optimization of design and operating parameters of a space-based optical-electronic system with a...

Optimization of design and operating parameters of aspace-based optical-electronic system with a

distributed aperture

Iouri Tcherniavski1,* and Mojtaba Kahrizi1,2

1Concordia University, Department of Electrical and Computer Engineering, 1455, de Maisonneuve Boulevard,West Montreal, Quebec, H3G 1M8, Canada.

[email protected]

*Corresponding author: [email protected]

Received 21 August 2008; accepted 2 October 2008;posted 8 October 2008 (Doc. ID 100373); published 13 November 2008

Using a gradient optimization method with objective functions formulated in terms of a signal-to-noiseratio (SNR) calculated at given values of the prescribed spatial ground resolution, optimization problemsof geometrical parameters of a distributed optical system and a charge-coupled device of a space-basedoptical-electronic system are solved for samples of the optical systems consisting of two and three annularsubapertures. The modulation transfer function (MTF) of the distributed aperture is expressed in termsof an average MTF taking residual image alignment (IA) and optical path difference (OPD) errors intoaccount. The results show optimal solutions of the optimization problems depending on diverse variableparameters. The information on the magnitudes of the SNR can be used to determine the number of thesubapertures and their sizes, while the information on the SNR decrease depending on the IA and OPDerrors can be useful in design of a beam combination control system to produce the necessary require-ments to its accuracy on the basis of the permissible deterioration in the image quality. © 2008 OpticalSociety of America

OCIS codes: 110.0110, 110.1220, 110.4850, 110.5100, 330.6130, 350.6090.

1. Introduction

The research on optical systems (OSs) with distribu-ted apertures is excited by an obvious desire toachieve a higher resolution capability without in-creasing the collecting area. It is well known thatthe resolution capability is proportional to the sizeof the receiving aperture. Consequently, the obviousway of increasing the resolution of mirror andmirror-lens OSs is by increasing the diameter of asystem head mirror. Unfortunately, technologicalproblems of manufacturing and adjustment of largeoptics impose some restrictions on the maximal sizeof optical details and, therefore, such increasing ofthe resolution is limited. Further, the increase in thediameter of the head mirror implies an increase in

the weight and overall dimensions of the OS as awhole, but its mass and size are limited by volumeand mass constraints imposed by a launch vehicle,

List of acronyms

AOTF average optical transfer functionCCD charge-coupled deviceGSD ground-sampled distanceIA image alignment

MRF MTF reduction factorMTF modulation transfer functionNIIRS National Imagery Interpretability Rating ScaleOES optical-electronic systemOPD optical path differenceOS optical systemPSF point spread functionSNR signal-to-noise ratioTDI time delay and integration

0003-6935/08/336159-18$15.00/0© 2008 Optical Society of America

20 November 2008 / Vol. 47, No. 33 / APPLIED OPTICS 6159

as well as by the scaling laws of manufacturing costs[1]. One of the ways to overcome the tendency of in-creasing the diameter of the monolithic head mirroris using the OSs with distributed apertures. The dis-tributed aperture OSs have no single monolithichead mirror—the aperture of such systems is formedby several subapertures which are either fixedrigidly at certain distances from each other ordynamically independent. The Space Interferometry

Mission [2] and the Terrestrial Planet Finder Inter-ferometer [3] are the samples of such systems.

Optimization of design and operating parametersof a space-based optical-electronic system (OES) is acomplex computation procedure including the con-sideration of influence of all the components of thesystem on the resulting values of interest. It is neces-sary to take numerous specific characteristics intoaccount. Among these are the characteristics of:assumed observation objects and spreading back-grounds, conditions of object and background illumi-nation, the atmosphere, orbital parameters anddynamics of carrying satellites, the OES, communi-cation, etc.

One of the first tasks is to determine an optimizedobjective function. Often the objective functions foroptimization of the aperture configurations arebased on a shape of the point spread function (PSF)[4–6], behavior of the modulation transfer function(MTF) [7–13], and the signal-to-noise ratio (SNR)[14–17].

Depending on the optimization criteria andimposed constraints, it is possible to consider eitheroptimization of only one part of the system, for exam-ple, only the OS to optimize the configuration of thereceiving aperture for the purpose of maximization ofthe optical MTF on predetermined intervals of spa-tial frequencies in the focal plane, or optimizationof two or more parts simultaneously, for example,the OS and photodetector to maximize the spatialground resolution which is one of the very important

Fig. 1. θR direction of resolution.

Fig. 2. Beam combination: (a) ideal ray paths, (b) eventual alignment ðxn; ynÞ and phasing ½ ~Δn ≡~Δnðxn; ynÞ� errors.

6160 APPLIED OPTICS / Vol. 47, No. 33 / 20 November 2008

operation parameters of the OES used for the earthsurface observation.The resolution of the system is always limited be-

cause of the presence of random noise. Therefore,imaging of objects with a prescribed spatial resolu-tion is possible only with some probability deter-mined by the signal and noise levels—SNR. Theoptimization procedure on the basis of a SNR criter-ion is more general than on the basis of OS para-meters only, and it is especially necessary at lowlight levels and contrast since noise can completelysuppress the signal.Herein we show the possible variants of the OES

optimization carried out in the context of ongoing re-search of the space-based OESs with distributedaperture dedicated to the earth surface observationwith a high spatial resolution. Note that the compu-tation methods used for optical sparse-array optimi-zation are mainly the so-called direct-search ones.The use of such methods for objective functions de-fined on the basis of the MTF is due to the possiblepresence of zero-value areas in the optical MTF.Meanwhile, we use a gradient optimization methodwith the objective function formulated on the basis ofthe SNR expressed in terms of the MTFs calculatedat the given values of the spatial ground resolution.

Such an opportunity became possible due to applyingthe step pupil function approximation approach [13]and the conception of the average optical transferfunction (AOTF) [18].

2. Criterion of the OES Estimation

The image-exploitation community often uses theNational Imagery Interpretability Rating Scale(NIIRS) [19] to quantify the interpretability or use-fulness of imagery. The NIIRS is expressed in termsof ground-sampled distance (GSD), MTF, SNR, etc.through the image quality equation [20] whose func-tion Ið�Þ determines the relationship between theNIIRS and physical imaging system design and op-erating parameters:

NIIRS ¼ IðGSD;MTF;SNR; etc:Þ: ð1ÞThe analysis [21] shows that the loss in image in-

terpretability measured as ΔNIIRS can be modeledas a linear relationship with the noise-equivalentchange in reflection. Just the change in reflectionof bar chart equivalents of the object is used in ourconsideration.

Regarding the signal and noise concepts accordingto the available definitions [22,23], we define the sig-nal in the object plane as the difference between theaverage value of energy collected during some time ofexposition from the object area and the average valueof the energy collected during the same time from theequal area of the homogeneous background. Noise isa mean-square deviation of the signal. Using a timedelay and integration charge-coupled device (TDI-CCD) as the image sensor of the OES, we calculatethe SNR at the output of the TDI-CCD (not takingthe horizontal registers into account).

