# Convolutional Neural Network to Restore Low-Dose Digital Breast
Tomosynthesis Projections in a Variance Stabilization Domain

Rodrigo B. Vimieiro  Department of Electrical and Computer Engineering, São
Carlos School of Engineering, University of São Paulo, São Carlos, Brazil
Department of Biomedical Engineering, School of Engineering, Center for
Biotechnology and Interdisciplinary Studies, Rensselaer Polytechnic Institute,
Troy, NY, USA  Chuang Niu  Department of Biomedical Engineering, School of
Engineering, Center for Biotechnology and Interdisciplinary Studies,
Rensselaer Polytechnic Institute, Troy, NY, USA  Hongming Shan  Institute of
Science and Technology for Brain-inspired Intelligence, Fudan University,
Shanghai, China  Shanghai Center for Brain Science and Brain-inspired
Technology, Shanghai, China  Lucas R. Borges  Department of Medical Imaging,
Hematology and Clinical Oncology, Ribeirão Preto School of Medicine,
University of São Paulo, Ribeirão Preto, Brazil  Ge Wang  Department of
Biomedical Engineering, School of Engineering, Center for Biotechnology and
Interdisciplinary Studies, Rensselaer Polytechnic Institute, Troy, NY, USA
Co-corresponding authors  Marcelo A. C. Vieira  Department of Electrical and
Computer Engineering, São Carlos School of Engineering, University of São
Paulo, São Carlos, Brazil  Co-corresponding authors

###### Abstract

Digital breast tomosynthesis (DBT) exams should utilize the lowest possible
radiation dose while maintaining sufficiently good image quality for accurate
medical diagnosis. In this work, we propose a convolution neural network (CNN)
to restore low-dose (LD) DBT projections to achieve an image quality
equivalent to a standard full-dose (FD) acquisition. The proposed network
architecture benefits from priors in terms of layers that were inspired by
traditional model-based (MB) restoration methods, considering a model-based
deep learning approach, where the network is trained to operate in the
variance stabilization transformation (VST) domain. To accurately control the
network operation point, in terms of noise and blur of the restored image, we
propose a loss function that minimizes the bias and matches residual noise
between the input and the output. The training dataset was composed of
clinical data acquired at the standard FD and low-dose pairs obtained by the
injection of quantum noise. The network was tested using real DBT projections
acquired with a physical anthropomorphic breast phantom. The proposed network
achieved superior results in terms of the mean normalized squared error
(MNSE), training time and noise spatial correlation compared with networks
trained with traditional data-driven methods. The proposed approach can be
extended for other medical imaging application that requires LD acquisitions.

_K_ eywords Digital breast tomosynthesis (DBT) $\cdot$ Convolutional neural
network (CNN) $\cdot$ deep learning (DL) $\cdot$ image denoising $\cdot$ image
restoration $\cdot$ radiation dose.

##  1 Introduction

Figure 1: Illustration of the proposed network architecture. From left to
right, ROIs represent the low-dose image, estimated noise-free image, restored
image and the respective full-dose.

Image restoration has been an active topic, especially in the field of medical
imaging systems where radiation dose is a concern. Digital breast
tomosynthesis (DBT) is an imaging modality currently being used for breast
screening. In this system, the x-ray tube moves along an arc acquiring several
projections of the breast. In the end, a pseudo-3D volume is reconstructed by
a combination of the set of projections, and radiologists may search for signs
of breast cancer [1].

Since this technique is used on a population-based breast cancer screening
procedure and the image acquisition process utilizes ionizing radiation, it is
desirable to achieve low radiation doses and preserve image quality for
satisfactory diagnosis. Restoring images from low-dose acquisitions has been
an active research topic and has been solved by different approaches, such as
traditional iterative reconstruction methods [2, 3, 4] and denoising
techniques [5, 6, 7]. More recently, data-based methods using neural networks
have also been applied to the field, achieving promising results [8, 9, 10,
11, 12, 13, 14, 15, 16, 17].

Specifically, in the DBT area, Gao, Fessler and Chan [18] proposed a
Wasserstein generative adversarial network (WGAN) for denoising DBT slices.
The network was trained on digital and physical phantoms, tested on clinical
images and both mean squared error (MSE) and adversarial loss were used in the
training step. Liu _et al._ [19] used two breast specimens to train a
convolutional neural network (CNN) model for the restoration of DBT images.
The low-dose (LD) and full-dose (FD) image pairs were acquired in the
equipment and the network was applied in the projections. Sahu _et al._ [20]
trained a GAN on synthetic images, generated through virtual clinical trials
(VCT) software, to denoise LD DBT projections. Both LD and HD images were
generated in the VCT software.

All these works rely on the fact that the network is expected to learn all the
restoration process strictly from the provided data. However, adding prior
knowledge on the network architecture might improve network performance. This
approach is known as model-based deep learning and has been investigated in
previous works [8, 13, 21, 22, 23, 24, 25, 17].

