 Research Article
 Open Access
 Published:
Experiment Design RegularizationBased Hardware/Software Codesign for RealTime Enhanced Imaging in Uncertain Remote Sensing Environment
EURASIP Journal on Advances in Signal Processing volume 2010, Article number: 254040 (2010)
Abstract
A new aggregated Hardware/Software (HW/SW) codesign approach to optimization of the digital signal processing techniques for enhanced imaging with realworld uncertain remote sensing (RS) data based on the concept of descriptive experiment design regularization (DEDR) is addressed. We consider the applications of the developed approach to typical singlelook synthetic aperture radar (SAR) imaging systems operating in the realworld uncertain RS scenarios. The software design is aimed at the algorithmiclevel decrease of the computational load of the largescale SAR image enhancement tasks. The innovative algorithmic idea is to incorporate into the DEDRoptimized fixedpoint iterative reconstruction/enhancement procedure the convex convergence enforcement regularization via constructing the proper multilevel projections onto convex sets (POCS) in the solution domain. The hardware design is performed via systolic array computing based on a Xilinx Field Programmable Gate Array (FPGA) XC4VSX3510ff668 and is aimed at implementing the unified DEDRPOCS image enhancement/reconstruction procedures in a computationally efficient multilevel parallel fashion that meets the (near) realtime image processing requirements. Finally, we comment on the simulation results indicative of the significantly increased performance efficiency both in resolution enhancement and in computational complexity reduction metrics gained with the proposed aggregated HW/SW codesign approach.
1. Introduction
In this paper, we address a new aggregated Hardware/Software (HW/SW) codesign approach to optimization of the digital signal and image processing techniques as required for enhanced remote sensing (RS) of the environment with the use of highresolution array radar and synthetic aperture radar (SAR) systems. At the algorithmdesign level, the RS imaging problem is treated as an illposed nonlinear inverse problem of reconstruction of the spatial spectrum pattern (SSP) of the backscattered field distributed over the remotely sensed scene via processing the SAR data signals distorted in the uncertain stochastic measurement channel [1–6]. The operational scenario uncertainties are attributed to inevitable random signal perturbations in inhomogeneous propagation medium [1, 2, 4], possible imperfect radar/SAR system calibration [1, 3], and SAR carrier trajectory deviations [3, 5, 6]. The unified approach that we address to solve such a problem is based on the recently proposed concept of descriptive experiment design regularization (DEDR) [7, 8]. The general DEDR method constructed in [7, 8] incorporates into the minimum risk (MR) nonparametric estimation strategy [4] the experiment designmotivated constraints of the image identifiably for the discreteform signal formation operator (SFO) specified by the employed signal modulation format [4–6]. On one hand, a considerable advantage of such DEDR paradigm relates to its flexibility in designing the desirable error metrics in the corresponding image representation space via defining different descriptive cost functions [7, 9]. On the other hand, the crucial limitations of the DEDR method relate to the necessity of performing simultaneously the solutiondependent SFO inversion operations and adaptive adjustments of the degrees of freedom of the overall DEDR image enhancement techniques ruled by the employed fixedpoint iterative process [8]. For the realworld largescale RS scenes, such adaptive fullformat DEDRoptimal method turns out to be computationally extremely consuming, therefore cannot be recommended as a practical technique realizable in (near) real time [10]. The innovative idea of this paper is to aggregate the DEDRoptimal fixedpoint iterative reconstruction/enhancement procedures developed in the previous studies [7, 8, 10] with the multilevel robustness and convergence enforcing regularization via constructing the proper projections onto convex sets (POCS) in the solution domain. The established POCSregularized iterative DEDR technique is performed separately along the range and azimuth directions over the scene frame making an optimal use of the rangeazimuth sparseness properties of the employed radar/SAR modulation format. Thus, at the SW codesign stage, we address two conceptually innovative propositions that distinguish our approach from the previous studies [7, 10]. First, two possible observation scenarios (instead of one) are unified now under the DEDR paradigm for the HW/SW codesign, namely, (i) regular case without model uncertaintiess and (ii) uncertain scenario with random perturbations in the SFO. Second, the POCS regularization is proposed to be performed in an aggregated multilevel fashion to make the optimal use of the nontrivial RS system model information for constructing the corresponding robustness and convergence enforcing POCS operators. In particular, the positivity and rangeazimuth orthogonalization projectors of [10] are aggregated with the point spread function (PSF) sparseness enforcing sliding window projectors acting in parallel over both range and azimuth image frames that set the corresponding PSF pixel values to zeroes outside their specified support regions. Such aggregated POCS regularization drastically speeds up the resulting fixedpoint iterative DEDR techniques making them exactly well fitted for the systolic computational implementation form; that is, provides the SW algorithmic base for the further HW codesign level of the problem treatment.
At the HW codesign stage, we propose to pursue the SystemonChip (SoC) single Field Programmable (FP) unit integration approach [9–14], which allows efficient coupling/integration of a number of predetermined complex components. Such a programmable unit is a viable solution for rapid prototyping and digital implementation of the radar/SAR image enhancement techniques developed at the SW codesign stage, in spite of designing the process in a common personal computer (PC) [11–14]. The main advantage of the proposed FPSoC platform is that all required component designs, including the embedded processor unit, memory, and peripherals are algorithmically "adapted" for the particular developed POCSregularized iterative fixedpoint DEDR image enhancement techniques. Therefore, at the HW design stage, the novel contribution of this study is twofold: first, the addressed HW/SW codesign methodology is aimed at an HW implementation of the developed software using systolic arrays as coprocessors units; second, the proposed systolicbased processing architecture is particularly adapted for computational implementation of the unified DEDRPOCS techniques in a computationally efficient fashion that meets the (near) realtime overall RS imaging system requirements. We resume this study with the analysis of the simulation results related to enhancement of the realworld degraded largescale SAR imagery (i.e., acquired in uncertain operational scenarios) indicative of the significantly increased reconstruction efficiency gained with the proposed HW/SW codesign approach.
