# PnP Restoration with Domain Adaptation for SANS

Shirin Shoushtari, Edward P. Chandler, Jialiang Zhang, Manjula Senanayake,
Sai Venkatesh Pingali, Marcus Foston, and Ulugbek S. Kamilov.
Washington University in St. Louis, MO 63130 USA
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

###### Abstract

Small Angle Neutron Scattering (SANS) is a non-destructive technique utilized
to probe the nano- to mesoscale structure of materials by analyzing the
scattering pattern of neutrons. Accelerating SANS acquisition for in-situ
analysis is essential, but it often reduces the signal-to-noise ratio (SNR),
highlighting the need for methods to enhance SNR even with short acquisition
times. While deep learning (DL) can be used for enhancing SNR of low quality
SANS, the amount of experimental data available for training is usually
severely limited. We address this issue by proposing a Plug-and-play
Restoration for SANS (PR-SANS) that uses domain-adapted priors. The prior in
PR-SANS is initially trained on a set of generic images and subsequently fine-
tuned using a limited amount of experimental SANS data. We present a
theoretical convergence analysis of PR-SANS by focusing on the error resulting
from using inexact domain-adapted priors instead of the ideal ones. We
demonstrate with experimentally collected SANS data that PR-SANS can recover
high-SNR 2D SANS detector images from low-SNR detector images, effectively
increasing the SNR. This advancement enables a reduction in acquisition times
by a factor of 12 while maintaining the original signal quality.

††This material is based upon work supported by the Gordon and Betty Moore
Foundation grant 11396.

##  1 Introduction

Small angle neutron scattering (SANS) is widely used technique for
investigating material structure at mesoscopic scale [1]. Long exposure to the
neutron beam is often necessary for detailed structural information from the
SANS experiments. There has been an increased demand for insights into the
dynamic behaviors of mesoscale structures, motivating the development of in-
situ and operando SANS. Traditionally, a limitation of SANS has been its lower
neutron flux compared to x-ray scattering, leading to longer acquisition times
and diminished temporal resolution. Recent advancements have seen the
introduction of SANS instruments equipped with larger and faster detectors,
enhanced guide coating technologies, and high-flux neutron sources, aiming to
overcome these challenges [2, 3]. Despite these improvements, obtaining
detailed structural information often requires significant signal averaging
and prolonged acquisitions times, complicating high-quality data acquisition
for applications like in-situ SANS. Consequently, attempts to expedite data
collection frequently compromise the temporal resolution or SNR of the
measurements [4].

The recovery of high-SNR SANS detector images from low-SNR SANS measurements
can be formulated as an image restoration problem. Deep learning (DL) has been
extensively used as a data-driven strategy to address a wide range of imaging
inverse problems [5, 6]. Traditional DL approaches are based on training a
neural network to map low-quality observations to high-quality images [7, 8,
9, 10, 11]. There is an increasing interest in leveraging DL for enhancing the
quality of SANS data [12], predicting structural geometry of sample material
[13], and performing data analysis and feature extraction from SANS [14, 15].
Despite the promising performance of DL in computationally enhancing quality
of SANS data [12], the lack of experimentally collected SANS data poses a
challenge for training effective DL models. Domain adaptation is a promising
strategy to address the issue of limited data availability for training DL
models, a challenge often faced in SANS experiments where acquiring large-
scale datasets is impractical. Domain adaptation involves pre-training a DL
model on one dataset (domain) and adapting it to perform effectively on a
different domain [16, 17, 18].

Plug-and-play (PnP) priors [19, 20] is a computational framework widely-used
for designing DL methods for solving imaging inverse problems. PnP methods
leverage powerful DL denoisers as priors by combining them with the forward
model of the imaging instruments. PnP has been successfully used in various
imaging inverse problems including super-resolution, phase retrieval,
microscopy, and medical imaging [7, 9, 21, 22] (see recent reviews [23, 24]).

Despite the success of PnP on a number of imaging inverse problems, it has
never been applied for restoring high-SNR SANS data. In this paper, we present
the first investigation into PnP for restoring high-SNR SANS data collected in
an accelerated fashion that results in low-SNR SANS detector images. Our PnP
restoration method, called PR-SANS, restores high-SNR detector images from
low-SNR measurements with shorter acquisition time, thereby mitigating the
need for long acquisition times in SANS experiments. To address the limited
amount of experimentally collected SANS datasets for training deep priors for
PnP, we propose a domain adaptation strategy that adapts a generic image prior
(trained on natural grayscale images) into a dedicated SANS prior. We present
a theoretical convergence analysis of PR-SANS that considers the errors
introduced by using adapted deep priors instead of ideal ones. We show on
experimentally collected SANS data that PR-SANS can restore high-SNR SANS
detector images while significantly reducing the acquisition times for
detector image of equivalent SNR ($12\times$ faster data acquisition). Our
results numerically show the relationship between the amount of training SANS
data and the final restoration performance.

Figure 1: Illustration of the proposed method: Initially, a deep prior is pre-
trained on a dataset of natural grayscale images. It is then adapted to the
target dataset (SANS detector images) using a limited number of samples. Once
adapted, this prior is plugged into the PR-SANS algorithm to effectively
restore the 2D SANS detector images.

##  2 Proposed Method

Image restoration. We propose to formulate the recovery of the clean image
${\bm{x}}\in\mathbb{R}^{n}$ from the noisy SANS measurement
${\bm{y}}\in\mathbb{R}^{n}$ as an image restoration problem of form

${\bm{y}}={\bm{x}}+{\bm{e}},$ (1)

where ${\bm{e}}\in\mathbb{R}^{n}$ is the noise. It is common to formulate
restoration problems as an optimization problem

${\widehat{\bm{x}}}=\mathop{\mathsf{arg\,min}}_{{\bm{x}}\in\mathbb{R}^{n}}f({\bm{x}})\quad\text{with}\quad
f({\bm{x}})=g({\bm{x}})+h({\bm{x}}),$ (2)

where $g$ denotes the data-fidelity term that measures the consistency of
recovered signal with the measurement ${\bm{y}}$ and $h$ is a regularizer that
imposes prior knowledge on the recovered signal. Widely-used data-fidelity and
regularization terms are least-squares
$g({\bm{x}})=\frac{1}{2}\|{\bm{x}}-{\bm{y}}\|_{2}^{2}$ and total variation
(TV) regularizer $h({\bm{x}})=\tau\|\bm{D}({\bm{x}})\|_{1}$, where $\bm{D}$ is
image gradient and $\tau>0$ is the regularization parameter [25].

