Alessandro Beghi,1 Angelo Cenedese,2 and Andrea Masiero3,*1Dipartimento di Ingegneria dell’Informazione, Università di Padova via Gradenigo 6/B, 35131 Padova, Italy
2Dipartimento di Tecnica e Gestione dei Sistemi Industriali,Università di Padova Stradella San Nicola 3, 36100 Vicenza, Italy
3Dipartimento di Ingegneria dell’Informazione, Università di Padova via Gradenigo 6/B, 35131 Padova, Italy
Motivated by the increasing importance of adaptiveoptics (AO) systems for improving the resolution oflarge ground telescopes, in this paper the problemof turbulence simulation is addressed to provide atest bed for the design of control strategies for AOsystems.
The presence of wind, local temperature, humidity,and pressure changes cause rapid variations in theatmosphere refraction index [1]. Thus, when thewavefront signal arriving from a star object entersthe Earth’s atmosphere, it is distorted proportionallyto the length of its optic path and depending on theencountered refraction index. Consequently, the at-mospheric turbulence effect is mainly to delay thelight beams of different phases. Therefore, the flatwavefront surface of a light beam arriving from a star
is no longer flat when it is detected on the telescopepupil: this significantly reduces the real resolution ofthe telescope.
The atmospheric turbulence effect can be modeledas a randomly changing phase delay added to thelight beam’s phase. Such delay, which will be alsocalled turbulent phase, can be statistically character-ized as a zero-mean second-order random process.Similarly, a phase screen can be defined as the setof phase values that affect the light beam’s wavefrontarriving on the telescope.
Commonly used methods for turbulent phase si-mulation are based on the fast Fourier transform(FFT). Such methods allow the quick generation ofturbulence samples that perfectly match the theore-tical turbulent phase statistical characteristics.However, since they generate all the samples to-gether, they can be used only for synthesizingfinite-dimensional phase screens.
The interest in studying the performances of AOcontrol algorithms in long exposure simulations
led to different approaches for turbulence simula-tion. Recently proposed procedures [2–4] considerthe turbulence values as the realization of a (zero-mean, wide-sense stationary) stochastic process.Then, a proper model for such stochastic process iscomputed to reproduce the second-order statisticalcharacteristics of the turbulence. References [2–4]exploit a suitably defined linear dynamic system tomodel the spatiotemporal dynamics of the turbulentphase along the wind direction. Then, such a dy-namic system, driven by white noise, is used to pro-duce possibly infinitely long sequences of phasescreen samples. The reader is referred to [5] for amore detailed introduction on models for stochasticprocesses.
In this paper, a new approach that integrates themethod of [2–4] with a multiscale stochastic model isproposed. A low-resolution version of the phasescreen is first computed as in [3], then the multiscalemodel is used to efficiently obtain the high-resolutionversion. The overall algorithm is computationallymore efficient than those in [2–4], and, as shownin Section 4, it accurately reproduces the desiredturbulence statistical characteristics.
A multiscale approach, usually called the midpointmethod, for the generation of phase screen has beenfirst proposed in [6] and recently generalized in [7] tobe usefully applied to the optical wavefront recon-struction problem. Similar to the methods in [6,7],the approach used in this paper computes each coef-ficient in the multiscale representation of the turbu-lence as the sum of two terms: a prediction termcomputed as a linear combination of previously gen-erated coefficients and an innovation term. Differentfrom [6,7], the values of the innovation term at thesame scale are not independently computed to accu-rately reproduce their theoretical correlations.
The multiscale approach proposed here is alsosimilar to that described in [8–12]. As in [10], a multi-resolution representation of the turbulence (e.g., awavelet decomposition [13,14]) is fixed a priori, andsimilar to [12], local spatial predictions are exploitedto reduce the spatial correlation of the consideredsignal. However, different from previously consid-ered approaches, the error process obtained afterprediction is modeled as a moving average process,leading to an efficient way for matching the theore-tical turbulence statistical characteristics at eachscale of the representation.
The paper is organized as follows. Section 2 dis-cusses the turbulence spatiotemporal statisticalcharacteristics. In Section 3, a multiscale representa-tion of the turbulent phase is introduced. Then, inSubsection 3.A the main results are presented, pro-posing a new multiscale stochastic model of theatmospheric turbulence. Finally, in Section 4 the per-formance of the algorithm is discussed by means ofsimulation results.
2. Turbulent Phase Modeling
The turbulent phase is assumed to be zero-mean sta-tionary and spatially homogeneous. Let u and v betwo unit vectors indicating two orthogonal spatialdirections, as in Fig. 1, and let ϕðu; v; tÞ be the valueof the turbulent phase on the point ðu; vÞ at time t onthe telescope aperture plane, where u and v are thepoint coordinates along u and v. Then, the covariancebetween two values of the turbulence, ϕðu; v; tÞ andϕðu0; v0; tÞ, depends only on the distance, ρ, betweenthe two points: cϕðρÞ ¼ E½ϕðu; v; tÞϕðu0; v0; tÞ�,∀ðu; v;u0; v0Þ, such that ρ ¼ jðu − u0; v − v0Þj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðu − u0Þ2 þ ðv − v0Þ2p
.According to the Von Karman theory, the shape of
the spatial covariance function, cϕð·Þ, is completelycharacterized by the values of two physical param-eters, r0, the Fried parameter, and L0, the outer scale[1,15]:
cϕðρÞ ¼�L0
r0
�5=3 η
2
�2πρL0
�5=6
K5=6
�2πρL0
�; ð1Þ
whereK:ð·Þ is theMacDonald function (modified Bes-sel function of the third type) and η is a constant.