2. Background
2.1. ContinuousForm Problem Model
The general formalism of the RS imaging problem presented in this paper is a structural extension of the problem considered in [4, 7, 8], hence some crucial model elements are repeated for convenience to the reader. Consider a coherent RS experiment in a random medium and the narrowband SAR assumption [1] that enables us to model the extended object backscattered wavefield in the baseband format [3] by imposing its time invariant complex scattering (backscattering) function in the object image domain (scattering surface) X ∋ x. The measurement radar/SAR data field consists of the echo signals s and additive noise n and is available for observations and recordings within the prescribed timespace observation domain , where defines the time(t)space() points in Y; The model of the data field u is defined by specifying the stochastic equation of observation (EO) that in the conventional integral form may be written as [4, 8]
The random kernel of the perturbed random signal formation operator (SFO) given by (1) defines the signal field formation model. Its mean is referred to as the nominal modulation law in the data formation channel defined by the timespace modulation of signals employed in a particular imaging radar/SAR system [3], and the variation about the mean models the stochastic perturbations of the wavefield at different propagation paths, where represents the zeromean multiplicative noise that models random propagation perturbations in the medium (the socalled general Rytov model [3, 5, 6]). The fields in (1) are assumed to be zeromean complex valued Gaussian random fields [3]. Next, we assume an incoherent nature of the backscattered field . This is naturally inherent to the RS experiments [1, 3, 5, 6] and leads to the form of the object field correlation function, , where (x) and b(x) are referred to as the object random complex scattering function and its average power scattering function or spatial spectrum pattern (SSP), respectively.
The problem of enhanced RS imaging is to develop a signal processing method for performing the efficient estimation of the SSP b(x) by processing the available radar/SAR measurements of the data wavefield u( y ). Such estimate of the SSP b(x) is referred to as the desired reconstructed RS image of the remotely sensed scene.
2.2. DiscreteForm Problem Model
Now we proceed from the stochastic integralform EO (1) to its finitedimensional discrete (vector) form approximation [8]
in which the disturbed SFO matrix
is the discreteform approximation of the integral SFO defined by the EO (1), and e, n, u represent zeromean vectors composed of the decomposition (sampling) coefficients and , respectively [7]. 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), , and + R _{ n }, respectively, where defines the averaging performed over the randomness of characterized by the probability density function p() unknown to the observer, and superscript ^{+} stands for Hermitian conjugate (conjugate transpose). Vector b is composed of the elements b _{ k }; k 1,, K, and is referred to as a KD vectorform representation of the SSP. The SSP vector b is associated with the socalled lexicographically ordered image pixels [7, 9]. The corresponding conventional K _{ y }K _{ x } rectangular frameordered scene image B relates to its lexicographically ordered vectorform representation b via the standard row by row expansion (socalled lexicographical reordering) procedure, BL [9]. Note that in the simple case of a certain operational scenario [1, 3, 7], the discreteform (i.e., matrixform) SFO S is assumed to be deterministic, in which case the random perturbation term in (3) is irrelevant, .
The digital enhanced RS imaging problem is formally stated as follows: to reconstruct the scene pixel frame image via lexicographical reordering L of the SSP vector estimate estimated from whatever available discrete measurements of the recorded radar/SAR data u. The reconstructed SSP vector is an estimate of the secondorder statistics of the scattering vector e observed through the perturbed SFO (3) contaminated with additive noise n and corrupted also with the signaldependent multiplicative noise, hence, the enhanced RS imaging problem at hand must be qualified and treated as a statistical nonlinear inverse problem with model uncertainties. The highresolution sensing implies formation of the RS image based on some statistically optimal solution of such an inverse problem robust against the problem model uncertainties. In this paper we propose to unify the POCS regularization with the DEDR method originally developed in [7, 8].
3. Unified DEDR Method
3.1. DEDR Strategy for Certain Operational Scenario
In the descriptive statistical formalism, the desired SSP vector is recognized to be the vector of a principal diagonal of the estimate of the correlation matrix R _{ e }(b); that is, . Thus one can seek to estimate given the data correlation matrix R _{ u } preestimated empirically via averaging recorded data vector snapshots ; for example, [7]
by determining the solution operator (SO) F such that
where defines the vector composed of the principal diagonal of the embraced matrix.
To optimize the search for F in the certain operational scenario, the DEDR strategy was proposed in [7] as
that implies the minimization of the weighted sum of the systematic and fluctuation errors in the desired estimate where the selection (adjustment) of the regularization parameter and the weight matrix A provide the additional experiment design degrees of freedom incorporating any descriptive properties of a solution if those are known a priori [3, 7]. It is easy to recognize that the strategy (6) is a structural extension of the statistical minimum risk estimation strategy [4] for the nonlinear spectral estimation problem at hand because in both cases the balance between the gained spatial resolution and the noise suppression in the resulting estimate is to be optimized.
3.2. Extended DEDR Strategy for Uncertain Scenario
To optimize the search for the desired SO F in the uncertain operational scenario with the randomly perturbed SFO (3), the extended DEDR strategy was proposed in [8] as
where the conditioning term (9) represents the worstcase statistical performance (WCSP) regularizing constraint imposed on the unknown secondorder statistics of the random distortion component of the SFO matrix (3), and the DEDR "extended risk" is defined by
where the regularization parameter and the metrics inducing weight matrix A compose the processing level "degrees of freedom" of the DEDR method.
To proceed with the derivation of the robust SO (8), in [8], the risk function (10) was next decomposed and evaluated for its the maximum value applying the CauchySchwarz inequality and Loewner ordering [9] of the weight matrix with the scaled Loewner ordering factor min. With these robustifications [8], the extended DEDR strategy (8) is transformed into the following optimization problem:
with the aggregated DEDR risk function [8]
where
3.3. DEDROptimal Solution Operators
Examining the DEDR strategies (6) and (11) one can deduce that those both are structurally similar and differ only by the definition of the second (i.e., noise) risk component terms in (7) and (12). In the certain operational scenario [5–7], the trace for the noise error measure is used, while in the uncertain scenario [8] the augmented measure tris employed with the diagonal loaded extension (13) of the composite noise correlation matrix . The established structural similarity (the socalled problem model homomorphism [5, 6]) of two DEDR problems (6) and (11) makes it possible to unify the solutions for both scenarios. Doing so, we specify the SOs for both considered operational scenarios, namely:

(1)
SO for certain operational scenario follows directly from the solution to the optimization problem (6) found in the previous study [7] that results in
(14)where
(15)represents the socalled regularized reconstruction operator [7]; is the noise whitening filter, and the adjoint (i.e., Hermitian transpose) SFO defines the matched spatial filter in the conventional signal processing terminology [1, 3];

(2)
SO for uncertain operational scenario follows as structural extension of (14) for the augmented (diagonal loaded) that yields [8]
(16)where
(17)represents the robustified reconstruction operator for the uncertain scenario.
