# Enhanced Radar Imaging in Uncertain Environment:  A Descriptive Experiment Design Regularization Approach

Authors: Yuriy Shkvarko; Hector Perez-Meana; Alejandro Castillo-Atoche
Journal: International Journal of Navigation and Observation
 (2008)

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## Abstract

A new robust technique for high-resolution reconstructive imaging is developed as required for enhanced remote sensing (RS) with imaging array radar or/and synthetic aperture radar (SAR) operating in an uncertain RS environment. The operational scenario uncertainties are associated with the unknown statistics of perturbations of the signal formation operator (SFO) in turbulent medium, imperfect array calibration, finite dimensionality of measurements, uncontrolled antenna vibrations, and random carrier trajectory deviations in the case of SAR. We propose new descriptive experiment design regularization (DEDR) approach to treat the uncertain radar image enhancement/reconstruction problems. The proposed DEDR incorporates into the minimum risk (MR) nonparametric estimation strategy the experiment design-motivated operational constraints algorithmically coupled with the worst-case statistical performance (WCSP) optimization-based regularization. The MR objective functional is constrained by the WCSP information, and the robust DEDR image reconstruction operator applicable to the scenarios with the low-rank uncertain estimated data correlation matrices is found. We report and discuss some simulation results related to enhancement of the uncertain SAR imagery indicative of the significantly increased performance efficiency gained with the developed approach.


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## Body

## 1. Introduction

Modern applied theory of reconstructive
radar imaging is now a mature and well-developed research field, presented and
detailed in many works (see, e.g., [1, 2] and references therein).
The classical imaging with array radar or SAR implies application of a method
called “matched spatial filtering” to process the recorded data signals [1, 3, 4]. Stated formally [1], such a method implies application of the adjoint signal
formation operator (SFO) to the recorded data, squared detection of the  filter outputs, and their averaging over the
actually recorded samples (the so-called snapshots [5]) of the independent data
observations. Although a number of authors have proposed different linear and
nonlinear postprocessing approaches to enhance the images formed using such
matched estimator (see, e.g., [3, 5–8]), all those are not a direct inference
from the Bayesian statistically optimal estimation theory [4]. Other approaches
had focused primarily on designing the constrained regularization techniques
for improving the resolution of the closely spaced components in the power spatial spectrum pattern (SSP) obtained by ways different from the matched spatial filtering [9–12], but again without aggregating the regularization principles with the minimum
risk estimation strategy. Although the existing theory offers a manifold of statistical
and descriptive regularization techniques for reconstructive imaging, in many
application areas there still remain some unresolved crucial theoretical and
processing problems related to large scale sensor array radar/SAR reconstructive imaging in uncertain operational scenarios.The predominant challenge of this study is
to solve the SSP reconstruction problem in context of the uncertain environment. Thus, the problem of enhanced imaging of the extended large-scale scenes
remotely sensed with an array radar/SAR operating in theuncertain remote sensing (RS) environment is stated and treated as
an ill-conditioned statistical nonlinear inverse problem. The operational
uncertainties are associated with the unknown statistics of perturbations of
the SFO in the turbulent medium, imperfect array calibration, finite
dimensionality of measurements, uncontrolled antenna vibrations, and random
carrier trajectory deviations in the case of SAR. New descriptive experiment
design regularization (DEDR) approach to radar imaging in the uncertain
environment is addressed to perform the enhanced reconstruction of the power spatial
spectrum pattern (SSP) of the scattered wavefield from the uncertain data
measurements. The proposed DEDR incorporates into the minimum risk (MR)
nonparametric estimation strategy the DEDR-motivated constraints of the
observability of the initial scene scattering wavefield algorithmically coupled
with the worst-case statistical performance (WCSP) optimization-based
regularization. The MR objective function is constrained by the WCSP
information, and the DEDR technique for robust image reconstruction applicable
to the scenarios with the low-rank uncertain estimated data correlation
matrices is found. Pursuing such an
approach, we establish a family of the robust DEDR-related estimators that
encompass a manifold of the imaging techniques ranging from traditional array
matched spatial filtering to new DEDR-related robust adaptive array
beamforming. We also present the robust DEDR-related imaging algorithms
that manifest enhanced resolution of the reconstructed array images with
substantially decreased computational load. The efficiency of two general DEDR-related
algorithms (the robust spatial filtering (RSF) algorithm and the robust adaptive
spatial filtering (RASF) algorithm) is illustrated through computer simulations
of reconstructing the digital images provided with the SAR operating in some
typical uncertain remote sensing scenarios.


## 2. Descriptive Experiment Design Regularization Formalism

### 2.1.
Problem Model

Consider a coherent RS experiment in a
random medium and the narrowband assumption [1, 4, 6] that enables us to
model the extended object backscattered field by imposing its time invariant
complex scattering (backscattering) function  e(x) in the scene domain (scattering  surface)   X∋x. The measurement data wavefield u(y)=s(y)+n(y) consists
of the echo signals s and additive noise n and is available for observations and
recordings within the prescribed
time-space observation domain Y=T×P,  where y=(t,p)T defines the time-space points in Y.
The model of the observation wavefield u is defined by specifying the stochastic equation of observation
(EO) of an operator form [1, 13]: u=S˜e+n;   e∈E; u,n∈U; S˜:E→U, in
the Hilbert signal spaces E and  U
with the metrics structures induced by the inner products,  [u1,u2]U=∫Yu1(y)u2∗(y)dy and  [e1,e2]E=∫Xe1(x)e2∗(x)dx, respectively. The
operator model of the stochastic EO in the conventional integral form [1, 13]
may be written as (1)u(y)=(S˜e(x))(y)+n(y)=∫XS˜(y,x)e(x)dx+n(y).The random functional kernelS˜(y,x) of the
stochastic integral SFO  S˜ given by (1) defines the signal wavefield
formation model. Its mean, S(y,x)=〈S˜(y,x)〉, is referred
to as the nominal SFO in the RS measurement channel specified by the time-space
modulation of signals employed in a particular radar system [3, 8] and the
variations about the mean   δS(y,x)=S˜(y,x)−S(y,x) model the model uncertainties and
random perturbations of the wavefield at different propagation paths (the
so-called extended Rytov’s model [1]).We assume an
incoherent nature of the backscattered fielde(x). This is naturally inherent to the RS experiments and leads to the δ-form of the object field correlation function, Re(x,x′)=b(x)δ(x−x′),
where e(x) and b(x)=〈|e(x)|2〉
are referred to as the scene random complex scattering function and its average
power scattering function or spatial spectrum pattern (SSP), respectively. The radar
imaging problem is to derive an estimate b^(x) of the SSP (2)b(x)=(ℬe)(x)=Aver(2){e(x)}=〈e(x)e*(x)〉  (referred to as the desired RS image) by processing the available finite
dimensional array radar/SAR measurements of the data wavefield u(y), where ℬ defines the second-order statistical averaging operator.