3. Main Component Characteristics of the SNR

The calculation of the SNR used for optimization isbased on photometric transforms of the light propa-gating from the object to the optical focal plane fornadir observation and on the properties of the TDI-CCD. Let us briefly discuss the main componentcharacteristics used for formulary calculation ofthe SNR, focusing our attention on the descriptionof the OS with a distributed aperture.

Fig. 3. Layout of the TDI-CCD elements.

Fig. 4. Spectral reflectance for the “object” (1 and 2) and the“background” (0).

Fig. 5. Initial aperture configurations.


A. Object and Background

We use the basic harmonic of the reflection coeffi-cient distribution of the hypothetical test object con-sisting of infinite number of alternate “light” and“dark” bars of the width of R and having the spectralreflection coefficients ρ1ðλÞ and ρ2ðλÞ, respectively.The test object lays on a horizontal Earth surfaceat the angle θR with respect to the TDI direction (xdirection, Fig. 1) of the CCD. The test-object is illu-minated by the sunlight, having a spectral irradiancethat depends on the solar zenith angle and meteor-ological conditions, and its reflectance propertiesobey the Lambert law.

B. Atmosphere

As is well known, the atmosphere influences both theamplitude and phase of the propagating light and in-troduces the haze radiation to received signals. Theadditive noise due to the scattering and self radiationis presented by the spectral haze radiance. The mul-tiplicative and nonlinear amplitude and phase dis-tortions due to the scattering and the amplitude andphase fluctuations of waves are taken into account bymeans of the spectral optical transmittance coeffi-cient and the spectral MTF reduction factor (MRF)MRFAðf ; λÞ associated with atmospheric turbulence.TheMRFof the turbulent atmosphere for a long-timeexposure is independent of the OS and can be writtenas [24,25]

MRFAðf ; λÞ ¼ exp½−0:5Dwðr; λÞ�; ð2Þwhere Dwðr; λÞ is the wave structure function of tur-bulence, r is a distance between two points in the pu-

pil plane, corresponding to the spatial frequencyf ¼ r=ðλFÞ, and F is the effective focal length of theOS. For satellite viewing, looking down from an alti-tude H, on the basis of a theory for isotropic turbu-lence developed by Kolmogorov [26], the wavestructure function is given by [27]

Dwðr; λÞ ¼ 2:91 k2r5=3ZH

0

C2nðxÞðx=HÞ5=3dx: ð3Þ

Using a model [28] of the vertical distribution ofC2

nðxÞ,

C2nðxÞ ¼ 6:7 × 10−14ð2x1=2Þ−2=3 expð−x=h0Þ; h0 ¼ 3200;

ð4Þ

and the relations: k ¼ 2π=λ and f ¼ H=ð2RFÞ, thewave structure function takes the form

DwðR; λÞ ¼ 2:3 × 10−4R−5=3λ−1=3γð7=3;H=3200Þ; ð5Þ

where γða; bÞ is the incomplete gamma function; alldimensions are in mks units.

C. Optical System

The OS operating as a Fizeau interferometer forms aunited image in a common focal plane: the beam com-bination system joins together the beams from thesubapertures. Because of a limited accuracy of thebeam combination control system, the beam combi-ner introduces some optical errors influencing the

Fig. 6. Optimal aperture configurations for H ¼ 400km, Object 1, Δλ ¼ 0:40–1:05 μm.



quality of the united image. The influence of the re-sidual random image alignment (IA) and optical pathdifference (OPD) errors is accounted for through themodulus of the AOTF [18] for a diffraction-limited,properly focused, but improperly phased and alignedoptical sparse array. Here we use the AOTF AOTFηðf ; λÞ depending on the step pupil function approxi-mation parameter η [13] (see Eq. (10)). The expres-sion for the η-dependent AOTF can be obtainedfrom the following.It is shown [18] that, for the Fraunhofer diffrac-

tion, the Fourier transform of the irradiance distribu-tion in the focal plane can be written as

Zþ∞

−∞

ZUðx; y; λÞU�ðx; y; λÞ exp½−i2πðf uxþ f vyÞ�dxdy

¼ ED

XNn¼1

½τnðλÞ�1=2 exp½−iαðxnuf þ ynvf Þ

þ ik ~Δnðxn; ynÞ� ×XNm¼1

½τmðλÞ�1=2

×ZZ

An∩Am

expfiα½ðxn − xmÞuþ ðyn − ymÞv�

− ik ~Δmðxm; ymÞgdudv; ð6Þ

where ED is the energy density in the pupil plane; Nis the number of the subapertures, τjðλÞ is the suba-perture spectral transmittance coefficient; α ¼ k=F;ðxj; yjÞ and ~Δjðxj; yjÞ are the IA and phasing errors,

respectively, corresponding to the jth subaperture,Fig. 2; uf ¼ λFf u, vf ¼ λFf v; f u ¼ f cos θR, f v ¼f sin θR; the integration area An∩Am is a nonemptyintersection of areas of the mth subaperture andthe nth one shifted by the vector λFf; j ¼ n;m. Thedynamical phasing error ~Δjðxj; yjÞ is the sum of thephasing error δjðxj; yjÞ, depending on xj and yj, andthe error Δj, which is the proper piston error ofthe beam combination system:

~Δjðxj; yjÞ ¼ ½B0P0P1� − ½B0P000� ¼ ½B0P0P1� − ½B0P0P2�

¼ δjðxj; yjÞ þΔj: ð7Þ

In bounds of the Fraunhofer approach:

uj ¼ u0j þ Lj=F · xj; vj ¼ v0j þ Lj=F · yj; ð8Þ

δjðxj; yjÞ ≈ −ð1 − Lj=FÞðu0j xj þ v0j yjÞ=½ðu0

j Þ2 þ ðv0j Þ2

þ F2�1=2; ð9Þ

where Lj is a lever of the beam combiner correspond-ing to the jth subaperture, ðuj; vjÞ are the coordinatesof the jth subaperture center at nonzero IA errors(the beam tilt γj ≠ 0), ðu0

j ; v0j Þ are the coordinates of

the subaperture center at zero IA errors (γj ¼ 0).For a diffraction-limited OS, we have approxi-

mated the annular subaperture step pupil functionpjðu; vÞwith a unity transmittance by the arctangent:




pjðu − uj; v − vjÞ ¼1π ðarctanfη½ðu − ujÞ2 þ ðv − vjÞ2

− r21j�g − arctanfη½ðu − ujÞ2 þ ðv − vjÞ2 − r22j�gÞ; ð10Þ

where parameter η determines the accuracy of ap-proximation.Let the random errors xj and yj obey the joint Gaus-

sian probability distribution Pðxj; yjÞ with mathema-tical expectations �xj and �yj, and rms deviations σxj and

σyj , respectively, and correlation coefficients ρxjyj ; theerrors Δj obey the Gaussian distribution PðΔjÞ withmathematical expectations �Δj, and rms deviationsσΔj

; j ¼ 1;…N. Then, denoting

βj ¼ ð1 − Lj=FÞ · k=½ðu0j Þ2 þ ðv0j Þ2 þ F2�1=2 ð11Þ

and takingEqs. (7)–(10) into account, Eq. (6) averagedover all xj, yj, and Δj takes the form