3.4. DEDRRelated Imaging Techniques
Here we exemplify three practically motivated DEDRrelated imaging techniques [7, 8], that will be used at the HW codesign stage, namely, the conventional matched spatial filtering (MSF) method, and two highresolution reconstructive imaging techniques: (i) the robust spatial filtering (RSF), and (ii) the robust adaptive spatial filtering (RASF) methods.

(1)
MSF. The MSF algorithm is a member of the DEDRrelated family [7] specified for , that is, the case of a dominating priority of suppression of noise over the systematic error in the optimization problem (6). In this case, the SO (14) is approximated by the matched spatial filter (MSF) [7]:
(18) 
(2)
RSF. The RSF method implies no preference to any prior model information (i.e., I) and balanced minimization of the systematic and noise error measures in (9), (11) by adjusting the regularization parameter to the inverse of the signaltonoise ratio (SNR). In that case the SO becomes the Tikhonovtype robust spatial filter (RSF) [7]:
(19)in which the RSF regularization parameter _{ RSF } is adjusted to a particular operational scenario model, namely, (/) for the case of a certain operational scenario [7], and (/) in the uncertain operational scenario case [8], respectively, where represents the white observation noise power density, is the average a priori SSP value, and + corresponds to the augmented noise power density in the correlation matrix specified by (13).

(3)
RASF. In the Bayesian statistically optimal problem treatment, and A are adjusted in an adaptive fashion following the Bayesian minimum risk strategy [8], that is, diag(), the diagonal matrix with the estimate at its principal diagonal, in which case the SOs (14), (16) become itself solutiondependent operators that result in the following robust adaptive spatial filters (RASFs):
(20)for the certain operational scenario [7], and
(21)for the uncertain operational scenario [8], respectively. Next, in all practical RS scenarios [1–3] (and specifically, in SAR uncertain imaging applications [2, 7, 8]), it is a common practice to accept the robust white additive noise model, that is, , attributing the unknown correlated noise component as well as multiplicative speckle noise to the composite uncertain noise term in (2), in which case I with the composite noise power density , the initial observation noise variance augmented by the loading factor specified by (13).
Using the defined above SOs, the DEDRrelated data processing techniques in the conventional pixelframe format can be unified now as follows:
with = ; , and , , respectively. Any other feasible adjustments of the DEDR degrees of freedom (the regularization parameters , , and the weight matrix A) provide other possible DEDRrelated SSP reconstruction techniques, that we do not consider in this paper.
4. POCS Regularized DEDR Method
Because of the extremely high dimension 10^{12} of the operator inversions required to form the corresponding SOs specified by (20), (21), it is questionable to recommend the generalform DEDRoptimal method (22) as a practical enhanced RS imaging technique realizable in (near) real computational time. Hence, one has to proceed from the conventionalform dimensional RSF and RASF algorithms (that require cumbersome operator inversions (20)–(22) to more computationally efficient iterative techniques that do not involve the largescale operator inversions and incorporate the convergence enforcement regularization into the DEDR procedure via constructing the proper projections onto convex sets (POCS) in the solution domain. In the considered here RS imaging applications, such POCS is aimed at performing the factorization of the overall procedures over the orthogonal range (y)azimuth (x) coordinates in the scene frame making also an optimal use of the sparseness properties of the employed radar/SAR modulation format. Thus, the innovative idea is to perform the POCS regularization in an aggregated multilevel fashion. In particular, we propose to aggregate the positivity and rangeazimuth orthogonalization projectors constructed previously in [10] with the point spread function (PSF) sparseness enforcing sliding window projectors acting in parallel over both range and azimuth image frames that set the corresponding PSF pixel values to zeroes outside their specified support regions. In this section, we address such a unified multilevel POCSregularized iterative DEDR method as an extension of the previously proposed singlelevel DEDRPOCS [10] that we develop here in two stages.
4.1. First Stage: FixedPoint Iterative DEDR Algorithm
The first stage is a structural extension of the fixedpoint method considered in [10], the extension being done for the case of the unified SOs specified now by (14) and (16). Thus, following the fixedpoint algorithm design scheme of [10, Section 3], we first, specify a sequence of the iterative DEDRPOCS estimates
where is a convergence enforcing projector (i.e., the POCSregularizing operator) that will be construct at the second design stage (in the next subsection). In (23),
represents the selfadjoint reconstruction operator at the i th iteration step, and
is the nominal system point spread function (PSF) operator (a matrix). Applying routinely the fixedpoint technique [9, 10] to (23), we derive the desired extended POCSregularized iterative SSP estimation algorithm
Here,
represents the solutiondependent matrixform iteration operator, in which
denotes the ShurHadamar (elementbyelement) matrix product, and the zerostep iteration
is formed as an outcome of the MSF algorithm from the DEDR family (22) specified for the adjoint SFO solution operator .
4.2. B. Second Stage: Multilevel POCS Regularization
Next, to specify the regularizing POCS projector operator in the fixedpoint algorithm (26) let us make the use of factorization of the PSM (25) over the azimuth (x) and range (y) coordinates valid for all existing imaging radar/SAR systems [2, 3, 9]. Such factorization is illustrated in Figure 1. We formalize this stage by introducing the rangeazimuth factorization operator , the same one as in the previous POCS regularization considered in [10]. Next, to make a use of the intrinsic sparseness properties of the SAR point spread functions over the range and azimuth frames, we propose to incorporate the new POCS regularization stage via constructing the xy factorized projection operator (algorithm) that acts as a composition of the orthogonal sliding windows [9] with the window apertures adjusted to the PSM widths: (i) 2_{ a } specifies the azimuth window frame adjusted to the effective pixel width of the nonzero strip _{ a }(x) of the azimuth PSM _{ a } along the x axis; (ii) 2_{ r } specifies the range window frame adjusted to the effective pixel width of the nonzero strip of the range PSM along the y axis, respectively, as illustrated in Figure 1. Such the sliding window projector is an easytoimplement numerical algorithm [9] that simply sets the pixels values to zero outside the support regions 2_{ a }K _{ x } and 2_{ r }K _{ y } around every particular pixel ; k _{ x }; k _{ v } in the rectangular image frame L separately reconstructed via (26) along the corresponding x and y axes, respectively. Last, following [10], to enforce prior knowledge on the intrinsic positivity of the SSP we impose, in addition to and , the positivity operator (algorithm) that has the effect of clipping off all negative values [8]. The defined above orthogonal projecting window and positivity operator are projectors onto convex sets, that is POCS operators [9], thus a composition
is a POCS operator as well. While this definition in the terms of the proposed aggregated projections sounds complicated, the algorithmic meaning of (31) is very simple and is easily established in the algorithmic form familiar to the signal processing and RS communities. Acting on a (that may be not a member of the convex set at a particular iteration ), the applied to produces the member of the convex cone set composed of nonnegative elements that is nearest to in the sense of minimization of the norm [9, Section ].