### 2.2. Projection Formalism for Data Representation

Viewing it as an approximation problem leads one to a projection concept
for a reduction of the data fieldu(y) to the M-D spatial-temporal data recordings: (3)u=vecm{um=[u,hm]U;m=1,…,M} composed of
the expansion/decomposition coefficients {um=[u,hm]U;m=1,…,M}, where {hm(y)}
defines the set of orthogonal normalized basis functions in the M-D data approximation subspace U(M)=PU(M)U [13]. These
are defined via corresponding compositions of the calibrated antenna array
tapering functions and sampling filters that explicitly specify the corresponding
data projection operator PU(M) (see [13–15] for details).In analogy to (3), one can
define now the K-D vector-form
approximation of the scene random scattering function as follows:(4)e=veck{ek=[e,gk]E;k=1,…,K}. The elements of vector (4) are composed of the decomposition
coefficients {ek=[e,gk]E;k=1,…,K} with respect to some chosen
normalized orthogonal set of expansion functions {gk(x)} that
span such K-D signal approximation
subspace E(K)=PE(K)E and specify the corresponding scene wavefield projection
operator PE(K).The descriptive experiment design (DED) aspects of the SSP
reconstruction problem involving the analysis of how to choose the basis
functions{gk(x)} that span the signal representation
subspace E(K)=PE(K)E=Span{gk}
for a given observation subspace U(M)=Span{hm} were investigated
in more details in the previous studies [13, 15]. Following [15], in the rest of this study, we consider the conventional (i.e., ordinary rectangular pixel
format) representation basis over a Kx2×Kx1 regular
pixel-formatted lattice [14, 16], where Kx1 defines the dimension
of the rectangular grid over the horizontal (azimuth) coordinate x1, and Kx2 defines its dimension over the orthogonal (range)
coordinate x2 (the number of the slant range gates projected onto the scene frame). Such
regular lattice of points is next specified by the ordered multi-index k=(kx1,kx2); kx1=1,…,Kx1; kx2=1,…,Kx2; k=1,…,K=Kx1×Kx2.


### 2.3. Uncertain Finite-Dimensional Observations

In the DED
formalism, an imperfect calibration of the array (due to displacements of some
array elements with respect to the presumed nominal positions, as well as
distorted antennas shapes [4, 9]) is attributed to the unknown disturbances {δhm;m=1,…,M} in the decomposition functions {h˜m=hm+δhm} in (3). In imaging SAR applications, such
disturbances incorporate the deviations of a carrier from the nominal
trajectory and antenna vibration [3, 17]. These disturbances and propagation
perturbations result in the uncertain SFO matrix:(5)S˜=S+Δ.In (5), the nominal M×K SFO
matrix S is composed of the elements {Smk=[Sgk,hm]U}, while all problem model uncertainties are
attributed to the distortion term, in which the elements of the uncertainty
matrix Δ are treated as unknown values (realizations of random variables)
with an unknown probability density function (pdf)  p(Δ).


### 2.4. Vector-Form
Equation of Observation

Now, we proceed from the stochastic integral-form EO (1) to its
finite-dimensional approximation (vector) form: (6)u=S˜e+n=Se+Δe+n,in which the disturbed SFO matrix is defined by (5), and e, n, u represent zero-mean vectors
composed of the decomposition coefficients  ek, nm, and um,  respectively. These vectors are characterized by the
correlation matrices: Re=D=D(b)=diag{b} (a diagonal matrix with vector b at
its principal diagonal), Rn, and Ru=〈S˜ReS˜+〉p(Δ)+Rn,
respectively, where 〈⋅〉p(Δ) defines the averaging performed over the
randomness of Δ
characterized by the unknown probability density function p(Δ). Vector b is composed of the elements, bk=ℬ(ek)=〈ekek*〉=〈|ek|2〉; k=1,…,K and is
referred to as a K-D vector-form representation
of the SSP.We refer to the estimate,b^,
as a discrete-form representation of the desired SSP, that is,  the brightness image of the wavefield sources
distributed over the pixel-formatted object scene remotely sensed with an employed
array radar/SAR. Thus, the uncertain SSP reconstruction
problem can be reformulated now as follows: to derive an estimator for
reconstructing the K-D approximation: (7)b^(K)(x)=∑k=1Kb^k|gk(x)|2=gT(x)diag{b^}g(x)of
the SSP distribution in the environment   X∋x. Note that in applications, we employ the ordinary pixel expansion format
[16], while all theoretical results are valid also for any feasible
decomposition function basis,  g(x)=veck{gk(x)},
in (7).


## 2.1.
Problem Model

Consider a coherent RS experiment in a
random medium and the narrowband assumption [1, 4, 6] that enables us to
model the extended object backscattered field by imposing its time invariant
complex scattering (backscattering) function  e(x) in the scene domain (scattering  surface)   X∋x. The measurement data wavefield u(y)=s(y)+n(y) consists
of the echo signals s and additive noise n and is available for observations and
recordings within the prescribed
time-space observation domain Y=T×P,  where y=(t,p)T defines the time-space points in Y.
The model of the observation wavefield u is defined by specifying the stochastic equation of observation
(EO) of an operator form [1, 13]: u=S˜e+n;   e∈E; u,n∈U; S˜:E→U, in
the Hilbert signal spaces E and  U
with the metrics structures induced by the inner products,  [u1,u2]U=∫Yu1(y)u2∗(y)dy and  [e1,e2]E=∫Xe1(x)e2∗(x)dx, respectively. The
operator model of the stochastic EO in the conventional integral form [1, 13]
may be written as (1)u(y)=(S˜e(x))(y)+n(y)=∫XS˜(y,x)e(x)dx+n(y).The random functional kernelS˜(y,x) of the
stochastic integral SFO  S˜ given by (1) defines the signal wavefield
formation model. Its mean, S(y,x)=〈S˜(y,x)〉, is referred
to as the nominal SFO in the RS measurement channel specified by the time-space
modulation of signals employed in a particular radar system [3, 8] and the
variations about the mean   δS(y,x)=S˜(y,x)−S(y,x) model the model uncertainties and
random perturbations of the wavefield at different propagation paths (the
so-called extended Rytov’s model [1]).We assume an
incoherent nature of the backscattered fielde(x). This is naturally inherent to the RS experiments and leads to the δ-form of the object field correlation function, Re(x,x′)=b(x)δ(x−x′),
where e(x) and b(x)=〈|e(x)|2〉
are referred to as the scene random complex scattering function and its average
power scattering function or spatial spectrum pattern (SSP), respectively. The radar
imaging problem is to derive an estimate b^(x) of the SSP (2)b(x)=(ℬe)(x)=Aver(2){e(x)}=〈e(x)e*(x)〉  (referred to as the desired RS image) by processing the available finite
dimensional array radar/SAR measurements of the data wavefield u(y), where ℬ defines the second-order statistical averaging operator.