PnP. PnP is one of the most popular approaches for solving imaging inverse
problems where image priors are specified using pre-trained image denoisers
[24]. Our proposed PR-SANS is a variant of PnP with the update rule

${\bm{x}}^{k}={\mathsf{prox}}_{\gamma
g}({\bm{x}}^{k-1}-\gamma\tau{\mathsf{\widehat{R}}}({\bm{x}}^{k-1}))\quad\text{with}\quad{\mathsf{\widehat{R}}}={\mathsf{I}}-{\widehat{\mathsf{D}}}_{\sigma},$
(3)

where ${\widehat{\mathsf{D}}}_{\sigma}$ is the domain-adapted deep prior with
parameter $\sigma>0$ for controlling the noise strength, $\tau>0$ is the
regularization parameter, and $\gamma>0$ is the penalty parameter. We denote
the ideal SANS prior as mapping
${\mathsf{D}}_{\sigma}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ pre-trained to solve
the following restoration problem

${\bm{z}}={\bm{x}}+{\bm{n}}\quad\text{with}\quad{\bm{n}}\sim{\mathcal{N}}(0,\sigma^{2}{\bm{I}}),$
(4)

where ${\bm{n}}\in\mathbb{R}^{n}$ is additive white Gaussian noise (AWGN) of
variance $\sigma^{2}$ [26, 27]. As common in PnP methods [24], we assume that
${\mathsf{D}}_{\sigma}$ performs MMSE estimation of
${\bm{x}}\in\mathbb{R}^{n}$ for the problem (4)

${\mathsf{D}}_{\sigma}({\bm{z}})=\mathbb{E}[{\bm{x}}|{\bm{z}}]=\int_{\mathbb{R}^{n}}{\bm{x}}p_{{\bm{x}}|{\bm{z}}}({\bm{x}};{\bm{z}}){\mathsf{d}}{\bm{x}},$
(5)

where $p_{\bm{x}}$ represent the distribution of target dataset. When the PnP
method converges, it converges to ${\bm{x}^{\ast}}\in\mathbb{R}^{n}$ that
satisfies

$\nabla g({\bm{x}^{\ast}})+\tau{\mathsf{R}}({\bm{x}^{\ast}})=\bm{0},$ (6)

where ${\bm{x}^{\ast}}\in\mathbb{R}^{n}$ is the fixed point. We discuss the
connection of fixed points with stationary points of an objective function
$f=g+h$ for some regularizer $h$ in Section 6.

Domain adaptation. In traditional DL frameworks for addressing image
restoration tasks, it is common to train convolutional neural networks (CNNs)
to map observations to desired images. In this approach, the training and
testing data are often drawn from the same distribution [28, 29, 30, 31].
However, domain shift can occur when training and testing data originate from
different distributions [32, 33, 34, 35, 36]. Domain adaptation techniques can
address this issue by generalizing a DL model trained on a source domain to a
domain of interest (target domain) [17, 37, 38]. Domain adaptation is
particularly beneficial in scenarios where the available training data is
insufficient [39]. Among adaptation techniques, a simple strategy involves
fine-tuning a pre-trained neural network using limited available data from the
target domain. This method relies on the premise that image features can be
transferred across different domains [39, 16, 18].

Our domain adaptation relies on a dataset of natural images
${\mathcal{D}}_{s}=\\{{\bm{x}}_{i}^{s},{\bm{z}}_{i}^{s}\\}_{i=1}^{N}$ as the
source dataset. ${\mathcal{D}}_{s}$ is used to train the initial denoiser
${\mathsf{D}}^{s}_{\sigma}$ by minimizing the MSE loss

${\mathcal{L}}({\mathsf{D}}^{s}_{\sigma})=\mathbb{E}\left[\|{\bm{x}}^{s}-{\mathsf{D}}^{s}_{\sigma}({\bm{z}}^{s})\|_{2}^{2}\right].$
(7)

The parameters of this initial denoiser, denoted by ${\bm{\theta}}^{s}$ , are
stored for the subsequent adaptation phase. The target dataset
${\mathcal{D}}_{t}=\\{{\bm{x}}_{i}^{t},{\bm{z}}_{i}^{t}\\}_{i=1}^{K}$ (where
$K\ll N$) includes a small number of SANS detector images. The adaptation
involves fine-tuning the initial denoiser using the target dataset
${\mathcal{D}}_{t}$ with the MSE loss

${\mathcal{L}}({\mathsf{\widehat{D}}}_{\sigma})=\mathbb{E}\left[\|{\bm{x}}^{t}-{\mathsf{\widehat{D}}}_{\sigma}({\bm{z}}^{t})\|_{2}^{2}\right],$
(8)

starting from the initial parameters ${\bm{\theta}}^{s}$ of the denoiser pre-
trained on source data.

##  3 Theoretical Analysis

In this section, we present the theoretical convergence of the iterates
generated by PR-SANS with adapted priors. Our analysis will require several
assumptions that act as sufficient conditions for our theoretical results.

######  Assumption 1.

The prior density $p_{{\bm{x}}}$ is non-degenerate over $\mathbb{R}^{n}$.

