**2. Remote sensing background**

The general formalism of the RS imaging problem presented in this study is a brief presentation of the problem considered in (Shkvarko, 2006, 2008), hence some crucial model elements are repeated for convenience to the reader.

134 Applications of Digital Signal Processing

(near) real time implementation with existing Digital Signal Processors (DSP) or Personal

To treat such class of real time implementation, the use of specialized arrays of processors in VLSI architectures as coprocessors or stand alone chips in aggregation with Field Programmable Gate Array (FPGA) devices via the hardware/software (HW/SW) co-design, will become a real possibility for high-speed Signal Processing (SP) in order to achieve the expected data processing performance (Plaza, A. & Chang, 2008), (Castillo Atoche et al., 2010a, 2010b). Also, it is important to mention that cluster-based computing is the most widely used platform on ground stations, however several factors, like space, cost and power make them impractical for on-board processing. FPGA-based reconfigurable systems in aggregation with custom VLSI architectures are emerging as newer solutions which offer enormous computation potential in both cluster-based systems and embedded systems area. In this work, we address two particular contributions related to the substantial reduction of the computational load of the Descriptive-Regularized RS image reconstruction technique based on its implementation with massively processor arrays via the aggregation of high-

First, at the algorithmic-level, we address the design of a family of Descriptive-Regularization techniques over the range and azimuth coordinates in the uncertain RS environment, and provide the relevant computational recipes for their application to imaging array radars and fractional imaging SAR operating in different uncertain scenarios. Such descriptive-regularized family algorithms are computationally adapted for their HWlevel implementation in an efficient mode using parallel computing techniques in order to

Second, at the systematic-level, the family of Descriptive-Regularization techniques based on reconstructive digital SP operations are conceptualized and employed with massively parallel processor arrays (MPPAs) in context of the real time SP requirements. Next, the array of processors of the selected reconstructive SP operations are efficiently optimized in fixed-point bit-level architectures for their implementation in a high-speed low-power VLSI architecture using 0.5um CMOS technology with low power standard cells libraries. The achieved VLSI accelerator is aggregated with a FPGA platform via HW/SW co-design

Alternatives propositions related to parallel computing, systolic arrays and HW/SW codesign techniques in order to achieve the near real time implementation of the regularizedbased procedures for the reconstruction of RS applications have been previously developed in (Plaza, A. & Chang, 2008), (Castillo Atoche et al., 2010a, 2010b). However, it should be noted that the design in hardware (HW) of a family of reconstructive signal processing operations have never been implemented in a high-speed low-power VLSI architecture

Finally, it is reported and discussed the implementation and performance issues related to real time enhancement of large-scale real-world RS imagery indicative of the significantly increased processing efficiency gained with the proposed implementation of high-speed

The general formalism of the RS imaging problem presented in this study is a brief presentation of the problem considered in (Shkvarko, 2006, 2008), hence some crucial model

speed low-power VLSI architectures with a FPGA platform.

based on massively parallel processor arrays in the past.

elements are repeated for convenience to the reader.

**2. Remote sensing background** 

low-power VLSI architectures of the descriptive-regularized algorithms.

achieve the maximum possible parallelism.

Computers (PC).

paradigm.

The problem of enhanced remote sensing (RS) imaging is stated and treated as an illposed nonlinear inverse problem with model uncertainties. The challenge 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 processing of finite recordings of the RS 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 (Henderson & Lewis, 1998), (Shkvarko, 2008). Such an operator is explicitly specified by the employed radar signal modulation and is traditionally referred to as the signal formation operator (SFO) (Shkvarko, 2006). The classical imaging with an array radar or SAR implies application of the method called "matched spatial filtering" to process the recorded data signals (Franceschetti et al., 2006), (Shkvarko, 2008), (Greco & Gini, 2007). 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 matched spatial filtering, e.g., (Franceschetti et al., 2006), (Shkvarko, 2006, 2008), (Greco & Gini, 2007), (Plaza, A. & Chang, 2008), (Castillo Atoche et al., 2010a, 2010b) but without aggregating the minimum risk descriptive estimation strategies and specialized hardware architectures via FPGA structures and VLSI components as accelerators units. In this study, we address a extended descriptive experiment design regularization (DEDR) approach to treat such uncertain SSP reconstruction problems that unifies the paradigms of minimum risk nonparametric spectral estimation, descriptive experiment design and worst-case statistical performance optimization-based regularization.