In this work, inspired by a model-based (MB) denoising pipeline [7], designed
to restore LD DBT projections, we propose a deep CNN architecture, that
benefits from prior knowledge, to restore LD DBT projections.

First, we pre-trained the network in a variance-stabilizing transformation
(VST) domain with a self-learning framework [26].

In common training strategies, local patches are extracted from the image and
feeded in the network. This makes it hard for the network to know the spatial
noise dependencies existing in the mammography image. In the VST domain, the
denoising task becomes easier as the transformation is intended to minimize
the signal and spatial dependencies. In other words, this will facilitate the
network learning process, such that it does not have to learn different
denoising strengths, depending on the input.

As we do not have noise-free clinical images to use as references in the VST
domain, we used the self-learning method to estimate this image. This allows
the use of clinical images in the training phase by the network in the VST
domain.

In the second step, we loaded the pre-trained weights in an architecture that
contains both the forward VST and its inverse as layers. Also, we added a
weighted sum module in the last residual layer, so the restored image matches
the expected mean and variance of the FD.

Zhang _et al._ [25] proposed VST-Net to denoise natural images corrupted by
Poisson noise. Their network performs the Anscombe transformation [27] and its
inverse as convolutional operations. Our work differs by the facts that we
used the generalized Anscombe transformation (GAT) due to the Poisson-Gaussian
noise nature of DBT projections, we used a properly unbiased inverse and our
transformations does not have learnable weights. Also, our last residual layer
induces the network to match the mean and variance of the target image. We
also proposed a loss function that yields a restored image residual noise (RN)
similar to the target FD, with minimal bias. Narage _et al._ [28] proposed a
similar loss function. Our approach differs in the fact that we directly
calculate bias and RN from different image realizations. Also, we restricted
this loss to fine-tune the final network so it can achieve specific denoising
properties.

The manuscript is divided as follows. Section 2 gives a general overview of
the image formation, the transformations and the respective network. Section 3
describe how the networks were trained, the loss functions used and the
datasets. Section 4 presents and discusses the results and finally, Section 5
concludes the work, states the limitations on the work and shows future works.

##  2 Theoretical Background

Noise in DBT projections is typically described by a Poisson-Gaussian model.
Quantum noise accounts for the signal-dependent part, whereas electronic or
thermal noise is described by the signal-independent part. The raw image
acquisition at the standard full-dose (FD) can be formulated as:

$z(x)=\alpha(x)p(x)+n(x)+\tau$ (1)

where $x$ indicates the pixel spatial coordinates,
$p(x)\sim\mathcal{P}(\alpha(x)^{-1}y(x))$ is a random variable with Poisson
distribution, $\alpha(x)$ is the spatially dependent quantum gain, $y(x)$ is
the (unknown) noise-free image, $n(x)\sim\mathcal{N}(0,\sigma^{2}_{e})$ is a
random variable with Gaussian distribution, $\sigma^{2}_{e}$ is the variance
of the electronic noise and $\tau$ is the pixel offset.

One can model a lower-dose (LD) projection $z_{\gamma}(x)$ by fixing the
radiographic factors as in $z(x)$ except for a reduction in the current-time
product (mAs):

$z_{\gamma}(x)=\alpha(x)p_{\gamma}(x)+n(x)+\tau$ (2)

where $p_{\gamma}\sim\mathcal{P}(\gamma\alpha(x)^{-1}y(x))$ and $0<\gamma<1$
is the corresponding dose reduction factor.

Such dose reduction directly impacts on the SNR, such as, for $z_{i}$:

$SNR=\frac{E\\{z_{i}\,|\,y_{i}\\}^{2}}{var\\{z_{i}\,|\,y_{i}\\}}=\frac{y_{i}^{2}}{\alpha_{i}^{-1}y_{i}+\sigma^{2}_{e}},$
(3)

and if we subtract the pixel offset and divide it by $\gamma$, it is clear
that the SNR is lower for $[z_{\gamma}]_{i}$:

$SNR_{\gamma}=\frac{E\\{[z_{\gamma}]_{i}\,|\,\gamma
y_{i}\\}^{2}}{var\\{[z_{\gamma}]_{i}\,|\,\gamma
y_{i}\\}}=\frac{y_{i}^{2}}{\gamma^{-1}\alpha_{i}^{-1}y_{i}+\sigma^{2}_{e}\gamma^{-2}}.$
(4)

In Borges _et al._ [7], an image restoration pipeline was proposed to restore
the LD DBT projections and achieve the FD SNR.

The restoration process can be treated by a modular approach, where first, a
VST is applied to approximate noise to a unitary variance. In the case of
Poisson-Gaussian distributions, the GAT [29] is commonly used:

$f(\dot{z}_{i})=\begin{cases}2\sqrt{\dot{z}_{i}+\frac{3}{8}+\sigma_{e}^{2}},&\text{
if }\dot{z}_{i}>-\frac{3}{8}-\sigma_{e}^{2}\\\ 0&\text{ if
}\dot{z}_{i}\leq-\frac{3}{8}-\sigma_{e}^{2},\end{cases}$ (5)

where $\dot{z}_{i}$ is the observed pixel after an affine transformation which
subtracts the offset and divides the quantum noise gain and
$\dot{\sigma}_{e}^{2}$ is the electronic noise variance after the affine
transformation.

