# DEEP DENOISING PRIOR-BASED SPECTRAL ESTIMATION FOR PHASELESS SYNTHETIC
APERTURE RADAR

###### Abstract

Incoherent processing for synthetic aperture radar (SAR) is a promising
approach that enables low implementation costs, simplified hardware designs
and operations in high frequency spectrum compared to the conventional imaging
methods using coherent processing. Existing non-convex phaseless imaging
algorithms offer recovery guarantees over limited range of forward models. In
recent years, several deep learning (DL) based techniques have been introduced
with the goal of extending applicability of phaseless imaging techniques to
wave-based imaging modalities by addressing fundamental challenges, such as,
lack of redundancy, non-uniqueness issues encountered commonly with inverse
scattering models. In this paper, we introduce a DL-based phaseless SAR
imaging approach that is designed under the premise that the spectral
estimation technique, widely used for initializing non-convex phase retrieval
algorithms, has significance far beyond generating good initial points. We
extend the iterative power method for spectral estimation by using deep
denoisers at each iteration, and subsequently design a deep imaging network
within the plug-and-play framework. Finally, we verify the feasibility of our
approach using synthetic SAR measurements.

Index Terms—  Synthetic aperture radar, phase retrieval, phaseless imaging,
deep learning

##  1 Introduction

Imaging an unknown quantity of interest from the magnitude or intensity values
of its linear mapping has gained significant interest in applications to wave-
based imaging using electromagnetic and acoustic waves [1, 2]. However, the
deployment of phaseless synthetic aperture radar (SAR) has been quite limited
in practice. For SAR imaging at high frequencies, it is challenging to
accurately measure the phase information [3]. Phase errors are commonplace due
to the presence of phase modulator noise, and temperature variations [4].
Moreover, while imaging using wide apertures, maintaining coherence across
phase measurements becomes demanding, requiring complex and sophisticated
receiver designs. Additionally, random fluctuation in the transmission medium
properties and uncompensated platform trajectory deviations also lead to phase
errors by giving rise to inaccurate model assumptions. Phaseless SAR, on the
other hand, only measures the magnitude or intensity values of back-scattered
waves, hence enables simple hardware designs and provides robustness to phase
errors that can occur during data acquisition.

Despite potential benefits, phaseless SAR imaging encounters a number of key
challenges in designing algorithms with recovery guarantees. The quadratic
nature of the problem implies that the true unknown is only recoverable up to
certain trivial ambiguities [5, 6, 7]. Moreover, for exact recovery, the loss
of phase information needs to be compensated either during the acquisition
phase via hardware modifications [3] or during imaging by advanced algorithms
[8, 9, 10]. Acquisition level compensations lead to using high number of
measurements, and designing optimal illumination strategies [11, 8, 3].
However, collecting prohibitively large number of measurements from spatially
distributed transceivers as well as manipulating sampling vectors via hardware
modifications are not desirable in practice. In this paper, we instead adopt
an algorithmic approach by utilizing prior information about the target class
of interest. Exact recovery guarantees of non-convex phase retrieval
algorithms [5, 12] rely on good initial estimates that reside within small and
locally strictly convex neighborhoods of the unknown of interest. This can be
accomplished under certain conditions by spectral estimation methods [13],
which use the leading eigenvector of back-projected intensity measurements, as
well as its variants [12, 9]. Sampling vectors for SAR typically do not
possess orthogonality or desirable statistical properties, such that, the
back-projected intensity measurements are sufficiently accurate with respect
to the underlying Kronecker scene [14], especially at low number of
measurements to number of unknown ratios.

In recent years, deep learning (DL) based phaseless imaging methods that
incorporate prior information with first order iterative algorithms [15, 16]
and with the power method for spectral estimation [17] have been introduced. A
challenge for phaseless SAR is the non-uniqueness issue in the far field due
to the translation invariance of the associated forward maps [18]. These DL
based methods, which implement generative, decoding or denoising type priors
[15, 16], restrict the estimated images to specific manifolds learned from
training data. These priors are particularly suitable for phaseless SAR as the
learned image manifolds can be optimized to eliminate undesirable translation
replicas. In [17], prior information is imposed while learning the leading
eigenvector of the back-projected intensity measurements by projecting onto
the range of a deep variational auto-encoder. However, theoretical analysis
relies on i.i.d. Gaussian distributed sampling vectors that are unsuitable for
SAR. In [19], we introduced a deep denoising prior-based spectral estimation
approach for SAR interferometric imaging problem using the plug-and-play (PnP)
[20] and regularization by denoising (RED) [21] frameworks. We designed end-
to-end imaging networks via unrolling [22], and hence enabled a supervised
denoiser training scheme that uses both the ground truth sample images and the
corresponding cross-correlated measurements to attain model aware denoising
operators.