#### **2.1 Problem statement**

Consider a coherent RS experiment in a random medium and the narrowband assumption (Henderson & Lewis, 1998), (Shkvarko, 2006) that enables us to model the extended object backscattered field by imposing its time invariant complex scattering (backscattering) function *e*(**x**) in the scene domain (scattering surface) *X* **x**. The measurement data wavefield *u*(**y**) = *s*(**y**) + *n*(**y**)consists of the echo signals *s* and additive noise *n* and is available for observations and recordings within the prescribed time-space observation domain *Y* = *T*u*P*, where **y** = (*t*, **p**)T defines the time-space points in *Y*. The model of the observation wavefield *u* is defined by specifying the stochastic equation of observation (EO) of an operator form (Shkvarko, 2008):

$$
\mu = \mathsf{S}e + n; \ e \in \mathsf{E}; \ \mathsf{u}, \ \mathsf{n} \in \mathsf{U}; \ \mathsf{S}: \mathsf{E} \to \mathsf{U} \ \mathsf{A} \tag{1}
$$

in the Hilbert signal spaces E and U with the metric structures induced by the inner products, [*u*1, *u*2]U = 1 2 () () *Y uud* ³ **y yy** , and [*e*1, *e*2]E = 1 2 () () *X eed* ³ **x xx** , respectively. The operator

model of the stochastic EO in the conventional integral form (Henderson & Lewis, 1998), (Shkvarko, 2008) may be rewritten as

$$u(\mathbf{y}) = (\tilde{S}e(\mathbf{x}))(\mathbf{y}) = \int\_{\mathcal{X}} \tilde{S}(\mathbf{y}, \mathbf{x}) \, e(\mathbf{x}) d\mathbf{x} + 4 \text{ } n(\mathbf{y}) = \int\_{\mathcal{X}} S(\mathbf{y}, \mathbf{x}) \, e(\mathbf{x}) d\mathbf{x} + \int\_{\mathcal{X}} \delta S(\mathbf{y}, \mathbf{x}) \, e(\mathbf{x}) d\mathbf{x} + n(\mathbf{y}) \tag{2}$$

High-Speed VLSI Architecture Based on Massively Parallel

perturbation term in (4) is irrelevant, **Ʀ** = **0**.

vector estimate ˆ

**2.3 DEDR method** 

matrix.

Thus one can seek to estimate ˆ

proposed in (Shkvarko, 2006)

in the desired estimate ˆ

Processor Arrays for Real-Time Remote Sensing Applications 137

simple case of certain operational scenario (Henderson & Lewis, 1998), (Shkvarko, 2008), the discrete-form (i.e. matrix-form) SFO **S** is assumed to be deterministic, i.e. the random

The digital enhanced RS imaging problem is formally stated as follows (Shkvarko, 2008): to

the second-order statistics of the scattering vector **e** observed through the perturbed SFO (4) and contaminated with noise **n**; hence, the RS imaging problem at hand must be qualified and treated as a statistical nonlinear inverse problem with the uncertain operator. The high-resolution imaging implies solution of such an inverse problem in some optimal way. Recall that in this paper we intend to follow the unified descriptive experiment design

**b** reconstructed from whatever available measurements of independent

**b** } of the SSP

**b** is an estimate of

**b** is recognized to be the

¦ **u u** , (5)

**Re** }diag given the data correlation matrix **Ru** pre-

**Re** }diag = {**FYF**+}diag (6)

(**F**)}, (7)

*j j*

**b** = { ˆ

**Re** }diag.

map the scene pixel frame image **B**ˆ via lexicographical reordering **B**ˆ = *L*{ ˆ

realizations of the recorded data vector **u.** The reconstructed SSP vector ˆ

regularized (DEDR) method proposed originally in (Shkvarko, 2008).

In the descriptive statistical formalism, the desired SSP vector ˆ

vector of a principal diagonal of the estimate of the correlation matrix **Re**(**b**), i.e. ˆ

**b** = { ˆ

**Ru** = aver *jJ*

> ˆ **b** = { ˆ

estimated empirically via averaging *J* t 1 recorded data vector snapshots {**u**(*<sup>j</sup>*)}

{ () () *j j*

where {ƛ}diag defines the vector composed of the principal diagonal of the embraced

To optimize the search for **F** in the *certain* operational scenario the DEDR strategy was

that implies the minimization of the weighted sum of the systematic and fluctuation errors

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 (Shkvarko, 2006). It is easy to recognize that the strategy (7) is a structural extension of the statistical minimum risk estimation strategy for the nonlinear spectral estimation problem at hand because in both cases the balance between the gained spatial resolution and the noise energy

**F** o min **F** {

 **u u** } = 1 () () 1

*J <sup>j</sup> J*

(**F**) *=* trace{(**FS – I**)**A**(**FS – I**)+} + D trace{**FRnF**+} (8)

**b** where the selection (adjustment) of the regularization parameter D

**2.3.1 DEDR strategy for certain operational scenario** 

**Y** = ˆ

by determining the solution operator (SO) **F** such that

in the resulting estimate is to be optimized.

