Resolution-enhanced radar/SAR imaging: an experiment design framework combined with neural network-adapted variational analysis regularization
- Yuriy Shkvarko^{1}Email author,
- Stewart Santos^{1} and
- Jose Tuxpan^{1}
https://doi.org/10.1186/1687-6180-2011-85
© Shkvarko et al; licensee Springer. 2011
Received: 11 May 2011
Accepted: 11 October 2011
Published: 11 October 2011
Abstract
The convex optimization-based descriptive experiment design regularization (DEDR) method is aggregated with the neural network (NN)-adapted variational analysis (VA) approach for adaptive high-resolution sensing into a unified DEDR -VA-NN framework that puts in a single optimization frame high-resolution radar/SAR image formation in uncertain operational scenarios, adaptive despeckling and dynamic scene image enhancement for a variety of sensing modes. The DEDR -VA-NN method outperforms the existing adaptive radar imaging techniques both in resolution and convergence rate. The simulation examples are incorporated to illustrate the efficiency of the proposed DEDR-VA-related imaging techniques.
Keywords
1. Introduction
In this article, we consider the problem of enhanced remote sensing (RS) imaging stated and treated as an ill-posed nonlinear inverse problem with model uncertainties. The problem at hand is to perform high-resolution reconstruction of the power spatial spectrum pattern (SSP) of the wavefield scattered from the extended remotely sensed scene via space-time adaptive processing of finite recordings of the imaging radar/SAR data distorted in a stochastic uncertain measurement channel. The SSP is defined as a spatial distribution of the power (i.e., the second-order statistics) of the random wavefield backscattered from the remotely sensed scene observed through the integral transform operator [1, 2]. Such an operator is explicitly specified by the employed radar/SAR signal modulation and is traditionally referred to as the signal formation operator (SFO) [2, 3]. The operational uncertainties are attributed to inevitable random signal perturbations in inhomogeneous propagation medium with unknown statistics, possible imperfect radar calibration, and uncontrolled sensor displacements or carrier trajectory deviations in the SAR case. The classical imaging with an array radar or SAR implies application of the method called "matched spatial filtering (MSF)" to process the recorded data signals [2, 3]. A number of approaches had been proposed to design the constrained regularization techniques for improving the resolution in the SSP obtained by ways different from the MSF, e.g., [1–9] but without aggregating the minimum risk (MR) descriptive estimation strategies with convex projection regularization. In [7], an approach was proposed to treat the uncertain RS imaging problems that unifies the MR spectral estimation strategy with the worst case statistical performance (WCSP) optimization-based convex regularization resulting in the descriptive experiment design regularization (DEDR) method. Next, the variational analysis (VA) framework has been combined with the DEDR in [2, 9] to satisfy the desirable descriptive properties of the reconstructed RS images, namely: (i) convex optimization-based maximization of spatial resolution balanced with noise suppression, (ii) consistency, (iii) positivity, (iv) continuity and agreement with the data. In this study, we extend the developments of the DEDR and VA techniques originated in [2, 7, 9] by performing the aggregation of the DEDR and VA paradigms and next putting the RS image enhancement/reconstruction tasks into the unified neural network (NN)-adapted computational frame addressed as a unified DEDR-VA-NN method. We have designed a family of such significantly speeded-up DEDR-VA-related algorithms, and performed the simulations to illustrate the effectiveness of the proposed high-resolution DEDR-VA-NN-based image enhancement/fusion approach.
The rest of the article is organized as follows. In Section 2, we provide the formalism of the radar/SAR inverse imaging problem at hand with necessary experiment design considerations. In Section 3, we adapt the celebrated maximum likelihood (ML) inspired amplitude phase estimation (APES) technique for array sensor/SAR imaging. The unified DEDR-VA framework for high-resolution radar/SAR imaging in uncertain scenarios is conceptualized in Section 4, adapted to the NN-oriented sensor systems/methods fusion mode in Section 5, next, is followed by illustrative simulations in Sections 6 and the conclusion in Section 7.