If the support of the probability density $p_{{\bm{x}}}$ is confined to space
with a lower dimension than $n$, it is degenerate over $\mathbb{R}^{n}$. We
use the assumption of non-degenerative prior density to establish a link
between the operator (5) and the following regularizer

$h({\bm{x}})=-\tau\sigma^{2}\log
p_{{\bm{z}}}({\bm{x}}),\quad{\bm{x}}\in\mathbb{R}^{n},$ (9)

where $\tau$ is the regularization parameter, $p_{\bm{z}}$ is the density of
observation (4), and $\sigma^{2}$ is the AWGN level used for training
${\mathsf{D}}_{\sigma}$ (the derivation is in Lemma 3). The link between MMSE
operators and and regularization beyond non-degenerate priors can be found in
[40]. It can be proven that function $h$ is infinitely continuously
differentiable [41, 40].

######  Assumption 2.

The adapted restoration operator ${\mathsf{\widehat{D}}}_{\sigma}$ satisfies

$\|{\mathsf{\widehat{D}}}_{\sigma}({\bm{x}}^{k})-{\mathsf{D}}_{\sigma}({\bm{x}}^{k})\|_{2}\leq\varepsilon_{k},\quad
k=0,1,2,3,\ldots$

where ${\mathsf{D}}_{\sigma}$ is given in (5).

This assumption serves as a bounding constraint on the discrepancy between the
domain adapted ${\widehat{\mathsf{D}}}_{\sigma}$ and ideal MMSE restoration
operator ${\mathsf{D}}_{\sigma}$ for SANS data at each iteration of PR-SANS
algorithm.

Our analysis assumes that at every iteration, PR-SANS uses the adapted MMSE
restoration operator, where the ideal MMSE restoration operator
${\mathsf{D}}_{\sigma}$ is unavailable. We consider the case where at
iteration $k$ of PR-SANS, the distance of the outputs of
${\mathsf{D}}_{\sigma}$ and ${\mathsf{\widehat{D}}}_{\sigma}$ is bounded by a
constant $\varepsilon_{k}$.

######  Assumption 3.

The function $g$ is continuously differentiable. Additionally, $\nabla g$ and
$\nabla h$ are Lipschitz continuous with constants $L>0$ and $M>0$,
respectively.

Lipschitz continuity is a standard assumption in the context of imaging
inverse problems [42].

######  Assumption 4.

The data-fidelity term $g$ and the implicit regularizer $h$ are bounded from
below.

This assumption implies that there exists $f^{\ast}=f({\bm{x}}^{*})>-\infty$
such that $f({\bm{x}})\geq f^{\ast}$ for all ${\bm{x}}\in\mathbb{R}^{n}$.

######  Theorem 1.

Run PR-SANS with the adapted MMSE restoration operator
${\mathsf{\widehat{D}}}_{\sigma}$ for $t\geq 1$ iterations under Assumptions
1-4 using a step-size $\gamma>0$. Then, for each iteration $1\leq k\leq t$, we
have

$\mathop{\mathsf{min}}_{1\leq k\leq t}\|\nabla
f({\bm{x}}^{k})\|^{2}_{2}\leq\frac{1}{t}\sum_{k=1}^{t}\|\nabla
f({\bm{x}}^{k})\|^{2}_{2}\leq\frac{B_{1}}{t}\left(f({\bm{x}}^{0})-f({\bm{x}}^{*})\right)+B_{2}{\overline{\varepsilon}}^{2}_{t},$

where $B_{1}>0$ and $B_{2}>0$ are iteration independent constants, and
${\overline{\varepsilon}}_{t}^{2}\,\coloneqq\,(1/t)\sum_{k=1}^{t}\varepsilon^{2}_{k-1}$
. Additionally, if the sequence of error terms $\\{\varepsilon_{i}\\}_{i\geq
0}$ is square-summable, we have that $\|\nabla f({\bm{x}}^{k})\|\to 0$ as
$k\to\infty$.

The expressions for constant $B_{1}$ and $B_{2}$ are given in the proof.
Theorem 1 shows that the iterates generated by PR-SANS with adapted priors
satisfy $\nabla f({\bm{x}}^{k})\to\bm{0}$ as $k\to\infty$ when the sequence of
error term $\varepsilon_{k}$ is square-summable. The theorem shows that if the
sequence of error is square-summable, PR-SANS asymptotically achieves a
stationary point of $f$. Additionally, if the sequence of error is _not_
square-summable, one can still precisely characterize the error due to the
discrepancy between adapted and ideal MMSE denoisers.

##  4 Experimental Results

We perform numerical validation of PR-SANS for restoring experimentally
collected SANS detector images. Our experiments are designed to evaluate the
effectiveness of the PR-SANS algorithm by comparing its performance against
baselines, such as the total variation (TV) denoising algorithm and end-to-end
CNNs. Furthermore, we highlight how the complexity of the sample in the domain
adaptation of deep priors influences the performance of PR-SANS.

Figure 2: Empirical evaluation of PR-SANS convergence to the true solution
${\bm{x}^{\ast}}$ with adapted and zero-start DL priors and end-to-end models
(DnCNN). The average SNR is plotted against iterations of the PR-SANS
algorithm. Note that the proposed algorithm outperforms the DL model. Also,
note the performance enhancement due to using adapted models instead of zero-
start ones.

We compare PR-SANS performance with several established methods including end-
to-end CNNs and TV denoising algorithm, which represents a non-learning-based
approach to noise reduction [43]. DnCNN [44], a specific type of CNN, serves
dual functions in our analysis: (a) it acts as an end-to-end network that
directly maps measurements to detector images, and (b) it functions as a deep
image prior in the PR-SANS algorithm. DnCNN models that are trained with a
limited amount of data are labeled as "zero-start," while those trained with
domain adaptation strategy are referred to as "adapted". Additionally, to
investigate the effect of sample complexity on the domain adaptation of deep
image priors, we evaluate the performance of PR-SANS with multiple adapted
priors that differ in the quantity of target domain samples used for
adaptation. Our evaluation metrics include SNR and root-square-mean-error
(RMSE) for the recovery of 2D detector images, alongside normalized-mean-
square-error (NMSE) and mean absolute error (MAE) for performance analysis of
1D intensity plots. We used a pre-trained DnCNN model, initially trained for
reducing AWGN with a noise level parameter of $\sigma=5$ on 400 natural
grayscale images [45]. For the training of both adapted and zero-start priors,
100 pairs of low- and high-SNR SANS detector images were used, with an extra
set of 10 SANS detector image pairs reserved for validation and 10 pairs
reserved for testing. The models were trained for 300 epoch and the model with
the best performance on validation set was chosen. The number of iteration for
PR-SANS was set to 20. The penalty parameter is set to $\gamma=0.7$ for all
the experiments. The regularization parameter $\tau$ was optimized for the
best performance. The data collection procedure for SANS experiments is
detailed in section 6.1.