In this paper, we present a PnP and spectral estimation based imaging method
for phaseless SAR by extending our earlier approach in [19]. Our unrolled
imaging network and the subsequent supervised denoiser training scheme is
motivated by analyzing the role that an optimal regularization term associated
with the denoiser should play towards attaining improved reconstructions
compared to the spectral estimation method. We verified the feasibility of our
proposed approach by using simulated SAR phaseless measurements and compared
performance with the Wirtinger flow (WF) algorithm for phaseless imaging [5,
7].

##  2 Problem Statement

Let $\mathbf{x}=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}$ and
$\bm{x}=(x_{1},x_{2})\in\mathbb{R}^{2}$ be the location variables in $3$-D and
$2$-D Cartesian coordinate systems, respectively. For ground topography
$\phi:\mathbb{R}^{2}\mapsto\mathbbm{R}$, location $\mathbf{x}$ on the ground
surface is represented as $(\bm{x},\phi(\bm{x}))$. We denote the temporal
frequencies by $\omega$ and the slow-time points by $s$. The transmitting and
receiving sensor locations at the various slow-time points are
$\bm{\gamma}_{T}(s)\in\mathbb{R}^{3}$ and
$\bm{\gamma}_{R}(s)\in\mathbb{R}^{3}$, respectively. We assume a homogeneous
background medium with electromagnetic wave speed $c_{0}$. Let
$\rho:\mathbb{R}^{2}\mapsto\mathbbm{R}$ denote the unknown reflectivity
function for the points on the ground topography . We further assume that the
target dimensions, scene reflectivities, and the transmission bandwidths are
such that, the Born approximation related model simplification is valid. Then,
under the start-stop signal model, SAR received signal $f(\omega,s)$ relates
to $\rho$ as follows:

$\displaystyle f(\omega,s)$
$\displaystyle=\int\exp\left(-\frac{\text{i}}{c_{0}}\omega\phi(s,\bm{x})\right)A(\omega,s,\bm{x})\rho(\bm{x})d\bm{x}.$
(1)

Phase term $\phi(s,\bm{x})$ refers to the total distance
$|\bm{\gamma}_{T}(s)-\bm{x}|+|\bm{x}-\gamma_{R}(s)|$ travelled by the
reflected signal at slow-time $s$. Amplitude $A(\omega,s,\bm{x})$ relates to
the fast-time frequencies, beam-patterns of the transmitting and receiving
antennas, and geometric spreading factor [14], and it is assumed to be
constant-valued here for simplification.

For phaseless SAR imaging, only the intensity value of $f(\omega,s)$ is
relevant which we denote by $d(\omega,s)$. From (1),

$\displaystyle d(\omega,s)$
$\displaystyle=\int\text{e}^{-\frac{\text{i}}{c_{0}}\omega(\phi(s,\bm{x})-\phi(s,\bm{x}^{\prime}))}\rho(\bm{x})\overline{\rho(\bm{x}^{\prime})}d\bm{x}d\bm{x}^{\prime}.$
(2)

Phase term $\phi(s,\bm{x})-\phi(s,\bm{x}^{\prime})$ can be simplified by using
the small scene and far field assumptions and the Taylor series expansion,
which leads to

$\displaystyle d(\omega,s)$
$\displaystyle=\int\text{e}^{-\frac{\text{i}\omega}{c_{0}}(\hat{\bm{\gamma}}_{T}(s)+\hat{\bm{\gamma}}_{R}(s)).(\bm{x}^{\prime}-\bm{x})}\rho(\bm{x})\overline{\rho(\bm{x}^{\prime})}d\bm{x}d\bm{x}^{\prime}.$
(3)

$\hat{(.)}$ notation over a vector is used to represent the unit vector along
its direction.

