Continuous and Orientation-preserving Correspondence via Functional Maps

This is a complete implementation for the paper "Continuous and Orientation-preserving Correspondence via Functinal Maps" by Jing Ren, Adrien Poulenard, Peter Wonka and Maks Ovsjanikov.

Update (2019 July)

Our method implicitly provides a tool for symmetry detection, for example, it is used as a baseline in the following paper:

• "Dense Point-to-Point Correspondences Between Genus-Zero Shapes" by Sing Chun Lee and Misha Kazhdan, Fig. 6
• "ZoomOut: Spectral Upsampling for Efficient Shape Correspondence" by Melzi et al, Fig. 9

If you would like to use our method for self-symmetry detection, please check the code run_selfSymm, where the default parameters are tested on the following datasets: FAUST, SCAPE, TOSCA, and SHREC19 (and worked reasonably). Please let us know if you have problems/questions of the code or parameter-tuning.

Main algorithms

• Compute a functional map with an orientation-preserving/reversing operator:

C12 = compute_fMap_regular_with_orientationOp(S1,S2,B1,B2,Ev1,Ev2,fct1,fct2,type)

% Input:
%   S1: the source mesh with the new basis B1, and the corresponding eigenvalues Ev1 (k1 Eigen-functions)
%   S2: the target mesh with the new basis B2, and the corresponding eigenvalues Ev2 (k2 Eigen-functions)
%   fct1: the descriptors of shape S1
%   fct2: the descriptors of shape S2
%   type: 'direct' or 'symmetric' (call the orientation preserving/reversing operator)
% Output:
%   C12: a functional map from S1 -> S2 (k2-by-k1 matrix)

• Refine the point-wise maps:

[T21, T12] = bcicp_refine(S1,S2,B1,B2,T21_ini,T12_ini,num_iter)

% Input: 
%   S1/S2 with corresponding Eigen-functions B1/B2
%   Initial point-wise maps from both directions: T12_ini: S1 -> S2 and T21_ini: S2 -> S1
%   num_iter: number of iterations to run BCICP refinement
% Output:
%   T12, T21: the refined point-wise maps with better accuracy, smoothness, bijectivity and coverage

Main parameters

• numTimes: the time-scale parameter to compute the WKS descriptors
• skipSize: the skip size of the computed WKS descriptors to save runtime
• k1(k2): the number of Eigen-basis used of mesh S1(S2)
• beta: the weight for the orientation-preserving/reversing term
• numIters: the number of iterations for the BCICP refinement step

Note that, for all the tests in the paper, k1 = k2 = 50, numIters = 10 and beta = 0.1. The rest two parameters for computing the WKS descriptors were (sloppily) tuned on a ramdon shape pair in a dataset, then applied to all the rest shape pairs in this collection. Our choices of parameters of each tested datasets are (also specified in the example scripts):

• For FAUST dataset, we set the parameters: numTimes = 100, skipSize = 10.
• For TOSCA isometric dataset, we set the parameters: numTimes = 100, skipSize = 20.
• For TOSCA non-isometric dataset, we set the parameters: numTimes = 50, skipSize = 10.

Examples

Self-symmetric maps

The script Example_selfSymm_Fig3.m reproduces the Fig.3 of the paper: for a given shape, we use the orientation-reversing operator to compute its self-symmetric map.

Note that here we used the same set of parameters to compute the self-symmetric maps. It would work much better if these parameters (especially numTimes and beta) are tuned per dataset. Moreover, BCICP can be added to refine the self-symmetric maps as well.

WKS (wave kernel signatures) initialization (on TOSCA dataset)

The script Example_WKSini_Fig14.m and Example_WKSini_Fig15.m reproduces Fig.14-15 of the paper: computing a map using +directOp/symmOp + BCICP.

Note that for TOSCA non-isometric dataset, the isometry assupmtion fails. The orientation-preserving/reversing operator still works to some extent but much less effective than the cases of isometric shape pair. In this case, more Eigen-basis are needed to compute the WKS desriptors and more BCICP iterations are needed to refine the maps.

SEG (segmentation) initialization (on FAUST dataset)

The script Example_SEGini.m shows an example of non-symmetric segmentation initialization (compare the ICP and BCICP refinement). For the segmentation part, please refer to the paper "Robust Structure-based Shape Correspondence" by Yanir Kleiman and Maks Ovsjanikov (and code: https://github.com/hexygen/structure-aware-correspondence).

Contact

Please let us know ([email protected]) if you have any question regarding the algorithms/paper (ฅωฅ*) or you find any bugs in the implementation (ÒωÓױ).