Furthermore, the turbulent phase is supposed tobe normally distributed [16]; hence, the second-orderstatistics are sufficient to completely describe itsstatistical properties.
In order to describe its temporal evolution, the tur-bulence is modeled as the superposition of a finitenumber L of layers. The lth layer models the atmo-sphere from an altitude of hl−1 to hl m, whereh0 ¼ 0 ≤ h1 ≤ � � � ≤ hl−1 ≤ hl ≤ � � � ≤ hL.
Let ψ lðu; v; tÞ be the value of the lth layer at pointðu; vÞ at time t. Then the total turbulent phase atðu; vÞ and at time t along the zenith direction is
ϕðu; v; tÞ ¼XLl¼1
γlψ lðu; v; tÞ; ð2Þ
where fγlg are suitable coefficients associated to thelayer energies. Without loss of generality we assumethat
PLl¼1 γ2l ¼ 1.
The layers are assumed to be stationary and char-acterized by similar spatial statistics; i.e., the covar-iance between two points at distance ρ of the lthturbulence layer can be written as
Fig. 1. (a) Coordinates on the telescope image domain. (b) Twopoints, ðu; vÞ and ðu0; v0Þ, separated by a distance ρ on the telescopeaperture plane. (c) Example of phase screen.
l ¼ 1;…;L, where νl ¼ νl;uuþ νl;vv, and kTs is a delaymultiple of the sampling period Ts.
Since the layers are assumed to be independent,the turbulent phase can be simulated by indepen-dently drawing temporal samples for each of thelayers and combining them through Eqs. (2) and(4). Hence, hereafter it is considered the problemof simulating a single layer. Furthermore, the phasescreen is simulated on a discrete set of locations, i.e.,on a grid. Then, a phase screen [Fig. 1(c)] can berepresented as anm × nmatrix containing the turbu-lent phase values. The physical dimension of eachpixel in the matrix is ps × ps. With a slight abuseof notation, hereafter u and vwill be integer numbersindicating the column and row index in the phasescreen [e.g., indicating the spatial position ðups; vpsÞ].
Without loss of generality, the wind direction asso-ciated to the considered layer is assumed to be par-allel to the u vector, i.e., νl ¼ νl;uu, νl;u ≠ 0. Then,simulating new values of the turbulence is equiva-lent to generating new columns of the phase screenmatrix and properly shifting the window correspond-ing to the telescope aperture. Thus, in the followingthe goal will be that of generating an m × n phasescreen, where n can go to infinity and m can be cho-sen arbitrarily large. To simplify the notation, here-after the time coordinate t is omitted from equations.
Recently proposed methods [2–4] consider theturbulent phase as a realization of a zero-meanwide-sense stationary stochastic process φ. Then,the problem of phase screen synthesis can be statedas the simulation of φ, which is done by using a lineardynamic system:
where φðuÞ ∈ Rm is a vector containing the values ofthe uth column of the synthesized turbulent phase,χ ∈ Rq is the state of the linear system (q is the sizeof χ), ξ is a zero-mean Gaussian white-noise processwith covariance Q0, and fA0;K0;C0g are parametersof the model, which can be computed as explained in[2,3], or [4].
The advantage of using Eq. (5) for turbulencesimulation is twofold: it accurately reproduces theturbulence spatiotemporal dynamic, and it can beused to produce infinite sequences of phase screens
(n → þ∞). On the other hand, since, in practical appli-cations, m can assume very large values (e.g.,m ≥ 1000) and q is typically linearly proportional tom, q ¼ OðmÞ, the computational complexity for gener-ating an m × n matrix of turbulent phase values isOðm2nÞ, which, for large values of m, may be quitetime demanding. Similar considerations hold alsofor the computational load andmemory requirementsfor computing the parameters of the system of Eq. (5).
Motivated by the above considerations, in Section 3a new approach will be introduced to reduce the com-putational complexity and memory requirements forthe generation of possibly infinite sequences of phasescreens of very large size.
3. Multiscale Approach for Turbulent PhaseSimulation
In this section a multiscale linear decomposition ofthe turbulentphase is considered: the resulting repre-sentation is formed by coefficients describing thecharacteristics of the phase screen at different scales.The considered multiscale decomposition is requiredto preserve homogeneity and isotropy of the originalprocess ϕ among the multiscale representations.
Among the possible multiresolution transforma-tions satisfying the above requirements, for simpli-city of exposition, in this paper the Haar waveletrepresentation is used: M scales of resolution areconsidered, and at each level of the representation,the turbulent phase is decomposed in four sets ofcoefficients, corresponding to the low-pass represen-tation of the turbulence and the higher frequency de-tails along the horizontal, vertical, and diagonaldirections, respectively (see Fig. 2).
