN = {1467-8659},

DOI = {
10.1111/cgf.13791}

}
                
@article{
10.1111:cgf.13792,

journal = {Computer Graphics Forum},
title = {{
A Convolutional Decoder for Point Clouds using Adaptive Instance Normalization}},

author = {
Lim, Isaak
 and 
Ibing, Moritz
 and 
Kobbelt, Leif
},
year = {
2019},

publisher = {
The Eurographics Association and John Wiley & Sons Ltd.},


ISSN = {1467-8659},

DOI = {
10.1111/cgf.13792}

}
                
@article{
10.1111:cgf.13793,

journal = {Computer Graphics Forum},
title = {{
Point Pattern Synthesis via Irregular Convolution}},

author = {
Tu, Peihan
 and 
Lischinski, Dani
 and 
Huang, Hui
},
year = {
2019},

publisher = {
The Eurographics Association and John Wiley & Sons Ltd.},


ISSN = {1467-8659},

DOI = {
10.1111/cgf.13793}

}
                
@article{
10.1111:cgf.13794,

journal = {Computer Graphics Forum},
title = {{
Unsupervised Cycle-consistent Deformation for Shape Matching}},

author = {
Groueix, Thibault
 and 
Fisher, Matthew
 and 
Kim, Vladimir G.
 and 
Russel, Bryan C.
 and 
Aubry, Mathieu
},
year = {
2019},

publisher = {
The Eurographics Association and John Wiley & Sons Ltd.},


ISSN = {1467-8659},

DOI = {
10.1111/cgf.13794}

}
                
@article{
10.1111:cgf.13795,

journal = {Computer Graphics Forum},
title = {{
Feature Preserving Octree-Based Hexahedral Meshing}},

author = {
Gao, Xifeng
 and 
Shen, Hanxiao
 and 
Panozzo, Daniele
},
year = {
2019},

publisher = {
The Eurographics Association and John Wiley & Sons Ltd.},


ISSN = {1467-8659},

DOI = {
10.1111/cgf.13795}

}
                
@article{
10.1111:cgf.13796,

journal = {Computer Graphics Forum},
title = {{
High Quality Refinable G-splines for Locally Quad-dominant Meshes With T-gons}},

author = {
Karciauskas, Kestutis
 and 
Peters, Jorg
},
year = {
2019},

publisher = {
The Eurographics Association and John Wiley & Sons Ltd.},


ISSN = {1467-8659},

DOI = {
10.1111/cgf.13796}

}
                
@article{
10.1111:cgf.13797,

journal = {Computer Graphics Forum},
title = {{
Parallel Globally Consistent Normal Orientation of Raw Unorganized Point Clouds}},

author = {
Jakob, Johannes
 and 
Buchenau, Christoph
 and 
Guthe, Michael
},
year = {
2019},

publisher = {
The Eurographics Association and John Wiley & Sons Ltd.},


ISSN = {1467-8659},

DOI = {
10.1111/cgf.13797}

}
                
@article{
10.1111:cgf.13798,

journal = {Computer Graphics Forum},
title = {{
On Evaluating Consensus in RANSAC Surface Registration}},

author = {
Hruda, Lukáš
 and 
Dvořák, Jan
 and 
Vasa, Libor
},
year = {
2019},

publisher = {
The Eurographics Association and John Wiley & Sons Ltd.},


ISSN = {1467-8659},

DOI = {
10.1111/cgf.13798}

}
                
@article{
10.1111:cgf.13799,

journal = {Computer Graphics Forum},
title = {{
Limit Shapes - A Tool for Understanding Shape Differences and Variability in 3D Model Collections}},

author = {
Huang, Ruqi
 and 
Achlioptas, Panos
 and 
Guibas, Leonidas
 and 
Ovsjanikov, Maks
},
year = {
2019},