Figure 3: Visual evaluation of several methods on restoring high-SNR SANS
detector images from low SNR data. Zero-start refers to the models that are
trained with limited experimental data and adapted refers to models with
domain adaptation. Note that generally, adapted models have better
performance. Also note the improvement acquired by using the proposed
restoration algorithm in comparison with just using DL models.  Figure 4:
Visual evaluation of several methods on recovering 1D intensity $\bf{I(Q)}$
plot against scattering vector $\bf{Q(\AA)}$ from low SNR SANS data. Note the
competitive performance of the proposed method against other approaches.

###  4.1 SANS detector image restoration

In this section, we compare the performance of PR-SANS using adapted and zero-
start priors against zero-start DnCNN, adapted DnCNN, and TV denoising
algorithm. Figure 3 illustrates the visual comparison of various methods,
evaluating their performance in terms of SNR and RMSE. Note the superior
performance of PR-SANS when using an adapted DnCNN prior, as opposed to the
adapted DnCNN functioning as an end-to-end mapping. Additionally, note the
performance improvement due to domain adaptation in PR-SANS algorithm and
DnCNN. Figure 4 illustrates visual comparison of 1D intensity $I(Q)$ plot
against scattering vector $Q(\AA)$ (Section 6.1 provides details on acquiring
$I(Q)$ from 2D detector images). Note that the proposed method PR-SANS with
adapted prior achieves the best performance. Table 1 presents the quantitative
evaluation of 2D detector images and 1D intensity plots restoration
performance, showing that PR-SANS outperforms other methods. Figure 2
illustrates the convergence of the PR-SANS with adapted and zero-start priors.

Figure 5: Visual evaluation of sample complexity used in domain adaptation of DL prior for PR-SANS algorithm. The performance is reported in terms of SNR (dB) and RMSE. Note the improvement of performance when the adaptation samples from the SANS dataset increases. Table 1: Numerical evaluation of 2D detector image and 1D intensity plot restoration. Table highlights the best and second best results. Note the suboptimal performance of DnCNN (Zero-start) due to the limited amount of training data, which highlights the importance of domain adaptation. Method | 2D |  | 1D
---|---|---|---
|  SNR $\uparrow$ |  RMSE $\downarrow$ |  |  NMSE $\downarrow$ |  MAE $\downarrow$
Low-SNR data | $18.44$ | $0.00680$ |  | $0.03683$ | $0.00316$
DnCNN (Zero-start) | $14.48$ | $0.01058$ |  | $0.03683$ | $0.00230$
PR-SANS (Zero-start) | $19.05$ | $0.00634$ |  | $0.01358$ | $0.00204$
Total Variation | $19.53$ | $0.00600$ |  | $0.01219$ | $0.00258$
DnCNN (Adapted) | $\bf{21.69}$ | $\bf{0.00468}$ |  | $\bf{0.00748}$ | $\bf{0.00162}$
PR-SANS (Adapted) | $\bf{22.36}$ | $\bf{0.00432}$ |  | $\bf{0.00635}$ | $\bf{0.00144}$

###  4.2 Sample Complexity of Domain Adaptation

In this section, we investigate the effect of sample complexity used for prior
adaptation on PR-SANS performance, particularly focusing on how the number of
detector images selected from the target domain (SANS) for prior adaptation
influences outcomes. Same pre-trained model is adapted using different number
of randomly chosen data pair from SANS dataset.

Figure 5 illustrates the visual comparison of 2D detector images restoration
using PR-SANS with different adapted priors . The performance is reported
using SNR (dB) and RMSE. In this context, “0 images” refers to PR-SANS with
the pre-trained base DnCNN without any domain adaptation. Note the progressive
improvement in the PR-SANS algorithm’s performance as the sample complexity
used in domain adaptation increases. Numerical results for the performance of
adapted models in recovering 2D detector image and 1D intensity plot, averaged
for all test samples, are reported in Table 2.

Figure 7 presents the empirical results comparing the performance PR-SANS with
adapted and zero-start priors. The performance in terms of SNR(dB) is plotted
against the number of samples used for adaptation and training of zero-start
priors.

Theorem 1 establishes that PR-SANS approximates the stationary points of
objective function with an error margin, where the error depends on the
difference between the ideal ${\mathsf{D}}_{\sigma}$ and adapted
${\widehat{\mathsf{D}}}_{\sigma}$ priors (see Assumption 2). Obtaining the
ideal prior for SANS images is not feasible in practice. Nonetheless, it can
be shown that increasing the number of images used for domain adaptation can
gradually bring the adapted ${\widehat{\mathsf{D}}}_{\sigma}$ closer to the
ideal ${\mathsf{D}}_{\sigma}$ prior, decreasing the restoration error margin.
Figure 7 illustrates the averaged error of 2D detector image restoration
against PR-SANS iterations for various priors. Note the reduction in error as
the number of images used for adaptation grows, a result that is consistent
with the result in Theorem 1.

