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We consider a problem of high-resolution array radar/SAR imaging formalized in terms of a nonlinear ill-posed inverse problem of nonparametric estimation of the power spatial spectrum pattern (SSP) of the random wavefield scattered from a remotely sensed scene observed through a kernel signal formation operator and contaminated with random Gaussian noise. First, the Sobolev-type solution space is constructed to specify the class of consistent kernel SSP estimators with the reproducing kernel structures adapted to the metrics in such the solution space. Next, the “model-free” variational analysis (VA)-based image enhancement approach and the “model-based” descriptive experiment design (DEED) regularization paradigm are unified into a new dynamic experiment design (DYED) regularization framework. Application of the proposed DYED framework to the adaptive array radar/SAR imaging problem leads to a class of two-level (DEED-VA) regularized SSP reconstruction techniques that aggregate the kernel adaptive anisotropic windowing with the projections onto convex sets to enforce the consistency and robustness of the overall iterative SSP estimators. We also show how the proposed DYED regularization method may be considered as a generalization of the MVDR, APES and other high-resolution nonparametric adaptive radar sensing techniques. A family of the DYED-related algorithms is constructed and their effectiveness is finally illustrated via numerical simulations.

Space-time adaptive processing (STAP) for high-resolution radar imaging with sensor arrays and synthetic aperture radar (SAR) systems has been an active research area in the environmental remote sensing (RS) field for several decades, and many sophisticated techniques are now available (see among others [

Another possible way to alleviate the ill-posedness of the nonlinear radar/SAR imaging problems is to incorporate a priori model considerations regarding the desired geometrical scene image properties into the STAP procedures via performing randomization of the SSP model and application of the Bayesian minimum risk (MR) or maximum

The reminder of the paper 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 compare the ML-APES approach with the DEED-related family of the SSP estimators. The performance guarantees are conceptualized in Section 4. An extension of the VA-based dynamic POCS regularization unified with the DEED paradigm that results in a new proposed DYED framework is addressed in Section 5 followed by some illustrative simulations and discussion in Sections 6 and conclusions in Section 7, respectively.

The general mathematical formalism of the problem at hand and the DEED regularization framework that we employ in this paper are similar in notation and structure to that described in [

In a general continuous-form (functional) formalism, a random temporal-spatial realization of the data field,

It is convenient in the RS applications to assume that due to the integral signal formation model (3), the central limit theorem conditions hold [

The formulation of the data discretization and sampling in this paper follows the experiment design formalism given in [_{l}_{i}_{m}_{l}_{i}_{m}_{(}_{M}_{)} = 𝒫_{𝕌(}_{M}_{)} 𝕌 = Span_{(}_{M}_{)}{_{m}_{𝕌(M)} defined by Equation (

In analogy to Equation (_{k}_{k}_{(}_{K}_{)} = 𝒫_{𝔼(}_{K}_{)} 𝔼 = Span_{(}_{K}_{)}{_{k}_{𝔼(}_{K}_{)}. Note, that to satisfy the observability requirement [_{𝔼(}_{K}_{)} defines a projector onto the _{𝕌(}_{M}_{)}𝒮 for any given/chosen 𝒫_{𝕌(}_{M}_{)} so that the observations (6) contain information of the observable _{𝕌(}_{M}_{)} and 𝒫_{𝔼(}_{K}_{)}. In this paper we adopt the technically inspired fine representation basis formed by a _{x}_{y}_{x}_{y}_{k}_{x}_{y}_{x}_{=} 1,.., _{x}_{y=}_{y}_{x}_{y}_{k}_{k}_{x}_{y}

With the specified decompositions (6), (7), the discrete (vector-form) approximation of the continuous-form EO (1), (3) is given by:
_{m}_{m}_{k}_{e}_{n}_{e}^{+} defines the Hermitian conjugate when stands with a matrix (or a vector). The vector

The nonlinear inverse problem of radar/SAR imaging with the discrete-form measurement data (8) is formulated now as follows: to derive an estimator for the SSP vector _{(}_{j}_{)};