Let xiðu; vÞ be the low-pass coefficient at scale i andspatial position ðups;i; vps;iÞ, where ps;i is the pixelphysical dimension at level i, ps;i ¼ ps2M−i, and iranges from 0 (lowest resolution) to M (highest reso-lution), i.e., xMðu; vÞ ¼ ϕðups; vpsÞ. Similarly, letwj
iðu; vÞ, j ¼ fh; v;dg be the value of the detail coeffi-cient for the horizontal, vertical, and diagonal direc-tions at scale i and spatial position ðups;i; vps;iÞ.
The coefficients at two successive scales, i andiþ 1, 0 ≤ i ≤ M − 1, of the two-dimensional (2D) Haartransform are linearly related as follows:
Fig. 2. Haar wavelet decomposition of a phase screen. i is thescale index, and M ¼ 2 (number of scales). Conventionally i ¼ Mcorresponds to the high-resolution turbulent phase, while in thewavelet decomposition i ranges between 0 and M − 1. The figuresof the ith row represent from left to right the low-pass version ofthe current phase screen (xi) and the details on the horizontal (wh
where ðu; vÞ is a spatial position in the Haar repre-sentation at level i,
C ¼ 12
2641 1 1 11 −1 1 −11 1 −1 −11 −1 −1 1
375;
and C−1 ¼ C⊤. By recursively applying Eq. (6),fxi;wh
i ;wvi ;w
di g, 0 ≤ i < M − 1 can be obtained as lin-
ear transformations of xM.Since the turbulent phase is assumed to be zero
mean and Gaussian, it is completely characterizedby its second-order statistics. Hence, the aim of a tur-bulence simulation algorithm is that of making thecovariances of xM reproduce the theoretical turbulentphase statistical characteristics that can be com-puted from Eq. (1). As shown by Eq. (6), the relationbetween the coefficients at different scales of the con-sidered multiresolution transformation is linear;hence, fxi;wh
i ;wvi ;w
di g, 0 ≤ i < M, are Gaussian and
their second-order statistical characteristics (e.g.,their covariances) can be recursively computed fromthose of xM using linear transformations [5]. Letcx;iðu; vÞ be the correlation between coefficients ofthe multiscale representation at scale i whose loca-tions are separated by the spatial vector ðu; vÞ, i.e.,cx;iðu; vÞ ¼ E½xiðu; vÞxið0; 0Þ�; then,
where u0 ¼ 2u and v0 ¼ 2v, ∀ðu; vÞ, 1 ≤ i ≤ M. Theabove equation can be derived from Eq. (6).
Linearity can be used to ensure that the multire-solution simulation algorithm reproduces the correctsecond-order statistics of the turbulence [Eqs. (7), (9),(8), and (10)] at all the considered scales. In the fol-lowing, such a goal is obtained by decomposing thesimulation procedure in two steps:
• Simulating the turbulent phase at low resolu-tion, m0 × n0, where m0 ≪ m and n0 ≪ n, withEq. (5), using previously proposed techniques [2–4].
• Using a multiscale stochastic model to producehigh-resolution turbulent phase values, xM, taking asinput the low-resolution samples, x0, computed at theprevious step.
The number of scales M used in the Haar multire-solution representation is set to the least integergreater than or equal to log2ðm=m0Þ, e.g., MðmÞ ¼⌈ log2ðm=m0Þ⌉. Assume that m0 is constant and in-dependent on m; then, the number of operationsrequired for simulating the low-resolution m0 × n0phase screen (associated to the high-resolutionm × n phase screen) with Eq. (5) is linearly propor-tional to m2
0n0: since q is constant and n0 ¼ n=2M ,then the computational complexity of the first stepof the simulation procedure is OðnÞ.
Notice that the second step of the procedure pro-duces a phase screen matrix of size m × n, so its com-putational complexity is at least OðmnÞ. Hence, thecomputational complexity of the overall procedure isgiven by that of the second step, while the computa-tional load of the first step is negligible.
A detailed description of the second step of the si-mulation procedure is provided in Subsection 3.A,while, for the first step, the reader is referred to[2–4].
A. Multiscale Stochastic Model
Multiscale stochastic models have been widely stu-died in the literature: the approach followed in thispaper is similar to those in [8–12], where multiscaleautoregressive (MAR) models are considered. Theaim is that of computing a multiscale model thatmatches the desired second-order statistics (7), (9),(8), and (10) at each scale.