The random functional kernel *S S S* ( , ) = ( , )+ ( , ) **yx yx yx** G of the stochastic signal formation operator (SFO) *S* given by (2) defines the signal wavefield formation model. Its mean, < ( , )> = ( , ) *S S* **yx yx** , is referred to as the nominal SFO in the RS measurement channel specified by the time-space modulation of signals employed in a particular radar system/SAR (Henderson & Lewis, 1998), and the variation about the mean G*S*(,) **y x** = P(**y**,**x**)*S*(**y**,**x**) models the stochastic perturbations of the wavefield at different propagation paths, where P(**y**,**x**) is associated with zero-mean multiplicative noise (so-called Rytov perturbation model). All the fields *enu* , , in (2) are assumed to be zero-mean complex valued Gaussian random fields. Next, we adopt an incoherent model (Henderson & Lewis, 1998), (Shkvarko, 2006) of the backscattered field *e*( ) **x** that leads to the G-form of its correlation function, *Re*(**x**1,**x**2) = *b*(**x**1)G(**x**1– **x**2). Here, *e*(**x**) and *b*(**x**) = <|*e*(**x**)|2> are referred to as the scene random complex scattering function and its average power scattering function or spatial spectrum pattern (SSP), respectively. The problem at hand is to derive an estimate ˆ *b*( ) **x** of the SSP *b*( ) **x** (referred to as the desired RS image) by processing the available finite dimensional array radar/SAR measurements of the data wavefield *u*(**y**) specified by (2).

#### **2.2 Discrete-form uncertain problem model**

The stochastic integral-form EO (2) to its finite-dimensional approximation (vector) form (Shkvarko, 2008) is now presented.

$$\mathbf{u} = \mathbf{\bar{S}e} + \mathbf{n} = \mathbf{S}e + \Delta \mathbf{e} + \mathbf{n} \tag{3}$$

in which the perturbed SFO matrix

$$
\tilde{\mathbf{S}} = \mathbf{S} + \Delta \mathbf{\varDelta} \tag{4}
$$

represents the discrete-form approximation of the integral SFO defined for the uncertain operational scenario by the EO (2), and **e**, **n**, **u** are zero-mean vectors composed of the decomposition coefficients 1 { }*<sup>K</sup> k k e* , 1 { }*<sup>M</sup> nm m* , and 1 { }*<sup>M</sup> um m* , respectively. These vectors are characterized by the correlation matrices: **Re** = **D** = **D**(**b**) = diag(**b)** (a diagonal matrix with vector **b** at its principal diagonal), **Rn**, and **Ru** = < **SR Se** <sup>&</sup>gt;*<sup>p</sup>*( **<sup>Ʀ</sup>** ) + **Rn**, respectively, where <>*<sup>p</sup>*( **Ʀ** ) defines the averaging performed over the randomness of **Ʀ** characterized by the *unknown* probability density function *p*( **Ʀ** ), and superscript + stands for Hermitian conjugate. Following (Shkvarko, 2008), the distortion term **Ʀ** in (4) is considered as a random zero mean matrix with the bounded second-order moment K t <sup>2</sup> || || **Ʀ** . Vector **b**  is composed of the elements, *bk =* ( ) *<sup>k</sup> e* = *ekek \**! = |*ek*|2!; *k* = 1, …, *K*, and is referred to as a *K*-D vector-form approximation of the SSP, where represents the second-order statistical ensemble averaging operator (Barrett & Myers, 2004). The SSP vector **b** is associated with the so-called lexicographically ordered image pixels (Barrett & Myers, 2004).

The corresponding conventional *Ky*u*Kx* rectangular frame ordered scene image **B** = {*b*(*kx*, *kx*); *kx*, = 1,…,*Kx*; *kv*, = 1,…,*Ky*} relates to its lexicographically ordered vector-form representation **b** = {*b*(*k*); *k* = 1,…,*K* = *Ky*u *Kx*} via the standard row by row concatenation (so-called lexicographical reordering) procedure, **B** = *L*{**b**} (Barrett & Myers, 2004). Note that in the perturbation term in (4) is irrelevant, **Ʀ** = **0**. The digital enhanced RS imaging problem is formally stated as follows (Shkvarko, 2008): to map the scene pixel frame image **B**ˆ via lexicographical reordering **B**ˆ = *L*{ ˆ **b** } of the SSP vector estimate ˆ **b** reconstructed from whatever available measurements of independent realizations of the recorded data vector **u.** The reconstructed SSP vector ˆ **b** is an estimate of the second-order statistics of the scattering vector **e** observed through the perturbed SFO (4) and contaminated with noise **n**; hence, the RS imaging problem at hand must be qualified and treated as a statistical nonlinear inverse problem with the uncertain operator. The high-resolution imaging implies solution of such an inverse problem in some optimal way. Recall that in this paper we intend to follow the unified descriptive experiment design regularized (DEDR) method proposed originally in (Shkvarko, 2008).

