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Supervised Reconstruction for Silhouette Tomography

Evan Bell, Michael T. McCann, Marc Klasky

TL;DR

silhouette tomography (ST) reframes X-ray CT as a geometry-only problem that uses binary per-ray occupancy via $y = S(x) = T_{>0}(H x)$, highlighting ill-posedness with multiple possible ${x}$ and the existence of a maximal solution ${x_{\max}}(y)$ given by ${x_{\max}}(y) = \neg T_{>0}(H^T (\neg y))$. To address this, the authors propose a supervised deep learning approach using a six-down / six-up U-Net to map $H^T y$ to $x$, trained on a ShapeNet-derived synthetic dataset. Results show the learned ST reconstructions substantially outperform the maximal baseline in MSE and PSNR, with binarization further boosting SSIM, and a linear tomography comparison indicates strong performance when the forward model is accurately known. The work demonstrates the viability of geometry-based silhouette tomography and outlines clear avenues for 3D extension, real-data validation, and automated per-view segmentation.

Abstract

In this paper, we introduce silhouette tomography, a novel formulation of X-ray computed tomography that relies only on the geometry of the imaging system. We formulate silhouette tomography mathematically and provide a simple method for obtaining a particular solution to the problem, assuming that any solution exists. We then propose a supervised reconstruction approach that uses a deep neural network to solve the silhouette tomography problem. We present experimental results on a synthetic dataset that demonstrate the effectiveness of the proposed method.

Supervised Reconstruction for Silhouette Tomography

TL;DR

silhouette tomography (ST) reframes X-ray CT as a geometry-only problem that uses binary per-ray occupancy via , highlighting ill-posedness with multiple possible and the existence of a maximal solution given by . To address this, the authors propose a supervised deep learning approach using a six-down / six-up U-Net to map to , trained on a ShapeNet-derived synthetic dataset. Results show the learned ST reconstructions substantially outperform the maximal baseline in MSE and PSNR, with binarization further boosting SSIM, and a linear tomography comparison indicates strong performance when the forward model is accurately known. The work demonstrates the viability of geometry-based silhouette tomography and outlines clear avenues for 3D extension, real-data validation, and automated per-view segmentation.

Abstract

In this paper, we introduce silhouette tomography, a novel formulation of X-ray computed tomography that relies only on the geometry of the imaging system. We formulate silhouette tomography mathematically and provide a simple method for obtaining a particular solution to the problem, assuming that any solution exists. We then propose a supervised reconstruction approach that uses a deep neural network to solve the silhouette tomography problem. We present experimental results on a synthetic dataset that demonstrate the effectiveness of the proposed method.
Paper Structure (5 sections, 1 theorem, 7 equations, 5 figures, 2 tables)

This paper contains 5 sections, 1 theorem, 7 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Silhouette Tomography
  3. Proposed Method: Supervised Reconstruction
  4. Experiments and Results
  5. Discussion and Conclusion

Key Result

Theorem 1

When any solution to eq:forward exists, the maximal solution of eq:forward is given by where $\neg$ denotes Boolean negation (i.e., swapping 0 and 1).

Figures (5)

  • Figure 1: Silhouette tomography generally has multiple solutions. For example, each ray intersecting the left object also intersects the right one and each ray not intersecting the left object also does not intersect the right one.
  • Figure 2: Schematic of the proposed neural network architecture. The internal structure of each downsampling and upsampling block is shown in the orange and green boxes respectively.
  • Figure 3: An object from the testing set and one slice from it.
  • Figure 4: Qualitative comparison of the proposed reconstruction methods on one slice of the test set.
  • Figure 5: Visual comparison of maximal and supervised approaches for 3D object reconstruction.

Theorems & Definitions (3)

  • Definition 1: Maximal solution
  • Theorem 1: Computing the maximal solution
  • proof