Now, the application of the constructed by (31) to the iteration process (26) with the corresponding lexicographical reordering L yields the desired resulting POCSregularized fixed point update rule
in which the zerostep iteration L is formed using the conventional (i.e., lowresolution) MSF imaging algorithm (30), the aggregated convergence enforcing POCS regularizing operator is constructed by (31), and the matrixform fixedpoint iteration operator is specified by (27).
We address such POCSregularized DEDR technique (32) as the unified DEDRPOCS method. Its general framework is presented in Figure 2. Note that the fixedpoint process (32) does not involve the cumbersome operator inversions (in contrast to the initial DEDR techniques defined by (5), (22) and, moreover, it is performed separately along the range (y) and azimuth (x) directions making an optimal use of the PSM sparseness properties . These features of the POCSregularized RSF and RASF algorithms generalized by (32) result in the drastically decreased algorithmic computational complexity (e.g, ~ 10^{3} 10^{4} times at each iteration for the typical largescale SAR image formats [1, 2]) that we will verify and analyze in more details further on in Section 6.
4.3. DEDRPOCS Convergence
We accomplish our algorithmic developments at the SW codesign stage with the analytical analysis of the convergence issues related to the developed unified DEDRPOCS method. Following the POCS regularization formalism [9], the convergence enforcing projectors in the iterated procedure (32) are to be constructed formally as
where 1, 2, 3 represent the relaxation (speedingup) regularization parameters and is the identity operator. The iteration rule (32) for the composed regularizing projectors (33) becomes
and is guaranteed to converge to the point in the intersection of the convex sets specified by provided for all 1, 2, 3 regardless of the initialization that is a direct sequence of the fundamental theorem of POCS [9, page 1066]. Note that the employed specifications of the projectors in (33), that is, ; ; ; with 1 for all and L, satisfy these POCS convergence conditions, in which case the formal convergent POCS procedure (34) becomes the developed above fixedpoint DEDRPOCS algorithm given by (32).
Now we are ready to proceed with the hardware codesign implementation stage of our development.
5. Hardware/Software Codesign Methodology
The allsoftware execution of the prescribed RS image formation and reconstruction operations in modern highspeed personal computers (PC) or any existing digital signal processors (DSP) may be intensively time consuming [15]. These high computational complexities of the generalform DEDRPOCS algorithms make them definitely unacceptable for real time PCaided implementation.
When a coprocessorbased solution is employed in the HW/SW codesign architecture, the computational time can be drastically reduced [16]. As an introductive example, consider computation of the matrix product AB, where A and B define matrices of sizes km and mp, respectively. Then to execute this product in a conventional sequential way, kmp multiply accumulation (MAC) operations are required. Therefore, the computational time required by a sequential processor or a highspeed PC for the allsoftware execution of the matrix product is of the order With the incorporation of a parallel and/or pipelined coprocessor alongside an embedded processor the required computational time is immediately reduced to O( where n defines the employed parallelism level.
In this section, we present a concise description of the proposed HW/SW codesign approach particularly adapted to the DEDRPOCS type algorithms, and demonstrate its flexibility in performing an efficient HW implementation of the SW processing tasks with systolic arrays as coprocessors units. In [10], we presented an initial version of the HW/SWarchitecture for implementing the digital processing of a largescale RS imagery in other operational context. The architecture developed in [10] did not involve systolic arrays and is considered here simply as a reference for the new pursued HW/SW codesign paradigm presented in Figure 3, where the corresponding blocks are to be designed to speedup the digital signal processing operations of the DEDRPOCSrelated algorithms developed at the previous SW stage of the overall HW/SW to meet the real time imaging system requirements. Our codesign methodology encompasses the following general stages: (i) algorithmic implementation (reference simulation in the MATLAB platform); (ii) computational tasks partitioning process (definition of the number of coprocessors), and (iii) operational mapping process employed to map the computation execution tasks onto the HW blocks (reconfigurable arrays).
In the HW design, we use the precision of 32 bits for performing all fixedpoint operations, in particular, 9bit integer and 23bits decimal for the implementation of each coprocessor. Such the precision guarantees numerical computational errors less than 10^{5} referring to the MATLAB Fixed Point Toolbox [17]. Using such the MATLAB fixedpoint toolbox we generated all the numerical test sequences required to verify computationally the proposed HW/SW codesign methodology (i.e., test sequences for performing the SW simulation and for the HW verifications). The results of such SW simulation and HW performance analysis will be presented and discussed further on in Sections 6.3 and 6.4. Finally, the host processor (the standard MicroBlaze embedded processor [18] in this study) performs the following functions: loading and storing of images, data transfer to the HW coprocessors, and data formatting for performing the correspondent mathematical operations.
5.1. Algorithmic Implementation
In this section, we develop the procedures for computational implementation of the DEDRPOCSrelated RSF and RASF algorithms in the MATLAB platform. This reference implementation scheme will be next compared with the proposed HW/SW codesign architecture based on the use of the single Field Programmable Gate Array chip.
To implement the iterative fixedpoint DEDRPOCSrelated RSF and RASF algorithms (32), we first, specify the corresponding computational procedures in the rectangular scene frame r∈ over the azimuth (horizontal axis, x) and range direction (vertical axis, y), respectively. Such multistage procedures are formalized via the unified algorithmic scheme presented in Table 1.
From the analysis of the algorithmic implementation scheme of Table 1, we outline the following important remarks regarding the possible HW/SW partitioning of the computational tasks required for implementing both RSF and RASF algorithms.

(i)
First, the PSMs (25), and factorized over the azimuth and range axes can be calculated concurrently that we refer to as , where symbol specifies now the concurrent execution of the corresponding computational operations.

