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Spatiotemporal Imaging with Diffeomorphic Optimal Transportation

Chong Chen

TL;DR

The paper tackles spatiotemporal imaging under large, mass-preserving deformations by integrating the Wasserstein distance with a flow of diffeomorphisms. It develops a variational model where the transport cost is given by the Benamou–Brenier formulation and the deformation flow is constrained to an admissible Hilbert space to guarantee a diffeomorphic evolution; an ODE-constrained (and equivalently PDE-constrained) reformulation is derived. A time-discretized alternating minimization algorithm is proposed to jointly reconstruct the template and estimate the velocity field, using a Gaussian RKHS for smooth diffeomorphisms and mass-preserving deformations. The method is validated on 2D space-time tomography with sparse views and noise, showing improved reconstruction accuracy, robust mass preservation, and smooth velocity fields compared to TV or plain $ abla$-based approaches, and it demonstrates relatively low sensitivity to regularization parameters. Overall, the work provides a rigorous, transport-based framework that unifies optimal transport with large-deformation registration to enhance spatiotemporal image reconstruction in challenging imaging modalities.

Abstract

We propose a variational model with diffeomorphic optimal transportation for joint image reconstruction and motion estimation. The proposed model is a production of assembling the Wasserstein distance with the Benamou--Brenier formula in optimal transportation and the flow of diffeomorphisms involved in large deformation diffeomorphic metric mapping, which is suitable for the scenario of spatiotemporal imaging with large diffeomorphic and mass-preserving deformations. Specifically, we first use the Benamou--Brenier formula to characterize the optimal transport cost among the flow of mass-preserving images, and restrict the velocity field into the admissible Hilbert space to guarantee the generated deformation flow being diffeomorphic. We then gain the ODE-constrained equivalent formulation for Benamou--Brenier formula. We finally obtain the proposed model with ODE constraint following the framework that presented in our previous work. We further get the equivalent PDE-constrained optimal control formulation. The proposed model is compared against several existing alternatives theoretically. The alternating minimization algorithm is presented for solving the time-discretized version of the proposed model with ODE constraint. Several important issues on the proposed model and associated algorithms are also discussed. Particularly, we present several potential models based on the proposed diffeomorphic optimal transportation. Under appropriate conditions, the proposed algorithm also provides a new scheme to solve the models using quadratic Wasserstein distance. The performance is finally evaluated by several numerical experiments in space-time tomography, where the data is measured from the concerned sequential images with sparse views and/or various noise levels.

Spatiotemporal Imaging with Diffeomorphic Optimal Transportation

TL;DR

The paper tackles spatiotemporal imaging under large, mass-preserving deformations by integrating the Wasserstein distance with a flow of diffeomorphisms. It develops a variational model where the transport cost is given by the Benamou–Brenier formulation and the deformation flow is constrained to an admissible Hilbert space to guarantee a diffeomorphic evolution; an ODE-constrained (and equivalently PDE-constrained) reformulation is derived. A time-discretized alternating minimization algorithm is proposed to jointly reconstruct the template and estimate the velocity field, using a Gaussian RKHS for smooth diffeomorphisms and mass-preserving deformations. The method is validated on 2D space-time tomography with sparse views and noise, showing improved reconstruction accuracy, robust mass preservation, and smooth velocity fields compared to TV or plain -based approaches, and it demonstrates relatively low sensitivity to regularization parameters. Overall, the work provides a rigorous, transport-based framework that unifies optimal transport with large-deformation registration to enhance spatiotemporal image reconstruction in challenging imaging modalities.

Abstract

We propose a variational model with diffeomorphic optimal transportation for joint image reconstruction and motion estimation. The proposed model is a production of assembling the Wasserstein distance with the Benamou--Brenier formula in optimal transportation and the flow of diffeomorphisms involved in large deformation diffeomorphic metric mapping, which is suitable for the scenario of spatiotemporal imaging with large diffeomorphic and mass-preserving deformations. Specifically, we first use the Benamou--Brenier formula to characterize the optimal transport cost among the flow of mass-preserving images, and restrict the velocity field into the admissible Hilbert space to guarantee the generated deformation flow being diffeomorphic. We then gain the ODE-constrained equivalent formulation for Benamou--Brenier formula. We finally obtain the proposed model with ODE constraint following the framework that presented in our previous work. We further get the equivalent PDE-constrained optimal control formulation. The proposed model is compared against several existing alternatives theoretically. The alternating minimization algorithm is presented for solving the time-discretized version of the proposed model with ODE constraint. Several important issues on the proposed model and associated algorithms are also discussed. Particularly, we present several potential models based on the proposed diffeomorphic optimal transportation. Under appropriate conditions, the proposed algorithm also provides a new scheme to solve the models using quadratic Wasserstein distance. The performance is finally evaluated by several numerical experiments in space-time tomography, where the data is measured from the concerned sequential images with sparse views and/or various noise levels.

Paper Structure

This paper contains 35 sections, 9 theorems, 95 equations, 7 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spatiotemporal imaging
  4. The Wasserstein distance
  5. The flow of diffeomorphisms
  6. Spatiotemporal imaging with diffeomorphic optimal transportation
  7. The proposed model
  8. Time discretization
  9. Comparison with several existing alternatives
  10. The joint variational model based on Wasserstein distance
  11. The joint variational model based on LDDMM
  12. Numerical implementation
  13. A specific example
  14. Template reconstruction
  15. Velocity field estimation
  16. ...and 20 more sections

Key Result

Theorem 1

Assume that the time-dependent density $f(t, x) \ge 0$ and velocity field $\boldsymbol{\nu}(t, x) \in \mathbb{R}^n$ are appropriately smooth, and $f_0$ and $f_1$ are compactly supported. The square of the $\mathscr{L}^2$ Wasserstein distance equals to such that

Figures (7)

  • Figure 1: Test suite 1. Reconstructed image by the TV-based method if the spatiotemporal problem is treated as a static one.
  • Figure 2: Test suite 1. Descent curve of the objective functional of the proposed model as the iteration grew.
  • Figure 3: Test suite 1. Reconstructed images of the multi-object phantom. The columns represent the different gates. For the noise-free data, the first three rows are the reconstructed spatiotemporal images by TV-based reconstruction method (row 1), the algorithm using $\mathscr{L}^2$-gradient descent method (row 2), and the proposed method (row 3). The last row (row 4) shows the ground truth for each gate.
  • Figure 4: Test suite 1. The computed optimal velocity field at the end time points $t=0$ (left) and $t=1$ (right) by the $\mathscr{L}^2$-gradient descent scheme (top) and the proposed method (bottom) in \ref{['Test_suite_1:multi_object_phantom']}, respectively.
  • Figure 5: Test suite 2. Data of the first projection view at Gate 1. The left, middle, and right figures show the data of the first view with $14.6$dB, $7.69$dB, and $5.53$dB noise levels, respectively. The blue curve denotes the noise-free data, and the red jagged curve shows the noisy data.
  • ...and 2 more figures

Theorems & Definitions (19)

  • Theorem 1: BeBr00
  • Definition 1: Yo10
  • Theorem 2: Yo10BrHo15
  • Remark 1
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Remark 2
  • Theorem 5
  • ...and 9 more