Unsupervised Learning for Inverse Problems in Computed Tomography

Abstract

Assume you encounter an inverse problem that shall be solved for a large number of data, but no ground-truth data is available. To emulate this encounter, in this study, we assume it is unknown how to solve the imaging problem of Computed Tomography (CT). An unsupervised deep learning approach is introduced, that leverages the inherent similarities between deep neural network training, deep image prior (DIP) and unrolled optimization schemes. We demonstrate the feasibility of reconstructing images from measurement data by pure network inference, without relying on ground-truth images in the training process or additional gradient steps for unseen samples. Our method is evaluated on the two-dimensional 2DeteCT dataset, showcasing superior performance in terms of mean squared error (MSE) and structural similarity index (SSIM) compared to traditional filtered backprojection (FBP) and maximum likelihood (ML) reconstruction techniques as well as similar performance compared to a supervised DL reconstruction. Additionally, our approach significantly reduces reconstruction time, making it a promising alternative for real-time medical imaging applications. Future work will focus on extending this methodology for adaptability of the projection geometry and other use-cases in medical imaging.

Figures (9)

• Figure 1: Gradient-based image reconstruction scheme. The colored boxes indicate the domain of the respective data ( projection domain, image domain). The boxes indicate the domain transform by FP ($A$) or BP ($A^T$). The steps to iteratively adapt the current image $f^k$ with the current gradient ($\nabla$) are indicated by arrows. The loss $\mathcal{L}$ is defined by Eq. \ref{['eq:loss']}.
• Figure 2: Proposed unsupervised training scheme. The colored boxes indicate the domain of the respective data ( projection domain, image domain). The boxes indicate the domain transform by FP ($A$) or BP ($A^T$). The steps to iteratively adapt the UNet++ weights are indicated by arrows.
• Figure 3: Exemplary visualization of (a) predicted projection data, (b) ground-truth projection data, and (c) difference. Image values are unit-less since they indicate relative attenuation behavior to the unattenuated beam intensity.
• Figure 4: MSE distribution of 300 projection predictions from test dataset w.r.t. ground-truth projection data. Values in the projection domain are unit-less.
• Figure 5: Quantitative results in image domain for radon inversion task on 300 test images. All methods are calculated based on forward projected data of ground-truth image. Deviations are calculated with respect to said ground-truth image. [MSE] = $1/\mathrm{cm}^2$