As the GAT is a non-linear transformation, removing noise on its domain may
introduce bias when using the algebraic inverse [30]. This one can be solved
by applying the asymptotically unbiased inverse, however, [30] demonstrated
that such approach is not sufficient for low-intensity images, and they
proposed a closed-form approximation of the exact unbiased inverse:

$\widetilde{\mathcal{I}}(d)=\frac{1}{4}d^{2}+\frac{1}{4}\sqrt{\frac{3}{2}}d^{-1}-\frac{11}{8}d^{-2}+\frac{5}{8}\sqrt{\frac{3}{2}}d^{-3}-\frac{1}{8}-\sigma_{e}^{2},$
(6)

where $d$ is the denoised image. Note that the affine inverse transformation
is applied after (6).

As our primary goal is to match the signal and noise properties of LD and FD
image, it is desired that their expected mean and variance match. So, after
denoising the image in the VST domain, we have an estimate of the noise-free
signal $[\widehat{y_{\gamma}}]_{i}$, then a weighted sum is performed between
the input image and the denoised one:

$\hat{z}=w_{i}\times([z_{\gamma}]_{i}-\tau)+\bar{w_{i}}\times([\widehat{y_{\gamma}}]_{i}-\tau)+\tau,$
(7)

where

$w_{i}=\sqrt{\frac{\alpha_{i}y_{i}+\sigma_{e}}{\gamma\alpha_{i}y_{i}+\sigma_{e}}},$
(8)

and

$\bar{w_{i}}=\frac{1}{\alpha_{i}}-w_{i},$ (9)

is sufficient to make the restored image expected mean and variance match the
FD, considering the denoising was successfully [7]. As we do not have access
to $y$, we approximate it to:

$\hat{y}=\frac{[\widehat{y_{\gamma}}]_{i}-\tau}{\gamma}$ (10)

Figure 1 illustrates the proposed network architecture, where GAT block
implements (5) within the affine transformation, BS normalize the signal from
0 to 1, iGAT implements (6) and wAdd implements (7) and (8) with a residual
layer. Note that, in clinical images, as we do not have access to $y_{i}$ from
(1), we used the self-learning framework Noise2Sim [26] to estimate it and
pre-train the network.

##  3 Materials & Methods

###  3.1 CNN Training

In general, two approaches were investigated in this work. The first one is
related to training the CNN to restore the LD image to the corresponding HD
with the slightly modified Residual Network (ResNet) presented before [31]. We
call it ResNet throughout the paper for the sake of simplicity. The second
approach involves training in two phases, where it first learns how to remove
all the noise from the image and, later, it is fine-tuned to closely match the
noisy properties to its corresponding FD.

####  3.1.1 Frameworks

We call the first framework the data-based learning (DBL) approach. Overall,
three networks were trained with different datasets to investigate their
performance. The first network was previously presented at [31], where it was
trained on FFDM images and at this work tested in the DBT dataset. For the
second approach, we used the same network as in the first case, however, fine-
tuning was performed in the DBT training dataset. The third network was
trained and tested only on DBT data. In all these approaches, no previous
information about the data was provided to the network, so it has to learn the
restoration only from the provided data.

The second framework provides previous knowledge for the network, so we call
it hybrid learning (HBL). In this domain, we investigated two different
methods. The first one, the network learns how to estimate the noise-free
signal inside the VST domain. In this task, we used the Noise2Sim (N2S)
framework [26], which uses non-local image information to estimate a noise-
free image in an unsupervised manner. Second, we used the previous network as
a pre-trained model and fine-tuned it with a bias-residual-noise (BRN) loss
function after the whole pipeline.

####  3.1.2 Loss Functions

The importance of the loss function on image restoration was previously
presented at [32] for natural images and at [31] for medical images. In this
work, we used two commonly known mean absolute error (MAE) and perceptual loss
(PL) functions. Also, we propose the bias-residual-noise loss function which
was inspired by the decomposition on the MNSE proposed at [7].

The BRN loss function tries to match the RN between the pair of LD and FD
images while it minimizes the signal bias of the restored image. (11) shows
the composition of these factors in the loss function:

$\mathcal{L}_{\mathcal{B}\mathcal{R}_{\mathcal{N}}}=\mathcal{B}^{2}+\lambda_{\mathcal{R}_{\mathcal{N}}}*|\mathcal{R}_{\mathcal{N}_{FD}}-\mathcal{R}_{\mathcal{N}_{RD}}|,$
(11)

where $\lambda_{\mathcal{R}_{\mathcal{N}}}$ is a weighting factor to control
the denoising strength by imposing a match between the RN of the FD and the
restored dose (RD).

