CT scans without X-rays: Parallel-beam imaging from nonlinear current flows

Figure 1.  Method overview. An illustration of our decomposition of the EIT problem into disjoint modules, with a representation of the virtual sinogram. Arrow type and color indicate the type of operation; red: ill-posed, black: well-posed, straight: linear, and curved: nonlinear. We flow through (A)-(G): (A) process EIT measurements into a DN map, (B) solve a boundary integral equation to yield CGO traces, (C$ ' $)-(C$ ''' $) apply a windowed Fourier transform and two simple linear operations, (D) remove higher-order scattering terms through machine learning, (E) apply 1D deconvolution in the pseudo-time domain, (F) apply FBP, TV, or other CT reconstruction algorithm, (G) apply a simple algebraic formula. In this work, (D) is achieved through generalization of (E). Ill-posedness is completely confined to linear steps (E) and (F), with (E) containing almost all of it. Process flows for D-bar and cost-function-based methods are included for comparison

Figure 2.  Visualization of the reasoning behind the Fourier windowing strategy used in (7). Black curves: $ \mbox{Re}(\omega^+(1,\tau)) $ calculated from the boundary integral equation (6) using many realizations of $ \boldsymbol{\Lambda}_\sigma $ with added simulated noise. Red curve: a suggested Gaussian window to be used in the Fourier transform, cutting away the unstable parts.

Figure 3.  Examples of sinograms corresponding to one representative training phantom, computed using one or more (odd) terms of the scattering series, all plotted on the same scale. (a) The ideal $ R_1\mu $ case is computed from $ \omega_1 $ only. (b) $ R_{1,3}\mu $ is computed from $ \omega_1 $ and $ \omega_3 $. (c) $ R_{1,3,5}\mu $ is computed from $ \omega_1, \omega_3, $ and $ \omega_5 $. (d) $ R_\text{odd}\mu $ is computed using all odd terms $ \omega_\text{odd} $

Figure 4.  Left: Column number 50 of each sinogram $ R_{1}\mu $, $ R_{1,3}\mu $, $ R_{1,3,5}\mu $, and $ R_{\text{odd}}\mu $ is plotted as a 1D profile function. Right: a zoomed-in plot of the same four curves, illustrating the interval where the differences are most apparent

Figure 5.  Examples of conductivity phantoms used to generate training data

Figure 6.  Neural network architecture. The size of the feature maps and number of channels is reported for each layer. BN stands for batch normalization

Figure 7.  Results from the numerically simulated datasets Ⅰ-Ⅲ. (a) The numerically simulated phantoms. White inclusions have high conductivity as compared to the gray background. (b) VHPT virtual sinograms, after performing process steps (D) and (E). (c) VHPT relative conductivity reconstructions via FBP. (d) VHPT relative conductivity reconstructions via TV regularization. (e) Comparative relative conductivity reconstructions computed using the well-established D-bar algorithm.

Figure 8.  Results from the experimental tank datasets Ⅳ-Ⅸ. (a) The experimental tank setup for EIT data collection. Red agar inclusions are conductive, and yellow inclusions (in Ⅶ-Ⅸ) are resistive, as compared to background. The ruler and ground wire visible in datasets Ⅳ-Ⅵ were above the tank and did not affect measurements. (b) VHPT virtual sinograms, after performing process steps (D) and (E). (c) VHPT relative conductivity reconstructions via FBP. (d) VHPT relative conductivity reconstructions via TV regularization. (e) Comparative relative conductivity reconstructions computed using the well-established D-bar algorithm.

Figure 9.  Virtual radiographs (yellow curves), superimposed on simulated conductivity phantoms (Ⅰ-Ⅲ) and cropped photos from data collection (Ⅳ-Ⅵ), with illustrations of virtual X-ray beams (red lines).

Figure 10.  Sinogram profiles, superimposed on cropped photos from data collection, with illustrations of virtual X-ray beams. A sampling of virtual radiographs for targets Ⅶ-Ⅸ from various directions (a), (b), (c) are plotted in yellow. These are the individual columns of the VHPT sinogram, representing the attenuation of virtual parallel X-ray beams (in red). In Ⅸ(b), a red arrow indicates a sharp "notch" in the radiograph corresponding to the open mouth of the Pac-Man

Figure 11.  Illustration of the propagation and reflection of singularities in pseudo-spacetime. Consider the conductivity $ \sigma(x) = 1+\chi_B(x) $, where $ B $ is the disc with center $ (1/5,0) $ and radius $ 0.3 $. The dark blue pillar is a vertical cylinder whose base is the singular support of $ \mu $, here the circle $ \partial B $. For fixed $ \varphi $, the map $ (x,t)\mapsto \widehat v_1(x,t,e^{i \varphi}) $ is singular on three planes. The magenta plane is the singular support of the incident wave $ \rho_{ \varphi}(x,t) = c\delta'(t+2 \text{Re}\,(e^{i \varphi}x)) $. We have $ \widehat v _1(x,t,e^{i \varphi}) = \overline{\partial}^{-1}_x(\mu\,\cdotp \rho_{ \varphi}) $ where $ \overline{\partial}^{-1}_x $ propagates the singularities of the product $ \mu\,\cdotp \rho_{ \varphi} $ in the light-blue horizontal planes $ t = 2r_0 $ and $ t = -2r_0 $