�Zþ∞

−∞

ZUðx; y; λÞU�ðx; y; λÞ exp½−i2πðf uxþ f vyÞ�dxdy

�¼ ED

Zþ∞

−∞

Z XNn¼1

½τnðλÞ�1=2Zþ∞

−∞

Zexpfiα½xnðu − uf Þ

þ ynðv − vf Þ� − iβnðu0nxn þ v0nynÞgPðxn; ynÞdxndyn ×

Zþ∞

−∞

expðikΔnÞPðΔnÞdΔn

×1π ðarctanfη½ðu − u0

n − uf Þ2 þ ðv − v0n − vf Þ2 − r21n�g − arctanfη½ðu − u0n − uf Þ2 þ ðv − v0n − vf Þ2 − r22n�gÞ

×XNm¼1

½τmðλÞ�1=2Z þ∞

−∞

Zexp½−iαðxmuþ ymvÞ þ iβmðu0

mxm þ v0mymÞ�Pðxm; ymÞdxmdym

×Zþ∞

−∞

expð−ikΔmÞPðΔmÞdΔm ×1π ðarctanfη½ðu − u0

mÞ2 þ ðv − v0mÞ2 − r21m�g

− arctanfη½ðu − u0mÞ2 þ ðv − v0mÞ2 − r22m�gÞdudv: ð12Þ

Integrating the expression in Eq. (12) over all xj, yj, and Δj, the AOTF, which is Eq. (12) normalized by thefactor π

PNj¼1 τjðλÞðr22j − r21jÞ, is

AOTFηðf ; λÞ ¼ 1=

�πXNj¼1

τjðλÞðr22j − r21jÞ�

×XNn¼1

½τnðλÞ=ðA1A2Þ�1=2=½2σxnσynð1 − ρ2xnynÞ1=2� exp½−ðSxn�x2n þ Syn�y

2nÞ þ Sxnyn�xn�yn�

×Zþ∞

−∞

Zðarctanfη½ðu − u0

n − uf Þ2 þ ðv − v0n − vf Þ2 − r21nÞ�g − arctanfη½ðu − u0n − uf Þ2 þ ðv − v0n − vf Þ2 − r22n�gÞ=π

×� XNm ¼ 1

m ≠ n

½τmðλÞ=ðA3A4Þ�1=2=½2σxmσymð1 − ρ2xmymÞ1=2� × exp½−ðSxm�x2m þ Sym�y

2mÞ þ Sxmym�xm�ym�

× exp�X4j¼1

ðB2j =AjÞ=4

�exp½−k2ðσ2Δn

þ σ2ΔmÞ=2þ ikð �Δn −

�ΔmÞ�

× ðarctanfη½ðu − u0mÞ2 þ ðv − v0mÞ2 − r21m�g − arctanfη½ðu − u0

mÞ2 þ ðv − v0mÞ2 − r22m�gÞ=πþ ½τnðλÞ�1=2 exp½ðB2

5=A1 þ B26=A2Þ=4� × ðarctanfη½ðu − u0

nÞ2 þ ðv − v0nÞ2 − r21n�g

− arctanfη½ðu − u0nÞ2 þ ðv − v0nÞ2 − r22n�gÞ=π

�dudv; ð13Þ


where

A1 ¼ Sxn ;

B1 ¼ 2Sxn�xn − Sxnyn�yn þ iaðu − uf Þ − iβnu0n;

C1 ¼ Sxnyn ;A2 ¼ Syn − C21=ð4A1Þ;

B2 ¼ 2Syn�yn − Sxnyn�xn þ iaðv − vf Þ − iβnv0nþ B1C1=ð2A1Þ;A3 ¼ Sxm ;

B3 ¼ 2Sxm�xm − Sxmym�ym − iauþ iβmu0m;

C3 ¼ Sxmym ;A4 ¼ Sym − C23=ð4A3Þ;

B4 ¼ 2Sym�ym − Sxmym�xm − iavþ iβmv0mþ B3C3=ð2A3Þ;

B5 ¼ 2Sxn�xn − Sxnyn�yn − iauf ;

B6 ¼ 2Syn�yn − Sxnyn�xn − iavf þ B5C1=ð2A1Þ;Sxn ¼ ½2σ2xnð1 − ρ2xnynÞ�−1;Sxnyn ¼ ρxnyn ½σxnσynð1 − ρ2xnynÞ�−1;Syn ¼ ½2σ2ynð1 − ρ2xnynÞ�−1;Sxm ¼ ½2σ2xmð1 − ρ2xmymÞ�−1;Sxmym ¼ ρxmym ½σxmσymð1 − ρ2xmymÞ�−1;Sym ¼ ½2σ2ymð1 − ρ2xmymÞ�−1: ð14Þ

D. Photodetector

The layout of the TDI-CCD elements is shown inFig. 3. In this figure px and py are pitches of theCCD elements, dx × dy is a pixel sensitivity region,M is the number of TDI lines, Vimg is the image ve-

locity in the focal plane, and βV is the angle betweenthe velocity vectors: Vimg and the average CCDcharge packets motion Vch.

The MTF of the TDI-CCD is basically determinedby five factors [29,30]: the charge integration, the dis-crete charge motion, the charge transfer inefficiency,the charge diffusion, and the velocities Vimg and Vchmismatch. The last factor defines the x and y compo-nents of the corresponding MTF:

MTFΔV ;xðf uÞ ¼ sinc½πpxMðαV cos βV − 1Þf u�; ð15Þ

MTFΔV;xðf vÞ ¼ sincðπpxMαV sin βVf vÞ; ð16Þ

here αV ¼ V img=Vch is the parameter of synchroniza-tion of V img and Vch. Denoting the total TDI-CCDMTFs along the x and y direction by MTFCCD;xðf u; λÞand MTFCCD;yðf v; λÞ, respectively, we define the TDI-CCD MTF in the arbitrary θR direction as

MTFCCDðf ; λ; θRÞ ¼ f½MTFCCD;xðf u; λÞ · cos θR�2þ ½MTFCCD;yðf v; λÞ · sin θR�2g1=2:

ð17Þ

E. Signal

Expressing the signal as the difference between thenumbers of electrons corresponding to the irradianceof the photodetector from the object [Nobðf ; θRÞ] andbackground [Nbgðf ; θRÞ], the signal is