2. Problem formalism
The general mathematical formalism of the problem at hand is similar in notation and structural framework to that described in [2, 7, 9] and some crucial elements are repeated for convenience to the reader. Following [1, 2, 9], we define the model of the observation RS wavefield u by specifying the stochastic equation of observation (EO) of an operator form u = $\mathcal{S}$e + n, where e = e(r), represents the complex scattering function over the probing surface R ∋ r, n is the additive noise, u = u(p), is the observation field, p = (t, ρ) defines the time (t)-space(ρ) points in the temporal-spatial observation domain p ∈ P = T × P (t ∈ T, ρ ∈ P) (in the SAR case, ρ = ρ(t) specifies the carrier trajectory [7]), and the kernel-type integral SFO $\mathcal{S}:E\left(R\right)\phantom{\rule{2.77695pt}{0ex}}\to \phantom{\rule{2.77695pt}{0ex}}U\left(P\right)$ defines a mapping of the source signal space $E\left(R\right)$ onto the observation signal space $U\left(P\right)$. The metrics structures in the corresponding Hilbert signal spaces $U\left(P\right)$$E\left(R\right)$ are imposed by scalar products, ${\left[u,{u}^{\prime}\right]}_{U}\phantom{\rule{2.77695pt}{0ex}}=\underset{P}{\int}u\left(\mathbf{p}\right){u}^{\prime}*\left(\mathbf{p}\right)d\mathbf{p},$, ${\left[e,{e}^{\prime}\right]}_{E}\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}\underset{R}{\int}e\left(\mathbf{r}\right){e}^{\prime}*\left(\mathbf{r}\right)d\mathbf{r},$ respectively [1]. The functional kernel S(p, r) of the SFO $\mathcal{S}$ is referred to as the unit signal[2] determined by the time-space modulation employed in a particular RS system. In the case of uncertain operational scenarios, the SFO is randomly perturbed [7], i.e. $\stackrel{\u0303}{\mathcal{S}}=\mathcal{S}+$$\Delta \mathcal{S}$ where $\Delta \mathcal{S}$ pertains to the random uncontrolled perturbations, usually with unknown statistics. The fields e, n, u. are assumed to be zero-mean complex valued Gaussian random fields [1, 7]. Next, since in all RS applications the regions of high correlation of e(r) are always small in comparison with the resolution element on the probing scene [1–3], the signals e(r) scattered from different directions r, r ' ∈ R of the remotely sensed scene R are assumed to be uncorrelated with the correlation function R_{ e }(r, r') = 〈e(r)e*(r') 〉 = b(r) δ(r-r');r,r'∈R where b(r) = 〈e(r)e*(r) 〉 = 〈|e(r)|^{2}〉; r∈R represents the power SSP of the scattered field [1]. The problem of high-resolution RS imaging is to develop a framework and related method(s) that perform optimal estimation of the SSP (referred to as a scene image) from the available radar/SAR data measurements. It is noted that in this study we are going to develop and follow the unified DEDR-VA-NN framework.
in which the disturbed M×K SFO matrix $\stackrel{\u0303}{\mathbf{S}}$ = S + Δ is the discrete-form approximation of the integral SFO for the uncertain operational scenario, and e, n, u represent zero-mean vectors composed of the sample (decomposition) coefficients {e_{ k } , n_{ m }, u_{ m } ; k = 1,...,K; m = 1,...,M}, respectively [1–3]. These vectors are characterized by the correlation matrices: R_{ e } = D = D(b) = diag(b) (a diagonal matrix with vector b at its principal diagonal), R_{ n }, and R_{ u } = < $\stackrel{\u0303}{\mathbf{S}}{\mathbf{R}}_{\mathbf{e}}{\stackrel{\u0303}{\mathbf{S}}}^{+}$ > _{p(}Δ_{)} + R_{ n }, respectively, where <·> _{p(Δ)}defines the averaging performed over the randomness of Δ characterized by the usually unknown probability density function p(Δ), and superscript "+" stands for Hermitian conjugate. Vector b composed of the elements, {b_{ k } =$\mathcal{B}\left\{{e}_{k}\right\}$ = <e_{ k }e_{ k }*> = <|e_{ k } |^{2}>; k = 1,...,K} is referred to as a K-D vector-form approximation of the SSP, where $\mathcal{B}$ represents the second-order statistical ensemble averaging operator [1, 2]. The SSP vector b is associated with the lexicographically ordered pixel-framed image [1, 7]. The corresponding conventional K_{ y } ×K_{ x } rectangular frame-ordered scene image B = {b(k_{ x } , k_{ y } ); k_{ x } , = 1,...,K_{ x } ; k_{ v } , = 1,...,K_{ y } } relates to its lexicographically ordered vector-form representation b = $\mathcal{L}\left\{\mathbf{B}\right\}$ = {b(k); k = 1,...,K = K_{ y } ×K_{ x } } via the standard row-by-row concatenation (i.e., lexicographical reordering) procedure, B = ${\mathcal{L}}^{-1}${b} [1]. It is noted that in the simple case of certain operational scenario [2, 3], the discrete-form (i.e., matrix-form) SFO S is assumed to be deterministic, i.e., the random perturbation term in (3) is irrelevant, Δ = 0.
