A Mini-Batch Quasi-Newton Proximal Method for Constrained Total-Variation Nonlinear Image Reconstruction

TL;DR

The paper tackles nonlinear, nonconvex inverse problems in 3D imaging with constrained TV regularization by introducing BQNPM, a mini-batch quasi-Newton proximal method that avoids full gradient evaluations. It couples a memory-efficient SR1 Hessian update with an efficient weighted proximal mapping for TV, solved via a dual formulation and a semismooth Newton method. The authors establish convergence guarantees under nonconvex settings and demonstrate superior iteration- and time-efficiency in 3D optical diffraction tomography on simulated and real data compared to ASPM, SQNPM, and FBQNPM. The framework is scalable to large measurement sets and can extend to other regularizers beyond TV, offering practical impact for high-quality, computationally feasible nonlinear reconstructions in imaging modalities like ODT, PET, CT, and MRI. Overall, BQNPM provides a robust, efficient approach for large-scale, constrained-TV inverse problems with nonlinear forward models.

Abstract

Over the years, computational imaging with accurate nonlinear physical models has garnered considerable interest due to its ability to achieve high-quality reconstructions. However, using such nonlinear models for reconstruction is computationally demanding. A popular choice for solving the corresponding inverse problems is the accelerated stochastic proximal method (ASPM), with the caveat that each iteration is still expensive. To overcome this issue, we propose a mini-batch quasi-Newton proximal method (BQNPM) tailored to image reconstruction problems with constrained total variation regularization. Compared to ASPM, BQNPM requires fewer iterations to converge. Moreover, we propose an efficient approach to compute a weighted proximal mapping at a cost similar to that of the proximal mapping in ASPM. We also analyze the convergence of BQNPM in the nonconvex setting. We assess the performance of BQNPM on three-dimensional inverse-scattering problems with linear and nonlinear physical models. Our results on simulated and real data demonstrate the effectiveness and efficiency of BQNPM, while also validating our theoretical analysis.

Figures (8)

• Figure 1: Principle of optical diffraction tomography. The arrows represent the wave vectors $\{\mathbf{k}^\mathrm{in}_l\in \mathbb R^3\}_{l=1}^L$ of the $L$ incident plane waves $\{u_\mathrm{in}^l\}_{l=1}^L$. The angles of illumination are limited to a cone around the optical axis. The refractive-index map of the sample $\eta(\mathrm{\mathbf r})$ is embedded in the domain $\Omega\subset\mathbb{R}^3$, and the recorded domain is denoted by $\Gamma$. The parameter $\bm r$ denotes the three-dimensional spatial coordinate.
• Figure 2: Performance of ASPM, SQNPM wang2019stochastic, BQNPM-I/II, and FBQNPM algorithms on the strongly scattering simulated sample using the LippS model. From top to bottom rows: Full cost and SNR values versus iteration and wall time.
• Figure 3: Orthoviews of the 3D refractive-index maps obtained by ASPM (iter. $k=100$) and BQNPM-I/II (iter. $k=38/100$) algorithms on the strongly scattering simulated sample using the Lippmann-Schwinger model. The SNR for each slice is displayed in the top-left corner of each image. SQNPM yielded the worst SNR, which we did not present here.
• Figure 4: Effect of $S$ on the convergence behavior of BQNPM-I/II with $\gamma=0.8$.
• Figure 5: Effect of $\gamma$ on the convergence behavior of BQNPM-I/II with $S=4$.

Theorems & Definitions (7)

• Definition 3.1: Weighted proximal mapping
• Theorem 3.2
• Lemma 4.1
• Remark 4.2
• Lemma 5.1
• Lemma 5.2
• Theorem 5.3