### **2.3 DEDR method**

136 Applications of Digital Signal Processing

< ( , )> = ( , ) *S S* **yx yx** , is referred to as the nominal SFO in the RS measurement channel specified by the time-space modulation of signals employed in a particular radar

(**y**,**x**)*S*(**y**,**x**) models the stochastic perturbations of the wavefield at different propagation

perturbation model). All the fields *enu* , , in (2) are assumed to be zero-mean complex valued Gaussian random fields. Next, we adopt an incoherent model (Henderson & Lewis,

as the scene random complex scattering function and its average power scattering function or spatial spectrum pattern (SSP), respectively. The problem at hand is to derive an estimate

*b*( ) **x** of the SSP *b*( ) **x** (referred to as the desired RS image) by processing the available finite dimensional array radar/SAR measurements of the data wavefield *u*(**y**) specified by (2).

The stochastic integral-form EO (2) to its finite-dimensional approximation (vector) form

represents the discrete-form approximation of the integral SFO defined for the uncertain operational scenario by the EO (2), and **e**, **n**, **u** are zero-mean vectors composed of the

characterized by the correlation matrices: **Re** = **D** = **D**(**b**) = diag(**b)** (a diagonal matrix with vector **b** at its principal diagonal), **Rn**, and **Ru** = < **SR Se** <sup>&</sup>gt;*<sup>p</sup>*( **<sup>Ʀ</sup>** ) + **Rn**, respectively, where <>*<sup>p</sup>*( **Ʀ** ) defines the averaging performed over the randomness of **Ʀ** characterized by the *unknown* probability density function *p*( **Ʀ** ), and superscript + stands for Hermitian conjugate. Following (Shkvarko, 2008), the distortion term **Ʀ** in (4) is considered as a

a *K*-D vector-form approximation of the SSP, where represents the second-order statistical ensemble averaging operator (Barrett & Myers, 2004). The SSP vector **b** is associated with the so-called lexicographically ordered image pixels (Barrett & Myers, 2004). The corresponding conventional *Ky*u*Kx* rectangular frame ordered scene image **B** = {*b*(*kx*, *kx*); *kx*, = 1,…,*Kx*; *kv*, = 1,…,*Ky*} relates to its lexicographically ordered vector-form representation **b** = {*b*(*k*); *k* = 1,…,*K* = *Ky*u *Kx*} via the standard row by row concatenation (so-called lexicographical reordering) procedure, **B** = *L*{**b**} (Barrett & Myers, 2004). Note that in the

system/SAR (Henderson & Lewis, 1998), and the variation about the mean

1998), (Shkvarko, 2006) of the backscattered field *e*( ) **x** that leads to the

G

**u** = **Se**

random zero mean matrix with the bounded second-order moment

G

given by (2) defines the signal wavefield formation model. Its mean,

(**y**,**x**) is associated with zero-mean multiplicative noise (so-called Rytov

of the stochastic signal formation

(**x**1– **x**2). Here, *e*(**x**) and *b*(**x**) = <|*e*(**x**)|2> are referred to

 **+ n** = **Se** + **Ʀe** + **n** , (3)

**S** = **S + Ʀ** , (4)

K

*\**! = |*ek*|2!; *k* = 1, …, *K*, and is referred to as

t <sup>2</sup> || || **Ʀ** . Vector **b** 

*k k e* , 1 { }*<sup>M</sup> nm m* , and 1 { }*<sup>M</sup> um m* , respectively. These vectors are

G

G

*S*(,) **y x** =


The random functional kernel *S S S* ( , ) = ( , )+ ( , ) **yx yx yx**

operator (SFO) *S*

paths, where

P

correlation function, *Re*(**x**1,**x**2) = *b*(**x**1)

(Shkvarko, 2008) is now presented.