## 2.2. Projection Formalism for Data Representation

Viewing it as an approximation problem leads one to a projection concept
for a reduction of the data fieldu(y) to the M-D spatial-temporal data recordings: (3)u=vecm{um=[u,hm]U;m=1,…,M} composed of
the expansion/decomposition coefficients {um=[u,hm]U;m=1,…,M}, where {hm(y)}
defines the set of orthogonal normalized basis functions in the M-D data approximation subspace U(M)=PU(M)U [13]. These
are defined via corresponding compositions of the calibrated antenna array
tapering functions and sampling filters that explicitly specify the corresponding
data projection operator PU(M) (see [13–15] for details).In analogy to (3), one can
define now the K-D vector-form
approximation of the scene random scattering function as follows:(4)e=veck{ek=[e,gk]E;k=1,…,K}. The elements of vector (4) are composed of the decomposition
coefficients {ek=[e,gk]E;k=1,…,K} with respect to some chosen
normalized orthogonal set of expansion functions {gk(x)} that
span such K-D signal approximation
subspace E(K)=PE(K)E and specify the corresponding scene wavefield projection
operator PE(K).The descriptive experiment design (DED) aspects of the SSP
reconstruction problem involving the analysis of how to choose the basis
functions{gk(x)} that span the signal representation
subspace E(K)=PE(K)E=Span{gk}
for a given observation subspace U(M)=Span{hm} were investigated
in more details in the previous studies [13, 15]. Following [15], in the rest of this study, we consider the conventional (i.e., ordinary rectangular pixel
format) representation basis over a Kx2×Kx1 regular
pixel-formatted lattice [14, 16], where Kx1 defines the dimension
of the rectangular grid over the horizontal (azimuth) coordinate x1, and Kx2 defines its dimension over the orthogonal (range)
coordinate x2 (the number of the slant range gates projected onto the scene frame). Such
regular lattice of points is next specified by the ordered multi-index k=(kx1,kx2); kx1=1,…,Kx1; kx2=1,…,Kx2; k=1,…,K=Kx1×Kx2.


## 2.3. Uncertain Finite-Dimensional Observations

In the DED
formalism, an imperfect calibration of the array (due to displacements of some
array elements with respect to the presumed nominal positions, as well as
distorted antennas shapes [4, 9]) is attributed to the unknown disturbances {δhm;m=1,…,M} in the decomposition functions {h˜m=hm+δhm} in (3). In imaging SAR applications, such
disturbances incorporate the deviations of a carrier from the nominal
trajectory and antenna vibration [3, 17]. These disturbances and propagation
perturbations result in the uncertain SFO matrix:(5)S˜=S+Δ.In (5), the nominal M×K SFO
matrix S is composed of the elements {Smk=[Sgk,hm]U}, while all problem model uncertainties are
attributed to the distortion term, in which the elements of the uncertainty
matrix Δ are treated as unknown values (realizations of random variables)
with an unknown probability density function (pdf)  p(Δ).


## 2.4. Vector-Form
Equation of Observation

Now, we proceed from the stochastic integral-form EO (1) to its
finite-dimensional approximation (vector) form: (6)u=S˜e+n=Se+Δe+n,in which the disturbed SFO matrix is defined by (5), and e, n, u represent zero-mean vectors
composed of the decomposition coefficients  ek, nm, and um,  respectively. These vectors are characterized by the
correlation matrices: Re=D=D(b)=diag{b} (a diagonal matrix with vector b at
its principal diagonal), Rn, and Ru=〈S˜ReS˜+〉p(Δ)+Rn,
respectively, where 〈⋅〉p(Δ) defines the averaging performed over the
randomness of Δ
characterized by the unknown probability density function p(Δ). Vector b is composed of the elements, bk=ℬ(ek)=〈ekek*〉=〈|ek|2〉; k=1,…,K and is
referred to as a K-D vector-form representation
of the SSP.We refer to the estimate,b^,
as a discrete-form representation of the desired SSP, that is,  the brightness image of the wavefield sources
distributed over the pixel-formatted object scene remotely sensed with an employed
array radar/SAR. Thus, the uncertain SSP reconstruction
problem can be reformulated now as follows: to derive an estimator for
reconstructing the K-D approximation: (7)b^(K)(x)=∑k=1Kb^k|gk(x)|2=gT(x)diag{b^}g(x)of
the SSP distribution in the environment   X∋x. Note that in applications, we employ the ordinary pixel expansion format
[16], while all theoretical results are valid also for any feasible
decomposition function basis,  g(x)=veck{gk(x)},
in (7).


## 3. Dedr Strategy

### 3.1. Formulation of DEDR Estimation Strategy

In the descriptive statistical formalism, the desired SSP vectorb^ is recognized to be a vector of
the principal diagonal of an estimate of the correlation matrix Re(b), that is, b^={R^e}diag.
Thus, one can seek to estimate the desired SSP b^(K)(x)=gT(x)diag{b^}g(x)  given the
data correlation matrix Ru pre-estimated
via averaging of some J independent
sampled correlations [6]:  (8)Y=R^u=averj∈J{u(j)u(j)+}=1J∑j=1Ju(j)u(j)+,and determining the solution operator (SO) F such  that (9)b^(K)(x)=gT(x)diag{{FYF+}diag}g(x).To optimize
the search for the desired SOF, we formulate here the following DEDR strategy: (10)F=argminF{ℜ(F)}(11)subject to〈∥Δ∥2〉p(Δ)≤δ, where the conditioning term represents the worst-case
statistical performance (WCSP) regularizing constraint imposed on the
unknown second-order statistics 〈∥Δ∥2〉p(Δ) of the random distortion component Δ of the SFO matrix (5), and the DEDR “generalized  risk” function is defined as (12)ℜ(F)=tr{〈(FS˜−I)A(FS˜−I)+〉p(Δ)}+αtr{FRnF+}, where superscript + defines conjugate transpose. The DEDR strategy
(10), (11) implies the minimization of the α-weighted
sum of the systematic error measure (specified by the first term in the risk
function (12)) and noise error (specified by the second term in the risk
function (12)) in the desired RSS estimate (9), in which the unknown
disturbances of the SFO are treated through the WCSP bounding constraint (11)
imposed onto the averaged squared norm of Δ. The selections (adjustments) of the regularization parameter α and the diagonal-form weight matrix A (the so-called metrics inducing matrix [13, 16]) with the diagonal composed
of positive numbers {akk>0;k=1,…,K} provide the additional DEDR “degrees of
freedom” assigning the weights akk to the particular SSP vector components bk.
These weights {akk} are the user-defined
parameters that may incorporate any descriptive metrics properties of a
solution [7, 8, 16]. In the simplest case of no preference to reconstruction
of particular SSP components over the observation scene frame, the uniform
metrics is typically induced by setting A=I, that is, the identity matrix.
In Section 3.2, we will consider the adaptive DEDR case and specify the
corresponding solution-dependent A. Nevertheless,
independent on any feasible choice of α, A in the risk function (12), the
conditional optimization problem  (10),
(11) can be reformulated as (13)F=argminFmax〈∥Δ∥2〉p(Δ)≤δ{ℜ(F)}.