publisher = {
The Eurographics Association and John Wiley & Sons Ltd.},


ISSN = {1467-8659},

DOI = {
10.1111/cgf.13799}

}

Browse

Recent Submissions

Now showing 1 - 16 of 16
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    Symposium on Geometry Processing 2019 - CGF38-5: Frontmatter
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Bommes, David; Huang, Hui; Bommes, David and Huang, Hui
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    Divergence-Free Shape Correspondence by Deformation
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Eisenberger, Marvin; Lähner, Zorah; Cremers, Daniel; Bommes, David and Huang, Hui
    We present a novel approach for solving the correspondence problem between a given pair of input shapes with non-rigid, nearly isometric pose difference. Our method alternates between calculating a deformation field and a sparse correspondence. The deformation field is constructed with a low rank Fourier basis which allows for a compact representation. Furthermore, we restrict the deformation fields to be divergence-free which makes our morphings volume preserving. This can be used to extract a correspondence between the inputs by deforming one of them along the deformation field using a second order Runge-Kutta method and resulting in an alignment of the inputs. The advantages of using our basis are that there is no need to discretize the embedding space and the deformation is volume preserving. The optimization of the deformation field is done efficiently using only a subsampling of the orginal shapes but the correspondence can be extracted for any mesh resolution with close to linear increase in runtime. We show 3D correspondence results on several known data sets and examples of natural intermediate shape sequences that appear as a by-product of our method.
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    Consistent Shape Matching via Coupled Optimization
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Azencot, Omri; Dubrovina, Anastasia; Guibas, Leonidas; Bommes, David and Huang, Hui
    We propose a new method for computing accurate point-to-point mappings between a pair of triangle meshes given imperfect initial correspondences. Unlike the majority of existing techniques, we optimize for a map while leveraging information from the inverse map, yielding results which are highly consistent with respect to composition of mappings. Remarkably, our method considers only a linear number of candidate points on the target shape, allowing us to work directly with high resolution meshes, and to avoid a delicate and possibly error-prone up-sampling procedure. Key to this dimensionality reduction is a novel candidate selection process, where the mapped points drift over the target shape, finalizing their location based on intrinsic distortion measures. Overall, we arrive at an iterative scheme where at each step we optimize for the map and its inverse by solving two relaxed Quadratic Assignment Problems using off-the-shelf optimization tools. We provide quantitative and qualitative comparison of our method with several existing techniques, and show that it provides a powerful matching tool when accurate and consistent correspondences are required.
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    Dense Point-to-Point Correspondences Between Genus-Zero Shapes
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Lee, Sing Chun; Kazhdan, Misha; Bommes, David and Huang, Hui
    We describe a novel approach that addresses the problem of establishing correspondences between non-rigidly deformed shapes by performing the registration over the unit sphere. In a pre-processing step, each shape is conformally parametrized over the sphere, centered to remove Möbius inversion ambiguity, and authalically evolved to expand regions that are excessively compressed by the conformal parametrization. Then, for each pair of shapes, we perform fast SO(3) correlation to find the optimal rotational alignment and refine the registration using optical flow. We evaluate our approach on the TOSCA dataset, demonstrating that our approach compares favorably to state-of-the-art methods.
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    Structured Regularization of Functional Map Computations
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Ren, Jing; Panine, Mikhail; Wonka, Peter; Ovsjanikov, Maks; Bommes, David and Huang, Hui
    We consider the problem of non-rigid shape matching using the functional map framework. Specifically, we analyze a commonly used approach for regularizing functional maps, which consists in penalizing the failure of the unknown map to commute with the Laplace-Beltrami operators on the source and target shapes. We show that this approach has certain undesirable fundamental theoretical limitations, and can be undefined even for trivial maps in the smooth setting. Instead we propose a novel, theoretically well-justified approach for regularizing functional maps, by using the notion of the resolvent of the Laplacian operator. In addition, we provide a natural one-parameter family of regularizers, that can be easily tuned depending on the expected approximate isometry of the input shape pair. We show on a wide range of shape correspondence scenarios that our novel regularization leads to an improvement in the quality of the estimated functional, and ultimately pointwise correspondences before and after commonly-used refinement techniques.