The enhanced RS imaging problem is stated generally as follows: to map the scene pixel-framed image $\widehat{\mathbf{B}}$ via lexicographical reordering $\widehat{\mathbf{B}}={\mathcal{L}}^{-1}\left\{\widehat{\mathbf{b}}\right\}$ of the SSP vector estimate $\widehat{\mathbf{b}}$ reconstructed from whatever available measurements of independent realizations of the recorded data (1). The reconstructed SSP vector $\widehat{\mathbf{b}}$ is an estimate of the second-order statistics of the scattering vector e observed through the perturbed SFO and contaminated with noise; hence, the imaging problem at hand must be qualified and treated as a statistical nonlinear uncertain inverse problem [1, 7, 9]. The enhanced high-resolution imaging implies solution of such inverse problem in some optimal way. We know that in this article we intend to develop and follow the unified DEDR-VA framework, next adapted to NN-based computational implementation.
3. Adaptation of APES technique for array sensor/SAR imaging
In the APES terminology (as well as in the minimum variance distortionless response (MVDR) and other ML-related approaches [1, 4, 6] etc.), s_{k} represents the so-called steering vector in the k th look direction, which in our notational conventions is essentially the k th column vector of the regular SFO matrix S. The numerical implementation of the APES algorithm (2) assumes application of an iterative fixed point technique by building the model-based estimate ${\widehat{\mathbf{R}}}_{\mathbf{u}}={\mathbf{R}}_{\mathbf{u}}\left({\widehat{\mathbf{b}}}_{\left[i\right]}\right)$ of the unknown covariance R_{ u } from the latest (i th) iterative SSP estimate ${\widehat{\mathbf{b}}}_{\left[i\right]}$ with the zero step initialization ${\widehat{\mathbf{b}}}_{\left[0\right]}={\widehat{\mathbf{b}}}_{MSF}$ computed applying the conventional MSF estimator [2].
where operator {·}_{diag} returns the vector of a principal diagonal of the embraced matrix. The algorithmic structure of the vector-form nonlinear (i.e., solution-dependent) APES estimator (3) guarantees positivity but does not guarantee the consistency. In the real-world uncertain (rank deficient) RS operational scenarios, the inconsistency inevitably results in speckle corrupted images unacceptable for further processing and interpretation. To overcome these limitations, in the next section we extend the unified DEDR-VA framework of [2, 9] for the considered here uncertain operational scenarios to guarantee consistency and significantly speed-up convergence.
4. Unified DEDR-VA framework for high-resolution radar/SAR imaging in uncertain scenarios
4.1. DEDR-VA approach
where $\mathbf{K}={\left({\mathbf{S}}^{+}{\mathbf{R}}_{\Sigma}^{-1}\mathbf{S}+\alpha {\mathbf{A}}^{-1}\right)}^{-1}$ defines the so-called reconstruction operator (with the regularization parameter α and stabilizer A^{-1}), and ${\mathbf{R}}_{\Sigma}^{-1}$ is the inverse of the diagonal loaded noise correlation matrix [7]R_{ Σ } = N_{Σ}I with the composite noise power N_{Σ} = N_{0}+β, the additive observation noise power N_{0} augmented by the loading factor β = γη/α ≥ 0 adjusted to the regularization parameter α, the Loewner ordering factor γ > 0 of the SFO S[1] and the uncertainty bound η imposed by the MR-WCSP conditional maximization (see [7, 8] for details).