For simplicity of exposition in this subsection,the process is taken to be one dimensional, and
the border effect is not considered; i.e., the domainof the process at each scale is assumed as an in-finite line. The process is decomposed with a one-dimensional Haar transform. At the ith scale thecoefficients are xi, the low-resolution representation,and wi, the details at level i. The domain L of therepresentation coefficients is the Z domain. ThenEq. (6) becomes
�xiðuÞwiðuÞ
�¼ C1
�xiþ1ð2uÞ
xiþ1ð2uþ 1Þ�; u ∈ L; ð11Þ
where
C1 ¼ 1ffiffiffi2
p�1 11 −1
�:
Similar to Eqs. (7)–(10), xi and wi are zero-meanwide-sense stationary processes, with covariancescomputable as follows:
cx;MðuÞ ¼ E½xMðuÞxMð0Þ� ¼ cϕðjujÞ; u ∈ L; ð12Þ
E�xi−1ðuÞwi−1ðuÞ
�½ xi−1ð0Þ wi−1ð0Þ �
¼ C1
�cx;ið2uÞ cx;ið2u − 1Þ
cx;ið2uþ 1Þ cx;ið2uÞ�C⊤
1 ; ð13Þ
where u ∈ L, 1 ≤ i ≤ M.The multiscale synthesis starts with samples of x0
and iteratively generates the values of the details atscale i ¼ 0;…;M − 1 in such a way to match theirtheoretical second-order statistical characteristics,Eqs. (12) and (13).
Since the low-resolution approximation and thedetail statistics are different at different scales, thenthe stochastic model derived in the following is scaledependent. Consider the model for level i: it takes asinput the low-resolution representation xi, which isassumed to match the corresponding ith scale theo-retical covariances, and produces as output wi suchthat xiþ1, computed inverting Eq. (11), reproducesthe second-order statistics (12) and (13) at level iþ 1.
By assumption, ϕ is homogeneous and isotropic,and cϕðρÞ ≈ 0 for large ρ. Since the relation betweenwavelet representation coefficients is Eq. (11), thenxi, and wi are homogeneous and isotropic as well.Furthermore, since cϕðρÞ ≈ 0, then it immediatelyfollows that
cx;iðuÞ ≈ 0; cw;iðuÞ ≈ 0;
cxw;iðuÞ ≈ 0; for large juj;ð14Þ
where cw;iðuÞ ¼ E½wiðuÞwið0Þ� and cwx;iðuÞ ¼E½wiðuÞxið0Þ�.
Motivated by Eq. (14), the following concept ofmultiscale Markovianity is assumed in this subsec-tion; the detail process at scale i and position u,
wiðuÞ is assumed to be conditionally independent[17] on far values in xi and wi given the values ofits spatial neighbors in xi:
where pð·j·Þ is the conditional density function andthe neighborhood of u, NiðuÞ is defined as
NiðuÞ ¼ fu0 ∈ Lj0 ≤ ju0 − uj ≤ �dig;
and �di is the neighborhood size, which can be scaledependent.
Then, the following multiscale stochastic model isconsidered to describe the local relation between thecoefficients at two successive scales:
wiðuÞ ¼X
u0∈NiðuÞai;ju0−ujxiðu0Þ þ eiðuÞ ¼ wiðuÞ þ eiðuÞ;
u ∈ L; ð16Þwhere
wiðuÞ ¼X
u0∈NiðuÞai;ju0−ujxiðu0Þ ð17Þ
is the best linear prediction of wiðuÞ given the valuesof xiðu0Þ, ∀u0 ∈ L and wiðu00Þ, ∀u00∉NiðuÞ. The coeffi-cients fai;·g can be computed as those that yield tothe minimum variance linear prediction [5] fromthe spatial covariances of Eq. (13). The prediction er-ror ei is a Gaussian wide-sense stationary (homoge-neous and isotropic) process, and it is assumed to beorthonormalizable; i.e., it can be represented asfollows:
eiðuÞ ¼Xþ∞
k¼−∞
θiðu − kÞϵiðkÞ; ð18Þ
where ϵi is a zero-mean Gaussian white-noise pro-cess with variance 1.
Similar to [12,18] for MAR and standard Markovrandom fields, the effect of using a (multiscale) spa-tial prediction is that of partially decorrelating wi inspace; i.e., the spatial correlations of ei vanishquickly. Usually eiðuÞ is not independent on eiðu0Þfor u0 ≠ u, i.e., ei is not spatially white; however, ifEq. (15) holds, then
From the stationarity of ei, ce;iðuÞ ¼E½eiðu0Þeiðu0 þ uÞ�, u ∈ L does not depend on u0.Furthermore, ce;iðuÞ ¼ ce;ið−uÞ, and ce;iðuÞ ¼ 0 foru > di. Since the spatial correlations of ei vanish atdistances larger than di, then ei is characterizedby a finite set of correlations, fce;ið−diÞ;…; ce;iðdiÞg,which can be computed as follows:
ce;iðuÞ ¼ E½eiðuÞeið0Þ�¼ E½wiðuÞwið0Þ� −
Xu0∈NiðuÞ
ai;ju0−ujE½xiðu0Þwið0Þ�
−X
u00∈Nið0Þai;ju00 jE½wiðuÞxiðu00Þ�
þX
u00∈Nið0Þai;ju00j
Xu0∈NiðuÞ
ai;ju0−ujcx;iðu0 − u00Þ; ð19Þ
where the values of the expectations can be obtainedfrom Eqs. (12) and (13).
