Adaptive optics (AO) uses wavefront sensors (WFSs)to detect the aberrations caused by the atmosphere.The sensors used in many AO applications are theShack–Hartmann WFS (SH-WFS) [1], the pyramidWFS [2], and the curvature WFS [3]. All these meth-ods indirectly measure the shape of the wavefrontaberration by measuring the gradient of the wave-front shape. The wavefront reconstruction from thegradient measurements has been considered ex-tensively in the literature, and several techniqueshave been proposed to obtain such reconstructions.Some of these techniques are based on the leastsquares solution [4–6], some on the Fourier trans-
form [7–11], one on the Fourier cosine series [12],and some are based on the multigrid conjugate gra-dient method [13–15].
Recently, a method was proposed by Hampton et al.[16], which is based on obtaining the wavelet decom-position of the wavefront directly from the gradientmeasurements. This method has computational com-plexity ofOðNÞ, and it allows an effective eliminationof waffle modes that are often dominated by noisecontamination [17]. The performance of this techni-que has been discussed in detail in [16]. However, thetechnique proposed in [16], as well as techniquesbased on the Fourier transform and cosine series,are based on gradient data being available ona square, which is not the case in astronomical appli-cations, where there is a circular pupil and thegradient data are available over this circularaperture.
For reconstruction using techniques that requiregradient measurements on a square, the measure-ments available on a circular aperture have to beextrapolated to a square. The extrapolated data mea-surements have to satisfy the zero curl condition, butbeyond this, the actual values of the extrapolatedphase are inconsequential because this extrapolatedregion will be ignored after reconstruction. There aretherefore many solutions to the extrapolation pro-blem, and the objective is to find such that they re-quire the least computations and also that they donot magnify measurement noise or waffle modes.Waffle is the highest frequency that can be repre-sented on a grid and resembles a checkerboard pat-tern. These modes are well known to be of lowmagnitude in atmospheric turbulence [18], and itis desirable to remove this and local waffle shapesfrom the reconstruction [17].One of the possible techniques proposed so far for
fast Fourier transform (FFT) is to assume that theextrapolation is periodic [7]. Another approach isto assume that the extrapolated region is flat [9].An advantage of this approach is that the gradientdata in the extrapolated region are zero. The onlysignificant computation cost is to determine the cor-rect transition from the pupil edge to the flat region.Another approach uses a polar coordinate systemwith a logarithmic radial axis to extrapolate the cen-tral circle of data that are often blocked by the sec-ondary mirror or the hole in the center of the primarymirror [12].It was reported by Poyneer et al. in [9] that waffle
modes can be excited when extrapolating in Friedalignment [4], and one solution presented is to con-vert the measured data to a modified-Hudgin geome-try as an intermediate step. One cause of the waffleexcitation in Fried alignment in [9] is that the extra-polation approach forces the gradients in the extra-polated region to be zero. This strict conditioncoupled with the invisibility of waffle may excite localwaffle shapes during the extrapolation process.In this paper newapproaches for extrapolating gra-
dient data from the circular aperture to a square arepresented so that the extrapolated gradient data canbe used for wavefront reconstruction. These extrapo-lation processes are based on desired properties of thereconstructed phase (i.e., continuous phase, wafflenot excited), rather than based on properties of thegradient data that leave thewaffle asambiguousuntilreconstruction occurs. The first one is based on extra-polating the gradient measurements so that theextrapolated gradients along the diagonals are zero(i.e., the reconstructed phase will be flat on the diag-onals). This is the “extension” method [9] applied tothe two diagonal grids of the Fried geometry. Thisallows a simple calculation of the extrapolated dataadjacent to the boundary and a simple extension be-yond it using only data copying. The other approachproposed here is based on considering the circularboundary as a mirror surface and extending the datausing the mirror equations. A curved mirror can be
used to extrapolate the data from the circle to an oc-tagon, and then flat mirrors can be used to extend thedata from the octagon to a square. The resultingextrapolated data will be one of the many possiblesolutions satisfying the zero curl condition.
The proposed approach has several advantages.When using reflections, the extrapolation is basedon data further in the interior of the pupil ratherthan only from the edge. This tends to attenuateany possible increase of the “waffle” modes causedby measurement noise since more measurementsare used in the process. Another advantage of the re-flection approach is that part of the extrapolation canbe performed using flat mirrors. In such a case, thegradient data can be simply copied outside the octa-gonal or rectangular data set with the proper signsand thus greatly reduce the computational cost.
