Noise-resilient approach for deep tomographic imaging

Zhen Guo1, Zhiguang Liu2, Qihang Zhang1, George Barbastathis2,3,7, Michael E.
Glinsky4

1Department of Electrical Engineering and Computer Science, Massachusetts
Institute of Technology, Cambridge, Massachusetts, 02139, USA

2Department of Mechanical Engineering, Massachusetts Institute of Technology,
Cambridge, Massachusetts, 02139, USA

3Singapore-MIT Alliance for Research and Technology (SMART) Centre, Singapore
138602

4qiTech, Inc., Santa Fe, New Mexico 87505, USA

7<EMAIL_ADDRESS>
###### Abstract

We propose a noise-resilient deep reconstruction algorithm for X-ray
tomography. Our approach shows strong noise resilience without obtaining noisy
training examples. The advantages of our framework may further enable low-
photon tomographic imaging.

## Main Text

X-ray tomography is a non-destructive imaging technique that visualizes the
interior features of solid objects, with applications in many disciplines. A
high quality reconstruction can be generated by densely-sampled, full-angle,
and noise-free measurements covering the Fourier space [1]. However, acquiring
such measurements can be a challenge in practice for an object with geometric
constraints or radiation sensitivity. For measurements that are limited-angle,
sparse-view and noisy, the resulting inverse problem becomes ill-conditioned
due to the deficits in the Fourier-space information [2, 3]. Under such
sampling conditions, it is essential to utilize a regularization prior on the
object distribution to retrieve a high-fidelity reconstruction [4].

Conventionally, the regularization prior is a general-purpose penalty function
included in the objective function of an iterative optimization algorithm. The
final reconstruction is regarded as the maximum a posteriori (MAP) estimate
given the prior distribution. Alternatively, regularization priors can also be
learned through supervised machine learning method [5, 6, 7]. Given a training
dataset, a machine learning algorithm can be trained to create an inverse map
that directly generates object reconstructions from the measurements, learning
the spatial correlations of the objects implicitly. Some efforts also utilize
a conventional algorithm that generates an approximate estimate of the object
for a deep neural network, achieving better reconstruction quality by
separating the forward model of tomography from the network training [8, 9].

One major assumption in previous studies of the learned prior is that the
noise statistics in the measurements are comparable between training and
testing dataset. Assuming measurements are corrupted by Poisson noise, the
reconstruction quality from the learned prior is generally accessed by testing
data with the same photons per ray as the training data. However, in practical
tomography systems, training and testing data might have different number of
photons per ray due to the variability in the light source and detector. When
the noise statistics in the dataset shifts between training and testing, the
generalization of learning is not guaranteed, leading to performance
degradation or failure in producing high-fidelity reconstruction [10, 11]. The
ability of maintaining reconstruction quality despite variations of the noise
statistics in the test data is called the “noise resilience” of the learned
prior.

Fig. 1: Quantitative comparison between different reconstruction algorithms
for tomographic simulations under different photon numbers per ray. The $x$
axis is the number of photons per ray, and the $y$ axis on the top figure is
$1-r$ where $r$ is the Pearson correlation coefficient. The $y$ axis on the
bottom is the $L^{2}$ distance in the scattering transform space. Here the
error-bars are the standard deviation in the log scale.

In this paper, we propose a noise-resilient deep-reconstruction algorithm for
X-ray tomography. Our method improves the noise resilience of the learned
prior by more computation, rather than more training data collected under
various noise levels. Instead of approximate reconstructions, the inputs to
the neural network are Maximum A Posteriori (MAP) Estimate reconstructions
from an iterative algorithm. For the special case when the prior distribution
is uniform, MAP can be regarded as Maximum Likelihood Estimate (MLE). By
enforcing data-fidelity and sparsity prior, the distribution shift between
input reconstructions from noise-free and noisy measurements is reduced.
Therefore, the noise resilience of the learned prior improves without
acquiring the noisy training data.

The quantitative comparison of different algorithms using simulation data is
shown in Fig. 1. The imaging condition is constrained to full-angle, sparse
sampling (32 projections) and low-photon tomographic simulations where the
ill-conditioning becomes severe without regularization. There are three
traditional reconstruction algorithms namely FBP, MLE, MAP with TV (total
variation), and three learning-based algorithms obtained by adding a UNet for
each, designated as FBP+UNet, MLE+UNet, and MAP+UNet. All the learning-based
algorithms share the same optimization parameters and UNet architecture. The
Pearson correlation coefficient is to evaluate the perceptual quality of the
reconstruction. The scattering transform is to evaluate the morphology of the
reconstruction [12]. We report the means and standard errors over the 1000
testing data in the plots. Assuming 0.9 as the acceptable quality for $r$,
MAP+UNet satisfies the requirement for the entire photon range, except for 32
photons per pixel. MLE+UNet is the second best algorithm, which can satisfy
the requirement down to 100 photons per pixel. FBP+UNet shows the least noise-
resilient among the learned algorithm, withstanding noisy measurements down to
500 photons per pixel. The scattering transform metrics shows similar trend as
the Pearson correlation metric, except that the distances between MLE and MAP
with TV are relatively small.

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