Since ei is orthonormalizable and it has a finite setof nonzero covariances {indeed, E½eiðu0ÞeiðuÞ� ¼ 0 forju0 − uj > di}, then Eq. (18) becomes
eiðuÞ ¼Xdi
k¼−di
θiðkÞϵiðu − kÞ: ð20Þ
Then, different from [8–12], the process ei is conveni-ently modeled as a moving average (MA) process[19]. From Eq. (20), the spatial correlationsfce;ið−diÞ;…; ce;iðdiÞg can be expressed as the convolu-tion product, denoted by �, of the finite kernel θi withitself considered in reverse order:
ce;iðkÞ ¼ ðθi � θ−i ÞðkÞ; ð21Þ
where fθið−diÞ;…; θiðdiÞg are the (real) kernel coeffi-cients and θ−i ðkÞ ¼ θið−kÞ, k ¼ −di;…;di.
This leads to the following MAR MA model:
wiðuÞ ¼X
u0∈NiðuÞai;ju0−ujxiðu0Þ þ
Xu0∈NiðuÞ
θiðu − u0Þϵiðu0Þ;
u ∈ L: ð22Þ
The values of the parameters fθið·Þg have to becomputed in such a way to make the covariancesof the MA process Eq. (20) match those in Eq. (19).This problem is similar to the minimum length cor-relation extension problem. The problem can besolved as in [20] by finding the sequence fθið·Þg ofminimal length ensuring the match. However, themethod presented in [20] is quite laborious in themultidimensional case; hence, the following proce-dure is proposed.
Let F ð·Þ be the discrete Fourier transform (FT) (ef-ficiently computable with the FFT algorithm) and letthe domain of ce;i be ½−2di;…; 2di�, where ce;iðhÞ ¼ 0∀u such that juj > di. Then, the MA coefficientsfθið·Þg can be computed similarly to [21].
• From the FT’s convolution property [22] ap-plied to Eq. (21):
F ðce;iÞðhÞ ¼ F ðθi � θ−i ÞðhÞ ¼ F ðθiÞðhÞ · F ðθ−i ÞðhÞ:
• Since θi is real, F ðθ−i ÞðhÞ ¼ ðF ðθiÞðhÞÞ�, and
F ðce;iÞðhÞ ¼ F ðθiÞðhÞ · ðF ðθiÞðhÞÞ� ≥ 0;
which, specifically, derives from the properties of sta-tionary processes.
• F ðθiÞð·Þ can be estimated by imposingF ðθiÞðhÞ ¼
• θi is obtained computing the inverse FTof F ðθiÞ.
Since ce;ið·Þ is even and real, then F ðce;iÞ, F ðθiÞ, andθi, computed as described above, are even and realtoo [22].
Notice that the above procedure does not necessa-rily lead to the minimum length sequence of param-eters that reproduces the desired covariances [23].
Let ½ui;1∶ui;2� be the domain of the multiscale phasescreen representation at scale i. Then, the synthesisprocedure iteratively executes the following steps.
1. System (5) produces new low-resolutionsamples, fx0ðuÞgu¼u0;1∶u0;2
.2. For i ¼ 0∶M − 1
– Compute the multiscale spatial predictionsfwiðuÞgu¼ui;1∶ui;2
as in Eq. (17).– Generate new values of ϵið·Þ by sampling from a
zero-mean white-noise random generator.– Compute the values of eið·Þ by filtering ϵið·Þ as
in Eq. (20).– Compute the details at scale i, as
wiðuÞ ¼ wiðuÞ þ eiðuÞ; u ¼ ui;1∶ui;2;
and combine fwiðuÞgu¼ui;1∶ui;2with fxiðuÞgu¼ui;1∶ui;2
toobtain fxiþ1ðuÞgu¼2ui;1∶2ui;2þ1.
End3. ϕðuÞ ¼ xMðuÞ, ∀u.
The above procedure can be adapted to the 2D casewith minor changes. Furthermore, since the multi-scale synthesis procedure involves only spatiallylocal operations, it can be easily modified to be itera-tively used after the generation of new phase screencolumns with Eq. (5), thus allowing the simulation ofinfinitely long sequences of turbulent phase values.
To take into account the border effect, which hasbeen neglected so far, the turbulence can be simu-lated on an oversized area, and only the internal partof the generated phase screen can be selected. Theaim of the algorithm is that of generating a sequenceof phase screens, where a singular phase screen is amatrix of m × �n pixels, while the total screen area tobe generated is an m × n matrix, with n ≫ �n. The
algorithm starts by simulating the first m × �n pixels:ϕðu; vÞ, u ¼ 1;…;m, v ¼ 1;…; �n, which, using aMatlab-like notation, will be indicated as ϕð∶; 1∶�nÞ.To properly generate ϕð∶; 1∶�nÞ, the algorithm simu-lates the turbulence on a larger area, i.e., a ðmþ �dÞ ×ð�nþ �dÞmatrix ϕð∶; 1 − �d∶�nþ �dÞ, where for simplicity�d ¼ di ¼ δi, ∀i.For simplicity assume that the algorithm at the
second step generates the phase screen ϕð∶; �nþ1∶2�nÞ formed by the columns with spatial index u ¼�nþ 1;…; 2�n [i.e., ϕð∶; �n − �dþ 1∶2�nþ �dÞ consideringalso the borders]. To preserve the spatial continuityof the turbulence, the second step inherits from thefirst the values of x0 and fϵigi¼0;…;M−1 close to theright border, i.e., x0ð∶; �n0 −
�d∶�n0 þ �dÞ, fϵið∶; �ni−�d∶�ni þ �dÞgi¼0;…;M−1, where �ni ¼ �n=2M−i for i ¼ 0;…;
M − 1. Then, the algorithm neglects the generatedphase screen border values. However, notice thatϕð∶; �n − �dþ 1∶�nÞ have already been generated atthe previous step, and ϕð∶; 2�nþ 1∶2�nþ �dÞ will begenerated at the following step. Hence, the fact thatonly the central part of each generated phase screenis considered does not limit the applicability of thealgorithm. The above considerations can be repeatedfor each step of the algorithm.