Table 1. Optimal Values for the Aperture Configurations in Fig. 6

Parameters Characteristics

Figure u0n (m) v0n (m) r1n (m) r2n (m) F (m) s12 (m) SNR

6(a) −0:392 0.221 0.055 0.130 13.867 0.328 7.26−0:065 0.221 0.055 0.130

6(b) −0:166 0.000 0.055 0.130 13.867 0.333 4.620.166 0.000 0.055 0.130

6(c) −0:001 0.005 0.055 0.130 13.867 0.328 9.11−0:001 −0:323 0.055 0.130

6(d) 0.000 0.166 0.055 0.130 13.867 0.333 5.790.000 −0:166 0.055 0.130




7(a) −0:330 0.084 0.055 0.130 13.867 0.331 8.520.001 0.084 0.055 0.130

7(b) −0:167 0.000 0.055 0.130 13.867 0.333 5.490.166 0.000 0.055 0.130

7(c) −0:069 −0:033 0.055 0.130 13.867 0.331 10.68−0:069 −0:364 0.055 0.130

7(d) 0.000 0.166 0.055 0.130 13.867 0.333 6.880.000 −0:167 0.055 0.130




8(a) −0:354 0.255 0.055 0.130 13.867 0.260 7.04−0:094 0.255 0.055 0.130

8(b) −0:130 0.000 0.055 0.130 13.867 0.260 3.180.130 0.000 0.055 0.130

8(c) 0.057 0.121 0.055 0.130 13.867 0.260 8.830.057 −0:139 0.055 0.130

8(d) 0.000 0.130 0.055 0.130 13.867 0.260 3.990.000 −0:130 0.055 0.130




9(a) −0:360 0.289 0.055 0.130 13.867 0.260 9.33−0:100 0.289 0.055 0.130

9(b) −0:130 0.000 0.055 0.130 13.867 0.260 4.280.130 0.000 0.055 0.130

9(c) −0:110 0.260 0.055 0.130 13.867 0.260 11.70−0:110 0.000 0.055 0.130

9(d) 0.000 0.130 0.055 0.130 13.867 0.260 5.360.000 −0:130 0.055 0.130


SðR; θRÞ ¼ jNobðf ; θRÞ −Nbgðf ; θRÞj: ð18Þ

Note that the signal is dependent on the total MTFofthe system components.

F. Noise

Denoting the number of electrons from the atmo-spheric haze by Nh, expressing the total noise interms of photon shot noise, dark current shot noiseNdc, and readout noise Nro, and taking the Poissondistribution of the emission process of photoelectrons(photon noise) and dark current noise, the Gaussiandistribution of the readout noise, and statistical in-dependence of the noises into account, the total noiseat the output of the TDI-CCD (without the horizontalregisters) is

NðθRÞ ¼ ½Nobð0; θRÞ þNbgð0; θRÞþ 2ðNh þMTelNdc þN2

roÞ�1=2; ð19Þ

where Tel is the time of integration by a single pixel.

4. Problems of Optimization

Let us define the objective function JðR;ΩRÞ in thefollowing form:

JðR;ΩRÞ ¼ZΩR

IfSNRðR; θRÞgdθR; ð20Þ

where ΩR is some domain of the resolution directionsθR of interest (see Fig. 1), and IfSNRðR; θRÞg is sometransform of the SNR, SNRðR; θRÞ ¼ SðR; θRÞ=NðθRÞ.If we are interested in the information on a set SR ofthe spatial resolutions, the objective function can bewritten as

JðSR;ΩRÞ ¼ZSR

ZΩR

IfSNRðR; θRÞgdθRdR: ð21Þ

The considered problems of the parametric optimi-zation of a space-based OES with a distributed aper-ture are to find the magnitudes of varied parametersof the OS and the CCD at the prescribed character-istics of the light irradiance on the earth surface, theobserved objects and background, the atmosphere,the orbit and dynamics of the satellites, the aperture,the beam combiner, the photodetector, the transfor-mation of the electronic SNR to the perceived one,etc. Since the main emphasis in this paper is on thesystems with distributed apertures, the main variedparameters are ones determining the following:



Figure u0n (m) v0n (m) r1n (m) r2n (m) F (m) s12 (m) s13 (m) s23 (m) SNRmin SNRmax

10(a) −0:133 0.413 0.055 0.130 13.867 0.312 0.317 0.312 3.30 8.04−0:401 0.254 0.055 0.130−0:132 0.096 0.055 0.130

10(b) 0.108 0.175 0.055 0.130 13.867 0.317 0.323 0.317 2.00 5.32−0:165 0.013 0.055 0.1300.108 −0:149 0.055 0.130

10(c) 0.388 0.307 0.055 0.130 13.867 0.315 0.321 0.315 3.23 9.810.117 0.147 0.055 0.1300.389 −0:014 0.055 0.130

10(d) 0.055 0.201 0.055 0.130 13.867 0.317 0.324 0.317 2.07 6.34−0:217 0.039 0.055 0.1300.055 −0:123 0.055 0.130




11(a) 0.040 0.025 0.055 0.130 13.867 0.260 0.260 0.260 5.57 8.67−0:185 −0:105 0.055 0.1300.040 −0:235 0.055 0.130

11(b) −0:079 0.131 0.055 0.130 13.867 0.260 0.260 0.260 2.65 4.040.146 0.001 0.055 0.130

−0:080 −0:129 0.055 0.13011(c) 0.350 −0:033 0.055 0.130 13.867 0.260 0.260 0.260 7.63 11.65

0.128 −0:169 0.055 0.1300.357 −0:293 0.055 0.130

11(d) −0:073 0.130 0.055 0.130 13.867 0.260 0.260 0.260 3.68 5.500.155 0.004 0.055 0.130

−0:068 −0:130 0.055 0.130


• aperture configuration,• effective focal length,• geometrical sizes of a CCD pixel,• number of CCD TDI lines.

For nadir observation, depending on the criterion ofoptimization, the optimized parameters, and con-straints, we will consider the following three optimi-zation problems.

A. Problem 1. OS’s Parameters Optimization: TheAperture Configuration and the Effective Focal Length

For an aperture consisting of N annular subaper-tures, the problem is to find the values of the suba-perture center coordinates ðu0

n; v0nÞ, inner and outersubaperture radii r1n and r2n, and the effective focallength F of the OS to provide the minimum of thefunction (20), subject to the following:

gn ≡ RA − f½ðu0nÞ2 þ ðv0nÞ2�1=2 þ r2ng ≥ 0;

n ¼ 1;…;N;ð22Þ

gnm ≡ ðu0n − u0

mÞ2 þ ðv0n − v0mÞ2 − ðr2n þ r2mÞ2 ≥ 0;

n ¼ 1;…;N − 1; m ¼ nþ 1;…;N;

ð23Þ

�RA ≤ u0n ≤ RA; n ¼ 1;…;N; ð24Þ

�RA ≤ v0n ≤ RA; n ¼ 1;…;N; ð25Þ

R11 ≤ r1n ≤ R12; n ¼ 1;…;N; ð26Þ

R21 ≤ r2n ≤ R22; n ¼ 1;…;N; ð27Þ

F1 ≤ F ≤ F2; ð28Þ

where RA is the aperture radius, R11, R12, R21, andR22 are the bounds of the subaperture radii, F1and F2 are the minimal and maximal possible focallength values determined by

F1 ¼ Hmaxfpx; pyg=ðKNqRÞ; ð29Þ

F2 ¼ CfHðRE þHÞ3=2px=½REðGMEÞ1=2 cos βV �: ð30Þ

The value of F1 is determined by the fact that theoperating spatial frequency f R ¼ H=ð2RFÞ in the fo-cal plane must not be greater than the minimal spa-tial Nyquist frequency f Nq ¼ minf1=ð2pxÞ; 1=ð2pyÞgwith the coefficient KNq ¼ f R=f Nq specifying the de-sirable relationship between f R and f Nq. The value of

Fig. 10. Optimal aperture configurations for H ¼ 400km, Object 1, Δλ: (a) and (b) 0:40–1:05 μm, (c) and (d) 0:75–1:05 μm.

