TRIPs-Py: Techniques for Regularization of Inverse Problems in Python

TL;DR

TRIPs-Py delivers a Python package for linear discrete inverse problems, unifying direct, projection-based, and generalized Krylov solvers under common regularization frameworks. It covers TSVD, Tikhonov, and TGSVD for small-scale problems and GMRES/LSQR/CGLS, GK, and MMGKS for large-scale, including ell_p–ell_q regularization via MMGKS and framelet-based operators. The package includes diverse test problems spanning 1D/2D deblurring, CT, and dynamic tomography (with both synthetic and real data), plus automatic parameter selection strategies such as the discrepancy principle and GCV. This work advances accessibility and comparability in inverse-problem research, enabling Python-based experimentation and didactic use, with open-source availability and future plans for more solvers and dynamic-imaging capabilities.

Abstract

In this paper, we describe TRIPs-Py, a new Python package of linear discrete inverse problems solvers and test problems. The goal of the package is two-fold: 1) to provide tools for solving small and large-scale inverse problems, and 2) to introduce test problems arising from a wide range of applications. The solvers available in TRIPs-Py include direct regularization methods (such as truncated singular value decomposition and Tikhonov) and iterative regularization techniques (such as Krylov subspace methods and recent solvers for $\ell_p$-$\ell_q$ formulations, which enforce sparse or edge-preserving solutions and handle different noise types). All our solvers have built-in strategies to define the regularization parameter(s). Some of the test problems in TRIPs-Py arise from simulated image deblurring and computerized tomography, while other test problems model realistic problems in dynamic computerized tomography. Numerical examples are included to illustrate the usage as well as the performance of the described methods on the provided test problems. To the best of our knowledge, TRIPs-Py is the first Python software package of this kind, which may serve both research and didactical purposes.

Figures (10)

• Figure 1: Overview of the TRIPs-Py's structure and contents. Most of the files available in the 'Utilities' directory are auxiliary functions that can be used by the TRIPs-Py solvers, such as functions to set the regularization operators, or to display data and reconstructions.
• Figure 2: 1D image deblurring test problem. (a) ${\bf x}_{\rm true}$ and data with $1\%$ Gaussian noise, (b) ${\bf x}_{\rm true}$ and naive, unregularized solution, (c) ${\bf x}_{\rm true}$, TSVD and TGSVD solutions, (d) ${\bf x}_{\rm true}$, standard and general form Tikhonov solutions.
• Figure 3: 2D image deblurring small-scale test problem. (a) True image of $50 \times 50$ pixels, (b) Blurred and noisy image with $1\%$ Gaussian noise. Approximate solutions by (c) TSVD, (d) Hybrid LSQR, and (e) MMGKS.
• Figure 4: 2D image deblurring large-scale test problem. (a) True image of $128 \times 128$ pixels. (b) Blurred and noisy image with $1\%$ Gaussian noise. Approximate solutions obtained by (c) Hybrid GMRES, (d) Hybrid LSQR, and (e) MMGKS.
• Figure 5: Tomography test problem. (a) True image of $256 \times 256$ pixels. (b) Sinogram. Approximate solutions obtained by (c) Hybrid LSQR, (d) GKS, and (e) MMGKS.

Theorems & Definitions (1)

• Remark 1