Table 2: Numerical evaluation of sample complexity for domain adaptation of DL prior in PR-SANS algorithm. Table highlights the  best results. Note how performance relates to the number of samples used for adaptation. Method | 2D |  | 1D
---|---|---|---
|  SNR $\uparrow$ |  RMSE $\downarrow$ |  |  NMSE $\downarrow$ |  MAE $\downarrow$
Adapted (0 imgs) | $18.46$ | $0.00679$ |  | $0.01566$ | $0.00316$
Adapted (20 imgs) | $20.17$ | $0.00556$ |  | $0.01047$ | $0.00228$
Adapted (40 imgs) | $20.25$ | $0.00553$ |  | $0.01038$ | $0.00215$
Adapted (60 imgs) | $21.26$ | $0.00492$ |  | $0.00827$ | $0.00193$
Adapted (80 imgs) | $\bf{21.67}$ | $\bf{0.00469}$ |  | $\bf{0.00749}$ | $\bf{0.00178}$

Figure 6: Empirical evaluation of performance of PR-SANS with adapted and
zero-start DL priors. Average SNR is plotted against number of samples used
for adaptation and training of zero-start priors. Note the performance
advantage of the domain adaptation over zero-start DL models.

Figure 7: Empirical evaluation of PR-SANS convergence to the true solutions
${\bm{x}^{\ast}}$ with multiple adapted priors. Note the error of restoration
decrease with increase of images used for adaptation.

##  5 Conclusion

In this paper, we introduce PR-SANS, a PnP algorithm leveraging deep image
priors to recover high-SNR SANS data. PR-SANS enhances the acquisition process
by restoring high-SNR detector images from low-SNR SANS data with shorter
acquisition time. Our findings reveal that domain adaptation successfully
addresses the challenge of limited training datasets, with a combination of
400 natural images and 100 SANS datasets proving adequate for obtaining
adapted priors that successfully restore SANS detector images through PR-SANS.
Comparative visual, empirical, and numerical analyses indicate that PR-SANS
surpasses both traditional restoration algorithms, such as Total Variation
(TV), and Convolutional Neural Networks (CNNs) in the restoration of SANS
data.

## Acknowledgment

Neutron scattering research conducted using the Bio-SANS instrument, a DOE
Office of Science, Office of Biological and Environmental Research resource
(FWP ERKP291), used resources at the High-Flux Isotope Reactor, a DOE Office
of Science, Scientific User Facility operated by the Oak Ridge National
Laboratory.

##  6 Appendix

###  6.1 Data collection procedure

SANS measurements were performed at the CG3-Bio-SANS instrument at the High
Flux Isotope Reactor (HFIR), at Oak Ridge National Laboratory (ORNL). The data
was collected in a single instrument configuration with the main detector
$15.5~{}m$ away from the sample; wing detector at $1.4^{\circ}$ rotational
angle from the beam to cover a scattering vector ($Q$) range
$0.003-0.8~{}\scriptstyle{\AA^{-1}}$ at a wavelength of
$6\scriptstyle{\AA^{-1}}(\Delta\lambda/\lambda=13.2\%)$; where
$Q=(4\pi/\lambda)\sin\theta$, $\theta$ is the scattering angle, and $\lambda$
is the neutron wavelength. The data were originally recorded for $1~{}hr$ to
obtain a high-resolution 2D image. The recorded 2D scattering patterns were
azimuthally averaged to obtain the high-resolution 1D SANS scattering curves
$I(Q)$ versus scattering wave vector $Q$ using the ORNL developed software
drtsans3 [46]. The high-resolution 2D image was sliced as a function of time
$5~{}min$ and azimuthally average to obtain low-resolution 1D data. The
reduction process applied corrections such as dark current, transmission, and
solid angle as well as subtracted parasitic scattering of the empty cell, and
the final output data was in absolute scale. Details regarding the data
collection and sample preparation processes are outlined in the appendices.

Materials. Hybrid poplar (NM6) wood chips, D2O, and $\gamma$-Valerolactone-d6.
Debarked Poplar wood chips were provided by the Great Lakes Bioenergy Research
Center (GLBRC). The poplar was milled and sieved through a $5~{}mm$ screen
before the lignin extraction. D2O was purchased from Cambridge Isotopes.
d6-$\gamma$-Valerolactone (d6-GVL) was self-synthesized from levulinic acid
(LA) $98\%$ (Sigma) following the reported method in [47].

Procedure for the $\mathbf{\gamma}$-Valerolactone-assisted lignin (GVL-lignin)
isolation. GVL-lignin was obtained by treatment of hybrid poplar wood chips
using a published method [48]. GVL-lignin was extracted at $100$ and
$120^{\circ}C$, referred to as $L100$ and $L120$.

Procedure for the synthesis of d6-$\mathbf{\gamma}$-Valerolactone (d6-GVL).
[3,3,4,5,5,5-d]GVL (d6-GVL) was synthesized according to the reported
methodology in [47]. Briefly, $16.4~{}g$ of LA and $0.5~{}mL$ of H2SO4
($99.9\%$ from Fisher Scientific) were dissolved in $200~{}mL$ of D2O ($99.9$
atom $\%$ D from Aldrich), the mixture was heated at $90^{\circ}C$ and stirred
for $72~{}hrs$ using a $450~{}mL$ Parr batch reactor. After $72~{}hrs$, the
D2O was removed by rotary evaporation, $200~{}mL$ of new D2O was added to the
reactor, and the deuteration continued for another $72~{}hrs$. This procedure
was repeated twice for a total reaction time of $9$ days. The product obtained
after the D2O removal was labeled as d6-LA and was analyzed by 1H and 2H-NMR
to determine the deuterium incorporation. The d6-LA obtained was used for the
synthesis of d6-GVL. $15.8~{}g$ of d6-LA and $0.13~{}g$ of $5$ wt$\%$ Ru/C
were added in $30~{}g$ of D2O and heated at $90^{\circ}C$ under 40 bar of
$25~{}mol\%$ of D2/N2 (Airgas). The reaction mixture was stirred at
$600~{}rpm$ for $3~{}hrs$ in a $75~{}mL$ Parr batch reactor and then quenched
using an ice bath.

SANS experiments. A customized reaction cell was used for the experiments.
$0.5$ and $10$ wt$\%$ lignin (L100/L120) were prepared in GVL-d6/D2O mixture
($90:10$ wt$\%$). The reaction cell was loaded with the pre-made lignin
solution and placed in the beamline and heated to the target temperate
($25,~{}50,~{}80$, and back to $25^{\circ}C$) at a heating rate of
$5^{\circ}C/min$. At each temperature stage, SANS data was collected for
$1~{}hr$.