###  3.2 Dataset

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

(a) LD

(b) FFDM

(c) FFDM∗

(d) DBT

(e) MB

(f) FD

Figure 2: Visualization of an ROI for a visual comparison of the DBL
restorations. Where (a) represents the low-dose image, (b) FFDM network
restoration, (c) FFDM with transfer-learning, (d) DBT network, (e) the MB
method and (f) the corresponding full-dose image.

We used different datasets to train and test DBL and HBL approaches.

####  3.2.1 Training

##### DBL

In this approach, we used two datasets. As, in this work, we have a reduction
factor as low as 5%, we had to train FFDM networks for these new values. For
that, we used the same FFDM dataset used in [31] for training. It consists of
400 retrospective clinical images acquired at the Barretos Cancer Hospital
(Brazil) under IRB approval (Certificado de Apresentação para Apreciação Ética
\- CAAE #78625417.1.1001.5437). For the DBT network, we used a new dataset,
consists of 1,982 retrospective clinical exams (29,730 projections) acquired
at the Institute of Radiology (InRad), Faculty of Medicine, University of São
Paulo (Brazil) under IRB approval (CAAE #56699016.7.0000.0065). Both datasets
were generated using the Hologic Selenia Dimensions equipment (Hologic,
Bedford, MA) and the images were used as raw data. All clinical images were
carefully anonymized to preserve patients’ medical records.

For FFDM and DBT images, we simulated LD acquisitions using the method
proposed at [33, 34], which injects quantum and electronic noise in the VST
domain.

##### HBL

In this procedure, we used the aforementioned DBT dataset for the N2S
framework and we also generated virtual clinical images using the virtual
clinical trial (VCT) framework developed at the University of Pennsylvania
[35]. We used that framework so that we could generate several acquisitions of
the same patient with different noise realizations. 7 patients were simulated
and we generated the corresponding noise-free DBT projections. Note that it is
possible to simulate as many patients as desired. We simulated five noise
realizations for each projection set using the method proposed at [36]. The
geometry of the Hologic Selenia Dimensions system was used and noise
simulation accounts for the quantum and electronic noise and also a
correlation kernel.

####  3.2.2 Testing

To test all the networks in a real scenario, we used a dataset of an
anthropomorphic phantom that was designed by the University of Pennsylvania
[37] and produced by CIRS, Inc. (Reston, VA). We performed 60 DBT acquisitions
that produced 20 images at the FD and 10 images at 50%, 25%, 15% and 5% of the
FD. The standard radiographic factors were obtained by using the automatic
exposure control (AEC), yielding 31 kVp and 60 mAs. In the system manual mode,
by reducing accordingly the mAs, it is possible to achieve LD images. This
dataset was used in both DBL and HBL approaches.

###  3.3 Implementation Details

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

(s)

(t)

(u)

(a) LD

(b) N2S

(c) VST-0.01

(d) VST-0.34

(e) VST-0.95

(f) MB

(g) FD

Figure 3: Visualization of an ROI for a visual comparison of the HBL
restorations. Where (a) represents the low-dose image, (b) N2S network
restoration, (c) to (e) VST networks with its respective $\lambda_{RN}$
values, (f) the MB method and (f) the corresponding full-dose image.

All neural networks were implemented using PyTorch Deep Learning library
111www.pytorch.org, trained and tested using an NVIDIA TITAN Xp with 12GB of
RAM. However, specific details are related to each restoration framework.

##### DBL

In this framework approach, one neural network is trained for each reduction
factor. We trained FFDM networks as done in [31], _i.e._ , first training a
network with MAE loss with 60 epochs and LR equals $1\times 10^{-4}$ and
subsequently training a network with PL4 for 60 epochs and the LR reduced by
ten times. The batch size was set to 256 in the pre-trained network and 64 in
the later training. The LR was reduced by half every 10 epochs. For fine-
tuning on DBT data, we used the aforementioned model as a pre-trained network
and trained for 10 epochs, PL4 and LR equal to $5\times 10^{-6}$. In this
approach, the LR was reduced by half every two epochs. DBT network was trained
in the same manner as the FFDM one, except for the fact that only DBT data was
used. We used Adam optimizer with running averages of gradient and its square
equal $0.5$ and $0.999$, respectively. All the networks were trained with
256,000 patches.

##### HBL

Different from the DBL, we restricted the network training procedure only for
the 50% of dose reduction. We observed that training at lower doses forces the
network to have strong denoising properties and yield to a bias increasing.
This model was tested on the images at the other reduction factors. In the N2S
model, we trained the ResNet in an unsupervised way. The target images are
generated by the N2S framework with a technique that was inspired by the non-
local means method. The N2S training was performed in the VST domain, _i.e._ ,
we performed the GAT on the data before training. We trained for 70 epochs,
batch size of 128, Adam optimizer with the same parameters as in the DBL, LR
which starts from 0, go up to $1\times 10^{-4}$ and return to 0 (default
method in the framework), MSE loss function and the number of simulated images
equal to 8.