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    Hierarchical Functional Maps between Subdivision Surfaces
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Shoham‬‏, ‪Meged; Vaxman, Amir; Ben-Chen, Mirela; Bommes, David and Huang, Hui
    We propose a novel approach for computing correspondences between subdivision surfaces with different control polygons. Our main observation is that the multi-resolution spectral basis functions that are often used for computing a functional correspondence can be compactly represented on subdivision surfaces, and therefore can be efficiently computed. Furthermore, the reconstruction of a pointwise map from a functional correspondence also greatly benefits from the subdivision structure. Leveraging these observations, we suggest a hierarchical pipeline for functional map inference, allowing us to compute correspondences between surfaces at fine subdivision levels, with hundreds of thousands of polygons, an order of magnitude faster than existing correspondence methods. We demonstrate the applicability of our results by transferring high-resolution sculpting displacement maps and textures between subdivision models.
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    A Family of Barycentric Coordinates for Co-Dimension 1 Manifolds with Simplicial Facets
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Yan, Zhipei; Schaefer, Scott; Bommes, David and Huang, Hui
    We construct a family of barycentric coordinates for 2D shapes including non-convex shapes, shapes with boundaries, and skeletons. Furthermore, we extend these coordinates to 3D and arbitrary dimension. Our approach modifies the construction of the Floater-Hormann-Kós family of barycentric coordinates for 2D convex shapes.We show why such coordinates are restricted to convex shapes and show how to modify these coordinates to extend to discrete manifolds of co-dimension 1 whose boundaries are composed of simplicial facets. Our coordinates are well-defined everywhere (no poles) and easy to evaluate. While our construction is widely applicable to many domains, we show several examples related to image and mesh deformation.
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    Structural Design Using Laplacian Shells
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Ulu, Erva; McCann, Jim; Kara, Levent Burak; Bommes, David and Huang, Hui
    We introduce a method to design lightweight shell objects that are structurally robust under the external forces they may experience during use. Given an input 3D model and a general description of the external forces, our algorithm generates a structurally-sound minimum weight shell object. Our approach works by altering the local shell thickness repeatedly based on the stresses that develop inside the object. A key issue in shell design is that large thickness values might result in self-intersections on the inner boundary creating a significant computational challenge during optimization. To address this, we propose a shape parametrization based on the solution to the Laplace's equation that guarantees smooth and intersection-free shell boundaries. Combined with our gradient-free optimization algorithm, our method provides a practical solution to the structural design of hollow objects with a single inner cavity. We demonstrate our method on a variety of problems with arbitrary 3D models under complex force configurations and validate its performance with physical experiments.
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    A Convolutional Decoder for Point Clouds using Adaptive Instance Normalization
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Lim, Isaak; Ibing, Moritz; Kobbelt, Leif; Bommes, David and Huang, Hui
    Automatic synthesis of high quality 3D shapes is an ongoing and challenging area of research. While several data-driven methods have been proposed that make use of neural networks to generate 3D shapes, none of them reach the level of quality that deep learning synthesis approaches for images provide. In this work we present a method for a convolutional point cloud decoder/generator that makes use of recent advances in the domain of image synthesis. Namely, we use Adaptive Instance Normalization and offer an intuition on why it can improve training. Furthermore, we propose extensions to the minimization of the commonly used Chamfer distance for auto-encoding point clouds. In addition, we show that careful sampling is important both for the input geometry and in our point cloud generation process to improve results. The results are evaluated in an autoencoding setup to offer both qualitative and quantitative analysis. The proposed decoder is validated by an extensive ablation study and is able to outperform current state of the art results in a number of experiments. We show the applicability of our method in the fields of point cloud upsampling, single view reconstruction, and shape synthesis.
