[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-128440-en":3,"doc-seo-128440-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},128440,962084925290,"Ophelia","https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",6,"Technology","Geodesic Deep Shape Optimization - Supplementary Material","Supplementary material expands on the handling of irregular vertices in CubeMesh CNNs, including an efficient unfolding-to-plane strategy and padding near cube corners so standard convolution kernels remain applicable. The document provides a quantitative efficiency comparison of convolution forward-pass time on unstructured versus structured meshes, using a Nvidia Titan X GPU and TensorFlow. It also gives analytical definitions for 2D and 3D deformation parameterizations and specifies flow conditions for the reported experiments.","View metadata, citation and similar [papers at ](papers at core.ac.uk)[core.ac.uk](papers at core.ac.uk) brought to you by CORE  \nprovided by Infoscience- École polytechnique fédérale de Lausanne  \nSupplementary material : Geodesic Deep Shape Optimization  \nPierre Baque * 1 Edoardo Remelli * 1 Franc¸ois Fleuret 2 1 Pascal Fua 1  \nIn this supplementary material, we first provide additional detail on the handling of the irregular vertices of the CubeMesh CNNs of Section 3.1 . We also report a quantitative comparison between the forward-pass time for convolution operations on unstructured meshes and similar operations running on structured meshes such as the ones produced by our re-meshing algorithm. We then give analytical definitions of the 2D and 3D deformation parameterizations of Sections 5.1 and 5.2 . Finally we specify the flow conditions for all reported experiments.  \n1. Handling Singular Points for Semi-Regular Quad-Meshes  \nAs discussed in Section 3.1 of the paper, when mapping a surface onto a cube-mesh, we have to deal with irregular vertices, which correspond to the corners of the cube and have three neighbors instead of four. To perform convolutions efficiently we first unfold the cube surface onto a plane. As illustrated by Fig. 1, we can then simply pad irregular corners with the feature values associated to cube edges. This enables us to use standard convolutional kernels even in the neighborhood of irregular vertices. Furthermore, since we use Geodesic Convolutions, the irregularity is naturally handled by the interpolation operation.  \n2. Computational Efficiency  \nWe provide a quantitative comparison between the forwardpass time of 5 convolution consecutive operations running on unstructured meshes and similar operations running on structured meshes such as the ones produced by our remeshing algorithm. We use a Nvidia Titan X GPU and a Tensorflow implementation.  \n􀀏 Standard Convolution: Exec. Time 0 .00098 s. Such a speed is possible thanks to the regular grid organization.  \n*Equal contribution 1 CVLab, EPFL, Lausanne, Switzerland 2Machine Learning Group, Idiap, Martigny, Switzerland. Correspondence to: Pierre Baque \u003Cpierre.baque@epfl.ch> .  \nProceedings of the 35 th International Conference on Machine Learning, Stockholm, Sweden, PMLR 80, 2018 . Copyright 2018 by the author(s) .  \nFeatures on the diagonal.  \nCube-Mapped Quad-Mesh  \nConvolutional filter. Dilation Factor = 1  \nConvolutional filter. Dilation Factor = 2  \nFigure 1 . Handling the singularities of the Quad-Mesh for convolution purposes.  \n􀀏 Graph Laplacian Convolution Sparse of (Kipf & Welling, 2016): Exec. Time 0.00889 s. The model of (Kipf & Welling, 2016) has limited expressivity because the convolutions are performed by averaging neighbouring features using Adjacency matrices.  \n􀀏 Geometric Convolution Dense of (Monti et al., 2016): Exec. Time 0.27200 s. It corresponds to the dense version as implemented in the public code of (Montiet al., 2016) . TensorFlow uses GPU computation with dense adjacency matrices. The main drawback is the memory requirements, which limit the graph size and the number of features.  \n3. Airfoil Parameterization in 2D  \nIn this section we will first briefly describe the standard NACA airfoil 4 digit parameterization (Jacobs et al., 1948), which, confusingly involves 3 degrees of freedom. We then discussed our extension to 19 degrees of freedom.  \nNACA 4 digit. Without loss of generality, we can assume that the airfoil is of unitary cord length and let 0 􀀔 x 􀀔 1 the coordinate that defines the position along that length. Let us further consider the airfoil thickness t, maximum camber m , along with its location p. To compute the airfoil  \nshape, we first define the mean camber line  \n mp2 􀀀2px 􀀀 x2 􀀁 ; 0 􀀔 x 􀀔 p  \nyc = \u003C>  m   \n:> (1 􀀀 p)2 􀀀(1 􀀀 2p) + 2px 􀀀 x2 􀀁; p 􀀔 x 􀀔 1  \nand the airfoil thickness to camber yt as  \n5t 􀀂0:2969 px 􀀀 0:1260x 􀀀 0:3516x2 + 0:2843x3 􀀀 0:1015x4􀀃 :  \nSince the thickness needs to be applied perpendicul","cbCaicyevzK4ayM0","https://ap.wps.com/l/cbCaicyevzK4ayM0","pdf",271056,1,3,"English","en",105,"# Handling Singular Points for Semi-Regular Quad-Meshes\n## Computational Efficiency\n# Airfoil Parameterization in 2D\n# Surface Parameterization in 3D","[{\"question\":\"How are irregular cube vertices handled for CubeMesh CNN convolutions?\",\"answer\":\"The surface is unfolded onto a plane, and irregular corners are padded using feature values associated with cube edges, enabling standard convolution kernels near these vertices.\"},{\"question\":\"What efficiency comparison is provided in the supplementary material?\",\"answer\":\"It compares forward-pass execution time for five consecutive convolution operations on unstructured meshes versus structured meshes generated by the remeshing algorithm, reporting GPU timings and implementation details.\"},{\"question\":\"What parameterizations are defined for 2D and 3D deformations?\",\"answer\":\"The material defines analytical deformation parameterizations for 2D airfoil geometry (including NACA-based formulations and extended degrees of freedom) and a 3D transformation function parameterized by a 21D vector applied to shape vertices.\"}]","Geodesic Deep Shape Optimization - 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