We, first, recall the conventional continuous-form kernel estimator [^{+}^{2} smoothed by the kernel window operator (WO) 𝒲 (^{*} (^{+}, that is, the adjoint SFO. Here, superscript + stands for the adjoint operator in the relevant signal spaces. In the Hilbert signal spaces introduced above, the MSF operator 𝒮^{+} adjoint to the SFO 𝒮 is defined via corresponding inner products (2) as follows, [^{+}_{𝔼} = [𝒮_{𝕌}. For a detailed analysis of this method and the corresponding synthesis of different windows with special scaling and smoothing properties we refer to [

In this Section, we extend the recently proposed high-resolution maximum likelihood-based amplitude phase estimator (ML-APES) [_{u}_{u}

In the APES terminology (as well as in the minimum variance distortionless response (MVDR) [_{k}_{u}_{u}_{[}_{i}_{]}) of the unknown covariance _{u}_{[}_{i}_{]}] with the zero step initialization _{[0]} = _{MSF}

Let us adapt the algorithm (13) to the considered here single snapshot/single look case (^{+}, taking into account the properties of the convergent MVDR estimates of the SSP, which in a coordinate/pixel form are given by
_{diag} returns the vector of a principal diagonal of the embraced matrix. Specifying the ML-APES matrix-form solution operator (SO):

The DEED regularization framework proposed and developed in [

The DEED-optimal SSP estimate _{DEED} represents the adaptive (_{DEED} in Equation (_{S}_{S}

Following the DEED framework [_{DEED} is formalized by the MR-WCSP optimization problem:
_{S}^{+}〉} that measures “how far” is the DEED-optimal SO from the pseudo inverse to the uncertain SFO ^{−1}), and
_{Σ} = _{Σ} (β) = _{n}_{n}_{0}_{S}e_{Σ} = _{0} + β, the additive observation noise power _{0} augmented by the loading factor β = γη/α ≥ 0 adjusted to the regularization parameter α, the Loewner SFO ordering factor γ > 0 of the SFO ^{+} and the self adjoint reconstruction operator _{Σ}^{−}^{1})^{−1} recognized to be the regularized inverse of the discrete-form ambiguity function (AF) matrix operator:

Putting ^{(2)}, ^{(3)} in Equation (^{(2)} and ^{(3)}, respectively. Note that other feasible adjustments of the processing level degrees of freedom {α, _{Σ}, _{Σ} specify the family of relevant POCS-regularized DEED-related (DEED-POCS) techniques (18) represented in the general form as follows:
^{(}^{p}^{)}; _{Σ}, ^{(}^{p}^{)} = ^{(}^{p}^{)}^{+};

The relationship between two high-resolution SSP estimators, the ML-APES (17) and the DEED-POCS (18), can now be established using the second equivalent form for representing the SO (16) given by (Appendix B, [_{DEED} = ^{(2)} specified by Equation (_{S}_{Σ} = _{n}_{0}

Following the DEED-POCS regularization formalism [^{+}, a composition of the MSF operator 𝒮^{+} and the DEED-optimal reconstruction operator 𝒦 given by the continuous-form assymptotic to (23) (subscript _{DEED} is omitted to simplify the notations). To analyze the consistency of the estimator (28), one should consider the large measure [_{Σ} → 0 the operator composition 𝒮 = 𝒦𝒮^{+}𝒮 tends to the identity operator [_{2} Hilbert space, and subscript 𝔐 indicates the measure of the observation domain for which the relevant estimate has been obtained. The ratio of the average fluctuation noise energy in the estimate (28) to the average fluctuation noise energy in the high-resolution sufficient statistics (SS),
_{𝔐} defines the covariance operator of the SS, _{𝔐}(_{𝔐}(_{𝔐}(_{𝔐}(_{𝔐}(_{𝔐}(_{𝔐}(_{𝔐}(^{2}(^{+}} < ∞ that provides

The next crucial performance issue relates to construction of convergent iterative scheme for efficient computational implementation of the POCS regularized DEED-related estimators. To convert such the technique to an iterative procedure we, first, transform the Equation (_{D}^{+} in the considered above rank-1 data covariance matrix case) applying the MSF SO ^{+}; _{[0]} =

Following the POCS regularization formalism [_{n}_{n}_{j}_{n}_{n}_{[0]} that is a direct sequence of the fundamental theorem of POCS (Sec. 15.4.5, [_{n}