where Q = S^{+}YS defines the MSF measurement statistics matrix independent on the solution $\widehat{\mathbf{b}}$, and different (say P) reconstruction operators {K^{(p)}; p = 1,...,P} specified for P different feasible assignments to the processing degrees of freedom {α, N_{Σ}, A} define the corresponding DEDR-POCS estimators (7) with the relevant SO's {F^{(p)}= K^{(p)}S^{+}; p = 1,...,P}.
4.2. Convergence guarantees
Following the VA regularization formalism [1, 7, 9], the POCS regularization operator $\mathcal{P}$ in (7) could be constructed as a composition of projectors ${\mathcal{P}}_{n}$ onto convex sets ${\u2102}_{n}$; n = 1,...,N with non-empty intersection, in which case the (7) is guaranteed to converge to a point in the intersection of the sets {${\u2102}_{n}$} regardless of the initialization ${\widehat{\mathbf{b}}}_{\left[0\right]}$ that is a direct sequence of the fundamental theorem of POCS (see [7, Part I, Appendix B]). Also, any operator that acts in the same convex set, e.g., kernel-type windowing operator (WO) can be incorporated into such composite regularization operator $\mathcal{P}$ to guarantee the consistency [1]. The RS system-oriented experiment design task is to make the use of the POCS regularization paradigm (5) employing the practical imaging radar/SAR-motivated considerations that we perform in the next section.
4.3. VA-motivated POCS regularization
The second sum on the right-hand side of (10) is recognized to be a 4-nearest-neighbors difference-form approximation of the Laplacian operator ${\nabla}_{\mathbf{r}}^{2}$ over the spatial coordinate r, while m^{(0)} and m^{(1)} represent the nonnegative real-valued scalars that control the balance between two metrics measures defined by the first and the second sums at the right-hand side of (10). In the equibalanced case, m^{(0)} = m^{(1)} = 1, the same importance is assigned to the both metrics measures, in which case (9) specifies the discrete-form approximation to the Sobolev metrics inducing operator $\mathcal{M}={m}^{\left(0\right)}\mathcal{I}+{m}^{\left(1\right)}{\nabla}_{\mathbf{r}}^{2}$ in the relevant continuous-form solution space $B\left(R\right)\u220db\left(\mathbf{r}\right)$, where $\mathcal{I}$ defines the identity operator [2]. Incorporating in (9) ${\mathcal{P}}_{1}$= $\mathcal{M}$ for the continuous model and ${\mathcal{P}}_{1}$ = M for the discrete-form image model, respectively, specifies the consistency-guaranteed anisotropic kernel-type windowing [2, 9] because it controls not only the SSP (image) discrepancy measure but also its gradient flow over the scene.
4.4. DEDR-VA-optimal dynamic SSP reconstruction
For the purpose of generality, instead of relaxation parameter τ and balancing coefficients m^{(0)} and m^{(1)} we incorporated into the PDE (15) three regularizing factors c_{0}, c_{1}, and c_{2}, respectively, to compete between noise smoothing and edge enhancement [2, 9]. These are viewed as additional VA-level user-controlled degrees of freedom.
4.5. Family of numerical DEDR-VA-related techniques for SSP reconstruction
- (i)
The simplest case relates to the specifications: c _{0} = 0, c _{1} = 0, c _{2} = const = -c, c > 0, and Φ (r, r';t) = δ(r - r') with excluded projector ${\mathcal{P}}_{+}$. In this case, the PDE (15) reduces to the isotropic diffusion (so-called heat diffusion) equation $\partial \widehat{b}(\mathbf{r};t)/\partial t=c{\nabla}_{\mathbf{r}}^{2}\widehat{b}(\mathbf{r};t)$. We reject the isotropic diffusion because of its resolution deteriorating nature [1].