### 3.2. Decomposition of DEDR Risk

To proceed with the
derivation of the estimator (9), (13), we now decompose
the risk (12) incorporating directly the WCSP uncertainty constraint into
the DEDR strategy. The first term in the risk function (12) specifies the
systematic error component as it measures “how far” the desired SO F is from the pseudoinverse of  S˜ in the averaged operator metrics.
We
next, decompose this term into the following: (14)tr{〈(FS˜−I)A(FS˜−I)+〉p(Δ)}=∥FS−I∥A2+〈∥FΔ∥A2〉p(Δ),where   ∥C∥A2=tr{CAC+} denotes the A-weighted squared operator norm of a
matrix, C. The second
term in (14) has the statistical meaning of the average noise energy in the
resulting solution (9); hence it specifies the fluctuation error measure. This
term can be bounded applying the Loewner ordering [16] of the weight matrix A≤yI   with the
Loewner ordering factor  γ=min{γ¯:A≤γ¯I}>0 that yields (15)〈∥FΔ∥A2〉≤γ〈∥FΔ∥2〉≤γ∥F∥2〈∥Δ∥2〉,where the second inequality follows from the Cauchy-Schwarz
inequality [16], and   ∥C∥2=∥C∥I2=tr{CC+} defines a conventional
squared norm of a matrix, C. Using the constraint (11), we next evaluate
the maximum value that may take the last term in the inequality (15), that is, (16)max〈∥Δ∥2〉p(Δ)≤δ{γ∥F∥2〈∥Δ∥2〉p(Δ)}=ε∥F∥2 valid for any given
bounding factor      ε=δγ≥0.
With this evaluation (16), the WCSP-constrained DEDR strategy (13) is transformed
into the following nonconstrained optimization problem: (17)F=argminF{ℜDEDR(F)}with the aggregated DEDR risk
functional: (18){ℜDEDR(F)}=tr{(FS−I)A(FS−I)+}+αtr{FRΣF+},where (19)RΣ=RΣ(β)=(Rn+βI);β=εα≥0.


## 3.1. Formulation of DEDR Estimation Strategy

In the descriptive statistical formalism, the desired SSP vectorb^ is recognized to be a vector of
the principal diagonal of an estimate of the correlation matrix Re(b), that is, b^={R^e}diag.
Thus, one can seek to estimate the desired SSP b^(K)(x)=gT(x)diag{b^}g(x)  given the
data correlation matrix Ru pre-estimated
via averaging of some J independent
sampled correlations [6]:  (8)Y=R^u=averj∈J{u(j)u(j)+}=1J∑j=1Ju(j)u(j)+,and determining the solution operator (SO) F such  that (9)b^(K)(x)=gT(x)diag{{FYF+}diag}g(x).To optimize
the search for the desired SOF, we formulate here the following DEDR strategy: (10)F=argminF{ℜ(F)}(11)subject to〈∥Δ∥2〉p(Δ)≤δ, where the conditioning term represents the worst-case
statistical performance (WCSP) regularizing constraint imposed on the
unknown second-order statistics 〈∥Δ∥2〉p(Δ) of the random distortion component Δ of the SFO matrix (5), and the DEDR “generalized  risk” function is defined as (12)ℜ(F)=tr{〈(FS˜−I)A(FS˜−I)+〉p(Δ)}+αtr{FRnF+}, where superscript + defines conjugate transpose. The DEDR strategy
(10), (11) implies the minimization of the α-weighted
sum of the systematic error measure (specified by the first term in the risk
function (12)) and noise error (specified by the second term in the risk
function (12)) in the desired RSS estimate (9), in which the unknown
disturbances of the SFO are treated through the WCSP bounding constraint (11)
imposed onto the averaged squared norm of Δ. The selections (adjustments) of the regularization parameter α and the diagonal-form weight matrix A (the so-called metrics inducing matrix [13, 16]) with the diagonal composed
of positive numbers {akk>0;k=1,…,K} provide the additional DEDR “degrees of
freedom” assigning the weights akk to the particular SSP vector components bk.
These weights {akk} are the user-defined
parameters that may incorporate any descriptive metrics properties of a
solution [7, 8, 16]. In the simplest case of no preference to reconstruction
of particular SSP components over the observation scene frame, the uniform
metrics is typically induced by setting A=I, that is, the identity matrix.
In Section 3.2, we will consider the adaptive DEDR case and specify the
corresponding solution-dependent A. Nevertheless,
independent on any feasible choice of α, A in the risk function (12), the
conditional optimization problem  (10),
(11) can be reformulated as (13)F=argminFmax〈∥Δ∥2〉p(Δ)≤δ{ℜ(F)}.


## 3.2. Decomposition of DEDR Risk

To proceed with the
derivation of the estimator (9), (13), we now decompose
the risk (12) incorporating directly the WCSP uncertainty constraint into
the DEDR strategy. The first term in the risk function (12) specifies the
systematic error component as it measures “how far” the desired SO F is from the pseudoinverse of  S˜ in the averaged operator metrics.
We
next, decompose this term into the following: (14)tr{〈(FS˜−I)A(FS˜−I)+〉p(Δ)}=∥FS−I∥A2+〈∥FΔ∥A2〉p(Δ),where   ∥C∥A2=tr{CAC+} denotes the A-weighted squared operator norm of a
matrix, C. The second
term in (14) has the statistical meaning of the average noise energy in the
resulting solution (9); hence it specifies the fluctuation error measure. This
term can be bounded applying the Loewner ordering [16] of the weight matrix A≤yI   with the
Loewner ordering factor  γ=min{γ¯:A≤γ¯I}>0 that yields (15)〈∥FΔ∥A2〉≤γ〈∥FΔ∥2〉≤γ∥F∥2〈∥Δ∥2〉,where the second inequality follows from the Cauchy-Schwarz
inequality [16], and   ∥C∥2=∥C∥I2=tr{CC+} defines a conventional
squared norm of a matrix, C. Using the constraint (11), we next evaluate
the maximum value that may take the last term in the inequality (15), that is, (16)max〈∥Δ∥2〉p(Δ)≤δ{γ∥F∥2〈∥Δ∥2〉p(Δ)}=ε∥F∥2 valid for any given
bounding factor      ε=δγ≥0.
With this evaluation (16), the WCSP-constrained DEDR strategy (13) is transformed
into the following nonconstrained optimization problem: (17)F=argminF{ℜDEDR(F)}with the aggregated DEDR risk
functional: (18){ℜDEDR(F)}=tr{(FS−I)A(FS−I)+}+αtr{FRΣF+},where (19)RΣ=RΣ(β)=(Rn+βI);β=εα≥0.