Fig. 11. Optimal aperture configurations for H ¼ 400km, Object 2, Δλ: (a) and (b) 0:40–1:05 μm, (c) and (d) 0:40–0:70 μm.


F2 is defined from the obvious inequality:Cf ≥ V img cos βV=px, where the constant Cf is themaximal clock frequency of the CCD along theTDI. For the image velocity, we used the formulafor a circular satellite orbit, and do not take Earth’srotation and gravitational field perturbations intoaccount, i.e., V img ¼ FREðGMEÞ1=2=½HðRE þHÞ3=2�,where RE is the mean Earth’s radius,G is the univer-sal gravitational constant, and ME is Earth’s mass.

B. Problem 2. CCD’s Parameters Optimization: The PixelSize and the Number of TDI Lines

Let us suppose that we have the prescribed apertureconfiguration and value of the effective focal length,and we want to calculate the optimal sizes of a CCDpixel and the number of TDI lines. In such a formu-lation, the problem is to find the size px × py, the CCDpixel sensitivity region dx × dy, and the number M ofTDI lines to provide theminimum of the function (20)subject to the following:

0 < πpxMðαV cos βV − 1Þf u < π; ð31Þ

0 < πpxMαV sin βVf v < π; ð32Þ

Hmaxfpx;pyg=ðKNqRÞ ≤ F; ð33Þ

F ≤ CfHðRE þHÞ3=2px=½REðGMEÞ1=2 cos βV �; ð34Þ

px1 ≤ px ≤ px2; ð35Þ

py1 ≤ py ≤ py2; ð36Þ

dx1 ≤ dx ≤ dx2; ð37Þ

dy1 ≤ dy ≤ dy2; ð38Þ

Fig. 12. SNR versus f Rx and f Ry for the configuration in Fig. 10(b): H ¼ 400km, Object 1, Δλ ¼ 0:40–1:05 μm.


Fig. 14. Variables for the 3D graphics.

Fig. 15. Dependence of the objective function (40) on F and dc

H ¼ 400km, Object 1, Δλ ¼ 0:40–1:05 μm, SNRTh ¼ 10.


M1 ≤ M ≤ M2: ð39ÞHere Eqs. (31) and (32) determine the fact that the

arguments of the sinc functions (15) and (16) must beinside the ð0; πÞ interval. Equations (33) and (34) aredetermined by the formulas (29) and (30), corre-spondingly. The parameters px1, px2, py1, py2, dx1,dx2, dy1, and dy2 are corresponding bounds.

C. Problem 3. Joint Optimization of OS’s and CCD’sParameters

Jointing Problems 1 and 2, a more general optimiza-tion problem is to find the values of the subaperturecenter coordinates ðu0

n; v0nÞ, inner and outer subaper-ture radii r1n and r2n, the effective focal length F ofthe OS, the geometrical sizes px, py, dx, and dy of aCCD pixel, and the numberM of TDI lines to providethe minimum of the function (20), subject to the con-strain functions and domains of parameters definedin Problems 1 and 2.

5. Numerical Results

The optimization problems were solved with the helpof the IMSL DNCONG routine solving a general non-linear programming problem using the successivequadratic programming algorithm and a user-sup-plied gradient. The results are shown in the figuresand tables.The solar spectral irradiance is taken in accor-

dance with the ASTM G 173-03, Tables for ReferenceSolar Spectral Irradiances, for the 1976 U.S. Stan-dard Atmosphere with the absolute air mass of 1.5.

As the “light” bar spectral reflectance coefficientsρ1ðλÞ, the coefficient of Olive Green Paint—“Object1” and the coefficient of Copper Metal 2—“Object2” are taken; as the “dark” bar reflectance coefficientρ2ðλÞ, the reflectance of Brown Sandy Loam is taken—as documented in the ASTER spectral library[31], Fig. 4.

The atmosphere transmittance coefficient dependson atmospheric conditions. We used the conventionalvalues [32] describing some particular case of thepropagation of radiation. The values of the haze ra-diance were taken in accordance with the data [33]calculated with the uniform field defined as 50% re-flective and viewed from a nadir sensor.

As to the image sensor, the Fairchild ImagingCCD525 [34] with 96 lines of TDI is used.

The axis control accuracy value of the TopSatspacecraft platform [35] was used for the angle βV be-tween the velocity vectors Vimg and Vch (Fig. 3).

It is known that today DigitalGlobe’s (Longmont,Colorado) QuickBird [36] is a spacecraft able to offersubmeter resolution imagery available commerciallywith 60 cm panchromatic imagery at nadir. It is alsoknown that DigitalGlobe’s WorldView-1 [37]launched in 2007 has 0:5m commercially availablepanchromatic resolution at nadir. Taking these factsin mind, we use the magnitude of the spatial resolu-tion equal to 0:5m.

Some parameters are as follows:

• object related: R ¼ 0:5m;• OS related: RA ¼ 1:0m, R11 ¼ 0:055m,

R12 ¼ 0:075m, R21 ¼ 0:1m, R22 ¼ 0:13m,τnðλÞ≡ 1:0, η ¼ 500, Ln ¼ 1:0m;

• satellite/CCD related: αV ¼ 1:01, βV ¼ 0:2°;• CCD related: M ¼ 96, KNq ¼ 0:75.

We used the objective function in the form of (20),changing the integration by the summation:

JðR;ΩRÞ ¼XNR

j¼1

fSNRTh

− SNR½R; 180°=NRðj − 1Þ�g2=ðNR − 1Þð40Þ

Fig. 16. Cross section curves of the surface in Fig. 15.

Fig. 17. Dependence of the objective function (40) on φA and dc forH ¼ 400km, Object 1, Δλ ¼ 0:40–1:05 μm, SNRTh ¼ 10.


for

�NR ¼ 18ΩR ¼ fθR∶θR ¼ 180°=NRðj − 1Þ; j ¼ 1;…;NRg;

ð41Þ

and

JðR;ΩRÞ ¼ ½SNRTh − SNRðR; θRÞ�2 ð42Þ

for

�NR ¼ 1ΩR ¼ fθR∶θR ¼ 0°g ; ð43Þ

�NR ¼ 1;ΩR ¼ fθR∶θR ¼ 90°g: ð44Þ

The threshold values SNRTh of the SNR were de-fined by reasons of the desirable inequality:

SNRth >maxθR∈ΩR

fSNRðR; θRÞg: ð45Þ

They were in the range of 5.0–13.0. The objectivefunctions have a form of variance of the SNR aroundthe threshold at a given value of the ground resolu-tion within some domain of the resolution directionsof interest.

We used two initial aperture configurations takenarbitrarily, shown in Fig. 5, with two and three an-nular subapertures.

The spectral range Δλ ¼ 0:4–1:05 μm was used foreach object. Besides,Δλ ¼ 0:75–1:05 μmwas used forObject 1, and Δλ ¼ 0:40–0:70 μm for Object 2.