In this paper, we propose an algorithm for recovering the unknown ground
reflectivities at a finite number of locations. To this end, we discretize the
scene into $N$ locations at $\\{\mathbf{x}_{n}\\}_{n=1}^{N}$, and define a
corresponding unknown or ground truth image vector
$\bm{\rho}^{*}\in\mathbb{C}^{N}$ whose $n^{th}$ component is calculated as
$\rho(\mathbf{x}_{n})$. We also assume that $M$ total measurements are
collected by sampling over the transmission bandwidth using $K$ equi-distant
points, $\\{\omega_{k}\\}_{k=1}^{K}$, at each of the $S$ slow time samples,
i.e., $M=SK$. Let $\mathbf{P}^{*}$ be the Kronecker scene for image vector
$\bm{\rho}^{*}$, i.e., $\mathbf{P}^{*}=\bm{\rho}^{*}{\bm{\rho}^{*}}^{H}$, and
$\mathbf{a}_{m}\in\mathbb{C}^{N}$ be the $m^{th}$ sampling vector whose
$n^{th}$ element $\mathbf{a}_{m}(n)$ is calculated as
$\text{e}^{\frac{-\text{i}\omega}{c_{0}}(\hat{\bm{\gamma}}_{T}(s)+\hat{\bm{\gamma}}_{R}(s)).\bm{x}_{n}}$.
We represent the $m^{th}$ intensity measurement by $d_{m}$, where
$m=(s-1)K+k$, $s\in[S]$ and $k\in[K]$. Then, from (3), we observe that $d_{m}$
can be expressed as $\mathbf{a}^{H}_{m}\mathbf{P}^{*}\mathbf{a}_{m}$. Let
$\mathbf{d}\in\mathbb{R}^{M}_{+}$ be the intensity measurement vector composed
of $\\{d_{m}\\}_{m=1}^{M}$, and $\mathcal{F}:\mathbb{C}^{N\times
N}\mapsto\mathbb{C}^{M}$ be a known lifted forward map such that

$\displaystyle\mathbf{d}$ $\displaystyle=\mathcal{F}(\mathbf{P}^{*}).$ (4)

Our objective is to estimate the unknown image $\bm{\rho}^{*}$, up to a global
phase factor, by using the known values of $\mathbf{d}$ and $\mathcal{F}$.

##  3 Proposed Approach

###  3.1 Spectral Estimation

Non-convex phaseless imaging methods employ spectral estimation or its
variants for initialization, and estimate directly at the signal level,
instead of calculating a lifted domain reconstruction from the linear data
model in (4). These methods are commonly equipped with theoretical recovery
guarantees for sampling vectors with ideal probability distributions [5, 12],
as well as for limited deterministic cases [7]. Spectral estimation sets the
initial image $\bm{\rho}_{0}\in\mathbb{C}^{N}$ as
$\sqrt{\lambda_{0}}\mathbf{u}_{1}$, where
$\lambda_{0}=\frac{1}{\sqrt{2M}}\|\mathbf{d}\|$ [7], and
$\mathbf{u}_{1}\in\mathbb{C}^{N}$ represents the leading eigenvector of the
spectral matrix $\hat{\mathbf{X}}\in\mathbb{C}^{N\times N}$, which is
calculated as

$\displaystyle\hat{\mathbf{X}}$
$\displaystyle=\frac{1}{M}\mathcal{F}^{H}\mathbf{d}.$ (5)

For phaseless SAR at low $M/N$ ratios, $\frac{1}{M}\mathcal{F}^{H}\mathcal{F}$
may deviate significantly from the ideal identity operator. To this end, we
observe from our approach in [19] that by applying a set of optimally learned
denoising operators, while estimating the leading eigenvector of the spectral
matrix using the power method, faster convergence rate and improved
reconstruction qualities are attained compared to using hand-crafted priors,
as well as when spectral estimation is followed by gradient descent updates
for further refinements.

###  3.2 Deep Network for Phaseless SAR imaging

Fig. 1: Imaging network for phaseless SAR.

In this paper, we extend our approach in [19] to the phaseless SAR imaging
problem. Unlike the generative prior based techniques in [15, 16], our
proposed DL-based method estimates the unknown image directly, instead of
determining its encoded representation, by modifying the manifold over which
the leading eigenvector of $\hat{\mathbf{X}}$ is searched for by using learned
prior information. This is accomplished by applying a denoiser after each
power method update to restrict the estimated image to the range of the
denoise similar to the PnP framework, and we model the denoiser by a deep
network (DN) with trainable parameters.