in which the perturbed SFO matrix

decomposition coefficients 1 { }*<sup>K</sup>*

is composed of the elements, *bk =* ( ) *<sup>k</sup> e* = *ekek*

**2.2 Discrete-form uncertain problem model** 

P

ˆ

#### **2.3.1 DEDR strategy for certain operational scenario**

In the descriptive statistical formalism, the desired SSP vector ˆ **b** is recognized to be the vector of a principal diagonal of the estimate of the correlation matrix **Re**(**b**), i.e. ˆ **b** = { ˆ **Re** }diag. Thus one can seek to estimate ˆ **b** = { ˆ **Re** }diag given the data correlation matrix **Ru** preestimated empirically via averaging *J* t 1 recorded data vector snapshots {**u**(*<sup>j</sup>*)}

$$\mathbf{Y} = \hat{\mathbf{R}}\_{\mathbf{u}} = \underset{j \in \mathcal{J}}{\text{aver}} \{ \mathbf{u}\_{(j)} \mathbf{u}\_{(j)}^{+} \} = \frac{1}{J} \sum\_{j=1}^{J} \mathbf{u}\_{(j)} \mathbf{u}\_{(j)}^{+} \,. \tag{5}$$

by determining the solution operator (SO) **F** such that

$$
\hat{\mathbf{b}} = \{\hat{\mathbf{R}}\_{\mathbf{e}}\}\_{\text{diag}} = \{\mathbf{F} \mathbf{Y} \mathbf{F}^\*\}\_{\text{diag}} \tag{6}
$$

where {ƛ}diag 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 (Shkvarko, 2006)

$$\mathbf{F} \to \min\_{\mathbf{F}} \left\{ \mathcal{H}(\mathbf{F}) \right\} \tag{7}$$

$$\mathcal{H}(\mathbf{F}) \equiv \text{trace}\{ (\mathbf{F}\mathbf{S} - \mathbf{I})\mathbf{A}(\mathbf{F}\mathbf{S} - \mathbf{I})^{\*} \} + \alpha \,\text{trace}\{ \mathbf{F}\mathbf{R}\_{\mathbf{n}}\mathbf{F}^{\*} \} \tag{8}$$

that implies the minimization of the weighted sum of the systematic and fluctuation errors in the desired estimate ˆ **b** where the selection (adjustment) of the regularization parameter D 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 (Shkvarko, 2006). It is easy to recognize that the strategy (7) is a structural extension of the statistical minimum risk estimation strategy for the nonlinear spectral estimation problem at hand because in both cases the balance between the gained spatial resolution and the noise energy in the resulting estimate is to be optimized.

High-Speed VLSI Architecture Based on Massively Parallel

**2.3.3 DEDR imaging techniques** 

adaptive spatial filtering (RASF) methods.

by the matched spatial filter (MSF):

correlation matrix specified by (16).

diagonal matrix with the estimate ˆ

adaptive spatial filters (RASFs):

for the certain operational scenario, and

6

conventional pixel-frame format can be unified now as follows

**B**ˆ = *L*{ ˆ

for the uncertain operational scenario, respectively.

F*RASF*

**F** becomes the Tikhonov-type robust spatial filter

Processor Arrays for Real-Time Remote Sensing Applications 139

Where **K**<sup>6</sup> = ( <sup>1</sup> **SR S**6 + D**A**–1)–1 (18)

In this sub-section, three practically motivated DEDR-related imaging techniques (Shkvarko, 2008) are presented that will be used at the HW co-design stage, namely, the conventional matched spatial filtering (MSF) method, and two high-resolution reconstructive imaging techniques: (i) the robust spatial filtering (RSF), and (ii) the robust


 **F***MSF* **= F**(1) | **S**+. (19) 2. *RSF*: The RSF method implies no preference to any prior model information (i.e., **A** = **I**) and balanced minimization of the systematic and noise error measures in (14) by adjusting the regularization parameter to the inverse of the signal-to-noise ratio (SNR), e.g. D = *N*0/*B*0, where *B*0 is the prior average gray level of the image. In that case the SO

 **F***RSF*= **F** (2) = (**S**+**S** + D*RSF***I**)–1**S**+. (20) in which the RSF regularization parameter D*RSF* is adjusted to a particular operational scenario model, namely, D*RSF* = (*N*0/*b*0) for the case of a certain operational scenario, and D*RSF* = (*N*6/*b*0) in the uncertain operational scenario case, respectively, where *N*<sup>0</sup> represents the white observation noise power density, *b*0 is the average a priori SSP value, and *N*6 = *N*0 + E corresponds to the augmented noise power density in the

3. *RASF*: In the statistically optimal problem treatment, D and **A** are adjusted in an adaptive fashion following the minimum risk strategy, i.e. D **A**–1 = **D**ˆ = diag( ˆ

(17) become itself solution-dependent operators that result in the following robust

**<sup>F</sup>***RASF* <sup>=</sup>**F**(3) <sup>=</sup>( <sup>1</sup> **SR Sn** <sup>+</sup> <sup>ˆ</sup> 1 1 ) **<sup>D</sup>** <sup>1</sup> **S Rn** (21)

Using the defined above SOs, the DEDR-related data processing techniques in the

**b** at its principal diagonal, in which case the SOs (9),

= F(4) = ( <sup>1</sup> **SR S**<sup>6</sup> <sup>+</sup><sup>ˆ</sup> 1 1 ) **<sup>D</sup>** <sup>1</sup> **S R**<sup>6</sup> (22)

**b** } = *L*{{**F**(*p*)**YF**(*<sup>p</sup>*)+}diag }; ); *p* = 1, 2, 3, 4 (23)

D>>

**b** ), the

1. *MSF*: The MSF algorithm is a member of the DEDR-related family specified for

represents the robustified reconstruction operator for the uncertain scenario.