(ii)
Second, the zero step iteration (MSF image) can be computed using the same factorized structure analogues to .

(iii)
Third, the reconstructed image , at the current (i1)st iteration step is an iteratively updated function of computed at the previous i th iteration that also admits the factorized computing.
5.2. Partitioning Phase
One of the challenging problems of the HW/SW codesign is to perform an efficient HW/SW partitioning of the computational tasks. The goal of the partitioning stage is to find which computational tasks can be implemented in an efficient parallelized HW/SW architecture seeking for balanced areatime tradeoffs between different admissible design solutions [18–20]. In this study, the iterative fixedpoint POCSDEDR regularized algorithm has been partitioned at the algorithmic level to minimize the overall signal processing (SP) time via transferring some required reconstructive SP functions from the SW to the HW. The solution to this problem requires, first, the definition of a partitioning model that meets all the specification requirements (functionality, goals and constraints).
The system partitioning is clearly influenced by the target architecture onto which the HW and the SW will be mapped. The target architecture proposed in this study consists of one 32 bits RISC instruction set embedded processor (MicroBlaze) running the software and three dedicated coprocessors implemented by systolic processor arrays.
We begin with the specifications of the systemlevel partitioning functions and detailing the selected design quality attributes for the HW/SW codesign aimed at the definition of the computational tasks that can be implemented in a systolic computing form, namely: hardware area (ha), hardware execution time (ht), software execution time (St), and the selected system resolution (n); where maxha, maxht and maxSt represent the upper bounds of these constraints. In particular, for implementing the iterative fixedpoint POCSregularized RSF and RASF algorithms, the partitioning process must satisfy the following performance requirements.

(i)
In order to ensure a viable solution, the system must always satisfy the constraints: max ha, max ht, for each i th hardware coprocessor and max St, for the embedded processor . These three hardware coprocessors and the embedded processor compose the target architecture , for the preselected FPGA with the corresponding predetermined architecture constraints : [18].

(ii)
Each block implementation must satisfy the predefined execution time performance requirements [18]: and conditioned by the specified above architecture constraints , and , correspondingly.
Next, the system architecture is to be specified to meet the desirable time consuming performances via bounding the total expected system processing time evaluated by
where represents the execution time required for implementing the corresponding DEDRPOCSrelated RSF and RASF algorithms in the standard MATLAB computational environment.
Following such partitioning paradigm, we decompose now the fixedpoint POCSregularized RSF and RASF algorithms developed at the SWdesign into the standard MicroBlaze embedded processor with three coprocessors as illustrated in Figure 4. The first coprocessor (referred to as MSF coprocessor) implements the zerostep iteration to form the MSF image specified by (30). The second coprocessor (referred to as PSM coprocessor) implements the computations of the PSM given by (25) concurrently over the azimuth and the range directions. The third coprocessor (referred to as Iterative POCS coprocessor) performs the required robust updating () for implementing the RSF algorithm and the adaptive updating () for implementing the POCSregularized RASF image reconstruction algorithm, respectively. All three coprocessors are next implemented as systolic processor arrays while the embedded processor executes all the required operational and control functions: loading and storing of the images, data transfer to the HW coprocessors, and data formatting for execution of all required numerical operations.
5.3. Mapping Phase
In this section, we proceed with the development of the procedure for mapping the corresponding algorithms onto array processors. A systolic array consists of a number of processor elements (PEs) with the corresponding interconnection links among the PEs, and the mapping technique transforms a space representation into a spacetime representation [21]. Systolic arrays are being used for matrix operations and required specific processing algorithms, such as, transform techniques, matrix multiplication, convolution, and so forth, [21, 22]. The methodology of mapping the algorithms onto array structures is depicted in Figure 5.
First, to achieve the desired maximal possible parallelism in an algorithm, we perform the analysis of the data dependencies in the corresponding computations. Then, the algorithm is transformed into a single assignment algorithm without global communication. A dependence graph (DG) is used to analyze these data dependencies of the corresponding algorithms [21]. Following [21], DG is defined as , where P represents a set of nodes and E is a set of arcs (or edges), that is, each edge connects the corresponding pair of nodes and the connection is formalized by .
Second, we employ the systolic design paradigm to map a high dimensional (N dimensional) DG to a lower dimensional Signal to Flow Graph (SFG) [21]. Recall that the systolic array is a spacetime representation, in which the function description defines the behavior within a node, whereas the structural description specifies the interconnections (edges and delays) between the nodes [21, 22]. In order to derive a regular systolic array architecture with minimum number of nodes, we employ the linear projection approach for processor assignment following the methodology developed in [21, 22], that is, the nodes of the DG in a certain straight line are projected onto the corresponding PEs in the processor array represented by the corresponding projection vector . Thus, we seek for a linear order reduction transformation [22]:
that maps the Ndimensional DG (G^{N}) onto the (N– 1)dimensional SFG ().
Such the desired linear transformation matrix admits the partitioning in two functions [22]
Here, defines a (1p) vector composed of the first row of that determines the time scheduling. This vector indicates the normal direction of the equitemporal hyperplanes in the DG, "equitemporal" being understood in the sense that all the nodes on the same hyperplane must be processed at the same time [22]. The submatrix of dimension (the rest rows of ), determine the space processor. With this mapping, we are now ready to proceed with the construction of the required regular (N – 1)dimensional systolic arrays.
5.4. HW Implementation
Once the HW/SW codesign has been defined, the three coprocessors employed in the architecture exemplified in Figure 4 can be implemented using the HW systolic arrays. In this study, we are oriented at the use of the Xilinx MicroBlaze soft processor that employs the On Chip Peripheral Bus (OPB) for transferring the data from/to the memory to/from the coprocessor [23]. Such the OPB is a fully synchronous bus that connects other separate 32 bit data buses. This system architecture (based on the FPGA XC4VSX3510ff668 with the embedded processor and the OPB buses) restricts the corresponding processing frequency to 100 MHz. The typical rate of the OPB bus is 133 MByte/s, providing that each data transfer of 32bits is accomplished at 30.05 ns [23]. Next, to avoid multiple data transfer from the embedded processor data memory to the coprocessors, a register file is to be implemented inside each coprocessor.