The main computational cost of the algorithm isdue to filtering operations. Since the number of coef-ficients in the Haar description of the phase screen isequal to the size of the phase screen itself and since�di does not depend on m and n, then the number ofoperations needed for generating an m × n matrix ofturbulent phases is OðmnÞ.
Notice that, in practical applications, Eq. (15) maybe only an approximation. Nevertheless, in this casethere exists an integer δi > 0 such that
E½eiðuÞeiðuþ δiÞ� ≈ 0; ð23Þ
thus, in this case the procedure can be applied ap-proximating the covariances of ce;iðuÞ with 0 forjuj > δi. In this case δi has to be chosen with somecare: indeed, too small a value for δi may lead to anonpositive covariance sequence f0;…; 0; ce;ið−δiÞ;…;ce;iðδiÞ; 0;…; 0g. However, as far as δi is sufficientlylarge, this approximation takes to very small repre-sentation errors.
In practical applications �di and δi can be consid-ered as design parameters, which have to be chosensufficiently large to grant an effective prediction, anadequate spatial decorrelation, and a good reproduc-tion of the correlations of ei. On the other hand, evenif they do not depend on m and n, �di and δi affect thevalues of constant factors in the computational com-plexity. From a practical point of view, too largevalues of �di and δi may slow down the simulation.Thus, it is necessary to choose trade-off values tohave both a good matching of the desired statisticsand a sufficiently low computational complexity.
4. Simulations and Discussion
In this section the performance of the proposedmeth-od is investigated in two case studies. In the first casethe atmospheric turbulence is simulated for a tele-scope with diameter d ¼ 8 meter, setting the pixelphysical dimension to ps ≃ 0:0021m. In particular,a 3840 pixel × 3840 pixel phase screen is generated,corresponding to an 8m× 8marea, as shown in Fig. 3(bottom).
The outer scale and Fried parameter are set toL0 ¼ 50m, r0 ¼ 0:2m, respectively, and, as in Sec-tion 3, the wind direction is assumed to be parallelto u.
The turbulence is simulated at low resolution,m0 × n0, using the algorithm described in [3], settingm0 ¼ 60, and consequently M ¼ 6 (so 3840 ¼ 2Mm0).In this simulation n0 ¼ 60; however, the method de-scribed in this paper can generate infinitely longsequences of columns; hence, the turbulence canbe simulated for any choice of n and n0. The low-resolution, 60 × 60, generated phase screen is shownin Fig. 3 (top left).
The multiscale model of Section 3.A generates thehigh-resolution phase screen, Fig. 3 (bottom left), re-taining the low spatial frequencies characteristicscomputed with [3] and adding high spatial frequen-cies details. To make clear the effect of the multiscalemodel, Fig. 3 shows also the zoom on a 2:1m× 2:1mwindow of Fig. 3 (left) comparing the generated phasescreen at low [Fig. 3 (top-right)] and high [Fig. 3(bottom-right)] resolution. Notice that the pixeldimensions at low and high resolutions are 0:13m ×0:13m and 0:0021m × 0:0021m, respectively. In the
Fig. 3. Phase screen synthesis: comparison of low- (xl0, top) andhigh-resolution (xl6, bottom) simulated turbulent phase on 8m×8m (left) and 2:1m× 2:1m (right) windows. Resolutions in pixelsare 60 × 60 (top left) and 16 × 16 (top right) pixels, 3840 × 3840(bottom left) and 1024 × 1024 (bottom right) pixels.
simulations of this section, the neighborhood size, �di,and theMA domain size, δi, for Eq. (22) are set to 5 foreach i ¼ 0;…;M − 1.
Astronomers commonly describe the turbulentphase spatial statistical characteristics by meansof the structure function, Dϕð·Þ, which can be com-puted as
DϕðρÞ ¼ 2ðcϕð0Þ − cϕðρÞÞ: ð24Þ
Then, Fig. 4 compares the sample estimates of thestructure function, obtained from the generatedphase screen, with the theoretical ones computedby means of Eq. (24). As shown in [3], it is more dif-ficult for methods based on Eq. (5) to correctly repro-duce the structure function along the wind directionthan along its orthogonal direction, so Fig. 4 com-pares the structure functions along the wind direc-tion. The sample structure function in Fig. 4 hasbeen estimated using just a 8m × 10698m phase
screen: since the estimates are obtained using a quitelarge number of phase samples, they are close to thetheoretical structure function values.
