From Magnetic Susceptibility Imaging

To Principled Learned Proximal Operators

Zhenghan Fang

Kavli NDI Breakfast Meeting

February 7, 2024

Imaging Tissue Magnetic Susceptibility in the Brain

Susceptibility Anisotropy

  • Fiber tracking
  • Disease characterization
\begin{bmatrix} \chi_{11} & \chi_{12} & \chi_{13} \\ \chi_{12} & \chi_{22} & \chi_{23} \\ \chi_{13} & \chi_{23} & \chi_{33} \end{bmatrix}

Susceptibility Tensor Imaging

Mean

Anisotropy

PEV

  • Acquire data at multiple head orientations
  • Time-consuming and uncomfortable

☹

Magnetic Susceptibility

M = {\color{green} \chi} H
  • Degree to which a material is magnetized in an external magnetic field

DeepSTI: Towards STI Using Fewer Orientations

Solving the Inverse Problem

Fang Z, Lai KW, van Zijl P, Li X, Sulam J. DeepSTI: Towards tensor reconstruction using fewer orientations in susceptibility tensor imaging. Medical image analysis. 2023 Jul 1;87:102829.

\min_x \|y - Ax \|_2^2 + {\color{darkorange}R}(x)

Regularizer

\hat{x}_{k+1} = {\color{darkorange} \mathrm{prox}_{\eta R}}(\hat{x}_k - \eta A^H(A\hat{x}_k - y))

Proximal Operator

\hat{x}_{k+1} = {\color{darkorange} f_\theta}(\hat{x}_k - \eta A^H(A\hat{x}_k - y)) \\

DeepSTI

Dipole Inversion

x
y

MRI phase measurements

STI image

Inverse Problem in STI

Dipole Convolution

y=Ax+v

In Vivo Tensor Reconstruction using DeepSTI

DTI

STIimag

[Li et al.]

MMSR

[Li and Van Zijl]

aSTI+

[Shi et al.]

DeepSTI

(ours)

[1] Li et al, NMRB 2017; [2] Li and van Zijl, MRM, 2014;

[3] Cao et al., MRM, 2021; [4] Shi et al., IEEE JBHI, 2022

[5] Fang et al. Medical Image Analysis, 2023

In Vivo Fiber Tractography using DeepSTI

[1] Li et al, NMRB 2017; [2] Li and van Zijl, MRM, 2014;

[3] Cao et al., MRM, 2021; [4] Shi et al., IEEE JBHI, 2022

[5] Fang et al. Medical Image Analysis, 2023

Susceptibility Source Separation

Susceptibility Source Separation: WaveSep and DeepSepSTI

In vivo Quantitative Susceptibility Mapping (QSM) Separation Result from WaveSep

In vivo STI Separation Result from DeepSepSTI on a Multiple Sclerosis Patient

  • WaveSep: decouple susceptibility estimation and source separation
  • DeepSepSTI: joint susceptibility estimation and source separation

What's in a Prior? Learned Proximal Networks for Inverse Problems

\hat{x}_{k+1} = {\color{darkorange} f_\theta}(\hat{x}_k - \eta A^H(A\hat{x}_k - y)) \\
  • Is there an \(R\) such that \(f_\theta = \mathrm{prox}_{\eta R}\)?
  • What is the regularizer/prior \(R\) learned by the neural network?
\min_x \|y - Ax \|_2^2 + {\color{darkorange}R}(x)
\hat{x}_{k+1} = {\color{darkorange} \mathrm{prox}_{\eta R}}(\hat{x}_k - \eta A^H(A\hat{x}_k - y))

Proposition (Learned Proxima Networks). Let \(\psi_\theta: \R^{n} \rightarrow \R\) be defined by \(z_{1} = g( \mathbf H_1 y + b_1), z_{k} = g(\mathbf W_k  z_{k-1} + \mathbf H_k y + b_k), \psi_\theta(y) = \mathbf w^T z_{K} + b\) with \(g\) a convex, non-decreasing nonlinear activation, and all \(\mathbf W_k\) and \(\mathbf w\) non-negative. Let \(f_\theta = \nabla \psi_{\theta}\). Then, there exists a function \(R_\theta\) such that  \(f_\theta(y) = \mathrm{prox}_{R_\theta}(y)\).

What's in a Prior? Learned Proximal Networks for Inverse Problems

Fang Z, Buchanan S., Sulam J. What's in a prior? Learned proximal networks for inverse problems. ICLR 2024

Proposition (Learned Proxima Networks). Let \(\psi_\theta: \R^{n} \rightarrow \R\) be defined by \(z_{1} = g( \mathbf H_1 y + b_1), z_{k} = g(\mathbf W_k  z_{k-1} + \mathbf H_k y + b_k), \psi_\theta(y) = \mathbf w^T z_{K} + b\) with \(g\) a convex, non-decreasing nonlinear activation, and all \(\mathbf W_k\) and \(\mathbf w\) non-negative. Let \(f_\theta = \nabla \psi_{\theta}\). Then, there exists a function \(R_\theta\) such that  \(f_\theta(y) = \mathrm{prox}_{R_\theta}(y)\).

What's in a Prior? Learned Proximal Networks for Inverse Problems

Theorem (Proximal Matching). Let
\[f^* = \argmin_{f} \lim_{\gamma \searrow 0} \mathbb{E}_{x,y} \left[ m_\gamma \left( \|f(y) - x\|_2 \right)  \right].\]

Then, almost surely (for almost all \(y\)),

\[f^*(y) = \argmax_{c} p_{x \mid y}(c) = \mathrm{prox}_{-\sigma^2\log p_x}(y).\]

\(m_{\gamma}(x) = 1 - \frac{1}{(\pi\gamma^2)^{n/2}}\exp\left(-\frac{x^2}{\gamma^2}\right)\)

Prox Matching Loss

Fang Z, Buchanan S., Sulam J. What's in a prior? Learned proximal networks for inverse problems. ICLR 2024

What's in a Prior? Learned Proximal Networks for Inverse Problems

Gaussian noise

LPN faithfully captures distribution of natural images

Fang Z, Buchanan S., Sulam J. What's in a prior? Learned proximal networks for inverse problems. ICLR 2024

What's in a Prior? Learned Proximal Networks for Inverse Problems

Deblurring on CelebA images

Fang Z, Buchanan S., Sulam J. What's in a prior? Learned proximal networks for inverse problems. ICLR 2024

Summary

  • DeepSTI enables in vivo magnetic susceptibility tensor reconstruction using much fewer orientations
  • WaveSep and DeepSepSTI achieves separation of paramagnetic and diamagnetic sources
  • Learned proximal networks (LPN) presents a principled and interpretable machine learning algorithm for inverse problems
  • NIH NIBIB (P41EB031771)
  • Distinguished Graduate Student Fellows program of the KAVLI Neuroscience Discovery Institute

Acknowledgements

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