## 4. Dedr Estimators of SSP

### 4.1. General-form SSP Estimator

Routinely solving the minimization problem (17), we obtain the desired
DEDR-optimal SO: (20)FDEDR=KA,α,βS+RΣ−1  with the
self-adjoint robust reconstruction operator: (21)KA,α,β=(S+RΣ−1(β)S+αA−1)−1 dependent on three degrees of freedom: α, β, and A.Note, that the derived robust
SO (20) involves the Hermitian conjugate S+ of the regular SFO S (i.e., it
satisfies the DED-observability requirements [15]) and does not involve the
inversion of Y (i.e., it is
applicable to the reconstructive SAR imaging problems with only one-recorded
realization of the trajectory data signal available for further processing, J=1).The general-form DEDR-optimal SO (20)
enables us now to derive the corresponding general-form robust SSP estimator
putting (20) into (9) that yields  (22)b^(K)(x)=b^(K)(x∣A,α,β)=gT(x)diag{{KA,α,βS+RΣ−1(β)YRΣ−1(β)SKA,α,β}diag}g(x).
This general-form DEDR estimator for the SSP can also be represented in
the alternative form as (23)b^(K)(x)=gT(x)diag{{KA,α,βaverj∈J{q(j)q(j)+}KA,α,β}diag}g(x),where  q(j)=S+RΣ−1u(j) is recognized to be an output
of the DEDR-regularized matched spatial processing algorithm with noise
whitening that assumes the given composed correlation matrix, RΣ=RΣ(β). In practical RS scenarios, it
is a common practice [3–5, 14] to accept the robust white observation
noise model, that is, Rn−1=(1/N0)I and treat the noise intensity N0 together with the uncertainty factor β in the composed model of RΣ  defined by (19).


### 4.2.  Family
of the DEDR-Related Algorithms

A family of the DEDR-related algorithms for estimating
the SSP can be derived now from (22) via controlling the regularization
parameters α, β, and the weight
matrix A that constitute the
degrees of freedom of the developed DEDR method.


#### 4.2.1.  Robust Spatial Filtering Algorithm

Consider the white zero-mean noise in
observations and no preference to any prior model information, that is, puttingA=I. Let the regularization
parameter be adjusted as the inverse of the signal-to-noise ratio (SNR), that is, α=(N0+β)/b0, where b0 is
the prior average gray level of the SSP, and the uncertainty factor β is
attributed to α. In
that case, the SO F is
recognized to be the Tikhonov-type robust spatial filter (RSF):(24)FRSF=(S+S+((N0+β)/b0)I)−1S+.


#### 4.2.2. Matched Spatial Filtering Algorithm

Consider the model from the previous
example for an assumption,α≫∥S+S∥, that is, the
case of a priority of the second error measure (suppression of noise)
over the systematic error in the optimization problem (17). In this case, we
can roughly approximate (20), (24) as the matched spatial filter (MSF): (25)FMSF≈const.S+,where the normalizing constant is
irrelevant as it specifies the constant image scaling factor that does not
influence the overall reconstructed image pattern.


#### 4.2.3. Robust Adaptive Spatial Filtering Algorithm

Consider the case of an arbitrary
zero-mean noise with the composed correlation matrixRΣ, equal importance of two error
measures in (18), that is, α=1, and the solution-dependent weight
matrix  A=D^=diag{b^}. In this case, the SO becomes the robust
adaptive (i.e., solution-dependent) spatial filter (RASF) operator:(26)FRASF=(S+RΣ−1S+D^−1)−1S+RΣ−1.The three SSP reconstruction
techniques that employ the SOs (24), (25), and (26) compose the family of the
DEDR-related estimators: (27)b^(K)(p)(x)=gT(x)diag{{F(p)YF(p)+}diag}g(x),p=1,2,3 with F(1)=FMSF, F(2)=FRSF, and F(3)=FRASF, respectively. Any other feasible adjustments of the
DEDR degrees of freedom (the regularization parameters α, β, and the weight matrix A) provide other
possible DEDR-related SSP reconstruction techniques numbered further on as p=4,…. As an important example, in
the sequential subsection, we show that such DEDR family encompasses also the
celebrated minimum variance distortionless response (MVDR) beamforming method
transformed into the high-resolution RSS estimation technique with the proper
MVDR SO FMVDR=F(4) specified further on by
(31).


### 4.3. Relationship with the Robust MVDR Beamformer

The conventional MVDR beamformer [7]
“reconstructs” the RS image by minimizing the power or variance of the adaptive
array output for all search directions, k=1,…,K, under the constraint
that the gain in the particular look direction is equal to a constant (one, for
simplicity). This results in the well-known conventional MVDR algorithm [7, 10]: (28)b^k=(sk+Y−1sk)−1;k=1,…,K,where sk represents the so-called “steering
vector” for the kth look direction,
which in our notations is essentially the kth
column vector of the nominal SFO matrix S.For the purposes of establishing a relationship between the MVDR
beamformer and the DEDR-related SSP estimators (27), we now rewrite the
conventional MVDR algorithm (28) as (29)b^MVDR={[diag{{S+Y−1S}diag}]−1}diag that can be considered as a solution
to the equation, D^=D^S+Y−1SD^.
Expressing now Y−1=Y−1YY−1 and using
the second-form representation [15] for the operator,  D^S+Y−1=(S+Rn−1S+D^−1)−1S+Rn−1, we obtain the alternative
representation form for the MVDR algorithm (28), that is, (30)b^MVDR={FMVDRYFMVDR+}diag with (31)FMVDR=F(4)=D^S+Y−1=(S+Rn−1S+D^−1)−1S+Rn−1.Examining the formulae (20), (21), and (31),
one may deduce that FMVDR=F(4) coincides with FDEDR for the nonrobust adaptive case, that
is, β=0,  A=D^.


## 4.1. General-form SSP Estimator

Routinely solving the minimization problem (17), we obtain the desired
DEDR-optimal SO: (20)FDEDR=KA,α,βS+RΣ−1  with the
self-adjoint robust reconstruction operator: (21)KA,α,β=(S+RΣ−1(β)S+αA−1)−1 dependent on three degrees of freedom: α, β, and A.Note, that the derived robust
SO (20) involves the Hermitian conjugate S+ of the regular SFO S (i.e., it
satisfies the DED-observability requirements [15]) and does not involve the
inversion of Y (i.e., it is
applicable to the reconstructive SAR imaging problems with only one-recorded
realization of the trajectory data signal available for further processing, J=1).The general-form DEDR-optimal SO (20)
enables us now to derive the corresponding general-form robust SSP estimator
putting (20) into (9) that yields  (22)b^(K)(x)=b^(K)(x∣A,α,β)=gT(x)diag{{KA,α,βS+RΣ−1(β)YRΣ−1(β)SKA,α,β}diag}g(x).
This general-form DEDR estimator for the SSP can also be represented in
the alternative form as (23)b^(K)(x)=gT(x)diag{{KA,α,βaverj∈J{q(j)q(j)+}KA,α,β}diag}g(x),where  q(j)=S+RΣ−1u(j) is recognized to be an output
of the DEDR-regularized matched spatial processing algorithm with noise
whitening that assumes the given composed correlation matrix, RΣ=RΣ(β). In practical RS scenarios, it
is a common practice [3–5, 14] to accept the robust white observation
noise model, that is, Rn−1=(1/N0)I and treat the noise intensity N0 together with the uncertainty factor β in the composed model of RΣ  defined by (19).