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    Point Pattern Synthesis via Irregular Convolution
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Tu, Peihan; Lischinski, Dani; Huang, Hui; Bommes, David and Huang, Hui
    Point pattern synthesis is a fundamental tool with various applications in computer graphics. To synthesize a point pattern, some techniques have taken an example-based approach, where the user provides a small exemplar of the target pattern. However, it remains challenging to synthesize patterns that faithfully capture the structures in the given exemplar. In this paper, we present a new example-based point pattern synthesis method that preserves both local and non-local structures present in the exemplar. Our method leverages recent neural texture synthesis techniques that have proven effective in synthesizing structured textures. The network that we present is end-to-end. It utilizes an irregular convolution layer, which converts a point pattern into a gridded feature map, to directly optimize point coordinates. The synthesis is then performed by matching inter- and intra-correlations of the responses produced by subsequent convolution layers. We demonstrate that our point pattern synthesis qualitatively outperforms state-of-the-art methods on challenging structured patterns, and enables various graphical applications, such as object placement in natural scenes, creative element patterns or realistic urban layouts in a 3D virtual environment.
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    Unsupervised Cycle-consistent Deformation for Shape Matching
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Groueix, Thibault; Fisher, Matthew; Kim, Vladimir G.; Russel, Bryan C.; Aubry, Mathieu; Bommes, David and Huang, Hui
    We propose a self-supervised approach to deep surface deformation. Given a pair of shapes, our algorithm directly predicts a parametric transformation from one shape to the other respecting correspondences. Our insight is to use cycle-consistency to define a notion of good correspondences in groups of objects and use it as a supervisory signal to train our network. Our method combines does not rely on a template, assume near isometric deformations or rely on point-correspondence supervision. We demonstrate the efficacy of our approach by using it to transfer segmentation across shapes. We show, on Shapenet, that our approach is competitive with comparable state-of-the-art methods when annotated training data is readily available, but outperforms them by a large margin in the few-shot segmentation scenario.
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    Feature Preserving Octree-Based Hexahedral Meshing
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Gao, Xifeng; Shen, Hanxiao; Panozzo, Daniele; Bommes, David and Huang, Hui
    We propose an octree-based algorithm to tessellate the interior of a closed surface with hexahedral cells. The generated hexahedral mesh (1) explicitly preserves sharp features of the original input, (2) has a maximal, user-controlled distance deviation from the input surface, (3) is composed of elements with only positive scaled jacobians (measured by the eight corners of a hex [SEK*07]), and (4) does not have self-intersections. We attempt to achieve these goals by proposing a novel pipeline to create an initial pure hexahedral mesh from an octree structure, taking advantage of recent developments in the generation of locally injective 3D parametrizations to warp the octree boundary to conform to the input surface. Sharp features in the input are bijectively mapped to poly-lines in the output and preserved by the deformation, which takes advantage of a scaffold mesh to prevent local and global intersections. The robustness of our technique is experimentally validated by batch processing a large collection of organic and CAD models, without any manual cleanup or parameter tuning. All results including mesh data and statistics in the paper are provided in the additional material. The open-source implementation will be made available online to foster further research in this direction.
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    High Quality Refinable G-splines for Locally Quad-dominant Meshes With T-gons
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Karciauskas, Kestutis; Peters, Jorg; Bommes, David and Huang, Hui
