1.

INTRODUCTION

X-Ray Computed Tomography (CT) is an essential technique that provides deep insight of a patient or an object of interest in a non-invasive manner. The forward model can be formulated as

where R represents the X-ray transform, f the quantity to be reconstructed (the absorption coefficients), while y represents the X-ray measurements and η some noise.

An important aspect of medical X-ray CT is the radiation exposure of the patients. One technique to lower the radiation dose is by lowering the number of projection views.¹ This is referred to as sparse-view or sparse X-Ray CT. The reconstructions of such sparse measurements tend to feature streak artifacts near edges tangent to the acquired X-rays.¹ With increasing sparsity, and thus with an increasing lack of measurement data, the severity of these streak artifacts increases as well. Hence a reconstruction approach alleviating the impact of these streak artifacts is highly desirable.

The visibility principle² tells us in essence that the visible part of an object is comprised of the set of edges tangent to the acquired X-rays, and the invisible part is comprised of the edges non-tangent to these X-rays. Moreover, which edges can or cannot be reconstructed is dependent on the acquisition geometry, therefore known before acquisition.

Following³ we leverage this principle by employing shearlets to resolve the wavefront set of such a signal. This enables us to properly reconstruct the visible edges using ℓ₁-regularization and to in-paint the invisible ones using deep learning. Using only model-based methods, by the visibility principle, we cannot retrieve the invisible information. On the other hand, using only deep learning on such an ill-posed problem, we might get satisfactory results up to a point, but we will not be able to certainly assert as to how much the original signal has changed.³

Therefore, our proposed approach, which we will term SDLX for the remainder of this work, is a hybrid method that benefits from the best aspects of both model-based and learning-based approaches for sparse-view X-ray CT reconstruction. The SDLX method is very closely related to,³ which was developed for limited-angle X-ray CT. In the following we will present the SDLX method in detail, along with first results on a publicly available data set.

2.

METHODS

2.1

SHEARLETS

Shearlets are a mathematical concept building on top of existing wavelet-theory components with distinct advantages. They represent a multi-scale framework that provides optimally sparse approximations of multivariate, anisotropic data. The approximation rate of shearlets is of O(N⁻²), comparatively better than the O(N⁻¹) of wavelets. Shearlets are constructed by applying three operations, translation, dilation, and shearing, see⁴ for details. They are applied to a single generating function ψ, resulting in a shearlet system

Here, a ∈ ℝ₊ dictates the dilation matrix A_a, s ∈ ℝ dictates the shearing matrix S_s, while t ∈ ℝ² represents the translations. The composite matrix M_as is then defined as . The continuous shearlet transform is then

For the discerete shearlet transform, we sample the parameter space ℝ₊ × ℝ × ℝ² at discrete points. This defines the regular discrete shearlet system as

The discrete shearlet transform is defined similarly to the continuous scenario.

Given the directional bias exhibited by regular shearlets,⁵ we will instead be using the cone-adapted shearlets as they provide a remedy to it. For a visual representation of the tiling that is generated in the Fourier domain, see Figure 1. Additionally, we want to explicitly specify and emphasize, that the cone-adapted discrete shearlet systems as mentioned above, under mild assumptions, form a Parseval frame.⁶ Based on this fact and the above statements, we know that

Figure 1:

Frequency tiling of the cone-adapted shearlet system. By Afg genzel - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=27761187

This equation represents a powerful reconstruction formula of the discrete shearlet transform, which is essential to our hybrid approach.

By far the most important property of the shearlet systems to the hybrid approach is their ability to resolve the wavefront set of the signals at hand.³ This allows us to differentiate the visible and invisible boundaries, and is accomplished by distinguishing different decay rates of the shearlet transform.

It is worth pointing out that we utilize classical shearlet systems, which are band-limited (compact support in the frequency domain⁴). Our implementation of such a discrete cone-adapted band-limited shearlet transform is based on,⁷ to which we refer the reader to for further details. We also utilize the shearlet transform available in⁸ as an intermediary operation after running the ℓ₁-regularization.

2.2

SPARSE REGULARIZATION WITH ADMM

Sparse regularization attempts to leverage the assumption that the output of a problem can be described by a fewer number of inputs, or put differently that for every output there exists a sparsifying representation system.³ More specifically for low-dose CT, it has been shown that such methods enable more accurate reconstructions given very few tomographic measurements.³ Therefore, such a paradigm is of interest to us for tackling sparse-view X-ray CT.