From the presented above DEDR strategie, one can deduce that the solution to the optimization problem found in the previous study (Shkvarko, 2006) results in

$$\mathbf{F} = \mathbf{K} \mathbf{S}^{+} \mathbf{R}\_{\mathbf{n}}^{-1} \; \; \; \; \; \tag{9}$$

$$\begin{array}{ll}\text{where} & \mathbf{K} = (\mathbf{S}^{+} \mathbf{R}\_{\mathbf{n}}^{-1} \mathbf{S} + \alpha \mathbf{A}^{-1})^{-1} \end{array} \tag{10}$$

represents the so-called regularized reconstruction operator; <sup>1</sup> **Rn** is the noise whitening filter, and the adjoint (i.e. Hermitian transpose) SFO **S**+ defines the matched spatial filter in the conventional signal processing terminology.

#### **2.3.2 DEDR strategy for uncertain operational scenario**

To optimize the search for the desired SO **F** in the *uncertain* operational scenario with the randomly perturbed SFO (4), the *extended DEDR* strategy was proposed in (Shkvarko, 2006)

$$\mathbf{F} = \arg\min\_{\mathbf{F}} \min\_{\substack{\preclearrowleft \ \left|\Lambda\right| \left|^{2} \right|^{2} >\_{p(\Lambda)} \leq \delta}} \{ \mathcal{H}\_{\text{ext}}(\mathbf{F}) \} \tag{11}$$

$$\text{subject to} \quad \le |\mid \Delta \mid |^2 \succ\_p (\Delta) \le \delta \tag{12}$$

where the conditioning term (12) represents the worst-case statistical performance (WCSP) regularizing constraint imposed on the unknown second-order statistics <|| **Ʀ** ||2>*<sup>p</sup>*( **Ʀ** ) of the random distortion component **Ʀ** of the SFO matrix (4), and the DEDR "extended risk" is defined by

$$\mathcal{H}\_{\rm crf}(\mathbf{F}) = \text{tr}\{\lhd(\mathbf{F}\widetilde{\mathbf{S}} - \mathbf{I})\mathbf{A}(\mathbf{F}\widetilde{\mathbf{S}} - \mathbf{I})^{\*} \rhd\_{\mathcal{V}}(\mathbf{A}\_{\cdot})\} + a\,\text{tr}\{\mathbf{F}\mathbf{R}\_{\mathbf{n}}\mathbf{F}^{\*}\}\tag{13}$$

where the regularization parameter D 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 SFO (11), the risk function (13) was next decomposed and evaluated for its the maximum value applying the Cauchy-Schwarz inequality and Loewner ordering (Greco & F. Gini, 2007) of the weight matrix **A** d J **I** with the scaled Loewner ordering factor J = min{ J : **<sup>A</sup>** <sup>d</sup> J **I** } = 1. With these robustifications, the extended DEDR strategy (11) is transformed into the following optimization problem

$$\mathbf{F} = \to \min\_{\mathbf{F}} \left\{ \mathcal{H}\_{\mathbf{\hat{z}}}(\mathbf{F}) \right\} \tag{14}$$

with the *aggregated* DEDR risk function

$$\mathcal{H}\_{\Sigma} \text{(F)} \{ \text{=tr} \{ (\text{FS} - \text{I}) \text{A(FS - I)}^{\*} \} + \text{attr} \{ \text{F} \, \mathbf{R}\_{\Sigma} \text{F}^{\*} \} \} \tag{15}$$

$$\text{Where}\tag{1}\tag{1}\tag{2}\tag{2}\tag{3}\tag{3}\tag{3}\tag{3}\tag{4}\tag{4}\tag{4}\tag{4}\tag{4}\tag{4}\tag{4}\tag{4}$$

The optimization solution of (14) follows a structural extension of (9) for the augmented (diagonal loaded) **R**6 that yields

$$\mathbf{F} = \mathbf{K}\_{\Sigma} \mathbf{S}^{+} \mathbf{R}\_{\Sigma}^{-1} \tag{17}$$

Where **K**<sup>6</sup> = ( <sup>1</sup> **SR S**6 + D**A**–1)–1 (18)

represents the robustified reconstruction operator for the uncertain scenario.