The paper is organized as follows: In Section 2,some preliminary material regarding the modelinggradient measurements using the Fried geometryis presented. In Section 3, the proposed techniquesfor extrapolation are presented. In Section 4, the ex-trapolation process is outlined, and the associatedcomputational complexity is discussed. The perfor-mance of the proposed approach is evaluated in con-nection with the reconstruction method presented in[16] in Section 5.
2. Preliminaries
Consider a continuous wavefront represented by Φand a WFS measuring the gradient of the wavefrontat discrete points. The continuous gradient of thewavefront is represented as
∇Φ ¼ ∂Φ∂x
ux þ∂Φ∂y
uy ¼ _Φxux þ _Φy uy ; ð1Þ
where x and y are the horizontal and vertical direc-tions and u represents a unit vector. The continuousinput to the SH-WFS is discretely sampled by an ar-ray of lenslets [1], and this discretization of [1] hasbeen modeled by Fried [4] in the following way.
ϕ is an element of matrix Φ representing thesampled wavefront, and _ϕx and _ϕy are the measuredgradient elements in the x and y directions, respec-tively. The Fried geometry modeling the gradient isgiven in Fig. 1. The arrows indicate the directionsof the gradients, and the gradientmeasurement pointiswhere they cross. This shows that themeasurementpoint for _ϕx and _ϕy is in the center of the four discretepixels. It can be seen from this figure that the gridpoints of ϕ are not the same as those for _ϕx and _ϕy.
It follows from this discrete model [4] that the aver-age change in the x direction is Eq. (2), and the aver-age change in the y direction is Eq. (3):
where the elements of matrices Φ, _Φx, and _Φy are ϕ,_ϕx, and _ϕy, respectively. Assuming grid spacing of 1unit, the indices representing the two grids, the ði; jÞfor the wavefront and the ðp; qÞ for the gradients, arerelated by
i ¼ p� 0:5; ð4Þ
j ¼ q� 0:5; ð5Þindicating that the grid of the wavefront elements, ϕ,is offset by half the pitch from the measurements ofthe x and y direction slopes, _ϕx and _ϕy, as shown inFig. 1. The positive vertical direction is downward,and the positive horizontal direction is to the right.The measured gradient data describe the spatial
rate of change of the wavefront and have some impor-tant properties that have to be satisfied when thesemeasurement data are extrapolated beyond the aper-ture used for measurements. A critical property ofsuch a gradient field is that the integral of the gra-dient along a path is equal to the difference in inten-sity between the end point and the starting point.This extends to the rule that the closed path integralof gradients (i.e., the curl) must equal zero since thestart and end points are equal. There are effectivelyinfinite possibilities to extend a gradient field so thatthese conditions are satisfied. The ones proposed inthis paper are chosen because of their low computa-tional complexity and their good performance in thepresence of measurement noise.
3. Extrapolation of Gradient Measurement
Given gradient measurements on a circular aper-ture, the goal of this paper is to provide a computa-tionally fast technique to extrapolate the gradientdata to a square aperture so that they can be usedfor phase reconstruction. This can be done usingthe same extrapolation method from the circularaperture to the square. An alternate approach pre-sented here is based on extrapolating from a circleto an octagon and from the octagon to the square.
The motivation for this is that the extrapolation fromthe octagon to any square is computationally negli-gible. The individual steps of such an extrapolationfrom circle to octagon to a square are illustrated inFig. 2, and the detailed description of the equationsare discussed in Subsections 3.A and 3.B.
The obtained gradients of Fig. 2(e) can be pro-cessed by any reconstruction algorithm that requiresa square set of gradient data. The reconstructed im-age shown in Fig. 3 is then masked to the originalcircular aperture shown in Fig. 4. The photographused here is to demonstrate the structure of thereflections, which is harder to visualize using atmo-spheric turbulence phase screens.
A. Diagonally Flat Extrapolation
As mentioned earlier, there are many possible ap-proaches to the extrapolation of gradient measure-ment data outside the circular aperture so that thezero curl condition is satisfied. In this section the “ex-tension” method of [9] is applied to the two diagonalgrids of the Fried geometry. In this approach, the gra-dients along the diagonal in the extrapolated regionare zero, i.e., the pixels along the diagonal in the ex-trapolated region have the same intensity. The diag-onal direction is the same in the upper left and lowerright quadrants and the upper right and lower leftquadrants, respectively. The case of extrapolatinggradient data in the upper left quadrant is discussedin more detail here.