The first systolic array (referred to as the MSF coprocessor) implements the zerostep iteration of the unified fixedpoint DEDRPOCS procedure (32) to form the MSF image as specified by (30). The function of this systolic array is to perform the triple matrix multiplication, where matrix S has the bandToeplitz structure [5, 6] with the width of the nonzero strip over the azimuth frame equal to Following the methodology addressed in the previous section, the triple matrix multiplication corresponding to the MSF function can be implemented using a cascade systolic array. First, the multiplication of a bandToeplitz matrix and a rectangular matrix is performed and then, the result is multiplied with another bandToeplitz matrix. Each slide of the DG in the multiplication of the bandToeplitz matrix and the rectangular matrix is employed using the following specifications in the transformations defined by (37): for the vector schedule, for the projection vector and for the space processor that determine the resulting transformation matrix (37). In Figure 6(a), we illustrate the triple matrix multiplication mapped into a cascade systolic array with the relevant MSF systolic array architecture exemplified in Figure 6(b). The corresponding computations require only O() fixedpoint operations, with where, as previously, 2_{ a } defines the width of the nonzero strip in the factorized bandToeplitz PSM (25) and is the original image dimension over the azimuth frame.
The MSF coprocessor systolic architecture of Figure 6(b) consists of identical linearlyconnected processing elements (PEs). In our case, the internal structure of each PE contains a multiplier and an adder. Each PE receives 32bits operands and generates 64bits product. Then, the product is truncated to 32bits with a fixedpoint adopted representation of 9 integers and 24 decimals. Next, since the bandToeplitz type matrix S_{ a } is preloaded, the incoming data are transmitted in parallel to the corresponding PEs. After 2 cycles of clock, the data outputs are produced and transferred to the registers (gray blocks in Figure 6(a)). Once the first of the triple matricial product is completed, the data transfer to the second array begins. The control unit block guarantees the correct synchronization between the arrival of the input data and the computations for each PE. The result buffer of Figure 6(b) consists of a shift buffer used to store the elements generated in parallel by the boundary PEs. Finally, the bus interface unit realizes the communication between the systolic array and the embedded processor.
The second systolic array (referred to as the PSM coprocessor) implements the computation of the Point Spread Matrix (PSM) function concurrently over the azimuth and range axes, that is , where, as previously, symbol specifies the concurrent execution of the corresponding computational operations. In the PSM function, both matrices and are bandToeplitz type matrices (dim; dim) with the widths of the nonzero strips equal to and , correspondingly, where due to the PSM sparseness, and . Thus, to perform the required reconstruction over the azimuth direction, it is possible to achieve full parallelism with only an rectangular array (as opposed to an original fulldimensional array in the general case [10, 21]). Due to the rangeazimuth factorization, the same parallelism is achievable in the range direction as well with the corresponding rectangular systolic array. The matrix multiplication of two bandToeplitz type matrices employs now the following specifications in the transformations defined by (37): for the vector schedule, for the projection vector, for the space processor that determine the resulting transformation matrix (37). The topological distribution of the processing elements (PEs) in such systolic structure is shown in Figure 7(a). The corresponding PSM systolic coprocessor architecture is presented in Figure 7(b) with three independent directions of data flow.
The third coprocessor (referred to as the Iterative POCS coprocessor) performs the adaptive updating of the iterative reconstruction operator in the corresponding fixedpoint DEDRPOCS procedure (32). The key operations of this coprocessor are to perform the standard 1D convolution and the vectormatrix multiplication. The systolic array for performing the 1D convolution employs now the following specifications in the transformations defined by (37): for the vector schedule, for the projection vector, and for the space processor that determine the resulting transformation matrix (37). Figure 8(a) illustrates the 1D convolution systolic array and Figure 8(b) presents the relevant systolic architecture.
In summary, the developed systolic architectures perform the parallel and pipelined schemes which exploit the proposed above mapping methodology. These architectures provide the necessary HWlevel implementation of the SWoptimized complex multipurpose RS imaging algorithms.
6. Simulations and Performance Analysis
6.1. Simulation Experiment Specifications
In the verification simulation experiments, we considered a conventional singlelook SAR with the fractionally synthesized aperture as an RS imaging system [1, 2]. Recall, that signal formation operator (SFO) of such a SAR is factored along two axes in the image plane [3]: the azimuth or crossrange coordinate (horizontal axis, x) and the slant range (vertical axis, y), respectively. We considered the conventional triangular SAR range ambiguity function (AF) [3] (y) and Gaussian approximation [5, 6], of the SAR azimuth AF with the adjustable fractional parameter, a. Note that in the imaging radar applications [3, 4], an AF is referred to as the continuousform approximation of the PSM defined by (25) and serves as an equivalent to the point spread function in the conventional image processing terminology [9]. The image degradation and noising effects were incorporated to simulate the process of formation of the degraded specklecorrupted MSF images. First, following [1, 3] the degradation in the spatial resolution due to the fractional aperture synthesis mode were simulated via blurring the original image with the range AF along the y axis and with the azimuth AF along the x axis, respectively. Next, the degradations at the imageformation level due to the propagation and calibration uncertainties were simulated using the statistical model of a SAR image defocusing [2, 3]. For a considered singlelook SAR, the conventional MSF image formation algorithm (30) implies, first, application of the regular adjoint SFO to the zeromean Gaussian data realization u, and second, performing the elementbyelement (i.e., pixelbypixel) squared detection of S^{+}u to compose the corresponding SSP pixel estimates . Consequently, the MSF pixel estimates are chisquared distributed with two degrees of freedom, and such a distribution is a negative exponential Rayleigh distribution [2, 9]. Thus, to comply with the technicallymotivated MSF image formation scheme, the composite multiplicative noise was simulated as a realization of the distributed random variables with the pixel mean value assigned to the actual degraded scene image pixel that directly obeys the statistical speckle model [2, 5, 6]. Such signaldependent multiplicative image noise dominates the additive noise component in the data in the sense that , hence the estimate performed empirically via the application of the local statistics method [2] was used to adjust the regularization degrees of freedom (regularization factors) in all simulated DEDRrelated SSP reconstruction procedures.
