Var_LDDMM

This is a MATLAB implementation of the papers Diffeomorphic Registration of Discrete Geometric Distributions and Metrics, quantization and registration in varifold spaces by Hsi-Wei Hsieh and Nicolas Charon. The package provides tools to permorm diffeomorphic registration, interpolation and compression of geometric shapes (such as point clouds, discrete cruves or triangulated surfaces) that are represented as discrete varifolds.

References

If you use this code for your research, please cite our papers:

@article{hsieh2019diffeomorphic,
  title={Diffeomorphic Registration of Discrete Geometric Distributions},
  author={Hsieh, Hsi-Wei and Charon, Nicolas},
  journal={Mathematics Of Shapes And Applications},
  volume={37},
  pages={45},
  year={2019},
  publisher={World Scientific}
}

@article{hsieh2019metrics,
  title={Metrics, quantization and registration in varifold spaces},
  author={Hsieh, Hsi-Wei and Charon, Nicolas},
  journal={arXiv},
  pages={arXiv--1903},
  year={2019}
}

Set up

To run functions without GPU acceleration, you will only need MATLAB and the HANSO library for L_BFGS optimization. We have included the HANSO routines in the folder /src/optimization/ for convenience.

Basic dependencies:

• MATLAB
• HANSO version 2.2: in /src/optimization/

Var_LDDMM also includes GPU acceleration implementations using the KeOps library (libkeops-master in folder '/src/'). It must be used on a machine equipped with an NVIDIA graphics card with recent CUDA drivers installed.

Optional dependencies (GPU acceleration):

• KeOps: libkeops-master in folder '/src/'

Usage

Registration:

The approach relies on the Large Deformation Diffeomorphic Metric Mapping (LDDMM) model with a geodesic shooting scheme. The cost function, which is the sum of a deformation penalty and a kernel fidelity metric between the transformed source and the target varifolds, is minimized with respect to the momenta variables of the deformation.
Use registration function for LDDMM varifolds matching algorithm:

[P_op,summary]= registration(Source,Target,defo,objfun,options)

• Input:

  • Source/Target: source/target shape represented as a discrete varifold (structure):

    • Source.center: a N-by-m array contains positions of N diracs in dimension m
    • Source.vector: a cell array with length d carries a frame for the grassmanian, each entry Source.vector{i} is a N-by-m array

  • defo: options for deformation and numerical ODE:

    • defo.kernel_size_mom: deformation kernel size
    • defo.nb_euler_steps: number of time steps in the forward and backward integration
    • defo.odemethod: numerical scheme for ODE integration, possible values: 'rk4' or 'middle_point'
    • defo.method: compute the deformation kernel operations using only matlab or GPU with keops, possible values: 'keops' or 'matlab'

  • objfun: options for varifold data attachment term:

    • objfun.kernel_geom: spatial kernel type, possible values:'gaussian' or 'cauchy'
    • objfun.kernel_size_geom: spatial kernel sizes in an array [a_1,...,a_K], successively runs K times optimizations with kernel sizes a_1,...,a_K, with each optimization using the estimated momemta from the previous run as initialization. The length of the array should be consistent with kernel_size_grass
    • objfun.kernel_grass: Grassmanian/orientation kernel type, possible values: 'linear', 'gaussian_oriented', 'gaussian_unoriented', 'binet'
    • objfun.kernel_size_grass: Grassmanian/orientation kernel sizes in an array [b_1,...,b_K]
    • objfun.method='keops': compute kernel operation in data attachment term using only matlab or GPU with keops, possible values: 'keops' or 'matlab'
    • objfun.lambda: weight parameter in front of the data attachment term

  • options: options for L-BFGS:

    • options.record_history: true or false
    • options.maxit: maximum number of iterations(default 1000) (applies to each starting vector)
    • options.nvec: 0 for full BFGS matrix update, otherwise specifies number of vectors to save and use in the limited memory updates (default: 0)
    • options.prtlevel: one of 0 (no printing), 1 (minimal), 2 (verbose) (default: 1)

• Output:

  • P_op: the optimized momenta stored in a structure
  • summary: summary of the optimization procedure

Compression/Quantization:

Use the function proj2_M_dirac to compress a discrete varifold to a sparser discrete varifold with exactly M diracs:

[Y,summary] = proj2_M_dirac(X,X_ini,objfun,options)

• Input:
  • X: full discrete varifold to be compressed
  • X_ini: initial discrete varifold with M diracs
  • objfun and options: same as in registration, the field objfun.lambda is not needed in compression.
• Output:
  • Y: compressed varifold
  • summary: summary of the optimization procedure

Examples:

See the two script files in the Demo scripts folder for some examples of basic use of the code. The first script script_Bone_Bottle.m computes the compressions and registrations of two curves for M= 25, 40, and 150.

Relative quantization errors	M=25	M=40	M=150
Difference to thegroundtruth optimal energy	M=25	M=40	M=150

The second script script_amygdala.m registers two amygdala surfaces using GPU acceleration:

Amygdala surfaces	Varifold representation

Licence

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.

Contacts

• Nicolas Charon ([email protected])
• Hsi-Wei Hsieh ([email protected])