SPLIT: Self-supervised Partitioning for Learned Inversion in Nonlinear Tomography
Authors: Markus Haltmeier, Lukas Neumann, Nadja Gruber, Gyeongha Hwang
Abstract
Machine learning has achieved impressive performance in tomographic reconstruction, but supervised training requires paired measurements and ground-truth images that are often unavailable. This has motivated self-supervised approaches, which have primarily addressed denoising and, more recently, linear inverse problems. We address nonlinear inverse problems and introduce SPLIT (Self-supervised Partitioning for Learned Inversion in Nonlinear Tomography), a self-supervised machine-learning framework for reconstructing images from nonlinear, incomplete, and noisy projection data without any samples of ground-truth images. SPLIT enforces cross-partition consistency and measurement-domain fidelity while exploiting complementary information across multiple partitions. Our main theoretical result shows that, under mild conditions, the proposed self-supervised objective is equivalent to its supervised counterpart in expectation. We regularize training with an automatic stopping rule that halts optimization when a no-reference image-quality surrogate saturates. As a concrete application, we derive SPLIT variants for multispectral computed tomography. Experiments on sparse-view acquisitions demonstrate high reconstruction quality and robustness to noise, surpassing classical iterative reconstruction and recent self-supervised baselines.
Self-supervised data splitting has emerged as a promising paradigm for sparse-view CT reconstruction, enabling training from incomplete measurements without fully sampled ground truth. However, the influence of key design choices, including partitioning strategy, preprocessing, and inference, remains insufficiently understood. In this work, we introduce a unified framework that decomposes splitting-based reconstruction into these three components, enabling controlled comparison of existing methods and two incremental extensions: multi-partition splitting and an alternative inference strategy. Experiments on simulated LoDoPaB-CT data under independent and correlated noise, together with validation on the real-world 2DeteCT dataset, show that the optimal partitioning strategy strongly depends on the measurement noise structure. Lattice-based splitting performs favorably under independent noise, whereas angular masking is more robust under correlated noise and real measured data. Multi-partition splitting consistently improves over pure projection-wise splitting in several settings. Complementary perceptual and structural metrics, including LPIPS and HaarPSI, reveal differences between masking strategies that are less apparent from PSNR and SSIM alone. These results provide practical guidelines for designing self-supervised sparse-view CT reconstruction methods and highlight the limitations of common independence assumptions in realistic imaging environments.
X-ray computed tomography reconstruction is an ill-posed inverse problem, particularly in low-dose and sparse-angle settings where measurements are noisy and incomplete. While learned reconstruction methods such as the Learned Primal-Dual algorithm achieve strong performance, they typically rely on supervised training with access to ground-truth data, which is often unavailable in practice. In this work, we propose a self-supervised reconstruction method by extending the Noise2Inverse framework to the Learned Primal-Dual algorithm. The resulting approach, called Noise2Inverse Learned Primal-Dual (N2I-LPD), enables training of a learned iterative reconstruction operator without ground-truth images by exploiting the statistical independence of noise in distinct measurements with respect to angular rotation of the CT-scan. We compare the proposed method with classical reconstruction methods, as well as neural network-based approaches such as a U-Net trained within the same N2I framework. The results demonstrate that N2I-LPD achieves improved reconstruction quality, highlighting the potential of combining learned reconstruction operators with self-supervised training strategies for practical CT imaging scenarios where ground-truth data is unavailable.
Antti Sällinen, Siiri Rautio, Santeri Kaupinmäki +1
Photoacoustic tomography (PAT) is an emerging imaging modality that combines the complementary strengths of optical contrast and ultrasonic resolution. A central task is image reconstruction, where measured acoustic signals are used to recover the initial pressure distribution. For ideal point-like or line-like detectors, several efficient and fast reconstruction algorithms exist, including Fourier methods, filtered backprojection, and time reversal. However, when applied to data acquired with finite-size detectors, these methods yield systematically blurred images. Although sharper images can be obtained by compensating for finite-detector effects, supervised learning approaches typically require ground-truth images that may not be available in practice. We propose a self-supervised reconstruction method based on Noisier2Inverse that addresses finite-size detector effects without requiring ground-truth data. Our approach operates directly on noisy measurements and learns to recover high-quality PAT images in a ground-truth-free manner. Its key components are: (i) PAT-specific modeling that recasts the problem as angular deblurring; (ii) a Noisier2Inverse formulation in the polar domain that leverages the known angular point-spread function; and (iii) a novel, statistically grounded early-stopping rule. In experiments, the proposed method consistently outperforms alternative approaches that do not use supervised data and achieves performance close to supervised benchmarks, while remaining practical for real acquisitions with finite-size detectors.