We have run the simulation experiments for both certain and uncertain operational scenarios. In the both scenarios, we considered the MSF, RSF and RASF algorithms from the DEDRPOCS family (22). Also, to compare the developed algorithms with the conventional SAR image enhancement techniques [1–3], the celebrated Lee adaptive despeckling filter based on the local statistics method [2] was simulated. The family of four simulated techniques were renumbered as . The first one (p1) relates to the conventional MSF estimator (30) that employs the adjoint SO . This degraded MSF image {} was then postprocessed applying the Lee adaptive despeckling filter [2] that we refer to as the adaptively despeckled MSF image , that is, p 2. Next, the nonadaptive RSF algorithm with the solution operator defined by (19) was applied to enhance the original MSF image employing the iterative DEDRRSF version of the unified fixedpoint iterative procedure (32); the resulting DEDRRSF enhanced image was specified as Land numbered as p3. Last, the fourth simulated technique corresponds to the adaptive DEDRRASF method (32) with the optimal solution operator given by (21); the resulting adaptively enhanced DEDRRASF image was specified as L and numbered correspondingly as . In the second (uncertain) simulated scenario, the system AF was additionally distorted over the azimuth frame within the realistic interval of that corresponds to the partially uncompensated carrier trajectory deviations interval [2, 10]. For both scenarios, the simulations were run for different composite signaltonoise ratios (SNR) μ defined as the ratio of the average signal component in the rough image formed using the MSF algorithm (30) to the relevant noise component in the same image,where represents the average gray level of the original scene image.
6.2. Performance Metrics
The first adopted quality metric was borrowed from the classical image reconstruction applications [9] defined as an improvement in the output signaltonoise ratio (IOSNR):
where represents the value of the k th element (pixel) of the original image , represents the value of the k th element (pixel) of the degraded image formed applying the MSF technique (37), and represents a value of the k th pixel of the image reconstructed with three simulated enhancement methods, where corresponds to the adaptive despeckling algorithm based on the local statistics method [2], corresponds to the POCSRSF algorithm and corresponds to the POCSRASF algorithm, that is, the best one from the developed DEDRPOCS family, respectively. The second adopted metric, the socalled mean absolute error (MAE), was employed as a metric suitable for quantification of edges and fine detail preservation in the reconstructed image defined as [15]
According to these quality metrics, the higher is the IOSNR, and the lower is the MAE, the better is the improvement of the image enhanced/reconstructed with the particular employed algorithm.
6.3. Simulations
In this study, the simulations were performed with a large scale (1Kby1K) pixelformat image borrowed from the realworld highresolution terrain SAR imagery (southwest Guadalajara region, Mexico [24]). The quantitative measures of the image enhancement/reconstruction performance gains achieved with the particular employed POCSRSF and POCSRASF techniques for different SNRs evaluated with two different quality metrics (38), (39) are reported in Table 2. Figure 9 shows the original scene image (not observable with the simulated SAR systems). Figure 10 illustrates the same original test scene represented in MATLAB pseudocolor scale.
The images of Figures 11(a) through 11(h) present the results of image formation and enhancement applying different DEDRrelated estimators without model uncertainties as specified in the figure captions. In the second simulated scenario, the fractional SAR system suffered from more severe degradations because of the additional system defocusing and multiplicative speckle noising due to the operational scenario uncertainties. Figures 12(a) through 12(h) present the results of image formation and enhancement applying different DEDRrelated estimators in the simulated uncertain operational scenario as specified in the figure captions. From the analysis of the reported simulation results, it is evident that the RASF method overperformed the robust nonadaptive RSF in both simulated scenarios. This demonstrates that employing the adaptive RASF technique from the DEDRPOCS family one could substantially improve the quality of the RS images (reconstructed from both certain and uncertain RS measurement data) approaching in the same time (near) realtime computational performances.
Next in Figure 13, we present the convergence curves related to the iterativeform implementation of the POCSRSF/RASF techniques for the test case SNR (i.e.,). From the analysis of these curves one can deduce that after 40 iterations both POCSRSF and POCSRASF algorithms begin to suffer from some numerical instabilities. This type of numerical instability is a subtle issue in constructing the regularized iterative techniques for different illconditioned problems, for example [1, 3, 9], and so forth. Moreover, the relationship between the resulting IOSNR quality metric and the visual reconstructed image quality is not fully understood, although, of course, one would expect a high degree of correlation between the two [9]. These observations are in concord with the similar observations from other studies of the inverse imaging problems in other illposed contexts, for example, [1, 2, 9, 10]. In our case, due to the POCS regularization, the appearance of the DEDR reconstructed images demonstrated substantial improvement up to 15 iterations from the MSF starting point. Next, the appearance of the reconstructed images changed very little from that of the 15 to 25 iterations. The changes became perceptually undistinguishable after 2530 iterations. This behavior can be used as a motivation for the empirical stopping rule at 2530 iterations of the POCSregularized iterative POCSRSF and POCSRASF algorithms.
6.4. HW/SW Codesign Performance Analysis
In this section, we complete our study with the comparative analysis of the computational complexities of the simulated iterative DEDRPOCS algorithms implemented using the systolic coprocessors constructed following the addressed HW/SW codesign approach. The synthesis metrics related to the implementation of the systolic arrays architectures as coprocessors are summarized in Table 3. First, we exemplify the MSF, PSM and iterative POCS coprocessor architectures for the following simplified specifications: data matrices of size and two BandToeplitz PSF matrices of the same pixel size with equal bandwidths of 2 and 2 pixels. The relevant SP performance analysis results are resumed in Table 3. Next, in Figures 14(a) through 14(c), we summarize the relevant HW synthesis performances for the realistic case of largescale processed RS scenes (e.g., to pixel size) and report the overall resource utilization performances attained with the proposed HW coprocessors architectures for different number of processing elements (PEs).