In the figures of optimal configurations, the partsdenoted by (a) and (c) present the configurationsfor zero values of the IA (almost zero: σxn ¼σyn ¼ 0:001 × 0:8 μm), and zero values of the OPD er-rors; the parts denoted by (b) and (d) present the con-figurations for the IA and OPD errors: �xn ¼ �yn ¼ 0:0,ρxnyn ¼ 0:0, the angle deviations of the beam tilts ofaround 0:07 arc seconds in the x and y directions,which correspond to σxn ¼ σyn ¼ 6 × 0:8 μm for F ¼13:867m, �Δn ¼ 0:0, and σΔn

¼ 0:1 × 0:8 μm for all n.

A. Results for Problem 1

1. Two Annular Subapertures

The aperture consists of two annular subaperturesas shown in Fig. 5(a). The objective function is de-fined by Eq. (42). The altitude of observationH ¼ 400km. The optimal configurations are shownin Figs. 6–9: (a) and (b) are for ΩR, defined byEq. (43), and (c) and (d) are for ΩR, defined byEq. (44). In corresponding Tables 1–4, s12 is the dis-tance between the subaperture centers.

These results show that the optimal aperture con-figuration depends on the observed object. The valueof the SNR in the y direction is greater than in the xdirection, i.e., the MTF of the CCD has differentvalues in the x and y directions. Some dependence

Fig. 18. Cross section curves of the surface in Fig. 17.

Fig. 19. Optimal aperture configurations for H ¼ 600km, Object 1, Δλ: (a) and (b) 0:40–1:05 μm; (c) and (d) 0:75–1:05 μm.


of the distance s12 on the values of the IA and OPDerrors—the greater errors, the greater distance—canbe explained by more influence of the errors in ashorter wavelength spectrum. Besides, the subaper-ture configurations with nonzero IA and OPD errorstend to the aperture center—in accordance withFigs. 8 and 9 in Ref. [18]—showing a greater opticalMTF decrease as subaperures are located far awayfrom the aperture center. It is also seen that theSNR can be increased by corresponding spectral fil-tration of the incoming radiation.The calculated optimal focal length of 13:867m

equals the minimal possible value F1 specifiedby Eq. (29).

2. Three Annular Subapertures

The aperture consists of three annular subaperturesas shown in Fig. 5(b). The objective function is de-fined by Eq. (40) and NR and ΩR are shown byEq. (41). The results are calculated for two altitudes:400 and 600km. The values SNRmin ¼min

θR∈ΩRfSNRðR; θRÞg and SNRmax ¼maxθR∈ΩR

fSNRðR; θRÞg arethe minimal and maximal values of the SNR overthe domain ΩR of the resolution directions θR. In cor-responding Tables 5, 6, 8, and 9, s12, s13, and s23 arethe distances between the centers of subapertures 1–2, 1–3, and 2–3, respectively.

Altitude H ¼ 400 kmThe optimal configurations are shown in Figs. (10)and (11) (corresponding Tables 5 and 6). For the un-touched subaperture configurations, the maximaldistances are between the first and third subaper-tures. It is also obvious that the subapertures withthe nonzero errors tend to the aperture center, whichcan be comprehended from Figs. 8 and 9 in Ref. [18]:the closer to the aperture center, the smaller theMTF decrease. The calculated F ¼ 13:867m equalsthe minimal possible values F1 specified by Eq. (29).

Figures 12 and 13 illustrate the 2D behavior of theSNR function versus the spatial frequencies f Rx ¼cos θR=ð2RÞ and f Ry ¼ sin θR=ð2RÞ on the Earth sur-face in the range bounded by the spatial Nyquist fre-quency. These graphics can be used for preliminaryestimations of the corresponding system spatialground resolution.

It is interesting to look at the 3D graphics of theobjective function (40). Figures 15 and 17 presentsuch graphics calculated for the three-subapertureconfigurations with the angle deviations of the beamtilts of 0:07 arc seconds and σΔn

¼ 0:1 × 0:8 μm for allthe subapertures, H ¼ 400km, Object 1, Δλ ¼ 0:40–1:05 μm. Figure 15 shows the dependence the objec-tive function (40) on the effective focal length F andthe distances dc between the subaperture centers co-inciding with the vertices of the equilateral triangleat its rotation angle φA ¼ 30:0°, see Fig. 14. The innerand outer radii of the subapertures are r1n ¼ 0:055m

Fig. 20. Optimal aperture configurations for H ¼ 600km, Object 2, Δλ: (a) and (b) 0:40–1:05 μm; (c) and (d) 0:40–0:70 μm.




and r2n ¼ 0:130m, respectively. The subapertures“move” simultaneously from the closest possible dis-tances dc ¼ 0:260m between their centers to the dis-tances dc ¼ 0:520malong their rays originating fromthe triangle center. The objective function has a sin-gle minimum corresponding to the minimal shownF ¼ 13:0m and dc ¼ 0:320m. The F and dc sectionsof the objective function surface are shown inFig. 16: (a)F ¼ 13:867m, and (b) dc ¼ 0:320m.Figure 17 shows the dependence of the objective

function (40) on the rotation angle φA varying from0° to 120° and the distances dc between the subaper-tures like in Fig. 15 at F ¼ 13:867m. The functionhas two equal minima corresponding to φA ¼ 30:0°and 90:0°, and dc ¼ 0:320m. The φA and dc sectionsof the objective function surface are shown inFig. 18: (a) φA ¼ 30:0° and 90:0°, and (b) dc ¼0:320m. The sections for φA ¼ 30:0° and 90:0° arethe same.Returning to the optimal configurations, Figs. 10

and 11, we see that the configurations in Figs. 10,11(a), and 11(c) correspond to φA ¼ 90:0°, while theconfigurations in Figs. 11(b) and 11(d) correspondto φA ¼ 30:0°. The existence of these preferable an-gles can be explained by a difference between theCCD MTF along the x and y axes. If the CCD MTFdid not depend on the direction, there would notbe preferable φA, i.e., the optimal configurationwould be determined by the subaperture dis-tances only.We can perform some conditional comparison of

the OESs with a distributed and monolithic aper-ture. Table 7 shows such a comparison for the mono-lithic annular aperture with inner and outer radii r1and r2, respectively, under the condition that

r2=r1 ¼ r2n=r1n ≡ 0:130=0:055 ¼ 2:364. All the otherparameters are the same. Column “r2=r1” containsthe values of the radii providing the SNR of theOES with a monolithic aperture equal to theSNRmax of the OES with the distributed apertureshown on the corresponding figure. AD and AM arethe collecting areas of the OS with the distributedand monolithic aperture, respectively. Their ratiopresents the decrease in the collecting area of the dis-tributed aperture with respect to the monolithic one.Column “SNRjAM¼AD

” contains the values of the SNRfor the OES with the monolithic aperture under thecondition that AM ¼ AD, i.e., r2jAM¼AD

¼ 0:225m.

Altitude H ¼ 600 kmThe optimal configurations are shown in Figs. 19 and20 (corresponding Tables 8 and 9). The maximal dis-tances between the subapertures and the character-istic of the subapertures with the nonzero errors totend to the aperture center are similar to ones forH ¼ 400km. The calculated F ¼ 20:8m equals theminimal possible values F1 specified by Eq. (29).