After properly training the ResNet in the N2S framework, we loaded its weights
in the VST network and performed a fine-tuning with the BRN loss function.
90,650 patches of the VCT projections were used in the training process. To
correctly measure the BRN loss, we stacked the five noise realizations in
sequence and set the batch size to 60. The loss was then calculated between
the LD image and the corresponding FD, outside of the VST domain. We fine-
tuned the model only for 2 epochs and set the LR to $1\times 10^{-5}$. The
$\lambda_{RN}$ was optimized and the results are presented in the following
sections.

##### MB

We also used as a benchmark the MB restoration technique that was presented in
[7]. This approach implements that pipeline mentioned in Section 2 and uses
the block-matching and 3D filtering (BM3D) [38] in the VST domain.

###  3.4 Figures of Merit

Table 1: MNSE analysis with its decomposition on bias and RN for the DBL approach and comparison with LD, FD and MB results. Values highlighted in blue mean the lowest bias for the DL restorations. In green, RN values which are closest to the value from FD images. mAs | Metric | Noisy | DL-FFDM | DL-FFDM* | DL-DBT | MB | Full-dose
---|---|---|---|---|---|---|---
30 | $\mathcal{B}^{2}$ | 0.15% |  0.50% | 0.62% | 0.87% | 0.26% | $\mathcal{B}^{2}$ | 0.09%
$\mathcal{R}_{\mathcal{N}}$ | 24.86% |  12.37% | 10.12% | 11.10% | 13.23%
MNSE | 25.02% | 12.88% | 10.74% | 11.97% | 13.50%
15 | $\mathcal{B}^{2}$ | 0.45% | 3.25% | 2.95% |  2.87% | 0.82%
$\mathcal{R}_{\mathcal{N}}$ | 50.80% | 8.47% |  8.73% | 7.53% | 14.82% | $\mathcal{R}_{\mathcal{N}}$ | 11.86%
MNSE | 51.25% | 11.72% | 11.68% | 10.40% | 15.65%
9 | $\mathcal{B}^{2}$ | 0.47% |  5.43% | 6.52% | 9.23% | 1.08%
$\mathcal{R}_{\mathcal{N}}$ | 87.94% |  10.89% | 18.71% | 7.27% | 17.43%
MNSE | 88.41% | 16.32% | 25.22% | 16.50% | 18.51% | MNSE | 11.95%
3 | $\mathcal{B}^{2}$ | 1.43% | 18.73% | 10.58% |  7.95% | 2.27%
$\mathcal{R}_{\mathcal{N}}$ | 284.05% | 8.66% | 26.20% |  14.29% | 47.98%
MNSE | 285.48% | 27.39% | 36.77% | 22.25% | 50.25%

In general, there is a trade-off between bias and residual noise in denoising
methods. When a clean image is the target, methods seek to reduce the RN as
much as possible whereas the bias increases. On the other hand, in dose
restoration, which is the case of this work, it is desirable to achieve the
same RN as the FD with the low bias as possible. To evaluate both these
metrics separately, we used the decomposition on the MNSE, as done before in
[7, 31], where RN is calculated as follow:

$\mathcal{R}_{\mathcal{N}}=\frac{1}{m\times n}\sum_{i}^{m\times
n}{\frac{\mathbb{V}\left(z^{{}^{\prime}}_{i}\right)}{y_{i}}}$ (12)

and bias:

$\mathcal{B}^{2}=\left[\frac{1}{m\times n}\sum_{i}^{m\times
n}{\frac{\left(\mathbb{E}\\{z^{{}^{\prime}}_{i}\\}-y_{i}\right)^{2}}{y_{i}}}\right]-\frac{\mathcal{R}_{\mathcal{N}}}{p}$
(13)

where $m$ is the number of rows, $n$ is the number of columns, $\mathbb{V}$
indicates the pixel-wise variance along the set of $z^{{}^{\prime}}$ image
realizations, $\mathbb{E}$ indicates the pixel-wise expectation, $y$ is the
noise-free image and $p$ is the number of realizations in the
$z^{{}^{\prime}}$ set.

We evaluated the MNSE in the anthropomorphic breast phantom. To generate the
pseudo-ground-truth, we used 10 images within the FD ones. For each dose
reduction factor, 10 images were used and also 10 for the FD. The metric was
only evaluated inside the breast phantom and we adjuted the mean value of all
images and restorations using an affine transformation between the input image
and the pseudo-ground-truth.

As the networks are based on the convolution operation, and it is known that
this operation might induce noise correlations in the final image. We measured
the power spectrum (PS) [39, 40] to investigate the impact of this correlation
in the restored image, in the following form:

$PS(u,v)=\frac{\Delta x\Delta
y}{n_{r}\,s_{rx}\,s_{ry}}\sum_{k}^{n_{r}}\left|\mathcal{F}\mathcal{F}\mathcal{T}\\{h(x,y)\times
u_{k}(x,y)\\}\right|^{2}$ (14)

where $\Delta x$ and $\Delta y$ are the pixel size in mm the $x$ and $y$
direction, respectively, $n_{r}$ is the number of extracted ROIs, $s_{rx}$ and
$s_{ry}$ are the ROI number of pixels in the $x$ and $y$ direction,
respectively, $h$ is a Hanning window and $\mathcal{F}\mathcal{F}\mathcal{T}$
indicates the discrete Fourier transform. We normalized the PS using the large
are signal, as following:

$NPS(u,v)=\frac{PS}{LAS^{2}}$ (15)

where $LAS$ is calculated as the mean pixel value inside the segmented breast.
Finally, the 1D plot was calculated as a radially mean of the 2D spectrum.