Alternating Direction Method of Multipliers (ADMM) is a general algorithm that works quite well in splitting the minimization of the sparsity-promoting ℓ₁-regularization term and the data fidelity term. Details on ADMM can be found in.⁹

Based on (1), we are now able to construct and utilize shearlet-based sparse regularization. Explicitly expressing the reconstruction problem built so far, we write

in which represents the weights to the regularization term. This allows us to split the wavefront set of the signal into the visible and invisible parts, as described in the visibility principle. In this equation we have also explicitly specified a constraint for non-negative solutions, as it leads to better reconstructions.³

We solve this minimization problem using ADMM.

3.

EXPERIMENTS AND RESULTS

The dataset used here is the one provided by the Mayo Clinic for the AAPM Low-Dose CT Grand Challenge.¹² It contains human abdomen scans with width and height of 512. We chose the low-dose scans, with pixel intensities in [0 : 255].

For training of the PhantomNet we selected 10 patients, with the IDs of L004, L006, L014, L019, L033, L049, L056, L057, L058, and L064, which comprised a total number of 1525 scans. For testing we chose another patient, with an ID of L071.

To generate training pairs for the PhantomNet, we first simulate sparse, 64 projection view sinograms (over an arc of 360 degrees) of the 1525 training images using,¹¹ adding 1% Gaussian noise. Then we compute ℓ₁-regularized reconstructions of those sinograms using ADMM as in (7), with 10 iterations of ADMM and 5 inner iterations of the conjugate gradient method on the normal equation. We manually selected the parameters of ADMM as ρ₁ = 1/2, ρ₂ = 1 (as in³), and w = 0.001. Afterwards, we apply the forward shearlet transform from.⁸

We train the PhantomNet for 100 epochs in single-batches (e.g. one (61, 512, 512) object as a batch) on a learning rate of 7e – 5 and weight decay of 1e – 7. The chosen optimizer is Adam. The loss function is the mean squared error loss from torch.nn.MSELoss.

For testing, we use the data from the patient with ID of L071, and simulate sparse-view sinograms with 64 projection views as above, adding 1% Gaussian noise. We execute the full SDLX method as in subsection 2.4, using the same ADMM parameters as for training the net. In the last step, we sum together the visible coefficients from the f₁-regularization and the estimated invisible coefficients that the trained PhantomNet predicts, and apply the inverse shearlet transform to it.

An example result f_SDLX of our proposed SDLX method is shown in Figure 2 for one of the slices of patient ID L071, which was not seen during training. We also compare with the ground truth and reconstructions using the same ADMM as in the first step of SDLX, f_ADMM, and the result of an unregularized CG reconstruction using 10 iterations, f_CG.

Figure 2:

64-view reconstruction results on patient L071. The ground truth is f, while f_SDLX is the output of our proposed SDLX algorithm. For comparison, we also show the reconstruction of the first step of the SDLX algorithm (f_ADMM) as well as an unregularized iterative CG reconstruction f_CG. The pixel intensities lie in [0 : 255]. For further details see section 3.

It is apparent that this hybrid method is capable of in-painting the missing singularities for sparse-view CT. SDLX outperforms the other methods, a claim also supported by the metrics, as displayed in Table 1. The metrics utilized are the Relative Error (RE), Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index Measure (SSIM), and Haar Wavelet-Based Perceptual Similarity Index (HaarPSI).

Table 1:

Metrics of the 64-view reconstruction results on patient L071 compared to the ground truth. The lower the RE, the better. The higher the PSNR, SSIM, HaarPSI, the better.

We also ran the same testing experiment without adding noise to the simulated sinograms. The trained model (on noisy simulated data) performed just as well on the unseen data, which serves as an indicator that SDLX is fairly robust towards noise.

4.

DISCUSSION AND CONCLUSION

The results from the experiment indicate that the hybrid approach works for sparse-view CT. However, a certain smoothing effect is also visible in the results. First experiments (not shown here) indicate that further tuning of the training of PhantomNet might negate this effect.

One aspect of the SDLX method that might introduce unexplained features is the deep learning step. Fortunately, this element is utilized here in a relatively controlled manner, given that it only handles the inference of the invisible coefficients. Further tuning of the hyper-parameters might be beneficial, as might be the study of more advanced models, such as transformers, instead of the U-net.

In summary, SDLX works because shearlets are capable of resolving the wavefront sets of the signals we are dealing with, and these decomposed coefficients adhere to certain rules which we can then learn. Adapting the work for limited-angle X-ray CT in,³ our first experiments for sparse-view X-ray CT on a publicly available data set show promising results.