Next, the reported metrics of Table 3 specify the area and time behaviors of the corresponding hardware systolic arrays, that is the corresponding MSF, the PSM, and the iterative POCS architectures specified above in Section 5. From the analysis of the data reported in Table 3 and Figures 14(a) through 14s(c), one can deduce the following: With the proposed HW/SW codesign architecture (in which the embedded processors iterate properly the corresponding SP procedures) the DEDRPOCSrelated algorithms can be efficiently implemented in an iterative fixedpoint fashion also for the realistic largescale scenes (e.g., pixel size). Pursuing the proposed systolic computing architecture concept, the increased scene dimensionality requires the proper segmentation of the scene frame with the parallelized computing performed over the partitioned segments followed by the relevant integration of the overall partial processed data. Such partitioned systolic HW/SW codesign computingoriented processing can be performed directly following the architecture design concept proposed and specified in the previous Section 5.4. Additionally, the scalability in terms of FlipFlops, Slices and LUTs (i.e., the HW resources of the FPGA) for the proposed MSF, PSM and iterative POCS coprocessors are reported in Figures 14(a) through 14(c). In fact, the corresponding DEDRrelated SP algorithms can be efficiently implemented in a Field Programmable Systems on Chip (FPSoC) mode in spite of employing conventional systems based on multiFPGAs or PCClusters [12–14, 16]. The latter is practically inspired and desirable for a wide range of RS and general SP applications due to the large range density of the existing FPGAs that incorporate huge resources of logical gates, block RAM memory modules and soft or hardembedded processors integrated on the same chip with the relevant custom coprocessing HW blocks, and so forth. For example, an alternative approach for highspeed computational implementation of the reconstructive RS image processing based on the use of clusters of PCs was presented in [12–14]. In [12], the cluster NSPO Parallel TestBed for performing parallel radiometric and geometrical corrections of the largescale 36002944pixel RS images was implemented. The reconstructive image processing was conducted using a PCCluster composed by three PCs each one with a PentiumIII 550 MHz with 128 MB of RAM connected with 100 Mbps FastEthernet LAN. The processing time achieved with such threePCs cluster was only 33.3 seconds (nearreal time for conventional RS users), while the corresponding processing performed with one single processor required 84.65 seconds. In [13, 14], another kind of parallel architecture was implemented for morphological classification of hyperspectral RS imagery at the NASA's Goddard Space Flight Center. The parallel classifier of [14] uses 256processor Beowulf cluster (Thunderhead cluster) with hybrid neural parallelism that enables such a system to perform an accurate classification of the hyperspectral RS scenes in only 17 seconds.
As a result, advances on high performance computing as well as on specialized high performance hardware modules are necessarily required to achieve the nearreal processing time performances for complex RS algorithms.
Last, we compared the required processing time of the generalform RSF/RASF DEDRrelated procedures (22) and the iterative fixedpoint DEDRPOCSregularized algorithm (32), both implemented using the conventional MATLAB software in a personal computer (PC) running at 3 GHz with a AMD Athlon (tm) 64 dualcore processor and 2 GB of RAM memory. Also, the same DEDRrelated algorithms were implemented using the proposed HW/SW codesign architecture (soft and hardware) without systolic and with systolic arrays employing the Xilinx FPGA XC4VSX3510ff668. The corresponding comparative results are reported in Table 4. Analyzing these reported results, one may deduce the following. The iterative fixedpoint DEDRPOCSregularized algorithm (32) manifests the (near) real time highresolution enhancement/reconstruction of the RS imagery. The implementation of the proposed HW/SW codesign architecture helps to reduce drastically the overall processing time. Particularly, the proposed implementation of the iterative POCSregularized RASF algorithm with systolic arrays takes only 2.56 seconds for each iteration of the image reconstruction. In total, it takes 64 sec for 25 iterations. This new computation time is approximately 3 times less than the previous implementation without systolic arrays [10], 8 times less than the corresponding processing time achievable with the MATLAB POCSbased implementation, and it is times less than the hypothetical processing time required for implementing the fullformat conventional generalform DEDRRASF algorithm (22) without POCS regularization and without systolic computing.
7. Concluding Remarks
The principal result of the undertaken study relates to the digital signal processingoriented solution of the RS image enhancement/reconstruction problems in a (near) real time computing mode (the "near real time" being understood in context of conventional RS users) via exploiting the aggregated hardware/software (HW/SW) codesign paradigm that results in an efficient hardware implementation architecture based on the use of systolic array processors. We have approached the goal of the (near) real time computational implementation of the enhancement/reconstruction of the RS imagery from two directions. First, we have analytically established that to alleviate the problem illposedness and reduce the overall computational load of the largescale image enhancement/reconstruction tasks at the algorithmic processing level, some special form of descriptive experiment design projectiontype numerical regularization must be employed. This stage was developed and addressed here as the unified DEDR method, and the efficient fixedpoint numerical iterative technique that incorporates the proper construction of the relevant orthogonally factorized regularizing projector onto convex sets (POCS) in the solution domain was designed and specified for the particular employed RS sensor system, namely, the sidelooking imaging synthetic aperture radar (SAR) operating in both certain and uncertain scenarios. We have also examined how such SARadapted POCSregularized fixedpoint iterative technique can be executed concurrently over the orthogonal rangeazimuth coordinates with optimal use of the sparseness properties of the overall SAR system point spread function characteristics. The algorithmiclevel advantages of such unified DEDRPOCSregularized RS image enhancement/reconstruction techniques relate to the theoretically guaranteed convergence of the corresponding fixedpoint iterative process with the proper factorization of the numerical reconstructive procedures over the orthogonal rangeazimuth directions in the representation image frame.
Second, we have examined that pursuing the proposed HW/SW codesign paradigm and employing the systolic arrays as coprocessing units, the (near) real time image processing requirements can be achieved due to performing the corresponding computations in an efficient systolic architecture mode. The unified algorithmic (softwarelevel, SW) and systematic (hardwarelevel, HW) codesign approach was verified via computer simulation experiments indicative of its efficiency for performing the RS image enhancement and reconstruction tasks in (near) real computational time. The tested DEDRPOCSrelated techniques implemented numerically using the proposed HW/SW codesigned computational architecture overperform the previously developed methods both in the attained reconstruction quality and the achievable computational complexity, that is manifest the substantially reduced overall computational time (e.g., up to three orders with respect to the schemes that do not aggregate the POCS regularization with the systolic computing). We do believe that pursuing the DEDRPOCSrelated HW/SW codesign paradigm with systolic array hardware accelerators one could approach definitely the real time computational requirements while performing the reconstructive processing of the largescale RS imagery attaining the enhancement/reconstruction performance gains close to the limiting bounds.
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Castillo Atoche, A., Torres Roman, D. & Shkvarko, Y. Experiment Design RegularizationBased Hardware/Software Codesign for RealTime Enhanced Imaging in Uncertain Remote Sensing Environment. EURASIP J. Adv. Signal Process. 2010, 254040 (2010). https://doi.org/10.1155/2010/254040
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DOI: https://doi.org/10.1155/2010/254040
Keywords
 Remote Sense
 Point Spread Function
 Synthetic Aperture Radar
 Systolic Array
 Remote Sense Imaging