### **2.3.3 DEDR imaging techniques**

138 Applications of Digital Signal Processing

From the presented above DEDR strategie, one can deduce that the solution to the

where **<sup>K</sup>** = ( <sup>1</sup> **SR Sn** + D**A**–1)–1 (10)

represents the so-called regularized reconstruction operator; <sup>1</sup> **Rn** is the noise whitening filter, and the adjoint (i.e. Hermitian transpose) SFO **S**+ defines the matched spatial filter in

To optimize the search for the desired SO **F** in the *uncertain* operational scenario with the randomly perturbed SFO (4), the *extended DEDR* strategy was proposed in (Shkvarko, 2006)

> max*p* G

 {

G

*ext*(**F**) *=* tr{<(**F – I**)**A**(**F – I**)+>*<sup>p</sup>*( **Ʀ** )} + D tr{**FRnF**+} (13)

<sup>6</sup> (F)} = tr{(FS – I)A(FS – I)+} + Dtr{F **R**<sup>6</sup> F+}, (15)

F = <sup>1</sup> **KSR** <sup>6</sup> <sup>6</sup> , (17)

and the metrics inducing weight matrix **A** compose

} = 1. With these robustifications, the

<sup>6</sup>(**F**)} (14)

'! d '

subject to <|| **Ʀ** ||2 >*<sup>p</sup>*( **Ʀ** ) d

where the conditioning term (12) represents the worst-case statistical performance (WCSP) regularizing constraint imposed on the unknown second-order statistics <|| **Ʀ** ||2>*<sup>p</sup>*( **Ʀ** ) of the random distortion component **Ʀ** of the SFO matrix (4), and the DEDR "extended risk"

To proceed with the derivation of the robust SFO (11), the risk function (13) was next decomposed and evaluated for its the maximum value applying the Cauchy-Schwarz

> : **<sup>A</sup>** <sup>d</sup> J**I**

Where (ǃ) **R R** 6 6 = (Rn + EI); E = G/D t 0. (16)

The optimization solution of (14) follows a structural extension of (9) for the augmented

inequality and Loewner ordering (Greco & F. Gini, 2007) of the weight matrix **A** d

extended DEDR strategy (11) is transformed into the following optimization problem

**F =** o min **F** {

**F =** arg min**<sup>F</sup>** <sup>2</sup> || || ( )

D

**S** <sup>~</sup> **<sup>S</sup>** ~

J = min{ J

the processing level "degrees of freedom" of the DEDR method.

F = <sup>1</sup> **KS Rn** , (9)

*ext* (**F**)} (11)

(12)

J**I** with

optimization problem found in the previous study (Shkvarko, 2006) results in

the conventional signal processing terminology.

where the regularization parameter

the scaled Loewner ordering factor

with the *aggregated* DEDR risk function

(diagonal loaded) **R**6 that yields

is defined by

**2.3.2 DEDR strategy for uncertain operational scenario** 

In this sub-section, three practically motivated DEDR-related imaging techniques (Shkvarko, 2008) are presented that will be used at the HW co-design stage, namely, the conventional matched spatial filtering (MSF) method, and two high-resolution 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 DEDR-related family specified for D >> ||**S+S**||, i.e. the case of a dominating priority of suppression of noise over the systematic error in the optimization problem (7). In this case, the SO (9) is approximated by the matched spatial filter (MSF):

$$\mathbf{F}\_{\rm MSF} = \mathbf{F}^{(1)} \approx \mathbf{S}^\*.\tag{19}$$

2. *RSF*: The RSF method implies no preference to any prior model information (i.e., **A** = **I**) and balanced minimization of the systematic and noise error measures in (14) by adjusting the regularization parameter to the inverse of the signal-to-noise ratio (SNR), e.g. D = *N*0/*B*0, where *B*0 is the prior average gray level of the image. In that case the SO **F** becomes the Tikhonov-type robust spatial filter

$$\mathbf{F}\_{\rm RSF} = \mathbf{F}^{\langle 2 \rangle} = (\mathbf{S}^\* \mathbf{S} + \alpha\_{\rm RSF} \mathbf{I})^{-1} \mathbf{S}^\*. \tag{20}$$

in which the RSF regularization parameter D*RSF* is adjusted to a particular operational scenario model, namely, D*RSF* = (*N*0/*b*0) for the case of a certain operational scenario, and D*RSF* = (*N*6/*b*0) in the uncertain operational scenario case, respectively, where *N*<sup>0</sup> represents the white observation noise power density, *b*0 is the average a priori SSP value, and *N*6 = *N*0 + E corresponds to the augmented noise power density in the correlation matrix specified by (16).