## 4.2.  Family
of the DEDR-Related Algorithms

A family of the DEDR-related algorithms for estimating
the SSP can be derived now from (22) via controlling the regularization
parameters α, β, and the weight
matrix A that constitute the
degrees of freedom of the developed DEDR method.


### 4.2.1.  Robust Spatial Filtering Algorithm

Consider the white zero-mean noise in
observations and no preference to any prior model information, that is, puttingA=I. Let the regularization
parameter be adjusted as the inverse of the signal-to-noise ratio (SNR), that is, α=(N0+β)/b0, where b0 is
the prior average gray level of the SSP, and the uncertainty factor β is
attributed to α. In
that case, the SO F is
recognized to be the Tikhonov-type robust spatial filter (RSF):(24)FRSF=(S+S+((N0+β)/b0)I)−1S+.


### 4.2.2. Matched Spatial Filtering Algorithm

Consider the model from the previous
example for an assumption,α≫∥S+S∥, that is, the
case of a priority of the second error measure (suppression of noise)
over the systematic error in the optimization problem (17). In this case, we
can roughly approximate (20), (24) as the matched spatial filter (MSF): (25)FMSF≈const.S+,where the normalizing constant is
irrelevant as it specifies the constant image scaling factor that does not
influence the overall reconstructed image pattern.


### 4.2.3. Robust Adaptive Spatial Filtering Algorithm

Consider the case of an arbitrary
zero-mean noise with the composed correlation matrixRΣ, equal importance of two error
measures in (18), that is, α=1, and the solution-dependent weight
matrix  A=D^=diag{b^}. In this case, the SO becomes the robust
adaptive (i.e., solution-dependent) spatial filter (RASF) operator:(26)FRASF=(S+RΣ−1S+D^−1)−1S+RΣ−1.The three SSP reconstruction
techniques that employ the SOs (24), (25), and (26) compose the family of the
DEDR-related estimators: (27)b^(K)(p)(x)=gT(x)diag{{F(p)YF(p)+}diag}g(x),p=1,2,3 with F(1)=FMSF, F(2)=FRSF, and F(3)=FRASF, respectively. Any other feasible adjustments of the
DEDR degrees of freedom (the regularization parameters α, β, and the weight matrix A) provide other
possible DEDR-related SSP reconstruction techniques numbered further on as p=4,…. As an important example, in
the sequential subsection, we show that such DEDR family encompasses also the
celebrated minimum variance distortionless response (MVDR) beamforming method
transformed into the high-resolution RSS estimation technique with the proper
MVDR SO FMVDR=F(4) specified further on by
(31).


## 4.2.1.  Robust Spatial Filtering Algorithm

Consider the white zero-mean noise in
observations and no preference to any prior model information, that is, puttingA=I. Let the regularization
parameter be adjusted as the inverse of the signal-to-noise ratio (SNR), that is, α=(N0+β)/b0, where b0 is
the prior average gray level of the SSP, and the uncertainty factor β is
attributed to α. In
that case, the SO F is
recognized to be the Tikhonov-type robust spatial filter (RSF):(24)FRSF=(S+S+((N0+β)/b0)I)−1S+.


## 4.2.2. Matched Spatial Filtering Algorithm

Consider the model from the previous
example for an assumption,α≫∥S+S∥, that is, the
case of a priority of the second error measure (suppression of noise)
over the systematic error in the optimization problem (17). In this case, we
can roughly approximate (20), (24) as the matched spatial filter (MSF): (25)FMSF≈const.S+,where the normalizing constant is
irrelevant as it specifies the constant image scaling factor that does not
influence the overall reconstructed image pattern.


## 4.2.3. Robust Adaptive Spatial Filtering Algorithm

Consider the case of an arbitrary
zero-mean noise with the composed correlation matrixRΣ, equal importance of two error
measures in (18), that is, α=1, and the solution-dependent weight
matrix  A=D^=diag{b^}. In this case, the SO becomes the robust
adaptive (i.e., solution-dependent) spatial filter (RASF) operator:(26)FRASF=(S+RΣ−1S+D^−1)−1S+RΣ−1.The three SSP reconstruction
techniques that employ the SOs (24), (25), and (26) compose the family of the
DEDR-related estimators: (27)b^(K)(p)(x)=gT(x)diag{{F(p)YF(p)+}diag}g(x),p=1,2,3 with F(1)=FMSF, F(2)=FRSF, and F(3)=FRASF, respectively. Any other feasible adjustments of the
DEDR degrees of freedom (the regularization parameters α, β, and the weight matrix A) provide other
possible DEDR-related SSP reconstruction techniques numbered further on as p=4,…. As an important example, in
the sequential subsection, we show that such DEDR family encompasses also the
celebrated minimum variance distortionless response (MVDR) beamforming method
transformed into the high-resolution RSS estimation technique with the proper
MVDR SO FMVDR=F(4) specified further on by
(31).


## 4.3. Relationship with the Robust MVDR Beamformer

The conventional MVDR beamformer [7]
“reconstructs” the RS image by minimizing the power or variance of the adaptive
array output for all search directions, k=1,…,K, under the constraint
that the gain in the particular look direction is equal to a constant (one, for
simplicity). This results in the well-known conventional MVDR algorithm [7, 10]: (28)b^k=(sk+Y−1sk)−1;k=1,…,K,where sk represents the so-called “steering
vector” for the kth look direction,
which in our notations is essentially the kth
column vector of the nominal SFO matrix S.For the purposes of establishing a relationship between the MVDR
beamformer and the DEDR-related SSP estimators (27), we now rewrite the
conventional MVDR algorithm (28) as (29)b^MVDR={[diag{{S+Y−1S}diag}]−1}diag that can be considered as a solution
to the equation, D^=D^S+Y−1SD^.
Expressing now Y−1=Y−1YY−1 and using
the second-form representation [15] for the operator,  D^S+Y−1=(S+Rn−1S+D^−1)−1S+Rn−1, we obtain the alternative
representation form for the MVDR algorithm (28), that is, (30)b^MVDR={FMVDRYFMVDR+}diag with (31)FMVDR=F(4)=D^S+Y−1=(S+Rn−1S+D^−1)−1S+Rn−1.Examining the formulae (20), (21), and (31),
one may deduce that FMVDR=F(4) coincides with FDEDR for the nonrobust adaptive case, that
is, β=0,  A=D^.