###  6.2 Proof of Theorem 1

######  Theorem.

Run PR-SANS with the adapted MMSE restoration operator
${\mathsf{\widehat{D}}}_{\sigma}$ for $t\geq 1$ iterations under Assumptions
1-4 using a step-size $0<\gamma\leq\mathop{\mathsf{min}}(1/M,1/L)$. Then, for
each iteration $1\leq k\leq t$, we have

$\mathop{\mathsf{min}}_{1\leq k\leq t}\|\nabla
f({\bm{x}}^{k})\|^{2}_{2}\leq\frac{1}{t}\sum_{k=1}^{t}\|\nabla
f({\bm{x}}^{k})\|^{2}_{2}\leq\frac{B_{1}}{t}\left(f({\bm{x}}^{0})-f({\bm{x}}^{*})\right)+B_{2}{\overline{\varepsilon}}_{t}^{2},$

where $B_{1}\,\coloneqq\,4(1+\gamma M)^{2}/(\gamma(1-\gamma M))$,
$B_{2}\,\coloneqq\,\lambda\alpha^{2}B_{1}/2+2\alpha^{2}(1+\gamma
L)^{2}/\gamma^{2}$, and
${\overline{\varepsilon}}_{t}^{2}\,\coloneqq\,(1/t)\sum_{k=1}^{t}\varepsilon^{2}_{k-1}$
are iteration independent constants. Additionally, if the sequence of error
terms $\\{\varepsilon_{i}\\}_{i\geq 0}$ is square-summable, we have that
$\|\nabla f({\bm{x}}^{k})\|\to 0$ as $k\to\infty$.

###### Proof.

The update rule for adapted operator ${\widehat{\mathsf{D}}}_{\sigma}$ and
ideal operator ${\mathsf{D}}_{\sigma}$ are
${\bm{x}}^{k}={\mathsf{prox}}_{\gamma
g}\left({\bm{x}}^{k-1}-\gamma\tau{\mathsf{\widehat{R}}}({\bm{x}}^{k-1})\right)$
and ${\overline{\bm{x}}}^{k}={\mathsf{prox}}_{\gamma
g}\left({\bm{x}}^{k-1}-\gamma\tau{\mathsf{R}}({\bm{x}}^{k-1})\right)$,
respectively. From the optimality conditions for the update rule of ideal
operator, we have

$\frac{1}{\gamma}({\overline{\bm{x}}}^{k}-{\bm{x}}^{k-1})+\nabla
h({\bm{x}}^{k-1})+\nabla g({\overline{\bm{x}}}^{k})=0.$

By using this equation, the gradient of the objective function $f$ in (2) can
be written as

$\displaystyle\|\nabla f({\bm{x}}^{k})\|_{2}$ $\displaystyle=\|\nabla
g({\bm{x}}^{k})+\nabla h({\bm{x}}^{k})\|_{2}=\|\nabla g({\bm{x}}^{k})-\nabla
g({\overline{\bm{x}}}^{k})+\nabla h({\bm{x}}^{k})-\nabla
h({\bm{x}}^{k-1})+\frac{1}{\gamma}({\bm{x}}^{k-1}-{\bm{x}}^{k})+\frac{1}{\gamma}({\bm{x}}^{k}-{\overline{\bm{x}}}^{k})\|_{2}$
$\displaystyle\leq\left(L+\frac{1}{\gamma}\right)\alpha\varepsilon_{k-1}+\left(M+\frac{1}{\gamma}\right)\|{\bm{x}}^{k}-{\bm{x}}^{k-1}\|_{2},$

where we used Lemma 2, $L$-Lipschitz continuity of $\nabla g$, and
$M$-Lipschitz continuity of $\nabla h$ from Assumptions 3. By squaring both
sides and using $(a+b)^{2}\leq 2a^{2}+2b^{2}$, we have

$\|\nabla f({\bm{x}}^{k})\|^{2}_{2}\leq
A_{1}\|{\bm{x}}^{k}-{\bm{x}}^{k-1}\|_{2}^{2}+A_{2}\varepsilon^{2}_{k-1},$

where $A_{1}\,\coloneqq\,2(M+1/\gamma)^{2}$ and
$A_{2}\,\coloneqq\,2\alpha^{2}(L+1/\gamma)^{2}$. By using the result from
Lemma 1 and averaging both sides of the bound over $t\geq 1$, we get the
desired result

$\mathop{\mathsf{min}}_{1\leq k\leq t}\|\nabla
f({\bm{x}}^{k})\|^{2}_{2}\leq\frac{1}{t}\sum_{k=1}^{t}\|\nabla
f({\bm{x}}^{k})\|^{2}_{2}\leq\frac{B_{1}}{t}\left(f({\bm{x}}^{0})-f({\bm{x}}^{t})\right)+\frac{B_{2}}{t}\sum_{k=1}^{t}\varepsilon^{2}_{k-1}\leq\frac{B_{1}}{t}\left(f({\bm{x}}^{0})-f({\bm{x}}^{*})\right)+B_{2}{\overline{\varepsilon}}_{t}^{2}$

where we used the fact that $f({\bm{x}}^{t})\geq f({\bm{x}^{\ast}})$ from
Assumption 4, and $B_{1}\,\coloneqq\,4(1+\gamma M)^{2}/(\gamma(1-\gamma M))$,
$B_{2}\,\coloneqq\,\lambda\alpha^{2}B_{1}/2+2\alpha^{2}(1+\gamma
L)^{2}/\gamma^{2}$, and
${\overline{\varepsilon}}_{t}^{2}\,\coloneqq\,(1/t)\sum_{k=1}^{t}\varepsilon^{2}_{k-1}$.
∎

Remark 1. If the sequence of error terms $\\{\varepsilon_{i}\\}_{i\geq 0}$ is
square-summable, we have ${\overline{\varepsilon}}_{t}^{2}\to 0$ as
$t\to\infty$. Consequently, $\|\nabla f({\bm{x}}^{t})\|\to 0$ as $t\to\infty$.