##  4 Results & Discussions

Table 2: MNSE analysis with its decomposition on bias and RN for the HBL approach and comparison with LD, FD and MB results. Values highlighted in blue mean the lowest bias for all restorations. In green, RN values which are closest to the value from FD images. mAs | Metric | Noisy | DL-N2S | DL-VST-0.01 | DL-VST-0.34 | DL-VST-0.95 | MB | Full-dose
---|---|---|---|---|---|---|---|---
30 | $\mathcal{B}^{2}$ | 0.15% | 0.34% |  0.24% | 0.30% | 0.39% | 0.26% | $\mathcal{B}^{2}$ | 0.09%
$\mathcal{R}_{\mathcal{N}}$ | 24.86% | 14.23% | 15.73% | 13.77% |  11.78% | 13.23%
MNSE | 25.02% | 14.57% | 15.98% | 14.07% | 12.16% | 13.50%
15 | $\mathcal{B}^{2}$ | 0.45% | 1.08% |  0.82% | 0.98% | 1.23% |  0.82%
$\mathcal{R}_{\mathcal{N}}$ | 50.80% | 16.83% | 21.92% | 17.07% |  12.76% | 14.82% | $\mathcal{R}_{\mathcal{N}}$ | 11.86%
MNSE | 51.25% | 17.92% | 22.74% | 18.05% | 13.98% | 15.65%
9 | $\mathcal{B}^{2}$ | 0.47% | 1.29% |  1.04% | 1.27% | 1.64% | 1.08%
$\mathcal{R}_{\mathcal{N}}$ | 87.94% | 21.57% | 30.01% | 22.21% |  15.99% | 17.43%
MNSE | 88.41% | 22.86% | 31.05% | 23.48% | 17.63% | 18.51% | MNSE | 11.95%
3 | $\mathcal{B}^{2}$ | 1.43% | 2.48% |  2.21% | 2.56% | 3.11% | 2.27%
$\mathcal{R}_{\mathcal{N}}$ | 284.05% |  41.36% | 65.43% | 48.23% | 42.30% | 47.98%
MNSE | 285.48% | 43.84% | 67.65% | 50.79% | 45.40% | 50.25%

In this section, we present the visual comparisons and the quantitative
results regarding both DBL and HBL approaches. We show the results as an
ablation study, _i.e._ , first DBL is presented such that no priors are added
and the network only learns from data. In sequence, the HBL is presented,
where we introduce the N2S network representing the training only in the VST,
_i.e._ , the forward VST, its inverse and the weighted sum were not taken into
account in the loss function. Finally, we present the VST network within the
$\lambda$ tuning, illustrating improvements in both visual and quantitative
results.

All the results are regarding the test set, where these cases represent a real
scenario, _i.e._ , the LD images were obtained directly in the DBT system.

###  4.1 DBL

####  4.1.1 Visual Comparison

Figure 2 illustrates the restoration of LD projections by different networks
in distinct dose reductions. From top to bottom, each row represents the
reduction factor at 50%, 25%, 15% and 5%, respectively. The first to last
column represents, respectively, the LD projections, FFDM, FFDM with transfer-
learning, DBT, MB and the FD images.

FFDM shows an interesting case, where the network was only trained on images
particularly for this system and even though performed well when applied in
the DBT test set. This indicates that the network generalizes well even for
different systems. Although DBT and FFDM modalities use the same equipment,
they have key differences such as detector size, radiation dose per image and
anti-scatter grid that affect image formation. In the two lowest dose cases,
the network performance is limited and the image quality is very degraded when
we compare with the MB approach.

In visual terms, the FFDM* network did not benefit from the transfer-learning
technique. No real improvements can be seen for neither dose reductions,
although its loss function decreased until DBT loss levels, as Fig. 4
illustrates.

DBT achieved better results for 25% when compared with FFDM cases, however, in
the 15% case, the underlying signal was degraded, as we can see in the small
bright point in the image center.

In all 50% cases, the restorations achieved similar results as the MB and the
FD. This was expected and showed in our previous work [31]. For other
reduction factors, the network did not perform well. We claim that for these
lower doses, PL4 does not preserve image details as the input becomes noisier,
although it reaches convergence as Fig. 4 shows. One option might be adding an
MSE or MAE loss in addition to the PL, so it enforces image fidelity. The
drawback of this procedure is that it would be more time-consuming since PL
already takes a long training time. Another option would be training the
networks after image reconstruction, as the resulting image would be a
combination of 15 projections, for the Hologic systems, yielding an image with
less noise.