Consider gradient data modeled using the Friedmodel as presented in Section 2. It has been shownin [9] that these gradients will satisfy the zero curlcondition in a closed path consisting of a squarerotated by 45° as shown in Fig. 5. For the upperleft quadrant, _ϕp;q is the gradient for the point tobe extrapolated using the other three gradientmeasurements.
For the zero curl condition the x and y componentsof the gradients, _ϕ, need to be rotated by 45° in thedirection of the integration path, and thus the scalingfactors in the x and y directions can be ignored.Therefore the equation for the closed path whenthe curl equals zero is equivalently representedby (6)
One possible choice is to require that the extrapo-lated gradient _ϕ is zero in the diagonal direction per-pendicular to the path shown in Fig. 5. This impliesthe condition
_ϕy;p;q ¼ − _ϕx;p;q ð7Þfor quadrant 2, which, when substituted in Eq. (6),leads to one equation for one unknown. The same ap-proach can be used for points in the other three quad-rants. The only difference is that condition (7)becomes
Fig. 1. Fried alignment of gradient measurements. Phase pixelsare represented by squares. Measurement is modeled as the dis-crete gradient of the phase pixels in each of the twoCartesian direc-tions. The measurement point is at the crossing of the two arrows.
for quadrants 1, 3, and 4, respectively.Once the extrapolation is extended beyond the
edge of the circular aperture, then in the upper left
Fig. 2. Extrapolation process of converting a circular gradientdata set to any larger sized square. White solid lines indicate si-mulated mirror locations. Black solid lines indicate the new dataset edge. The figures display the logarithm of the magnitude of thegradients and tend to look like pencil sketches for image gradientdata. Reflections are to be processed on each of the two data setsindividually.
Fig. 3. Reconstruction of data represented by Fig. 2(e). The resultresembles a kaleidoscope.
Fig. 4. Pupil masked representation of the image shown in Fig. 2.
quadrant, see Fig. 5, it is _ϕx;p;q and _ϕy;p;q that must beextrapolated and _ϕx;pþ1;qþ1 and _ϕy;pþ1;qþ1 are the pre-viously extrapolated data. Further, since _ϕpþ1;q and_ϕp;qþ1 are outside the aperture, they are zero inthe diagonal direction and thus contribute zero alongthe diagonal integration path. It follows then that_ϕx;p;q ¼ _ϕx;pþ1;qþ1 and _ϕy;p;q ¼ _ϕy;pþ1;qþ1 is a choicethat satisfies the zero curl condition. Clearly extra-polation beyond the edge of the aperture requiresmerely data copying and no further calculations. Asimilar method is used for all other quadrants. Thisextrapolation technique is used only up to an areaequal to or smaller than the smallest rectangle,and further extrapolation is conducted with the re-flection based extrapolation technique in the follow-ing section.
B. Extrapolation by Reflection
An alternative approach to obtain a valid gradientfield outside the circular aperture is based on usingthe mirror equations of optics. This is explained inthe rest of this subsection.
1. Reflection Using a Curved Mirror
Lemma 1: Given gradient data ∇Φðrin; θÞ measuredinside a circular aperture of radius R, then the gra-dient data can be extrapolated by reflecting the mea-sured data (the inside region) to the unmeasuredregion (the outside region) using
∇Φðrout; θÞ ¼R2
r2out
�−∂Φðrin; θÞ
∂rinur þ
∂Φðrin; θÞrin∂θ
uθ
�;
ð11Þ
where R is the radius of the curved mirror,rin ∈ r ≤ R, rout ∈ r > R. This extrapolation, whichis based on the standard curvedmirror equation withfocal length equal to R, leads to a gradient field thatsatisfies the zero curl condition.
Proof: From the curved mirror equation in optics,the relationship between the inner and outer radiusis obtained in Eq.(12) and is shown in the diagram ofFig. 6:
rin ¼ R2
rout: ð12Þ
The reflection using a curved mirror is describedusing the simple equality of
Φinðrin; θÞ ¼ Φoutðrout; θÞ; ð13Þ
and the polar gradient ofΦoutðrout; θÞ can be obtainedusing its partial derivative,
∇Φðrout; θÞ ¼∂Φðrout; θÞ
∂routur þ
∂Φðrout; θÞrout∂θ
uθ: ð14Þ
By substituting Eq. (13) into Eq. (14),
∇Φðrout; θÞ ¼∂Φðrin; θÞ
∂routur þ
∂Φðrin; θÞrout∂θ
uθ; ð15Þ
which leads to
∇Φðrout; θÞ ¼�∂rin∂rout
�∂Φðrin; θÞ
∂rinur
þ�rinrout
�∂Φðrin; θÞ
rin∂θuθ: ð16Þ
This is a scaled version of the gradients inside thecircular aperture. Substituting Eq. (12) and the deri-vative of Eq. (12) with respect to rout into Eq. (16),followed by factoring, leads directly to Eq. (11). Thisrelationship is determined as a gradient of a scalarfunction Φ and therefore automatically satisfiesthe condition that the curl equals zero for any closedpath.
