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Time-Harmonic Optical Flow with Applications in Elastography

Oleh Melnyk, Michael Quellmalz, Gabriele Steidl, Noah Jaitner, Jakob Jordan, Ingolf Sack

TL;DR

This work develops a Fourier-domain framework for time-harmonic optical flow in elastography, enabling recovery of the velocity amplitude $a(x)$ from multi-frame sequences given a known frequency $\omega$ via $v(t,x)=\mathrm{Re}(a(x)e^{i\omega t})$. It introduces three variational models: Model I with a quadratic data term and smoothness (leading to a period-independent linear system), Model II with an $L_1$ data term and TV-like regularization solved by IRLS, and Model III combining $L_1$ data fidelity with $L_2$ regularization; all are formulated in the time-frequency domain and solved efficiently. The paper demonstrates that Model I performs best in low-noise settings, while Model III provides robustness under heavy noise, with IRLS-based approaches offering favorable trade-offs for non-smooth objectives. These results enable efficient estimation of time-harmonic velocity fields and lay groundwork for subsequent reconstruction of material properties such as the shear modulus in elastography.

Abstract

In this paper, we propose mathematical models for reconstructing the optical flow in time-harmonic elastography. In this image acquisition technique, the object undergoes a special time-harmonic oscillation with known frequency so that only the spatially varying amplitude of the velocity field has to be determined. This allows for a simpler multi-frame optical flow analysis using Fourier analytic tools in time. We propose three variational optical flow models and show how their minimization can be tackled via Fourier transform in time. Numerical examples with synthetic as well as real-world data demonstrate the benefits of our approach. Keywords: optical flow, elastography, Fourier transform, iteratively reweighted least squares, Horn--Schunck method

Time-Harmonic Optical Flow with Applications in Elastography

TL;DR

This work develops a Fourier-domain framework for time-harmonic optical flow in elastography, enabling recovery of the velocity amplitude from multi-frame sequences given a known frequency via . It introduces three variational models: Model I with a quadratic data term and smoothness (leading to a period-independent linear system), Model II with an data term and TV-like regularization solved by IRLS, and Model III combining data fidelity with regularization; all are formulated in the time-frequency domain and solved efficiently. The paper demonstrates that Model I performs best in low-noise settings, while Model III provides robustness under heavy noise, with IRLS-based approaches offering favorable trade-offs for non-smooth objectives. These results enable efficient estimation of time-harmonic velocity fields and lay groundwork for subsequent reconstruction of material properties such as the shear modulus in elastography.

Abstract

In this paper, we propose mathematical models for reconstructing the optical flow in time-harmonic elastography. In this image acquisition technique, the object undergoes a special time-harmonic oscillation with known frequency so that only the spatially varying amplitude of the velocity field has to be determined. This allows for a simpler multi-frame optical flow analysis using Fourier analytic tools in time. We propose three variational optical flow models and show how their minimization can be tackled via Fourier transform in time. Numerical examples with synthetic as well as real-world data demonstrate the benefits of our approach. Keywords: optical flow, elastography, Fourier transform, iteratively reweighted least squares, Horn--Schunck method
Paper Structure (17 sections, 5 theorems, 89 equations, 9 figures, 1 algorithm)

This paper contains 17 sections, 5 theorems, 89 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Outline of the paper.
  3. Time-harmonic optical flow
  4. Time-harmonic approach with $v$ versus $u$.
  5. Variational models for time-harmonic optical flow
  6. Model I
  7. Model II
  8. Model III
  9. Numerical Implementation
  10. Numerical Results
  11. Generation of synthetic data
  12. Synthetic example
  13. Real-world data
  14. Conclusion
  15. Periodicity of $u$ versus $v$
  16. ...and 2 more sections

Key Result

Theorem 1

For a given frequency $\omega \in \mathbb{R}$ and $p \in \mathbb{N}$, set $T \coloneqq 2\pi p /\omega$. Let $I\colon [0,T] \times \Omega \to \mathbb{R}$ be an image sequence in $C^1$ and $v\colon [0,T] \times \Omega \to \mathbb{R}^d$ a time-harmonic velocity in $C^2$ of the form eq:va. Then, the mi where $\overline{a}$ denotes the complex conjugate of $a$ and $\Delta$ is the Laplace operator.

Figures (9)

  • Figure 1: Trajectories $\varphi(t,x)$ corresponding to the velocity amplitude $a(x)=0.2\,x$ from \ref{['ex:periodic']} with different starting values $x$ for $\omega=1$.
  • Figure 2: Two-dimensional images $I(t,\cdot)$ corresponding to a harmonic velocity \ref{['eq:va']} with a real-valued amplitude function $a$ and period $T=2000$. The displacement $u$ is periodic. The second row is the zoom-in of the first row to the red rectangle. The red $\color{red}\pmb\bullet$ depicts the position $\varphi(t,x)$ of a single point $x$ at time $t$ and the red curve is its trajectory.
  • Figure 3: Trajectories $\varphi(t,x)$ of \ref{['ex:nonperiodic']} for different starting values $x$ with $\omega=1$.
  • Figure 4: Two-dimensional images $I(t,\cdot)$ corresponding to a harmonic velocity \ref{['eq:va']} with an amplitude function of the form $a(x)\in\mathrm{i}\mathbb{R}\times\mathbb{R}$ and period $T=2000$. The resulting displacement $u$ is not periodic. The second row is the zoom-in of the first row to the red rectangle. The red $\color{red}\pmb\bullet$ depicts the position $\varphi(t,x)$ of a single point $x$ at time $t$ and the red curve is its trajectory.
  • Figure 5: (a), (b): Absolute values of the two components of the ground truth amplitude $\boldsymbol{a}$, (c): noiseless initial image $I_0$, (d): initial image $I_0$ corrupted by Poisson and salt-and-pepper noise.
  • ...and 4 more figures

Theorems & Definitions (9)

  • Example 1: Periodic deformation
  • Example 2: Non-periodic deformation
  • Theorem 1
  • Proof 1
  • Lemma 1
  • Lemma 2: KumVerSto21
  • Theorem 2
  • Proof 2
  • Theorem 3