######  Lemma 1.

Run PR-SANS with adapted restoration operator
${\mathsf{\widehat{D}}}_{\sigma}$ for $k\geq 1$ iterations under Assumptions
1-4 using a step-size $0<\gamma\leq\mathop{\mathsf{min}}(1/M,1/L)$. Then we
have

$f({\bm{x}}^{k})\leq f({\bm{x}}^{k-1})-\frac{1-\gamma
M}{2\gamma}\|{\bm{x}}^{k}-{\bm{x}}^{k-1}\|_{2}^{2}+\frac{\lambda\alpha^{2}\varepsilon^{2}_{k-1}}{2}.$

###### Proof.

Consider the iteration $k\geq 1$ of the algorithm

${\bm{x}}^{k}={\mathsf{prox}}_{\gamma
g}\left({\bm{x}}^{k-1}-\gamma\tau{\mathsf{\widehat{R}}}({\bm{x}}^{k-1})\right)~{}\text{with}~{}{\mathsf{\widehat{R}}}\,\coloneqq\,{\mathsf{I}}-{\mathsf{\widehat{D}}}_{\sigma},$

${\mathsf{\widehat{D}}}_{\sigma}$ is the adapted restoration operator. Since
${\mathsf{D}}_{\sigma}$ is the ideal MMSE restoration operator specified in
(5), ${\overline{\bm{x}}}^{k}={\mathsf{prox}}_{\gamma
g}\left({\bm{x}}^{k-1}-\gamma\tau{\mathsf{R}}({\bm{x}}^{k-1})\right)$
minimizes

$\phi({\bm{x}})\,\coloneqq\,\frac{1}{2\gamma}\|{\bm{x}}-\left({\bm{x}}^{k-1}-\tau\gamma\left({\bm{x}}^{k-1}-{\mathsf{D}}_{\sigma}({\bm{x}}^{k-1})\right)\right)\|_{2}^{2}+g({\bm{x}})=\frac{1}{2\gamma}\|{\bm{x}}-\left({\bm{x}}^{k-1}-\gamma\nabla
h({\bm{x}}^{k-1})\right)\|_{2}^{2}+g({\bm{x}}),$

where we used the result of Lemma 3. From Assumption 3, we know that $\nabla
g$ is $L-$Lipschitz continuous, which implies

$\|\nabla\phi({\bm{v}})-\nabla\phi({\bm{u}})\|_{2}\leq\lambda\|{\bm{v}}-{\bm{u}}\|_{2}~{}\text{with}~{}\lambda\,\coloneqq\,\frac{1}{\gamma}+L.$

By using the fact that $\phi$ is a smooth function from Assumption 3 and since
${\overline{\bm{x}}}^{k}$ minimize it, we have

$\displaystyle\phi({\bm{x}}^{k})$
$\displaystyle\leq\phi({\overline{\bm{x}}}^{k})+\nabla\phi({\overline{\bm{x}}}^{k})^{\mathsf{T}}({\bm{x}}^{k}-{\overline{\bm{x}}}^{k})+\frac{\lambda}{2}\|{\bm{x}}^{k}-{\overline{\bm{x}}}^{k}\|_{2}^{2}\leq\phi({\overline{\bm{x}}}^{k})+\frac{\lambda\alpha^{2}\varepsilon^{2}_{k-1}}{2}$
$\displaystyle\leq\mathop{\mathsf{min}}_{{\bm{x}}}\left\\{\frac{1}{2\gamma}\|{\bm{x}}-\left({\bm{x}}^{k-1}-\gamma\nabla
h({\bm{x}}^{k-1})\right)\|_{2}^{2}+g({\bm{x}})\right\\}+\frac{\lambda\alpha^{2}\varepsilon^{2}_{k-1}}{2}$
$\displaystyle\leq\frac{1}{2\gamma}\|{\bm{x}}^{k-1}-\left({\bm{x}}^{k-1}-\gamma\nabla
h({\bm{x}}^{k-1})\right)\|_{2}^{2}+g({\bm{x}}^{k-1})+\frac{\lambda\alpha^{2}\varepsilon^{2}_{k-1}}{2},$

where we used the result of Lemma 2 in the first inequality. By expanding the
first term on the left side of the inequality and simplifying it, we have

$g({\bm{x}}^{k})\leq g({\bm{x}}^{k-1})-\nabla
h({\bm{x}}^{k-1})^{\mathsf{T}}({\bm{x}}^{k}-{\bm{x}}^{k-1})-\frac{1}{2\gamma}\|{\bm{x}}^{k}-{\bm{x}}^{k-1}\|_{2}^{2}+\frac{\lambda\alpha^{2}\varepsilon^{2}_{k-1}}{2}.$

From the $M-$Lipschitz continuity of $\nabla h$, we have

$h({\bm{x}}^{k})\leq h({\bm{x}}^{k-1})+\nabla
h({\bm{x}}^{k-1})^{\mathsf{T}}({\bm{x}}^{k}-{\bm{x}}^{k-1})+\frac{M}{2}\|{\bm{x}}^{k}-{\bm{x}}^{k-1}\|_{2}^{2}.$
(10)

By combining (6.2) and (10), we have

$\displaystyle f({\bm{x}}^{k})=g({\bm{x}}^{k})+h({\bm{x}}^{k})$
$\displaystyle\leq g({\bm{x}}^{k-1})+h({\bm{x}}^{k-1})-\frac{1-\gamma
M}{2\gamma}\|{\bm{x}}^{k}-{\bm{x}}^{k-1}\|_{2}^{2}+\frac{\lambda\alpha^{2}\varepsilon^{2}_{k-1}}{2}$
$\displaystyle\leq f({\bm{x}}^{k-1})-\frac{1-\gamma
M}{2\gamma}\|{\bm{x}}^{k}-{\bm{x}}^{k-1}\|_{2}^{2}+\frac{\lambda\alpha^{2}\varepsilon^{2}_{k-1}}{2}.$
(11)

∎

######  Lemma 2.

Run PR-SANS with adapted restoration operator
${\mathsf{\widehat{D}}}_{\sigma}$ for $k\geq 1$ iterations under Assumptions
1-4 using a step-size $0<\gamma\leq\mathop{\mathsf{min}}(1/M,1/L)$. Then we
have

$\|{\bm{x}}^{k}-{\overline{\bm{x}}}^{k}\|_{2}\leq\alpha\varepsilon_{k-1},$

where $\alpha\,\coloneqq\,\gamma\tau/(1-\gamma\tau)$ is a positive constant.