(a)

(b)

(c)

Figure 4: Illustration of the loss function for the (a) FFDM network, (b) FFDM
with transfer-learning and (c) DBT network. We used the exponential moving
average with the weight equals 0.999 to better show the losses.

####  4.1.2 Quantitative Analysis

Table 1 provides an objective analysis, in terms of MNSE and its decomposition
on bias and RN, for LD and FD acquisitions, DL restorations and also the MB
method. We highlighted in blue the lowest value of bias within DL restorations
and in green the closest RN to the FD value.

The network trained only on FFDM data had better results for both bias and RN
for 50% and 15% reduction factors. This could be partially explained by the
fact that FFDM images have a higher signal-to-noise ratio (SNR) in comparison
with individual DBT projections, so even though the FFDM network was trained
on LD images, _e.g._ , 50%, these LD images contain an SNR higher than the
individuals 50% DBT projections. This makes the learning process easier, so
the network does not have to perform aggressive denoising to achieve FD image
properties.

DBT network achieved the lowest bias in the 25% and 5% cases. However, it only
yield the closest RN in the lowest dose. FFDM with with transfer learning only
achieved good results in the RN for the 25% reduction factor.

In general, MB performed better compared with DL methods in DBL mode. We also
can see that in this mode, there is no control over how the network behaves
regarding denoising properties.

###  4.2 HBL

Although the MSE loss function for the N2S network reached the same values for
all reduction factors, as illustrated by Fig. 5, we selected the network that
was trained with 50% dose reduction, as this network yield less bias.

Figure 5: Illustration of the MSE loss function for the N2S network. We used
the exponential moving average with the weight equals 0.992 to better show the
losses.

####  4.2.1 $\lambda_{RN}$ Optimization

As previously mentioned, there is a trade-off in image restoration when
denoising an image between bias and RN. We optimized the $\lambda_{RN}$ hyper-
parameter to evaluate the values where the network seeks low bias, RN
matching, or an intermediate point.

Figures 6 and 7 show the results of $\mathcal{B}^{2}$ and
$\mathcal{R}_{\mathcal{N}}$ for different $\lambda_{RN}$ values, when training
for 1 and 2 epochs, respectively. In general, we can observe, objectively, the
trade-off between the measured metrics and also that the $\lambda_{RN}$
regarding the intermediate point decreased in the second epoch. This point
represents the hyper-parameter value where the network achieves the lowest
bias and RN. Approximately, for epoch 1, $\lambda_{RN}=0.42$ and
$\lambda_{RN}=0.38$ for epoch 2. It can also be inferred that when RN is
preferred, the network has strong denoising properties because the loss forces
a noise variance math between input and target images. On the other hand, and
the loss is bias seek, the network prioritizes the underlying signal, yielding
cautious denoising and higher noise variance. Theoretically, the lowest bias
that could be achieved is when the output image is the same as the input. In
terms of RN, the lowest it can be achieved would be performing total denoising
seeking the underlying signal. In DBT image restoration, we seek a match
between RN of FD and LD images, with low bias as possible.

We also measured two different network operation points, where the loss
function only contains the bias term, _i.e._ , $\lambda_{RN}=0$, and the other
point where it only contains the RN term, _i.e._ , $\lambda_{RN}=\infty$.

Figure 6: $\lambda_{RN}$ tuning plot on the first epoch. The graphic
illustrates the network operation points in terms of bias and RN for each
$\lambda_{RN}$ value. Figure 7: $\lambda_{RN}$ tuning plot on the second
epoch. The graphic illustrates the network operation points in terms of bias
and RN for each $\lambda_{RN}$ value..

####  4.2.2 Visual Comparison

Figure 3 illustrates the results regarding the HBL approach. The same ROI from
Fig. 2 is shown, now for N2S and VST DL methods. For the VST network, we
choose three $\lambda_{RN}$ values (0.01, 0.34 and 0.95) to show different
points of operation. These points were considered because they represent the
case with the lowest bias, intermediate point and closest RN. In general, all
DL methods achieve satisfactory results which are favorable or comparable with
MB ones.

(a) 50%

(b) 25%

(c) 15%

(d) 05%

Figure 8: Graphics illustrating the NPS for all the restorations in DBL mode,
respective to the reduction factors of (a) 50%, (b) 25%, (c), 15% and (d) 5%.

(a) 50%

(b) 25%

(c) 15%

(d) 05%

Figure 9: Graphics illustrating the NPS for all the restorations in HBL mode,
respective to the reduction factors of (a) 50%, (b) 25%, (c), 15% and (d) 5%.

N2S by itself benefited from training in the VST domain, where noise has
approximately unit variance and already achieved good results. Although we
used this framework for training the network, it is possible to use any other
architecture and framework for the denoising job. The limitation of this
method is that it is not possible to control the network operation point in
terms of bias and RN.