## 5. Simulations and Discussions

We simulated
a conventional side-looking SAR with the fractionally synthesized aperture, that
is, the array was synthesized by the moving antenna. The regular SFO of such
SAR is factored along two axes in the image plane [17, 18]: the azimuth or
cross-range coordinate (horizontal axis, x1) and the slant range (vertical axis, x2). In the simulations, we
considered the conventional triangular SAR range ambiguity function (AF) Ψr(x2) and two approximations of
the SAR azimuth AF: (i) “sinc” approximation, Ψa(1)(x1)=|sinc(x1/a)|, and (ii) Gaussian
“bell” approximation, Ψa(2)(x1)=exp(−(x1)2/a2), with the adjustable fractional
parameters a [15]. Note that in the imaging radar theory [3, 8], the
AF is referred to as the continuous-form approximation of the ambiguity
operator matrix Ψ=S+S and serves as an equivalent to the
point spread function in the conventional image processing terminology [16, 19].
In this paper, we present the simulations performed with two characteristic
scenes. The fist one of the 512-by-512 pixel format was artificially generated.
The second one of the same 512-by-512 pixel format was borrowed from the real world
high-resolution terrain SAR imagery (south-west Guadalajara region, Mexico
[20]).  The first scene was used as a
test for adjustment of the degrees of freedom of the developed RSF and RASF
algorithms to attain the desired improvement in the image enhancement
performances (the IOSNR defined
below). In the reported simulations, the representation formats along the x2 (slant range) and x1 (cross range, i.e., azimuth) directions
were adjusted to the same effective pixel width. In the x1 direction, the fractional
parameter a was controlled to adjust different effective widths ΔΨa(x1) of the azimuth AF. The corresponding
adjustment of different effective width of the range AF ΔΨr(x2) was performed over the slant
range direction (x2).For the
purpose of objectively testing the performances of different DEDR-related SSP
estimation algorithms, a quantitative evaluation of the improvement in the SSP
estimates (gained due to applying the DEDR-related reconstructive solution
operatorsF(p);  p=2,…, instead
of the MSF, i.e., the adjoint operator F(1)=S+) was accomplished.
In analogy to image reconstruction quality metrics [16, 19], we adopt here the
quality metric defined as an improvement in the output signal-to-noise ratio (IOSNR):  (32)IOSNR(p)=10log10∑k=1K(b^k(MSF)−bk)2∑k=1K(b^k(p)−bk)2,p=2,3,4,5,where bk represents a value of the kth element
(pixel) of the original SSP b, b^k(MSF) represents a pixel value of the  kth
element (pixel) of the rough SSP estimate b^MSF formed applying the matched spatial filtering
technique (conventional matched beamformer with F(1)=S+),
and b^k(p) represents a value of the kth pixel of the SSP reconstructed
from the matched b^MSF applying one of the particular developed
DEDR-related SOs. In the simulation studies, four different DEDR-related
estimators were tested, renumbered here as p=2, 3, 4, and 5. The F(2) corresponds to the nonconstrained FRSF, that is, to the RSF method adjusted incorrectly to the scenario assuming
no uncertainties in the data (β=0).
The F(3) corresponds to the
constrained FRSF
with the SFO uncertainty factor kΔ=β/N0 correctly adjusted
to two different uncertain scenarios (as specified in Tables 1 and 2). The F(4) corresponds to the nonconstrained RASF, that is, the RASF
method adjusted incorrectly to the scenario with no uncertainties in the data (β=0). Last, the F(5) corresponds to the constrained FRASF with the SFO uncertainty factor kΔ=β/N0 correctly adjusted
to two different uncertain scenarios (as specified in Tables 1 and 2), that is, the WCSP-optimized DEDR
estimator. According to the quality metric (32),
the higher the IOSNR, the better the improvement in the SSP estimate is,
that is, the closer the estimate is to the original SSP.Table 1
IOSNR gained with
different DEDR-related reconstruction algorithms (results are
reported for the first uncertain operational scenario and second scene).
Scenario 1:ΔΨa(1)(x1)=10; ΔΨr(x2)=3; kΔ=β/N0=0.1μ [dB]Nonconstrained RSFConstrained RSFNonconstrained
RASFConstrained RASF
(WCSP-optimized)IOSNR(2)IOSNR(3)IOSNR(4)IOSNR(5)51.852.1582.22.45102.42.682.322.89152.562.762.673.4202.733.373.024.2253.474.233.15.32303.854.953.645.46Table 2
IOSNR gained with
different DEDR-related reconstruction algorithms (results are
reported for the second uncertain operational scenario and second scene).
Scenario 2:ΔΨa(1)(x1)=14; ΔΨr(x2)=6; kΔ=β/N0=0.05μ [dB]Nonconstrained RSFConstrained RSFNonconstrained RASFConstrained
RASF (WCSP-optimized)IOSNR(2)IOSNR(3)IOSNR(4)IOSNR(5)51.712.171.92.41101.852.611.922.88151.92.92.23.45201.933.42.184.16252.013.782.64.56302.114.33.085.32In this section, we report
the qualitative simulation results and the relevant quantitative performances
evaluated via the IOSNRs (32) (in the dB scale) gained with these four robust
DEDR-related  estimators, in particular: IOSNR(2) gained using the nonconstrained
RSF in the uncertain scenario; IOSNR(3) gained  applying the constrained RSF
in the same uncertain scenario; IOSNR(4) gained using the
nonconstrained RASF; and IOSNR(5) gained applying the constrained RASF (WCSP-optimized estimator) in the same
uncertain scenario. The simulation experiments were run for two typical SAR
systems that operate under different SNRs levels μ=b0/N0,
different fractionally synthesized apertures (characterized by the width of the
azimuth AFs ΔΨa(x1)), and different uncertainty factors kΔ=β/N0 (as specified in
Tables 1 and 2) that bound via (11),
(19) the impact of the uncertainty SFO term. In particular,
the simulated scenarios are specified as follows.
(i)
First uncertain operational scenario
(simulation experiment specifications):(a)
fractional azimuth AF width,ΔΨa(1)(x1)=10 pixels of the 512×512 scene pixel format(at the 0.5 from the peak value of
the “sinc-type” AF,Ψa(1)(x1)=|sinc(x1/a)|);(b)
range AF
width,ΔΨr(x2)=3 pixels (at the 0.5 from the peak value of the triangular Ψr(x2));(c)
SNRs range,μ=b0/N0=5dB,…,30dB;(d)
SFO uncertainty factor,kΔ=β/N0=0.1.(ii)
Second uncertain operational scenario
(simulation experiment specifications):(a)
fractional
azimuth AF width,ΔΨa(2)(x1)=14 pixels(at the 0.5 from the peak value of the
“bell-type” AF,Ψa(2)(x1)=exp(−(x1)2/a2));(b)
range AF
width,ΔΨr(x2)=6 pixels (at the 0.5 from the peak value of the triangular Ψr(x2));(c)
SNRs range,μ=b0/N0=5dB,…,30dB;(d)
SFO uncertainty factor,kΔ=β/N0=0.05. These specifications correspond to two typical uncertain scenarios with
airborne SAR sensor
trajectory deviations modelled in [17].Figures1(a) and 2(a) show the same artificially synthesized test
scene. Figures 3(a) and 4(a) show the second tested original scene
(borrowed from the real world high-resolution SAR imagery [20]). The remaining
images of Figure 1 through Figure  4
present the results of image formation applying different DEDR-related SSP estimators
as specified in the figure captions. Figures 1(b) through 4(b) demonstrate the
images formed applying the conventional MSF for the uncertain fractionally
synthesized SAR scenarios. According to the EO (6), the overall uncertain data
degradations  n˜=Δe+n
were composed of a mixture of conventional white additive
observation noise n and correlated
(scene-dependent) multiplicative noise Δe. Following the DEDR methodology (detailed in
Section 3), the SFO uncertainty cannot
be factorized into separate terms caused by the environmental perturbations,
SAR trajectory deviations, or antenna vibrations. Thus, the composed
multiplicative degradation effect was modeled via simulating the MSF scene
image corrupted by the speckle noise via incorporating into (9) with the SO (25),
the uncertain operational scenario factors, in particular, the uncertain data
model correlation matrix Y that corresponds to the degraded EO (6) with
the diagonal loaded noise augmented correlation matrix (19). Figures 1(c) through
4(c) show the enhanced images formed applying the unconstrained RSF, that is,
the RSF incorrectly adjusted to the uncertain scenario via ignoring the
uncertainty factor (β=0).
Figures 1(d) through 4(d) present the enhanced images formed using the
constrained RSF properly adjusted to the particular uncertain scenario (kΔ=β/N0=0.1
for the first scenario, and kΔ=β/N0=0.05
for the second scenario, respectively). The images enhanced with the unconstrained RASF (β=0) are shown in
Figures 1(e)–4(e), and the corresponding images
reconstructed with the constrained RASF (WCSP-optimized  method) are presented in
Figures 1(f)–4(f),
respectively.First operational scenario, first scene (μ=20dB): (a) artificially synthesized
original scene; (b) degraded uncertain scene image formed applying the MSF
method; (c) image reconstructed applying the nonconstrained RSF algorithm; (d)
image reconstructed with the constrained RSF algorithm; (e) image reconstructed
applying the nonconstrained RASF algorithm; and (f) image
reconstructed applying the constrained RASF (WCSP-optimized) algorithm.
(a)(b)(c)(d)(e)(f)Second operational scenario, first scene (μ=20dB): (a) artificially synthesized
original scene; (b) degraded uncertain scene image formed applying the MSF
method; (c) image reconstructed applying the nonconstrained RSF algorithm; (d)
image reconstructed with the constrained RSF algorithm; (e) image reconstructed
applying the nonconstrained RASF algorithm; and (f) image
reconstructed applying the constrained RASF (WCSP-optimized) algorithm.
(a)(b)(c)(d)(e)(f)First operational scenario, second scene (μ=20dB): (a) original scene; (b) degraded
uncertain scene image formed applying the MSF method; (c) image reconstructed
applying the nonconstrained RSF algorithm; (d) image reconstructed with the constrained
RSF algorithm; (e) image reconstructed applying the nonconstrained RASF algorithm;
and (f) image reconstructed applying the constrained RASF (WCSP-optimized)
algorithm.
(a)(b)(c)(d)(e)(f)Second operational scenario, second scene (μ=20dB): (a) original scene; (b) degraded
uncertain scene image formed applying the MSF method; (c) image reconstructed
applying the nonconstrained RSF algorithm; (d) image reconstructed with the constrained
RSF algorithm; (e) image reconstructed applying the nonconstrained RASF algorithm;
and (f) image reconstructed applying the constrained RASF (WCSP-optimized)
algorithm.
(a)(b)(c)(d)(e)(f)From the presented simulation results, the advantage of the well-designed
imaging experiments (constrained RSF and WCSP-optimized RASF) over the case of
badly designed experiment (nonrobust MSF and unconstrained RSF) is evident. Due
to the performed regularized inversions, the resolution was substantially
improved in all simulated scenarios (as reported in Tables1 and 2). The
higher values of IOSNR(3)>IOSNR(2) as well as IOSNR(5)>IOSNR(4) were obtained with the constrained
DEDR-related estimators, that is, with the DEDR techniques adopted to the
uncertain scenarios. Note that IOSNR (32) is basically a square-type
error metric. Thus, it does not qualify quantitatively the “delicate” visual
features in the reconstructed images; hence, small differences in the
corresponding IOSNRs reported in Tables 1 and 2. In addition, both
enhanced robust estimators manifest the higher IOSNRs in the case of more smooth azimuth AFs (larger ΔΨa(x1)) and higher SNRs μ. For the DEDR-optimized RASF method, in
addition, the ringing (image speckle) effect was substantially reduced, while
the nonadaptive constrained RSF estimator requires considerably less
computational load. These results qualitatively demonstrate that with proper
adjustment of the degrees of freedom in the developed DEDR estimators (24), (27),
one could approach the quality of the DEDR-optimal image formation method (22) avoiding
the cumbersome adaptive computations required to implement the DEDR-optimal algorithm
[10, 15].


