# Topology‐Aware Surface Reconstruction for Point Clouds

@article{Gabrielsson2020TopologyAwareSR, title={Topology‐Aware Surface Reconstruction for Point Clouds}, author={Rickard Br{\"u}el Gabrielsson and Vignesh Ganapathi-Subramanian and Primoz Skraba and Leonidas J. Guibas}, journal={Computer Graphics Forum}, year={2020}, volume={39} }

We present an approach to incorporate topological priors in the reconstruction of a surface from a point scan. We base the reconstruction on basis functions which are optimized to provide a good fit to the point scan while satisfying predefined topological constraints. We optimize the parameters of a model to obtain a likelihood function over the reconstruction domain. The topological constraints are captured by persistence diagrams which are incorporated within the optimization algorithm to… Expand

#### 15 Citations

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- 2021

A novel method is presented aimed to endow deep generative models with physical reasoning, in particular, a loss and a learning framework that promote two key characteristics of the generated shapes: their connectivity and physical stability. Expand

Topological Regularization for Dense Prediction

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- ArXiv
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Experimental results show that the output topology can also appear in the internal activations of trained neural networks which allows for a novel use of topological regularization to the internal states of neural networks during training, reducing the computational cost of the regularization. Expand

Optimizing persistent homology based functions

- Computer Science
- ICML
- 2021

This work proposes a general framework that allows us to define and compute gradients for persistence-based functions in a very simple way, and provides a simple, explicit and sufficient condition for convergence of stochastic subgradient methods for such functions. Expand

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A novel gradient descent algorithm extending the well-known Gradient Sampling methodology to the class of stratifiably smooth objective functions, which are defined as locally Lipschitz functions that are smooth on some regular pieces of the ambient Euclidean space, achieves a sub-linear convergence rate. Expand

Persistent Homology-based Projection Pursuit

- Computer Science
- 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW)
- 2020

This work formulate topology-preserving dimensionality reduction as finding the optimal orthogonal projection to the lower-dimensional subspace which minimizes discrepancy between persistent diagrams of the original data and the projection. Expand

Robust Persistence Diagrams using Reproducing Kernels

- Computer Science, Mathematics
- NeurIPS
- 2020

This work develops a framework for constructing robust persistence diagrams from superlevel filtrations of robust density estimators constructed using reproducing kernels using an analogue of the influence function on the space of persistence diagrams to be less sensitive to outliers. Expand

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This work provides a study of the effectiveness and computational cost of several `1-minimization optimization procedures for constructing homological cycle bases for persistent homology with rational coefficients in dimension one, including uniform-weighted and length- Weighted edge-loss algorithms as well as uniform- weighted and area-weighting triangle-lossgorithms. Expand

A note on stochastic subgradient descent for persistence-based functionals: convergence and practical aspects

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- 2020

This article studies the differentiability of a general functional associated with the most common topological construction, that is, the persistence map, and proves a convergence result of stochastic subgradient descent for such a functional. Expand

A Topology Layer for Machine Learning

- Computer Science, Mathematics
- AISTATS
- 2020

A differentiable topology layer that computes persistent homology based on level set Filtrations and distance-bases filtrations is presented that can serve as a regularizer directly on data or the weights of machine learning models. Expand

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The proposed method is successfully applied to distinguish porous structures on a novel data set of porous models to obtain topological patterns of objects represented by finite-dimensional vectors for machine learning classification tasks. Expand

#### References

SHOWING 1-10 OF 58 REFERENCES

Interactive topology-aware surface reconstruction

- Mathematics
- SIGGRAPH 2007
- 2007

The reconstruction of a complete watertight model from scan data is still a difficult process. In particular, since scanned data is often incomplete, the reconstruction of the expected shape is an… Expand

Morfit: interactive surface reconstruction from incomplete point clouds with curve-driven topology and geometry control

- Computer Science, Mathematics
- ACM Trans. Graph.
- 2014

An interactive technique for surface reconstruction from incomplete and sparse scans of 3D objects possessing sharp features is presented and a novel skeleton-driven morph-to-fit, or morfit, scheme is introduced which reconstructs the shape as an ensemble of generalized cylinders. Expand

State of the Art in Surface Reconstruction from Point Clouds

- Computer Science
- Eurographics
- 2014

A holistic view of surface reconstruction is considered, providing a detailed characterization of the field, highlights similarities between diverse reconstruction techniques, and provides directions for future work in surface reconstruction. Expand

Competing Fronts for Coarse–to–Fine Surface Reconstruction

- Computer Science
- Comput. Graph. Forum
- 2006

A deformable model to reconstruct a surface from a point cloud based on an explicit mesh representation composed of multiple competing evolving fronts that enables adaptive handling of non‐homogenous sample density, including robustness to missing data in defected areas. Expand

Poisson surface reconstruction

- Mathematics, Computer Science
- SGP '06
- 2006

A spatially adaptive multiscale algorithm whose time and space complexities are proportional to the size of the reconstructed model, and which reduces to a well conditioned sparse linear system. Expand

Robust optimization for topological surface reconstruction

- Computer Science
- ACM Trans. Graph.
- 2018

This paper proposes a novel and general optimization method for surface reconstruction under topological constraints and demonstrates the benefit of topology control over classical topology-oblivious methods such as Marching Cubes. Expand

Surface reconstruction from unorganized points

- Computer Science, Mathematics
- SIGGRAPH
- 1992

A general method for automatic reconstruction of accurate, concise, piecewise smooth surfaces from unorganized 3D points that is able to automatically infer the topological type of the surface, its geometry, and the presence and location of features such as boundaries, creases, and corners. Expand

Removing excess topology from isosurfaces

- Computer Science
- TOGS
- 2004

This article presents a practical method for removing handles in an isosurface by making an axis-aligned sweep through the volume to locate handles, compute their sizes, and selectively remove them, and demonstrates topology simplification on several complex models, and shows its benefits for subsequent surface processing. Expand

Topology-controlled reconstruction of multi-labelled domains from cross-sections

- Computer Science, Mathematics
- ACM Trans. Graph.
- 2017

This work presents the first algorithm for reconstructing multi-labeled material interfaces the allows for explicit topology control, defining a space of topology-varying material interfaces, which extends the family of level sets in a scalar function, and developing discrete methods for sampling distinct topologies in this space. Expand

Reconstruction with Voronoi centered radial basis functions

- Mathematics, Computer Science
- SGP '06
- 2006

This work considers the problem of reconstructing a surface from scattered points sampled on a physical shape by using as centers of basis functions a set of points located on an estimate of the medial axis, instead of the input data points. Expand