Our proposed algorithm estimates the $\ell_{2}$ normalize ground truth image
vector, which we denote by $\bm{\rho}^{*}_{n}\in\mathbb{C}^{N}$, i.e.,
$\bm{\rho}^{*}_{n}=\bm{\rho}^{*}/\|\bm{\rho}^{*}\|$. We represent the
estimated quantity by $\hat{\bm{\rho}}_{n}\in\mathbb{C}^{N}$. The algorithm
starts from a fixed initial vector $\bm{\rho}_{0}\in\mathbb{C}^{N}$, which is
followed by iterative update steps. At the $l^{th}$ iteration, the updated
image is calculated as follows:

$\displaystyle\mathbf{w}_{l}$ $\displaystyle=\hat{\mathbf{X}}\bm{\rho}_{l-1},$
(6) $\displaystyle\mathbf{z}_{l}$
$\displaystyle=\mathcal{D}_{l}(\mathbf{w}_{l}),$ (7)
$\displaystyle\bm{\rho}_{l}$
$\displaystyle=\mathbf{z}_{l}/\|\mathbf{z}_{l}\|.$ (8)

$\mathcal{D}_{l}:\mathbb{C}^{N}\mapsto\mathbb{C}^{N}$ represents the denoising
operator applied at the $l^{th}$ iteration. When same denoising operator,
$\mathcal{D}:\mathbb{C}^{N}\mapsto\mathbb{C}^{N}$, is applied at each update,
we can interpret it as the proximal operator associated with an implicit
regularization term $\mathcal{R}:\mathbb{C}^{N}\mapsto\mathbb{C}^{N}$, i.e.,

$\displaystyle\mathcal{D}(\mathbf{w}_{l})=\operatorname*{argmin}_{\mathbf{z}\in\mathbb{C}^{N}}\|\mathbf{z}-\mathbf{w}_{l}\|^{2}+\mathcal{R}(\mathbf{z}).$
(9)

PnP framework directly applies $\mathcal{D}$ without explicitly determining a
corresponding $\mathcal{R}$ term [20]. DL-based implementations model
$\mathcal{D}$ by a DN, whose optimal parameter values are learned using a set
of noisy training images and the corresponding ground truth noise-free ones
without considering the data model for the inversion task at hand. When
applied to ill-posed inversion tasks, such as, imaging for phaseless SAR under
low sample complexities, such model independent denoiser designs may not be
sufficient to generate good reconstructions.

To illuminate this point further in the context of the spectral estimation
task for phaseless SAR, we next state the optimization problem that our
algorithm in (6) to (8) addresses when $\mathcal{D}_{l}$ is replaced by
$\mathcal{D}$:

$\displaystyle\hat{\bm{\rho}}_{n}$
$\displaystyle=\operatorname*{argmin}_{\bm{\rho}\in\mathbb{C}^{N}}\mathcal{J}_{S}(\bm{\rho})+\mathcal{I}_{n}(\bm{\rho})+\mathcal{R}(\bm{\rho}).$
(10)

$\mathcal{I}_{n}$ denotes an indicator whose value is large if the $\ell_{2}$
norm of its argument is not $1$. Data fidelity term
$\mathcal{J}_{S}(\bm{\rho})$ is defined as follows:

$\displaystyle\mathcal{J}_{S}(\bm{\rho})$
$\displaystyle=-\bm{\rho}^{H}\hat{\mathbf{X}}\bm{\rho}+\|\bm{\rho}\|^{2}.$
(11)

By expanding $\mathcal{J}_{S}$, it is straight-forward to verify that

$\displaystyle\mathcal{J}_{S}(\bm{\rho})=-|\bm{\rho}^{H}\bm{\rho}^{*}|^{2}+\|\bm{\rho}\|^{2}-\bm{\rho}^{H}\delta(\bm{\rho}^{*}{\bm{\rho}^{*}}^{H})\bm{\rho},$
(12)

where

$\displaystyle\delta=\frac{1}{M}\mathcal{F}^{H}\mathcal{F}-\mathcal{I}.$ (13)

This indicates that an ideal regularization term, imposed indirectly by the
denoiser, should reduce the contribution of the undesirable third term in (12)
for all possible $\bm{\rho}^{*}$ from the image class of interest. This term
captures the perturbation present in the spectral matrix from the ideal one,
$\mathbf{P}^{*}$, due to the deviation of
$\frac{1}{M}\mathcal{F}^{H}\mathcal{F}$ operator from identity. For SAR
phaseless imaging with limited number of slow-time points, this perturbation
term can become significant, and this explains the resulting poor spectral
estimation quality when no prior information is utilized.