In the VST network, we can observe that the $\lambda_{RN}$ directly affects
image results. For the highest value, it is possible to see that the network
has strong denoising properties. Differently, in the lowest value, we can see
much more noise when compared to the previous case. In terms of bias, _i.e._ ,
signal preservation, we can notice that the tinny bright spot is sharper in
VST-0.01 oppositely to the VST-0.95 where it is more blurred. VST-0.34 shows a
network case that compromises in terms of preservation of the underlying
signal and RN matching.

####  4.2.3 Quantitative Analysis

Table 2 contains the MNSE, bias and RN results concerning the HBL mode. Once
more, we highlighted the values with the lowest bias and closest RN for all
reduction factors. Different from Tab. 1, now we considered the MB method.

With respect to bias, the VST-0.01 network achieved the lowest values for all
reduction factors, except in the 25% case, where it achieves the same as the
MB. In terms of RN, the VST-0.01 network accomplished RN values closest to the
FD one, except for the lowest reduction factor. Both facts demonstrate the
effectiveness of the BRN loss function to fine-tune the network. Also, it
shows that it is possible to control the network operation concerning
denoising strength, accomplishing the specific results, unlike the DBL mode.
Moreover, adding prior knowledge to the training process enhanced the
restoration quantitative results, as shown by N2S by itself.

Another advantage of HBL mode is the training time, as illustrated by Tab. 3.
For DBL, using the PL4, an average of 86.5h were spent for training purposes,
among 7.5h for pre-training with L1 and 79 for training with PL4. PL4*
illustrates the time for the transfer learning technique. However, this
network was previously trained with FFDM data, also spending around 86.5h. HBL
only takes approximately 8h for training in the N2S framework, as it uses the
low-time cost MSE loss function. Finally, the fine-tunning process with BRN
loss only takes 3min for an epoch. It is also worth noting that PL4 requires
the VGG network to be loaded in GPU memory, demanding more memory and,
consequently decreasing image batch size.

Table 3: Training time, in hours, for both DBL and HBL modes. L1 represents the pre-training process in DBL, PL4 the training process in HBL, PL$4^{*}$ the fine-tuning for FFDM, N2S the training in the VST domain and BRN the fine-tuning for the VST network. Model | Time (h)
---|---
L1 | 7.5
PL4 | 79
PL4* | 13
N2S | 8.5
BRN | 0.1

As observed in Fig. 2, some DL restorations impose correlations on the
resulted image. To objectively measure it, we evaluated the PS in all
restorations, as illustrated by Fig. 8 and 9 for DBL and HBL, respectively.

We can see that DBL methods diverge from LD and FD curves, especially for
high-frequency (HF) values. Even for low frequency, the curves start to move
away in 25% and lower doses. Also, in HF, the curve does not have a flat
behavior as LD and FD do, indicating that the network correlates the noise.

Different from DBL mode, HBL approximates well from FD and LD curves, both in
low and HF values. N2S network had the best performance, due to its training
in the VST domain, where noise has approximately unity variance. VST networks
introduce some correlation as the fine-tuning was performed in the image
domain, where noise does not have a well-defined behavior in terms of
variance. However, this correlation does have wave tendency as observed in DBL
HF values. It is also worth noting that the MB method achieved the best
performance in terms of PS for all frequencies.

##  5 Conclusion

In conclusion, CNNs designed with prior knowledge on its architecture benefit
from it and have superior results from those without it. For the VST network,
we achieved comparable or superior results from the MB method in terms of bias
and RN. When HBL was evaluated against DBL mode, it achieved superior results
in training time, bias, RN, PS and also requires less GPU memory.

This work has some limitations. We did not evaluate DBL with different losses
and combinations of them. This would require tuning weight parameters between
these losses. Also, we restricted the network architecture for the modified
ResNet presented in our previous work. Further evaluation with different
architectures such as U-net [41], original ResNet [42], WGAN [43],
Transformers [44], etc., is fundamental. Also, we could evaluate different
frameworks for training the network in the VST domain. Another limitation
respects the equipment parameters estimation necessity for the GAT and
weighted sum layers. Future works could investigate performing the GAT, its
inverse and the wSum as trainable layers and compare the results. Also, there
is a necessity to inspect the $\lambda_{RN}$ parameter in terms of lesion
diagnosis with radiologists. This would answer the question of whether is a
low bias image with a slightly high RN is preferred or an RN match between
restored image and FD, with a loss of bias.

Finally, we claim that the presented framework for training networks for image
restoration can be expanded for other areas in the medical image field and
also for different areas such as natural images and video processing.

## Acknowledgment

This work was supported in part by the National Council for Scientific and
Technological Development (CNPq), by the São Paulo Research Foundation (FAPESP
grant 2021/12673-6 and 2018/19888-5), by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior (CAPES finance code 001), and National Institute of
Health (R01EB026646, R01CA233888, R01CA237267, and R01HL151561).

The authors would like to thank the Institute of Radiology (InRad), in
particular Dra. Denise Yanikian Nersissian and Éric Francisco Scolastici, for
providing the clinical data. The authors also would like to thank Dr. Andrew
D. A. Maidment, from the University of Pennsylvania, for making the
anthropomorphic breast phantom images available for this work.

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