In the second case the atmospheric turbulence issimulated on a 320m× 80080m area. The dimensionof each phase screen is 320m × 320m; the pixel phys-ical dimension is ps ¼ 1=32≃ 0:031m, m0 ¼ 80, andM ¼ 7. The outer scale, Fried parameter, and winddirection are as in the first case study. Figure 5 com-pares the sample estimates of the structure functionalong the wind direction, obtained from the gen-erated phase screen, with the theoretical ones com-puted by means of Eq. (24).
To conclude, computational complexity andmemory requirements of the proposed synthesisalgorithm are examined in the following. Since thedimension of telescopes is ever growing and an accu-rate spatial description of the values of the turbu-lence phase allows to more precise evaluations ofthe AO system performances, then in the following
0 1 2 3 4 5 60.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Pha
se S
truc
ture
Fun
ctio
n [r
ad2 ]
Separation [m] Separation [m]0 1 2 3 4 5 6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0x 10−3
Pha
se S
truc
ture
Fun
ctio
n E
rror
[rad
2 ]
Fig. 4. Comparison of the theoretical structure function (dashed curve) with the sample structure function (solid curve) evaluated alongthe wind direction computed from a 8m× 10698m phase screen. The values of the parameters are set to L0 ¼ 50m, r0 ¼ 0:2m, d ¼ 8m,ps ¼ 0:0021m.
0 10 20 30 40 500.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Pha
se S
truc
ture
Fun
ctio
n [r
ad2 ]
Separation [m] Separation [m]0 10 20 30 40 50
−20
−15
−10
−5
0
5x 10−4
Pha
se S
truc
ture
Fun
ctio
n E
rror
[rad
2 ]
Fig. 5. Comparison of the theoretical structure function (dashed curve) with the sample structure function (solid curve) evaluated alongthe wind direction computed from a 320m× 80080mphase screen. The values of the parameters are set toL0 ¼ 50m, r0 ¼ 0:2m, d ¼ 42m,ps ¼ 0:031m.
the computational complexity of the overall algo-rithm is derived with respect to bothm and n, assum-ing to be interested in generating long sequences ofhigh-resolution phase screens (i.e., m is large and npossibly goes to infinity). Since the value of m0 isfixed, then, as shown in Section 3, the computationalcomplexity of generating the m0 × n0 low-resolutionphase screen with Eq. (5) is OðnÞ. On the other hand,the synthesis of the multiscale coefficients is OðmnÞ(Subsection 3.A); thus, the overall computationalcomplexity of the algorithm is OðmnÞ. Thus, theproposed approach significantly reduces the com-putational load of [2–4] for the synthesis of high-resolution phase screens (see Fig. 6).
Similar considerations can be repeated for the pro-blem of computing the parameters of the multiscalemodel: at each scale the computation of the param-eters of Eqs. (16) and (20) involve only the use of localstatistics; thus, without requiring large amounts ofmemory nor of time.
The values of �di and δi affect only constant factorsin the asymptotical computational complexity of thesynthesis procedure; however, in practical applica-tions they can have a significative influence on thecomputational time. Showing the dependence on �di(and assuming δi ¼ �di ¼ �d, ∀i), the computationaltime [24] is Oð�d2mnÞ. This consideration motivatesan accurate choice of their values and thus of themultiscale linear decomposition to be used. In thispaper the Haar wavelet representation has beenused; however, different multiscale transforms maylead to more efficient spatial predictions and conse-quently to lower values of �di and δi.
Because of the quadratic dependence on �d, for val-ues of typical interest of �d, FFT methods are moreefficient than the approach described in this paperfor generating square phase screens of not very largedimension. Instead, this multiscale approach be-comes convenient when dealing with long phasescreens (which are typically generated iteratively,
e.g., as n=�n successive phase screens of dimension�m × �n, and �m ¼ m in this example). In actualfact, it has been the scientists’ need for a procedureto synthesize long phase screens that motivatedthis study.
The convenience for long sequences of phasescreens is confirmed by its memory requirements.The amount of memory for storing model parametersis usually negligible with respect to that for storing aphase screen, which, by doing operations in place, is�α �m �n, where �α ≈ 1.When considering the generation of the overall
m × n screen, the required memory becomes αmn,where α ≈ 1:1 (α depends on the values of �d, �m, �n).Such a value of α is obtained for the Haar wavelet.However, using different wavelets typically takesto larger values of α. Hence, there is a trade-off inthe choice of the multiscale transform between itsmultiscale prediction ability (reducing �d) and itscomputational complexity and memory require-ments [25]. A comparison between performances ledby different multiscale representations is currentlyunder investigation.
5. Conclusions
In this paper a new method for simulating high-resolution phase screens has been proposed. Sucha method is based on the combination of a dynamicsystem [3,2,4], which simulates the low-resolutiontemporal dynamic of the turbulence, with a multi-scale stochastic model, which generates the high-resolution details of the turbulent phase.
The resulting procedure, which can be used for si-mulating infinite sequences of turbulent phases, en-sures that thegenerated samples reproducewithhighaccuracy the theoretical statistics of the turbulence.
On the other hand, the overall computationalcomplexity of the synthesis procedure is particu-larly appealing thanks to the great ability ofmultiscale models to capture the turbulence spatialstatistical characteristics in a compact and efficientrepresentation.