The dependence of the ideal regularization term, i.e.
$\bm{\rho}^{H}\delta(\bm{\rho}^{*}{\bm{\rho}^{*}}^{H})\bm{\rho}$, on the
unknown ground truth images makes it challenging to apply it directly by
deriving an associated proximal operator. Instead we implement it indirectly
by using DL-based modelling of the denoisers at the various iterations of our
algorithm. Additionally, to facilitate model-aware denoiser training, we
implement the unrolling technique [22]. Unrolling refers to the mapping of $L$
iterations of an algorithm into $L$ stages of a DN. The estimated image
$\hat{\bm{\rho}}_{n}$ is then set to $\bm{\rho}_{L}$. This results in an end-
to-end imaging network with input $\mathbf{d}$ and output $\bm{\rho}_{L}$, and
hence it can be trained in a supervised manner by using a set of ground truth
training images $\\{\bm{\rho}^{*}_{t}\\}_{t=1}^{T}$ and the associated
intensity measurement vector $\\{\mathbf{d}_{t}\\}_{t=1}^{T}$. Our overall
phaseless SAR imaging network is shown in Fig. 1.

##  4 Numerical Simulations

Fig. 2: Ground truth image with a rectangular object.

(b) WF (5dB)

(c) Proposed approach (5dB)

(b) WF (10dB)

(c) Proposed approach (10dB)

Fig. 3: Reconstructed images using the WF algorithm with $150$ iterations and
the proposed approach for intensity measurements with $5$ dB and $10$ dB
additive noise.

In this section, we verified the feasibility of our approach by using
simulated SAR scenes and intensity measurements. We considered a dataset of
images, each with a single extended and randomly located rectangular target of
random dimensions. Our training set is composed of $5000$ samples, each having
the ground truth image vector and the corresponding SAR intensity measurements
available, while the test set contains $50$ samples, each having only the
intensity measurements information. For each sample, we assume that the area
within the field of view of the SAR, that contains the rectangular target, has
dimensions $62$ m $\times 62$ m. The SAR is mono-static and operates in the
spot-light mode with its moving platform having a circular trajectory of
radius $10$ km, aperture length $\pi$ radian and an altitude of $7$ km. The
central frequency and the transmission bandwidth are $9.9$ GHz and $75$ MHz,
respectively. We reconstruct the scenes from each set of measurements as
$31\times 31$ pixels images. The $M/N$ ratio used is $2$ with the number of
slow time points, distributed at equi-distant locations over the aperture,
being $62$ and the number of frequency samples at each slow time point being
$31$. We additionally considered that additive Gaussian distributed noise of
$5$ dB and $10$ dB are present on the measured intensity values of the
scattered signals.

We implemented the denoisers by using convolutional neural networks (CNN) with
$16$ convolution layers and $\text{relu}(.)$ type nonlinear activation
functions. We used $4$ iterations for constructing our unrolled imaging
network from Fig. 1. Also, to limit the total number of trainable parameters,
we gradually increased this value and used the same set of denoiser parameter
values for two consecutive layers. For the example ground truth image in Fig.
2, the reconstructed images are shown in Fig. 3 using our proposed approach,
along with the ones obtained using the WF algorithm [5, 7] for measurements
with two different noise levels. For the given $M/N$ ratio, our approach out-
performs the WF algorithm in terms of mean squared error (MSE) metric at both
SNR levels. Average computation times per test sample using our approach and
the WF algorithm to attain these MSE levels are approximately $0.004$ seconds
and $21.5$ seconds, respectively, indicating an improvement in computation
time by our approach.

##  5 Conclusion

In this paper, we presented a DL-based SAR phaseless imaging method by
modifying the spectral estimation approach using learned prior information.
Our approach is empirically observed to out-perform the iterative non-DL based
WF algorithm using a simulated SAR dataset. Theoretically studying sufficient
conditions on the forward map and the denoiser architecture for guaranteeing
exact recovery, as well as, its implementation using complex scenes and real
SAR measurements are two important open problems that we leave for future
work.

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