###### Proof.

By using the ideal ${\mathsf{D}}_{\sigma}$ and adapted
${\widehat{\mathsf{D}}}_{\sigma}$ restoration operators, we have the following
expressions:

${\overline{\bm{x}}}^{k}={\mathsf{prox}}_{\gamma
g}({\bm{x}}^{k-1}-\gamma\tau{\mathsf{R}}({\bm{x}}^{k-1}))\quad\text{and}\quad{\bm{x}}^{k}={\mathsf{prox}}_{\gamma
g}({\bm{x}}^{k-1}-\gamma\tau{\mathsf{\widehat{R}}}({\bm{x}}^{k-1})).$

By using these expressions, the optimality conditions for proximal operators
and Assumption 2, we have

$\|{\bm{x}}^{k}-{\overline{\bm{x}}}^{k}+\gamma(\nabla g({\bm{x}}^{k})-\nabla
g({\overline{\bm{x}}}^{k}))\|_{2}=\gamma\tau\|{\mathsf{R}}({\bm{x}}^{k-1})-{\mathsf{\widehat{R}}}({\bm{x}}^{k-1})\|_{2}=\gamma\tau\|{\mathsf{D}}_{\sigma}({\bm{x}}^{k-1})-{\widehat{\mathsf{D}}}_{\sigma}({\bm{x}}^{k-1})\|_{2}\leq\gamma\tau\varepsilon_{k-1}.$
(12)

By this equality and $\|a\|_{2}-\|b\|_{2}\leq\|a+b\|_{2}$ and $L$-Lipschitz
continuity of $\nabla g$, we have

$\|{\bm{x}}^{k}-{\overline{\bm{x}}}^{k}\|_{2}-\gamma\|(\nabla
g({\bm{x}}^{k})-\nabla g({\overline{\bm{x}}}^{k}))\|_{2}\geq(1-\gamma
L)\|{\bm{x}}^{k}-{\overline{\bm{x}}}^{k}\|_{2}.$ (13)

From equations (12) and (13) and the fact that $\gamma L\leq 1$, we have

$\|{\bm{x}}^{k}-{\overline{\bm{x}}}^{k}\|_{2}\leq\alpha\varepsilon_{k-1},$

where $\alpha\,\coloneqq\,\gamma\tau/(1-\gamma L)$. ∎

######  Lemma 3.

Let ${\mathsf{D}}_{\sigma}$ be the ideal MMSE restoration operator in 5
corresponding to restoration problem 4 under Assumption 1 and 3. Then we have

$\nabla_{\bm{z}}h({\bm{z}})=\tau({\bm{z}}-{\mathsf{D}}_{\sigma}({\bm{z}})),$

where $h$ is the regularization term in 2.

###### Proof.

The density $p_{{\bm{z}}}$ is defined as

$p_{\bm{z}}({\bm{z}})=\int
p_{{\bm{z}}|{\bm{x}}}({\bm{z}};{\bm{x}})p_{\bm{x}}({\bm{x}}){\mathsf{d}}{\bm{x}}=\int
G_{\sigma}({\bm{z}}-{\bm{x}})p_{\bm{x}}({\bm{x}}){\mathsf{d}}{\bm{x}},$

where $G_{\sigma}$ denotes the Gaussian density function with the standard
deviation $\sigma>0$. The gradient of $h({\bm{z}})=-\tau\sigma^{2}\log
p_{{\bm{z}}}({\bm{z}})$ can be expressed as

$\nabla_{\bm{z}}h({\bm{z}})=\frac{-\tau\sigma^{2}\nabla_{\bm{z}}p_{\bm{z}}({\bm{z}})}{p_{\bm{z}}({\bm{z}})}=\frac{\tau\int({\bm{z}}-{\bm{x}})G_{\sigma}({\bm{z}}-{\bm{x}})p_{\bm{x}}({\bm{x}}){\mathsf{d}}{\bm{x}}}{p_{\bm{z}}({\bm{z}})}=\tau({\bm{z}}-{\mathsf{D}}_{\sigma}({\bm{z}})).$

where we used (9), (6.2),and (5) in the first, second and, last equality,
respectively. ∎

Remark 2. For the fixed points of PR-SANS algorithm using ideal operator
${\mathsf{D}}_{\sigma}$, we have

${\bm{x}^{\ast}}={\mathsf{prox}}_{\gamma
g}({\bm{x}^{\ast}}-\gamma\tau{\mathsf{R}}({\bm{x}^{\ast}})).$

By using the results from Lemma 3 and the optimality condition for proximal
operators, we have

$\frac{1}{\gamma}\left({\bm{x}^{\ast}}-{\bm{x}^{\ast}}+\gamma\tau{\mathsf{R}}({\bm{x}^{\ast}})\right)+\nabla
g({\bm{x}^{\ast}})=\nabla
g({\bm{x}^{\ast}})+\tau{\mathsf{R}}({\bm{x}^{\ast}})=\nabla
g({\bm{x}^{\ast}})+\nabla h({\bm{x}^{\ast}})=\bm{0},$

where ${\mathsf{R}}={\mathsf{I}}-{\mathsf{D}}$. The result states that any
fixed-point ${\bm{x}^{\ast}}\in\mathbb{R}^{n}$ of the PR-SANS with ideal
restoration operator is indeed a stationary point of objective function $f$.

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