References and Notes1. F. Roddier, Adaptive Optics in Astronomy (Cambridge
University, 1999).2. F. Assemat, R. Wilson, and E. Gendron, “Method for simulat-
ing infinitely long and non stationary phase screens with op-timized memory storage,” Opt. Express 14, 988–999 (2006).
3. A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realizationapproach to the efficient simulation of phase screens,” J. Opt.Soc. Am. A 25, 515–525 (2008).
4. D. Fried and T. Clark, “Extruding Kolmogorov-type phasescreen ribbons,” J. Opt. Soc. Am. A 25, 463–468 (2008).
5. P. S. Maybeck, Stochastic Models, Estimation, and Control(Academic, 1979), Vol. 1.
6. R. G. Lane and A. Glindemann and J. C. Dainty, “Simulationof a Kolmogorov phase screen,” Waves Random Media 2,209–224 (1992).
7. E. Thiebaut and M. Tallon, “Fast minimum variance wave-front reconstruction for extremely large telescope,” J. Opt.Soc. Am. A 27, 1046–1059 (2010).
0 500 1000 1500 2000 2500 3000 3500 400010
4
105
106
107
108
109
1010
1011
Num
ber
of o
pera
tions
Dynamic model (5)Multiscale procedure
Fig. 6. Computational complexity of turbulent phase synthesis:comparison between dynamic model (5) used as in [2–4] [solidcurve, Oðm2nÞ] and procedure of Section 3 [dotted–dashed line,OðmnÞ]. n ¼ 1000; m varies between 40 and 4000.
8. M. Luettgen, W. Karl, A. Willsky, and R. Tenney, “Multiscalerepresentations ofMarkov random fields,” IEEETrans. SignalProcess. 41, 3377–3396 (1993).
9. A. Benveniste, R. Nikoukhah, and A. Willsky, “Multiscale sys-tem theory,” IEEE Trans. Circuits Syst. I 41, 2–15 (1994).
10. K. Daoudi, A. Frakt, and A. Willsky, “Multiscale autoregres-sive models and wavelets,” IEEE Trans. Inf. Theory 45, 828–845 (1999).
11. W. Irving and A.Willsky, “A canonical correlations approach tomultiscale stochastic realization,” IEEE Trans. Autom. Con-trol 46, 1514–1528 (2001).
12. A. Frakt and A. Willsky, “Computationally efficient stochasticrealization for internal multiscale autoregressive models,”Multidimens. Syst. Signal Process. 12, 109–142 (2001).
13. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF RegionalConference Series in Applied Mathematics 61 (SIAM, 1992).
14. S. Mallat, A Wavelet Tour of Signal Processing (Academic,1999).
15. R. Conan, “Modelisation des effets de l’echelle externe de co-herence spatiale du front d’onde pour l’observation a hauteresolution angulaire en astronomie,” Ph.D. thesis (UniversitéNice Sophia Antipolis, 2000).
16. F. Roddier, “The effects of atmospheric turbulence in opticalastronomy,” Prog. Opt. 19, 281–376 (1981).
17. From Eq. (14),wiðuÞ andw0iðuÞ are practically uncorrelated for
large ju − u0j. Furthermore, intuitively, the conditioned valueof wiðuÞ, given the local values of xi (i.e., the process at lowerresolution), is much more uncorrelated with wiðu0Þ, u ≠ u0,with respect to wiðuÞ.
18. J. Woods, “Two-dimensional discrete Markovian fields,” IEEETrans. Inf. Theory 18, 232–240 (1972).
19. M. B. Priestley, Spectral Analysis and Time Series (Academic,1982), Vol. 1.
20. A. Steinhardt, “Correlation matching by finite length se-quences,” IEEE Trans. Acoust. Speech Signal Process. 36,545–559 (1988).
21. M. Le Ravalec, B. Noetinger, and L. Hu, “The FFT movingaverage (FFT-MA) generator: an efficient numerical methodfor generating and conditioning Gaussian simulations,”Math.Geol. 32, 701–723 (2000).
22. A. Oppenheim and R. Schafer, Digital Signal Processing(Prentice-Hall, 1975).
23. Imposing F ðθiÞðhÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF ðce;iÞðhÞ
p, the solution is restricted to
be symmetric. However, there might exist a shorter nonsym-metric sequence of coefficients leading to the same covar-iances fce;ið−diÞ;…; ce;iðdiÞg.
24. To be more precise, let �m and �n be the dimensions of a singlephase screen (if the wind velocity is parallel to u, thentypically �m ¼ m and �n ≪ n) and assume m0 ≪ m andn0 ≪ n; then the number of operations computed by the algo-rithm to generate an �m × �n phase screen are approximately64�d2 �m�n. When iteratively generating an m × n screen, withn ≫ �n, dividing it in approximately n=�n phase screens, thereis an extra computational load due to the extra computationsto ensure the continuity between successive phase screens.Anyway, such extra computational load is usually a minorterm in the overall complexity (lower than 10% in our simula-tions). Similar considerations can be repeated for the memoryrequirements.
25. Wavelets different from the Haar transform have bettermultiscale prediction ability but worse complexity and mem-ory requirements.
