Deep Learning of truncated singular values for limited view photoacoustic tomography

TL;DR

This work tackles the severely ill-posed problem of limited-view photoacoustic tomography by integrating a data-driven regularization network with a classical truncated SVD. The method first computes a low-frequency reconstruction via truncated SVD and then trains a CNN to recover the missing high-frequency components within the small-singular-value subspace, effectively performing approximate null-space learning. Numerical results on a discretized PAT model show that the CNN-enhanced reconstruction substantially improves over pure truncated SVD, achieving lower relative errors and better detail at realistic noise levels. The approach is generalizable to other inverse problems where the forward operator's SVD is known or computable, and it provides a convergent regularization framework when combined with data-driven components.

Abstract

We develop a data-driven regularization method for the severely ill-posed problem of photoacoustic image reconstruction from limited view data. Our approach is based on the regularizing networks that have been recently introduced and analyzed in [J. Schwab, S. Antholzer, and M. Haltmeier. Big in Japan: Regularizing networks for solving inverse problems (2018), arXiv:1812.00965] and consists of two steps. In the first step, an intermediate reconstruction is performed by applying truncated singular value decomposition (SVD). In order to prevent noise amplification, only coefficients corresponding to sufficiently large singular values are used, whereas the remaining coefficients are set zero. In a second step, a trained deep neural network is applied to recover the truncated SVD coefficients. Numerical results are presented demonstrating that the proposed data driven estimation of the truncated singular values significantly improves the pure truncated SVD reconstruction. We point out that proposed reconstruction framework can straightforwardly be applied to other inverse problems, where the SVD is either known analytically or can be computed numerically.

Figures (5)

• Figure 1: Basic principles of PAT. The absorption of short optical pulses inside a semitransparent sample causes thermoelastic expansion, which in turn induces acoustic pressure waves. The acoustic waves propagate outwards, where they are measured and used to reconstruct an image of the interior.
• Figure 2: Considered limited view measurement geometry. Left: The phantom is allowed to take non-vanishing values in the square $[-1,1]$. Measurements are made on a semi circle of radius one, resulting in a severally ill-posed reconstruction problem. Right: solid black curve shows the singular values of the corresponding discrete system matrix using $16384$ basis functions and $150400$ data points. A large fraction of the singular values is zero or close to zero, reflecting the ill-posedness. The red dashed curve indicates the used singular values for 7% additive Gaussian noise.
• Figure 3: Phantom and noisy data. Left: Test phantom generated by the method described above not contained in the training set. Right: Limited view data with $7%$ additive Gaussian noise.
• Figure 4: Reconstructions result for $7%$ noise. Top row: Optimally truncated SVD (left), truncated SVD used as input to the CNN (center), and proposed CNN continuated SVD (right). Bottom row: Absolute differences from the ground truth corresponding to the reconstruction in the top row.
• Figure 5: Reconstructions result for $4%$ noise. Top row: Optimally truncated SVD (left), and proposed CNN continuated SVD (right). Bottom row: Absolute differences from the ground truth corresponding to the reconstruction in the top row.