- (ii)
The previous assignments but with the anisotropic conduction factor, -c _{2} = c(r; t) ≥ 0 specified as a monotonically decreasing function of the magnitude of the image gradient distribution [4], i.e., a function $c\left(\mathbf{r},\mid {\nabla}_{\mathbf{r}}\widehat{b}\left(\mathbf{r};t\right)\mid \right)$ ≥ 0, transforms the (15) into the anisotropic diffusion (AD) PDE, $\partial \widehat{b}(\mathbf{r};t)/\partial t=c(\mathbf{r};|{}_{\nabla}^{\mathbf{r}}\widehat{b}(\mathbf{r};t)|){\nabla}_{\mathbf{r}}^{2}\widehat{b}(\mathbf{r};t)$, which specifies the celebrated Perona-Malik AD method [4] that sharpens the edge map on the low-resolution MSF images.
- (iii)
For the Lebesgue metrics specification c _{0} = 1 with c _{1} = c _{2} = 0, the PDE (15) involves only the first term at the right-hand side resulting in the locally selective robust adaptive spatial filtering (RASF) approach investigated in details in our previous studies [7, 9].
- (iv)
The alternative assignments c _{0} = 0 with c _{1} = c _{2} = 1 combine the isotropic diffusion with the anisotropic gain controlled by the Laplacian edge map. This approach is addressed as a selective information fusion method [5] that manifests almost the same performances as the DEDR-related RASF method [7].
- (v)
The aggregated approach that we address here as the unified DEDR-VA method involves all the three terms at the right-hand side of the PDE (15) with the equibalanced c _{0} = c _{1} = c _{2} = const (one for simplicity), hence, it combines the isotropic diffusion (specified by the second term at the right-hand side of (16)) with the composite anisotropic gain dependent both on the evolution of the synthesized SSP frame and its Laplacian edge map [2]. This produces a balanced compromise between the anisotropic reconstruction-fusion and locally selective image despeckling with adaptive anisotropic kernel windowing that preserves and even sharpen the image edge map [2].
Next, several RS images formed by different sensor systems or applying different image formation techniques can be aggregated into an enhanced fused RS image employing the NN computational framework [10]. We are now ready to proceed with construction of such NN-adapted DEDR-VA-related techniques.
5. Radar/SAR image enhancement via sensor and method fusion
5.1. Fusion problem formulation
which model the data {q^{(p)}} acquired by P RS imaging systems that employ the image formation methods from the DEDR-VA-related family specified in the previous section. In (17), b represents the original K-D image vector, {Φ^{(p)}} are the RS image formation operators referred to as the PSF operators of the corresponding DEDR-VA-related imaging systems (or methods) where we have omitted the sub index D for notational simplicity, and {ν^{(p)}} represent the system noise with further assumption that these are uncorrelated from system to system.
for the assigned values of the regularization parameters λ. A proper selection of λ is next associated with parametrical optimization [10] of such the aggregated fusion process.
5.2. NN-adapted fusion algorithm
∀k, i = 1,...,K, where we redefined {x_{ k } = b_{ k } } and ignored the constant term E_{const} in E(x) that does not involve the state vector x. The regularization parameters {λ_{ p } } in (22), (23) should be specified by an observer o pre-estimated invoking, for example, the VA inspired resolution-over-noise-suppression balancing method developed in [10, Section 3]. In the latter case, the result of the enhancement-fusion becomes a balanced tradeoff between the gained spatial resolution and noise suppression in the resulting fused enhanced image with the POCS-based regularizing stabilizer.
6. Simulations
where b_{ k } represents the value of the k th element (pixel) of the original SSP, ${\widehat{b}}_{k}^{\left(\mathsf{\text{MSF}}\right)}$ represents the value of the k th element (pixel) of the rough SSP estimate formed applying the conventional low-resolution MSF technique (12), and ${\widehat{b}}_{k}^{\left(p\right)}$ represents the value of the k th element (pixel) of the enhanced SSP estimate formed applying the p th enhanced imaging method (p = 1,...,P), correspondingly. We consider and compare here five (i.e., P = 5) RS image enhancement/reconstruction methods, in which case p = 1 corresponds to the Lee's local statistics-based adaptive despeckling technique [2], p = 2 corresponds to the Perona-Malik AD method [5], p = 3 corresponds to the DEDR-related locally selective RASF technique [7], p = 4 corresponds to the APES method [6], and p = 5 corresponds to the NN-fused RSF and DEDR-VA methods, respectively.
