SPLIT: Self-supervised Partitioning for Learned Inversion in Nonlinear Tomography

1. Problem Statement

The paper addresses the challenge of tomographic image reconstruction under nonlinear, incomplete, and noisy conditions, specifically targeting Multispectral Computed Tomography (MSCT).

• The Core Challenge: Standard supervised deep learning requires paired datasets of ground-truth images ($x$) and measurements ($y$), which are often unavailable in medical and industrial settings. While self-supervised learning (SSL) has succeeded in linear inverse problems (e.g., denoising, sparse-view CT) and linear tomography, it has not been effectively applied to nonlinear inverse problems.
• Nonlinearity in MSCT: In MSCT, the forward operator is nonlinear due to the polychromatic nature of X-ray sources and energy-dependent material attenuation. The problem is modeled as:
  $$y = A(x) + \eta$$
  where $A$ is a nonlinear forward map combining the Radon transform and a spectral nonlinearity, $x$ represents material density maps, and $\eta$ is noise (typically Poisson-distributed photon noise plus additive electronic noise).
• Limitations of Existing Methods: Classical iterative methods struggle with artifacts (e.g., beam hardening) and computational cost. Existing self-supervised methods (like Noise2Self or Sparse2Inverse) are designed for linear operators and fail to handle the specific null-space and nonlinearity issues inherent in MSCT.

2. Methodology: The SPLIT Framework

The authors propose SPLIT (Self-supervised Partitioning for Learned Inversion in Nonlinear Tomography), a framework that learns to reconstruct images using only measurement data $y$, without ground truth $x$.

Key Components:

1. Multi-Partitioning Strategy:
   Unlike previous SSL methods that use a single data split, SPLIT employs multiple partitions of the measurement domain.

   • Data Splitting: The measurement vector $y$ is partitioned into subsets $\Omega_{i,j}$ and their complements $\Omega^c_{i,j}$ across $K$ different partition schemes (e.g., splitting by angular view and by detector offset).
   • Partial Reconstructions: For each partition, "partial" reconstructions are generated using fixed, linear reconstruction operators ($B_{i,j}$ and $B^c_{i,j}$) applied to the split data. These serve as auxiliary inputs and targets.

2. Self-Supervised Loss Functions:
   The framework trains a neural network $f$ (typically a U-Net) to map the partial reconstruction from the complement set to the partial reconstruction of the target set. Two loss functions are defined:

   • Image-Space Loss ($L_X$): Minimizes the difference between network outputs in the image domain (similar to Noise2Inverse).
   • Measurement-Space Loss ($L_Y$): Minimizes the difference between the network's output projected back into the measurement domain and the actual measurements.
   • Key Innovation: The paper argues that the Measurement-Space Loss is superior for nonlinear problems. It allows the network to operate within the null space of the forward operator, effectively removing undersampling artifacts that fixed linear reconstructions cannot resolve.

3. Theoretical Guarantee:
   The authors prove Theorem 1, showing that under mild conditions (pairwise independent, zero-mean noise), minimizing the self-supervised measurement-space loss is equivalent in expectation to minimizing the supervised loss (difference from ground truth). This justifies the approach theoretically for nonlinear operators.

4. Automatic Stopping Rule:
   To prevent overfitting without ground truth, SPLIT uses a cross-partition consistency metric. It monitors the Peak Signal-to-Noise Ratio (PSNR) between reconstructions derived from different data partitions. Training stops when this metric saturates or diverges, indicating the onset of overfitting.

Application to MSCT (Double-SPLIT):

For MSCT, the authors implement Double-SPLIT, which utilizes two specific partitions:

• Partition 1 (Angular): Splits projection angles (e.g., odd vs. even views).
• Partition 2 (Detector): Splits detector elements (e.g., odd vs. even detector bins).
  The network performs joint image formation and material decomposition in a single step, unlike previous two-step pipelines.

3. Key Contributions

• First Self-Supervised Nonlinear Tomography: SPLIT is the first framework to successfully apply self-supervised learning to nonlinear inverse problems in tomography.
• Multi-Partitioning: Introduces the concept of using multiple, complementary data partitions (angular and detector) to address both noise and undersampling simultaneously, a strategy not previously explored in SSL imaging.
• Measurement-Domain Loss for Nonlinearity: Demonstrates that enforcing fidelity in the measurement domain (rather than image domain) is crucial for handling the null-space issues in nonlinear reconstruction.
• Joint Reconstruction: Achieves simultaneous image reconstruction and material decomposition for MSCT without requiring paired training data.
• Theoretical Equivalence: Provides a rigorous proof that the self-supervised objective converges to the supervised objective in expectation.

4. Experimental Results

The method was tested on synthetic Multispectral CT data involving three materials (Water, Iodine, Gadolinium) across 5 energy bins.

• Baselines: Compared against:
  • CP-Fast: A classical iterative one-step algorithm.
  • X-space Loss: A self-supervised baseline using image-space loss (adaptation of Noise2Inverse).
  • Single-SPLIT: Self-supervised with a single partition.
• Performance Metrics: Evaluated using PSNR and SSIM across varying levels of angular sparsity (725, 145, and 29 views).
• Findings:
  • Superiority: Both Single-SPLIT and Double-SPLIT significantly outperformed the classical CP-Fast and the X-space loss baseline across all metrics.
  • Robustness to Sparsity: Double-SPLIT achieved the best overall performance, particularly in ultra-sparse scenarios (29 views), demonstrating high robustness to noise and missing data.
  • Material Decomposition: The method successfully reconstructed distinct material maps (Iodine, Gadolinium, Water) with high fidelity, effectively mitigating beam-hardening artifacts common in MSCT.
  • Quantitative Gains: In the 29-view scenario, Double-SPLIT improved PSNR by approximately 10 dB over the X-space loss baseline for Iodine and Gadolinium channels.

5. Significance

This work bridges a critical gap in medical imaging AI by enabling high-quality, data-efficient reconstruction for complex nonlinear tomographic problems where ground truth is impossible to obtain.

• Clinical Impact: It offers a pathway to reduce radiation dose (via sparse-view acquisition) and improve material specificity in MSCT without the need for expensive phantom scans or simulations for training.
• Generalizability: While demonstrated on MSCT, the SPLIT framework is general and applicable to any nonlinear inverse problem admitting multiple data partitions (e.g., photoacoustic tomography, diffuse optical tomography).
• Future Direction: The paper sets the stage for integrating physics-informed priors and uncertainty quantification into self-supervised nonlinear inversion, moving beyond the current reliance on purely data-driven or linear approximations.