The DEED-POCS framework offers a possibility to design the POCS regularization operator 𝒫 in such a way that to preserve high spatial resolution performances of the resulting DEED-related consistent SSP estimates. Following the VA-based image enhancement approach [_{(}_{K}_{)} ∋ ^{(0)} and ^{(1)} are the nonnegative real-valued scalars that control the balance between two metrics measures in Equation (^{(0)} = 1, ^{(1)} = 0, then (36) reduces to the conventional Lebesgue 𝕃_{2} metrics in the Hilbert space 𝔹(^{(0)} = 0, ^{(1)} = 1, the metrics Equation (^{(0)} = ^{(1)} = 1, the same importance is assigned to the both metrics measures specified by the kernel metrics inducing operator . Incorporation of such metrics inducing operator as the WO into the general DEED-optimal technique (28),

To proceed with designing the related WO adapted to the discrete problem model, the relevant 𝔹_{(}_{K}_{)} ∋ _{k}_{x}_{y}_{x}_{y}_{x=}_{x}_{y=}_{y}; k_{x}_{y}

The second sum on the right hand side of Equation (^{2} is the numerically approximated Laplacian operator
^{2} to a vector ^{2}_{x}_{y}_{x}_{y}^{(0)} = ^{(1)} = 1.

Last, we restrict the solution subspace (the so-called active solution set or correctness set in the DEED terminology [_{+} ⊂ 𝔹_{(}_{K}_{)} of SSP vectors with nonnegative elements (as power is always nonnegative). This is formalized by specifying the projector 𝒫_{+} onto such convex set 𝔹_{+} _{1} = _{2} = 𝒫_{+} (in the function image space, 𝔹(_{1} = ).

With the model (40), the discrete-form contractive progressive mapping iterative process (33) transforms into:
_{[0]} =

Associating the iterations _{b̂}_{D}_{[}_{i}_{]} in Equation (

Three practically inspired versions of Equation (

= specifies the conventional Lebesgue metrics, in which case the evolution process (44) does not involve control of the image gradient flow over the scene.

For the purpose of generality, instead of two metrics balancing coefficients ^{(0)} and ^{(1)} we incorporated into the PDE (45) three regularizing factors _{0}, _{1} and _{2}, respectively, viewed as VA-level user-controllable degrees of freedom to compete between smoothing and edge enhancement. Although due to the solution-depended nature the dynamic DEED-VA scheme in its continuous PDE form (45) cannot be addressed as a practically realizable procedure, the undertaken theoretical developments are useful for establishing the relationship between the general-form VA scheme (45) and the already existing dynamic image enhancement approaches [

Different feasible assignments to the processing level degrees of freedom in the PDE (45) specify different VA-related procedures. Here beneath we consider the following ones:

The simplest case relates to the specifications: _{0} = 0, _{1} = 0, _{2} = const = – _{+}. In this case, the PDE (45) reduces to the

The previous assignments but with the anisotropic factor, − _{2} = _{r}

For the Lebesgue metrics specification _{0} = 1 with _{1} = _{2} = 0, the PDE (45) involves only the first term at its right hand side. This case leads to the locally selective

The alternative assignments _{0} = 0 with _{1} = _{2} = 1 combine the isotropic diffusion with the anisotropic gain controlled by the Laplacian edge map. This approach addressed in [

The VA-based approach that we address here as the DEED-VA-fused DYED method involves all three terms at the right hand side of the PDE (45) with the equibalanced weights, _{0} = _{1} = _{2} =

The discrete-form approximation of the PDE (45) in “iterative time” {_{[0]} = ^{2}{·} applied to the embraces quantity is defined by the 4-nearest-neighbors difference-form approximation of the continuous Laplacian operator
_{1} = _{2} = 0 in Equation (

_{+}. Nevertheless (as it is frequently observed with nonlinear iterative processes [

In the simulation experiment, we considered a fractional SAR as a sensor system, analogous to a single look fraction of a multi look focused SAR [_{a}_{θ}_{θ} = arctan(Δ_{s}_{s}_{r}_{τ}_{τ} = 2Δ_{s}_{s}_{s}_{s}_{Ψ} is a normalizing constant, not essential in the simulations [_{s}_{a}_{0} is the wavelength of the radar signal transmitted, and _{a}_{a}_{a}_{max}, a fraction _{a}_{max} ≈ _{0}_{s}_{A}_{A}_{a}_{a}_{a}_{a}_{s0} [_{r}_{r}_{r}_{s}_{s}^{2}-type _{r}_{s}_{s}_{r}_{r}_{r}_{a}_{r}_{a}

Next, to comply with the technically motivated MSF fractional image formation mode, the blurred scene image was degraded with the composite (signal-dependent) noise simulated as a realization of
_{SAR} defined as the ratio of the average signal component in the degraded image ^{(1)} formed using the MSF algorithm (32) to the relevant composite noise component in that same speckle corrupted MSF image.