The quality metrics specified by (26) and (27) allow us to quantify the performance of the developed DEDR-VA-related high-resolution reconstructive methods (enumerated above by p = 1,...,P = 5) and, also, the NN fusion quality.
IOSNR values provided with three methods, p = 3, 4, 5
SNR (dB) | IOSNR^{(p)}; p= 3, 4, 5 | |||||
---|---|---|---|---|---|---|
First scenario: κ _{ r } = 10; κ _{ a } = 30 | Second scenario: κ _{ r } = 7; κ _{ a } = 20 | |||||
IOSNR ^{ (3) } | IOSNR ^{ (4) } | IOSNR ^{ (5) } | IOSNR ^{ (3) } | IOSNR ^{ (4) } | IOSNR ^{ (5) } | |
5 | 3.58 | 6.21 | 10.36 | 4.75 | 7.27 | 11.74 |
7 | 4.37 | 7.46 | 12.54 | 5.69 | 8.74 | 12.36 |
10 | 5.45 | 8.27 | 13.23 | 5.94 | 9.57 | 14.75 |
15 | 7.36 | 8.83 | 15.27 | 7.58 | 10.35 | 16.27 |
MAE values provided with three simulated methods, p = 3, 4, 5
SNR (dB) | MAE^{(p)}; p= 3, 4, 5 | |||||
---|---|---|---|---|---|---|
First scenario: κ _{ r } = 10; κ _{ a } = 30 | Second scenario: κ _{ r } = 7; κ _{ a } = 20 | |||||
MAE ^{ (3) } | MAE ^{ (4) } | MAE ^{ (5) } | MAE ^{ (3) } | MAE ^{ (4) } | MAE ^{ (5) } | |
5 | 16.46 | 14.68 | 11.48 | 14.87 | 13.85 | 11.74 |
7 | 14.75 | 13.84 | 10.74 | 13.11 | 11.32 | 9.36 |
10 | 13.48 | 12.27 | 9.66 | 12.47 | 10.86 | 8.75 |
15 | 13.04 | 11.75 | 9.19 | 10.75 | 9.69 | 7.38 |
7. Concluding remarks
The extended DEDR method combined with the dynamic VA regularization has been adapted to the NN computational framework for perceptually enhanced and considerably speeded up reconstruction of the RS imagery acquired with imaging array radar and/or fractional SAR imaging systems operating in an uncertain RS environment. Connections have been drawn between different types of enhanced RS imaging approaches, and it has been established that the convex optimization-based unified DEDR-VA-NN framework provides an indispensable toolbox for high-resolution RS imaging system design offering to observer a possibility to control the order, the type, and the amount of the employed two-level regularization (at the DEDR level and at the VA level, correspondingly). Algorithmically, this task is performed via construction of the proper POCS operators that unify the desirable image metrics properties in the convex image/solution sets with the employed radar/SAR motivated data processing considerations. The addressed family of the efficient contractive progressive mapping iterative DEDR-VA-related techniques has particularly been adapted for the NN computing mode with sensor systems/method fusion. The efficiency of the proposed fusion-based enhancement of the fractional SAR imagery has been verified for the two method fusion example in the reported simulation experiments. Our algorithmic developments and the simulations revealed that with the NN-adapted POCS-regularized DEDR-VA techniques, the overall RS imaging performances are improved if compared with those obtained using separately the most prominent in the literature despeckling, AD or locally selective RS image reconstruction methods that do not unify the DEDR, the VA and the NN-adapted method fusion considerations. Therefore, the developed unified DEDR-VA-NN framework puts in a single optimization frame, radar/SAR image formation, speckle reduction, and adaptive dynamic scene image enhancement/fusion performed in the rapidly convergent NN-adapted computational fashion.
Declarations
Authors’ Affiliations
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