Figures 21 and 22, like Figs. 12 and 13, illustratethe 2D behavior of the SNR function versus the spa-tial frequencies f Rx and f Ry, and also can be used forpreliminary estimations of the corresponding systemspatial resolution.

Table 10, like Table 7, shows conditional compar-ison parameters of the OESs with a distributed andmonolithic aperture.

B. Results for Problem 2

We have used the calculated optimal aperture config-urationspresentedinFigs.10(b),11(b),19(b),and20(b)with nonzero IA and OPD errors, and the correspond-ing values of the calculated optimal effective focallength. The spectral range Δλ ¼ 0:40–1:05 μm. Be-sides, in addition to the constraints of Eqs. (31)–(39),the following equalities were used:

dx ¼ px; dy ¼ py ð46Þ

The values used for the bounds are px1 ¼ py1 ¼ 1 μm,px2 ¼ py2 ¼ 200 μm,M1 ¼ 1,M2 ¼ 2048.

Table 7. Conditional Comparison Parameters of the OESs with aDistributed and Monolithic Aperture for H ¼ 400 km, F ¼ 13:867m

Object Δλ Fig. SNRmax r2=r1 AD=AM SNRjAM¼AD

1 0.40–1.05 10(a) 8.04 0.248/0.105 0.824 5.790.75–1.05 10(c) 9.81 0.252/0.107 0.798 6.55

2 0.40–1.05 11(a) 8.67 0.263/0.111 0.733 7.120.40–0.70 11(c) 11.65 0.253/0.107 0.792 9.82




19(a) 0.169 0.069 0.055 0.130 20.8 0.476 0.485 0.475 0.02 4.69−0:243 −0:170 0.055 0.1300.163 −0:416 0.055 0.130

19(b) 0.149 0.252 0.055 0.130 20.8 0.482 0.491 0.482 0.01 3.04−0:267 0.007 0.055 0.1300.147 −0:239 0.055 0.130

19(c) −0:189 0.071 0.055 0.130 20.8 0.481 0.491 0.480 0.03 5.800.226 −0:172 0.055 0.130

−0:186 −0:420 0.055 0.13019(d) 0.150 0.252 0.055 0.130 20.8 0.483 0.493 0.482 0.02 3.72

−0:267 0.007 0.055 0.1300.147 −0:241 0.055 0.130


Table 11 shows characteristics of the OES with re-spect to the standard parameters of the CCD manu-factured sample, and the calculated ones for Objects1 and 2. Parameters px, py, andM in the first line foreach object are equal to ones of the manufacturedsample, while the second line contains the calculatedoptimal values. We see that the sizes of a pixel havenot been changed, while the standard number of TDIlines,M ¼ 96, is not an optimal value with respect tothe objective function (40) and the given magnitudesof the considered OES parameters. The optimalvalue is 200 for the aperture configurations whosesubapertures do not touch each other.It is interesting to look at the 3D graphics of the

objective function versus the pixel sizes and the effec-tive focal length, and versus the number of TDI linesand the effective focal length. Figures 23 and 24 showsuch dependences for the optimal aperture config-uration presented in Fig. 10(a) with zero-value IAand OPD errors, H ¼ 400km, and Δλ ¼ 0:40–1:05 μm for Object 1. The applicable domains of theobjective function (nonflat surface for Fig. 23, and F ≥

10:4m for Fig. 24) are determined by Eq. (29) atKNq ¼ 1 and R ¼ 0:5m. Figure 23 uses M ¼ 200.We see a practical straight dependence of the optimalfocal length versus the pixel sizes for an optimal so-lution. Figure 24 uses px ¼ py ¼ dx ¼ dy ¼ 13 μm.Such pictures stimulate a joint optimization of theOS and CCD parameters.

C. Results for Problem 3

The results of the joint OS and CCD optimizationgive the optimal aperture configurations with thesubapertures whose relative positions are similarto ones shown in Subsection 5.A. The difference is

only in the values of the distances between the sub-aperture centers. As a result, we do not show the op-timal configurations but provide the necessarycorresponding numeric information in the tables.

Here we compare the distances between the suba-perture centers, the effective focal length, the pixelsizes, the number of TDI lines, and the minimaland maximal SNR depending on some variable para-meters located in the first column of each table. Allthe numerical values are the same as taken in Sub-sections 5.A and 5.B for H ¼ 400km, Object 1,Δλ ¼ 0:75–1:05 μm, and the nonzero IA and OPD er-rors. Besides, we have imposed an additional con-straint on the maximum possible effective focallength that can result from the overall dimension re-quirements. As a reasonable value, we have chosen5:0m, although any desirable value can be assigned.All the results show the maximum possible magni-tude of the focal length.

Table 12 shows independence of the optimal CCDparameters on the observed objects, and alreadyknown dependence of the subaperture distances onthe object spectral reflectance (see Subsections 5.Aand 5.B).

Table 13 shows the linear inverse dependence ofthe pixel size on the altitude, i.e., 600=400≡4:6875=3:125, the independence of the number ofTDI lines, and already known dependence of the




20(a) −0:088 0.269 0.055 0.130 20.8 0.307 0.313 0.308 1.57 3.92−0:351 0.109 0.055 0.130−0:084 −0:044 0.055 0.130

20(b) 0.088 0.170 0.055 0.130 20.8 0.317 0.327 0.316 0.56 1.55−0:184 0.008 0.055 0.1300.085 −0:157 0.055 0.130

20(c) 0.052 0.011 0.055 0.130 20.8 0.310 0.311 0.310 2.00 4.89−0:213 −0:150 0.055 0.1300.059 −0:300 0.055 0.130

20(d) 0.089 0.171 0.055 0.130 20.8 0.321 0.331 0.321 0.72 2.04−0:188 0.007 0.055 0.1300.086 −0:160 0.055 0.130

Table 10. Conditional Comparison Parameters of the OESs with aDistributed and Monolithic Aperture for H ¼ 600 km, F ¼ 20:8m

Object Δλ Fig. SNRmax r2=r1 AD=AM SNRjAM¼AD

1 0.40–1.05 19(a) 4.69 0.322/0.136 0.489 0.580.75–1.05 19(c) 5.80 0.328/0.139 0.471 0.0

2 0.40–1.05 20(a) 3.92 0.241/0.102 0.873 2.960.40–0.70 20(c) 4.89 0.240/0.102 0.880 3.69

Table 11. Characteristics of the OES with Respect to the Standard andCalculated Parameters of the CCD for Objects 1 and 2, SNRTh ¼ 13:0


H(km)

F(m) Object

px

(μm)py

(μm) MJ

ðR;ΩRÞ SNRmin SNRmax

400 13.867 1 13.0 13.0 96 99.81 2.00 5.32400 13.867 1 13.0 a 13.0 a 200 a 88.30 1.36 6.87400 13.867 2 13.0 13.0 96 94.42 2.65 4.04400 13.867 2 13.0 a 13.0 a 181 a 83.69 1.87 5.37600 20.8 1 13.0 13.0 96 148.95 0.01 3.04600 20.8 1 13.0 a 13.0 a 200 a 142.20 0.004 3.92600 20.8 2 13.0 13.0 96 153.69 0.56 1.55600 20.8 2 13.0 a 13.0 a 200 a 148.39 0.41 2.21aOptimal parameters.