For objective evaluating of the reconstructive imaging quality, we have adopted two quality metrics traditionally used in image restoration/enhancement [_{x}_{y}^{(}^{p}^{)} (_{x}_{y}

The second employed metrics is the so-called improvement in the output signal-to-noise ratio (_{2}^{(1)} is the low-resolution speckle-corrupted image formed by a fractional SAR system that employs the conventional MSF method (32), and {^{(}^{p}^{)}} represents the SSP reconstructed from the corrupted MSF image

_{r}_{a}_{r}_{a}

Next, _{0} = _{1} = _{2} = 1, after the same 30 iterations are presented in

From the reported simulation results, the advantage of the well-designed imaging experiments (cases of the ML-APES and the optimal DEED-VA techniques) over the poorer design enhancement experiments (MSF and anisotropic diffusion (VA-AD) without ML-APES reconstruction) is evident for both scenarios. Due to the performed regularized inversions, the resolution was substantially improved. Quantitative performance improvement measures are reported in

The highest values of the

In this paper, we have addressed the unified DYED method for nonparametric high-resolution adaptive sensing of the spatially distributed scenes in the uncertain RS environment that extends the previously developed DEED regularization framework via its aggregation with the dynamic VA-based enhanced imaging approach. We have treated the RS imaging problem in an array radar/SAR adapted statement. The scene image is associated with the estimate of the SSP of the scattered wavefield observed through the randomly perturbed kernel SFO under severe snapshot limitations resulting in a degraded speckle corrupted RS image. The crucial issue in treating such a nonlinear ill-posed inverse problem relates to the development of a statistical SSP estimation/reconstruction method that balances the resolution enhancement with noise suppression and guaranties consistency, convergence and robustness of the resulting STAP procedures. In the addressed experiment design setting, all these desirable performance issues have been formalized via inducing the corresponding Sobolev-type metrics structure in the solution/image space, next, constructing the DEED-balanced resolution-enhancement-over-noise-suppression objective measures and, last, solving the relevant SSP reconstruction inverse problem incorporating the two-level regularization (the DEED level and the VA level, respectively). Furthermore, the incorporation of the second-level VA-based dynamic POCS regularization not only speeds up the related iterative processing procedures but provides also perceptually enhanced imagery. Also, the developed DYED method is user-oriented in the sense that it provides a flexibility in specifying some design (regularization) parameters viewed as processing-level degrees of freedom, which control the type, the order and the amount of the employed two-level regularization producing a variety of DYED-related techniques with different operational performances and complexity. Simulations verified that the POCS-regularized DEED-VA-optimal DYED technique outperforms the most prominent methods in the literature based on the ML and VA approaches that do not unify the DEED framework with the POCS-based convergence enforcing dynamic regularization in the corresponding applications.

Ambiguity function

Adaptive spatial filtering

Descriptive experiment design

Dynamic experiment design

Equation of observation

Maximum likelihood

Matched spatial filtering

Partial differential equation

Projections onto convex sets

Point spread function

Robust ASF

Remote sensing

Synthetic aperture radar

Signal formation operator

Spatial spectrum pattern

Space-time adaptive processing

Variational analysis

Window operator

Original scene (not observable in the radar imaging experiment).

Simulation results for the first scenario: (_{r}_{a}_{SAR} = 15 dB); (

Simulation results for the second scenario: (_{r}_{a}_{SAR} = 10 dB); (

Comparative analysis of the 1-D imaging results: (_{r}_{a}_{SAR} = 15 dB); (

Comparative analysis of the 1-D imaging results: (_{r}_{a}_{SAR} = 10 dB); (

Quantitative reconstructive imaging performances for the first simulated operational scenario: (

Quantitative reconstructive imaging performances for the second simulated operational scenario: (

Convergence rates evaluated via: (_{SAR} = 15 dB.