    Polyhedral modeling and re-meshing algorithms use T-junctions to add or remove feature lines in a quadrilateral mesh. In many ways this is akin to adaptive knot insertion in a tensor-product spline, but differs in that the designer or meshing algorithm does not necessarily protect the consistent combinatorial structure that is required to interpret the resulting quad-dominant mesh as the control net of a hierarchical spline - and so associate a smooth surface with the mesh as in the popular tensor-product spline paradigm. While G-splines for multi-sided holes or generalized subdivision can, in principle, convert quad-dominant meshes with T-junctions into smooth surfaces, they do not preserve the two preferred directions and so cause visible shape artifacts. Only recently have n-gons with T-junctions (T-gons) in unstructured quad-dominant meshes been recognized as a distinct challenge for generalized splines. This paper makes precise the notion of locally quad-dominant mesh as quad-meshes including t-nets, i.e. T-gons surrounded by quads; and presents the first high-quality G-spline construction that can use t-nets as control nets for spline surfaces suitable, e.g., for automobile outer surfaces. Remarkably, T-gons can be neighbors, separated by only one quad, both of T-gons and of points where many quads meet. A t-net surface cap consists of 16 polynomial pieces of degree (3,5) and is refinable in a way that is consistent with the surrounding surface. An alternative, everywhere bi-3 cap is not formally smooth, but achieves the same high-quality highlight line distribution.
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    Parallel Globally Consistent Normal Orientation of Raw Unorganized Point Clouds
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Jakob, Johannes; Buchenau, Christoph; Guthe, Michael; Bommes, David and Huang, Hui
    A mandatory component for many point set algorithms is the availability of consistently oriented vertex-normals (e.g. for surface reconstruction, feature detection, visualization). Previous orientation methods on meshes or raw point clouds do not consider a global context, are often based on unrealistic assumptions, or have extremely long computation times, making them unusable on real-world data. We present a novel massively parallelized method to compute globally consistent oriented point normals for raw and unsorted point clouds. Built on the idea of graph-based energy optimization, we create a complete kNN-graph over the entire point cloud. A new weighted similarity criterion encodes the graph-energy. To orient normals in a globally consistent way we perform a highly parallel greedy edge collapse, which merges similar parts of the graph and orients them consistently. We compare our method to current state-of-the-art approaches and achieve speedups of up to two orders of magnitude. The achieved quality of normal orientation is on par or better than existing solutions, especially for real-world noisy 3D scanned data.
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    On Evaluating Consensus in RANSAC Surface Registration
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Hruda, Lukáš; Dvořák, Jan; Vasa, Libor; Bommes, David and Huang, Hui
    Random Sample Consensus is a powerful paradigm that was successfully applied in various contexts, including Location Determination Problem, fundamental matrix estimation and global 3D surface registration, where many previously proposed algorithms can be interpreted as a particular implementation of this concept. In general, a set of candidate transformations is generated by some simple procedure, and an aligning transformation is chosen within this set, such that it aligns the largest portion of the input data. We observe that choosing the aligning transformation may also be interpreted as finding consensus among the candidates, which in turn involves measuring similarity of candidate rigid transformations. While it is not difficult to construct a metric that provides reasonable results, most approaches come with certain limitations and drawbacks. In this paper, we investigate possible means of measuring distances in SE(3) and compare their properties both theoretically and experimentally in a model RANSAC registration algorithm. We also propose modifications to existing measures and propose a novel method of locating the consensus transformation based on Vantage Point Tree data structure.
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    Limit Shapes - A Tool for Understanding Shape Differences and Variability in 3D Model Collections
    (The Eurographics Association and John Wiley & Sons Ltd., 2019) Huang, Ruqi; Achlioptas, Panos; Guibas, Leonidas; Ovsjanikov, Maks; Bommes, David and Huang, Hui
    We propose a novel construction for extracting a central or limit shape in a shape collection, connected via a functional map network. Our approach is based on enriching the latent space induced by a functional map network with an additional natural metric structure. We call this shape-like dual object the limit shape and show that its construction avoids many of the biases introduced by selecting a fixed base shape or template. We also show that shape differences between real shapes and the limit shape can be computed and characterize the unique properties of each shape in a collection - leading to a compact and rich shape representation. We demonstrate the utility of this representation in a range of shape analysis tasks, including improving functional maps in difficult situations through the mediation of limit shapes, understanding and visualizing the variability within and across different shape classes, and several others. In this way, our analysis sheds light on the missing geometric structure in previously used latent functional spaces, demonstrates how these can be addressed and finally enables a compact and meaningful shape representation useful in a variety of practical applications.