subaperture distances on the altitude (see Subsec-tion 5.B).Table 14 shows the dependence of the subaperture

distances on βV : a decrease in the differences as βVincreases. The greater βV is, the smaller the CCDMTF in the y direction. Since this MTF is greaterthan one in the x direction, the equating of the xand y MTFs occurs, and, as a result, the distancestend to be equal. We also see an independence ofthe pixel size and a decrease in the number of TDIlines as βV increases.Table 15 shows the dependence of the subaperture

distances on αV : the stretching of the configuration inthe y direction as αV increases. This situation is in-verse to the previous one with respect to the apertureconfiguration, and is similar with respect to the pixelsize and the number of TDI lines.Table 16 shows the already known dependence of

the subaperture distances on the IA and OPD errors(see Subsections 5.A and 5.B), and an independenceof these distances on the atmosphere turbulence. Be-sides, we see an independence of the CCD para-meters on a presence or an absence of the IA andOPD errors.Table 17 shows the linear dependence of the pixel

size on the effective focal length, i.e., 5:00=10:00≡

4:6875=9:375 (see also Fig. 23), and a practical inde-pendence of the aperture configuration on F. Thenumber of TDI lines is independent too.

Table 18 shows an interesting decrease in the sub-aperture distances as R22 increases, while all theother parameters remain unchanged. The optimalvalues of the inner radii equal 0:055m.

Table 19, like Table 18, shows a similar depen-dence of the subaperture distances as the activeaperture area increases; all the other parameters re-main unchanged. The optimal values of the outer ra-dii equal 0:130m.

Now let us compare the results for R ¼ 0:5m andR ¼ 0:4m. We have used the second resolution valuesince one of the high resolution imaging satellitesplanned for launch in 2008 (GeoEye-1 [38] (ex Orb-View-5) of GeoEye, Dulles, Virginia) is required tohave a panchromatic spatial resolution of 0:41m.Table 20, like F2 dependence in Table 17, shows a lin-ear dependence of the pixel size on the spatial reso-lution, i.e., 0:5=0:4≡ 4:6875=3:75. But the ratios ofthe corresponding subaperture distances are inverseproportional to the ratio of the spatial resolutionswith some greater coefficients in the range of1:26–1:27.

6. Conclusion

The presented results show the opportunity of sol-ving the considered optimization problems of theaperture configuration, the effective focal length,the CCD pixel size, and the number of TDI linesfor the space-based OES, using a fast convergent gra-dient method with the objective functions formulatedon the basis of the SNR depending on the MTF of adistributed aperture. This is demonstrated on thesamples of the OS consisting of two and three annu-lar subapertures, using the SNR criterion calculatedat the given value of the spatial ground resolution.The results show optimal solutions of the optimiza-tion problems depending on diverse variable para-meters. The information on the magnitudes of theSNR presented in the tables can be used for determi-nation of the number of the subapertures and theirsizes while the information on the SNR decrease canbe useful in design of a beam combination controlsystem to produce the necessary requirements toits accuracy on the basis of permissible deteriora-tions in the image quality.

Fig. 23. Dependence of the objective function (40) on the CCD pix-el sizes and F for H ¼ 400km, Object 1, Δλ ¼ 0:40–1:05 μm,SNRTh ¼ 18.

Fig. 24. Dependence of the objective function (40) onM and F forH ¼ 400km, Object 1, Δλ ¼ 0:40–1:05 μm, SNRTh ¼ 18.

Table 12. Optimal Parameters Versus the Object Spectral Reflectance(H ¼ 600 km, Δλ ¼ 0:40–1:05 μm)

Object s12 s13 s23 F px py M SNRmin SNRmax

1 0.476 0.499 0.476 5.00 3.125 3.125 200 0.0004 2.792 0.309 0.346 0.307 5.00 3.125 3.125 200 0.33 2.16

Table 13. Optimal Parameters Versus the Altitude H

H s12 s13 s23 F px py M SNRmin SNRmax

400 0.312 0.332 0.311 5.00 4.6875 4.6875 200 1.09 6.45600 0.477 0.503 0.477 5.00 3.125 3.125 200 0.002 3.49


Table 14. Optimal Parameters Versus the Angle βV ½°� Between Vimg and Vch

βV s12 s13 s23 F px py M SNRmin SNRmax

0.2 0.312 0.332 0.311 5.00 4.6875 4.6875 200 1.09 6.450.4 0.316 0.326 0.316 5.00 4.6875 4.6875 157 1.54 5.28

Table 15. Optimal Parameters Versus the V img and V ch Synchronization Parameter αV

αV s12 s13 s23 F px py M SNRmin SNRmax

1.01 0.312 0.332 0.311 5.00 4.6875 4.6875 200 1.09 6.451.02 0.310 0.334 0.310 5.00 4.6875 4.6875 100 0.72 4.69

Table 16. Optimal Parameters Versus the Presence (þE) and Absence (−E) of the IA and OPD Errors (E), and the Presence (þT) and Absence (−T)of the Atmosphere Turbulence (T)

E=T s12 s13 s23 F px py M SNRmin SNRmax

þE=þ T 0.312 0.332 0.311 5.00 4.6875 4.6875 200 1.09 6.45þE= − T 0.311 0.332 0.312 5.00 4.6875 4.6875 200 1.15 6.75−E=þ T 0.311 0.327 0.311 5.00 4.6875 4.6875 200 1.76 10.18−E= − T 0.311 0.327 0.311 5.00 4.6875 4.6875 200 1.85 10.67

Table 17. Optimal Parameters Versus the Upper Bound F 2½m� of the Effective Focal Length

F2 s12 s13 s23 F px py M SNRmin SNRmax

5.00 0.312 0.332 0.311 5.00 4.6875 4.6875 200 1.09 6.4510.00 0.312 0.331 0.312 10.00 9.375 9.375 200 1.29 7.58

Table 18. Optimal Parameters Versus the Upper Bound R22½m� of the Outer Subaperture Radii

R22 s12 s13 s23 F px py M SNRmin SNRmax

0.130 0.312 0.332 0.311 5.00 4.6875 4.6875 200 1.09 6.450.150 0.310 0.327 0.310 5.00 4.6875 4.6875 200 1.63 7.90

Table 19. Optimal Parameters Versus the Lower Bound R11½m� of the Inner Subaperture Radii

R11 s12 s13 s23 F px py M SNRmin SNRmax

0.055 0.312 0.332 0.311 5.00 4.6875 4.6875 200 1.09 6.450.045 0.309 0.333 0.310 5.00 4.6875 4.6875 200 1.09 6.85

Table 20. Optimal Parameters Versus the Spatial Resolution R½m�

R s12 s13 s23 F px py M SNRmin SNRmax

0.5 0.312 0.332 0.311 5.00 4.6875 4.6875 200 1.09 6.450.4 0.394 0.419 0.394 5.00 3.75 3.75 200 0.21 3.93


The research is supported in part by NaturalSciences and Engineering Research Council ofCanada (NSERC) and the Faculty of Engineeringand Computer Science at Concordia University.

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