## 6. Conclusion

New descriptive experiment design
regularization (DEDR) approach for estimation of the spatial spectrum pattern
(SSP) of the wavefield power distribution in the uncertain remotely sensed
environment has been proposed as required for the conventional array imaging
radar, side-looking airborne radar, and SAR. Unifying the DEDR and the
worst-case statistical performance (WCSP) optimization into the aggregated
WCSP-constrained minimum risk technique, the inverse problem ill-posedness has
been alleviated in a statistically grounded fashion. The derived general-form
DEDR estimator does not involve
the inversion of the estimated data correlation matrix. This principal
algorithmic-level result of the undertaken study constitutes the crucial
advantage of the developed family of the DEDR-related estimators that makes
them applicable to the uncertain operational scenarios with ill-conditioned
(e.g., low-rank) estimates of the array data correlation matrices, in
particular, to the SAR imaging scenarios where only one realization of the
trajectory data signal degraded due to the uncontrolled random carrier
trajectory deviation and antenna vibration is available for further processing.
Being nonlinear and solution-dependent, the DEDR-optimal robust adaptive
spatial filtering (RASF) estimator requires rather complex signal processing.
The computational complexity arises due to the necessity to perform
simultaneously the solution-dependent operator inversion operations and
adaptive adjustments of the degrees of freedom of the overall RASF technique.
To reduce the computational load, the simplified constrained robust spatial
filtering (RSF) algorithm was proposed and employed, which manifests almost the
same reconstruction performances as the RASF in typical uncertain operational
scenarios that was verified in the simulation experiment.



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*Source: 810816-2008-08-13.xml*