A.

Deep Scatter Estimation (DSE)

The basic idea of the DSE approach is illustrated in figure 1. Here, DSE uses a U-net-like architecture to predict scatter as a function of the acquired projection data. To learn the corresponding mapping, DSE is trained to reproduce Monte Carlo simulations, i.e. the U-net’s weights are determined by minimizing the following loss function:

Fig. 1.

Schematic of the DSE and the proposed uDSE approach. DSE uses a U-net-like architecture to predict scatter as a function of the acquired projection data. In a supervised setup DSE is trained to reproduce a ground truth Monte Carlos scatter distribution. uDSE, in contrast, uses a Wasserstein GAN (WGAN) setup. Here, a scatter correction followed by a CT reconstruction is performed. Subsequently, the correction is evaluated by a critic network that is trained simultaneously to recognize scatter artifact-free images. By optimizing the weights of the scatter estimation network to fool the critic network, scatter estimation can be learned without labels.

where θ denotes the parameter vector, n the sample number within a batch of size B, I_n the flat field-corrected intensities, and S_n the Monte Carlo scatter estimate. It has to be noted that the first layer of the DSE network performs a “pep”-transform to be consistent with our DSE publication [14].

B.

Unsupervised Deep Scatter Estimation (uDSE)

The proposed uDSE approach, shown in figure 1, extends the concept of DSE to cases where labeled data are not available. To do so, it is composed of a generator network and a critic network. Here, the generator combines the DSE network with a scatter correction layer and a Feldkamp reconstruction layer, such that it is able to map acquired intensities I of a CT scan to scatter-corrected CT reconstructions. The critic network, in turn, is designed to distinguish between the generator’s output and true scatter-free CT reconstructions. Thus, letting the critic network act as loss function for the generator allows to learn CT scatter estimation without labeled or paired data, respectively.

Here, the corresponding optimization is performed using a WGAN setup in which the generator netork G_θg (I) and the critic network C_θc (f) are optimized in an alternating manner according to the following loss functions:

where θ_c and θ_g denote the parameter vectors of the generator and the critic network, n is the sample number within a batch of size B, and f_real corresponds to a sample from a set of almost scatter-free clinical CT reconstructions.

C.

Datasets

In the present study, DSE as well as uDSE were trained and tested on simulated CBCT data. Therefore, clinical CT reconstructions of 65 patients were used as prior. Based on the corresponding voxel volumes f_prior, CBCT scans with 360 views and an angular coverage of 360° were simulated at five different z-positions within the abdomen region using a tube voltage of 120 kV, a source-to-isocenter distance of 700 mm, a source-to-detector distance of 1100 mm, and a 1024×768 flat detector with an isotropic pixel spacing of 0.39 mm. Furthermore, a shifted detector was used to increase the field of measurement to about 380 mm. Using this setup, three datasets were generated using different patients: one scatter-corrupted dataset (30 patients) from which input data for the generator network were sampled during training, one scatter-free dataset (30 patients) that was used to provide ideal reference CT reconstructions for the critic network, and a scatter-corrupted dataset (5 remaining patients) for testing. In any case, the scatter-corrupted data were simulated as I = I_P + S_MC, where I_p corresponds to a polychromatic forward projection of the prior volume and S_MC is a scatter distribution that was generated using our in-house Monte Carlo simulation. The ideal reference data, on the other hand, correspond to a CBCT reconstruction of only the primary intensities, i.e. f_real = X^‒1(‒ln(I_p)), with X^‒1 being the CBCT reconstruction operator. It has to be noted that f_real is not sampled directly from the set of clinical CT reconstructions, but is generated via forward and back-projection, to have the same spatial resolution and the same field of measurement as the reconstructions provided by the generator network.

D.

Training and Evaluation

The uDSE approach was trained using the datasets described in section II-C. However, to avoid memory issues as well as to increase the computational performance, a 2-fold angular downsampling to 180 views and an 8-fold spatial downsampling to a detector size of 128×96 was performed in advance. This downsampling can be justified by the fact that scatter distributions are known to be of low frequency. Therefore, we only expect a minor degradation of accuracy compared to a training that uses the full size projection data. Given the small detector size, the internal reconstruction operations were also performed on a low resolution grid with 144×144×48 voxels and an isotropic spacing of 2.7 mm.

The corresponding optimization was implemented according to section II-B using the Tensorflow frame-work. Here, all hyperparameters except for the batch size (this study: B = 16) and the number of iterations (this study: N_itr = 5000) were chosen following to the original WGAN publication, i.e. five updates of the critic network are followed by one update of the generator network using an RMSProp optimizer with a learning rate of 0.00005 and a weight clipping to [−0.01, 0.01] in the critic network.

Finally, uDSE was applied to the test data and compared against DSE which was trained in a supervised setting using the Monte Carlo scatter distributions a labels. In contrast to the training data, the scatter correction was performed on the full size projection data. Therefore, all scatter predictions were upsampled to the original detector size of 1024 × 768 pixels prior to scatter correction.