3. *RASF*: In the statistically optimal problem treatment, D and **A** are adjusted in an adaptive fashion following the minimum risk strategy, i.e. D **A**–1 = **D**ˆ = diag( ˆ **b** ), the diagonal matrix with the estimate ˆ **b** at its principal diagonal, in which case the SOs (9), (17) become itself solution-dependent operators that result in the following robust adaptive spatial filters (RASFs):

$$\mathbf{F\_{RASF}} = \mathbf{F^{(3)}} = \left(\mathbf{S^{+}R\_{n}^{-1}S} + \hat{\mathbf{D}}^{-1}\right)^{-1} \mathbf{S^{+}R\_{n}^{-1}}\tag{21}$$

for the certain operational scenario, and

$$\mathbf{F}\_{\rm RASF} = \mathbf{F}^{(4)} = \left(\mathbf{S}^{+}\mathbf{R}\_{\Sigma}^{-1}\mathbf{S} + \hat{\mathbf{D}}^{-1}\right)^{-1}\mathbf{S}^{+}\mathbf{R}\_{\Sigma}^{-1} \tag{22}$$

for the uncertain operational scenario, respectively.

Using the defined above SOs, the DEDR-related data processing techniques in the conventional pixel-frame format can be unified now as follows

$$\hat{\mathbf{B}} = L\{\hat{\mathbf{b}}\} = L\{\{\mathbf{F}(\boldsymbol{\psi})\mathbf{Y}\mathbf{F}(\boldsymbol{\psi})^{\*}\}\_{\text{diag}}\}; \quad p = 1, 2, 3, 4 \tag{23}$$

High-Speed VLSI Architecture Based on Massively Parallel

approaches.

system requirements.

Processor Arrays for Real-Time Remote Sensing Applications 141

HW/SW co-design is a hybrid method aimed at increasing the flexibility of the implementation and improvement of the overall design process (Castillo Atoche et al., 2010a). When a co-processor-based solution is employed in the HW/SW co-design architecture, the computational time can be drastically reduced. Two opposite alternatives can be considered when exploring the HW/SW co-design of a complex SP system. One of them is the use of standard components whose functionality can be defined by means of programming. The other one is the implementation of this functionality via a microelectronic circuit specifically tailored for that application. It is well known that the first alternative (the software alternative) provides solutions that present a great flexibility in spite of high area requirements and long execution times, while the second one (the hardware alternative) optimizes the size aspects and the operation speed but limits the flexibility of the solution. Halfway between both, hardware/software co-design techniques try to obtain an appropriate trade-off between the advantages and drawbacks of these two

In (Castillo Atoche et al., 2010a), an initial version of the HW/SW- architecture was presented for implementing the digital processing of a large-scale RS imagery in the operational context. The architecture developed in (Castillo Atoche et al., 2010a) did not involve MPPAs and is considered here as a simply reference for the new pursued HW/SW co-design paradigm, where the corresponding blocks are to be designed to speed-up the digital SP operations of the DEDR-POCS-related algorithms developed at the previous SW stage of the overall HW/SW co-design to meet the real time imaging

i. Algorithmic implementation (reference simulation in MATLAB and C++ platforms);

iv. Architecture design procedure of the addressed reconstructive SP computational tasks

In this sub-section, the procedures for computational implementation of the DEDR-related robust space filter (RSF) and robust adaptive space filter (RASF) algorithms in the MATLAB and C++ platforms are developed. This reference implementation scheme will be next

Having established the optimal RSF/RASF estimator (20) and (21), let us now consider the way in which the processing of the data vector **u** that results in the optimum estimate ˆ

can be computationally performed. For this purpose, we refer to the estimator (20) as a multi-stage computational procedure. We part the overall computations prescribed by the

At this stage the a priori known value of the data mean ¢**u Sm** ² **b** is subtracted from the

the unknown deviations **b**<sup>D</sup> = (**b – mb**) of the vector **b** from its prescribed (known) mean

<sup>D</sup> contains all new information regarding

**b**

compared with the proposed architecture based on the use of a VLSI-FPGA platform.

The proposed co-design flow encompasses the following general stages:

ii. Partitioning process of the computational tasks; iii. Aggregation of parallel computing techniques;

onto HW blocks (MPPAs);

**3.1.1 Algorithmic implementation** 

estimator (16) into four following steps. a. First Step: Data Innovations

b. Second Step: Rough Signal Estimation

value **mb** .

data vector **u**. The innovations vector **u u Sm <sup>b</sup>**

with **F** (1) = **F***MSF*; **F**(2) = **F***RSF*, and **F**(3) = **F***RASF*, **F**(4) = **F***RASF*6,respectively.

Any other feasible adjustments of the DEDR degrees of freedom (the regularization parameters D, E, and the weight matrix **A**) provide other possible DEDR-related SSP reconstruction techniques, that we do not consider in this study.