Equation (11) defines the gradient field outside thecircular aperture using the gradient data measuredinside the circular aperture. In the next subsection it
Fig. 5. Smallest closed path of measured gradients in Fried align-ment. Example of process in quadrant 2. The variable nameconvention holds for all quadrants.
Fig. 6. Fundamental relationships of curved mirror reflection.
is shown how this relationship can be used to extra-polate the gradient measurements when the mea-surements are carried out using a rectangular grid.
2. Curved Mirror Reflection on aRectangular Grid
Lemma 2: Given gradient data ∇Φðrin; θÞ measuredon a rectangular grid inside a circular aperture of ra-dius R, then the data can be extrapolated by reflect-ing the measured data (the inside region) to theunmeasured region (the outside region) by using
� ∂Φout∂x
∂Φout∂y
�¼ B−1GBAd
⇀; ð17Þ
where d⇀
is a vector of the nearest four gradient mea-surements to the location inside the circular aper-ture as shown in Fig. 7:
d ¼ ½ _ϕx;p;q_ϕx;p;qþ1
_ϕx;pþ1;q_ϕx;pþ1;qþ1
_ϕy;p;q_ϕy;p;qþ1
_ϕy;pþ1;q_ϕy;pþ1;qþ1 �T : ð18Þ
Matrix A is a 2 × 8 interpolation matrix given by
A ¼24 a
⇀null
null a⇀
35; ð19Þ
where “null” is a 1 × 4 zero vector and
a¼½ð1−ΔxÞð1−ΔyÞ Δxð1−ΔyÞ ð1−ΔxÞΔy ΔxΔy �:ð20Þ
Matrix B represents the rotation matrix from Carte-sian to polar coordinates given by
B ¼�
cos θ sin θ− sin θ cos θ
�; ð21Þ
and matrix G is the gain factor applied to the innerdata to scale it appropriately for use in the regionoutside the circular aperture:
G ¼ R2
r2out
�−1 00 1
�: ð22Þ
Equation (17) follows when Eq. (11) is used to ob-tain the gradient values on the rectangular grid out-side the circular aperture. The resulting values forrin, for a given rout, on the grid, will in general notbe on the inside rectangular grid and thereforethe corresponding inside gradient values will beobtained using an interpolation given by
d⇀
in ¼ Ad⇀; ð23Þ
where A and d are defined in Eqs. (18) and (19), re-spectively. An example of the relative locations of the
inside measurement points (·), outside extrapolatedpoints (x), and the inside interpolated points (o) isgiven in Fig. 8. The circular aperture in Fig. 8 isthe largest circle inscribed in the square.
After the gradient data have been interpolated, itmust be rotated from Cartesian coordinates to polarcoordinates. A single point example for the relation-ship between these coordinate systems is given inFig. 9, and the corresponding rotation matrix is givenby B in Eq. (21).
This leads to
� ∂Φðr;θÞ∂r
∂Φðr;θÞr∂θ
�¼ B
� ∂Φðr;θÞ∂x
∂Φðr;θÞ∂y
�: ð24Þ
Once in polar coordinates, lemma 1 indicates that thegradient must be scaled by the factor of ðR=routÞ2, andthe sign of the radial partial derivative changes. Thisis carried out using matrix G in Eq. (21). Finally, thereflected gradient data are converted back to Carte-sian coordinates using the inverse of matrix B.
For implementation, B−1GB can be implementedas a single 2 × 2 matrix for each extrapolation pointthat is computed off line. Matrix A would also becomputed off line and implemented in a way to avoidall multiplications by 0. This reduces the computa-tion cost to 12 multiplications and 9 additions per ex-trapolated grid point (i.e., 21 flops per extrapolation).
Remark: Extrapolation using curved mirrors can-not be used to compute the gradient for data pointsimmediately adjacent to the curved boundary. Sincethe overall technique uses reflections, the gradientperpendicular to the mirror will be near zero, andthe gradient tangential to the mirror will be nearly
Fig. 7. Example of desiredmeasurement point between fourmea-surement grid positions. Indices for measurement points increasetoward the right for p and downward for q in Eq. (19).
the same as the tangential gradient values near themirror edge. A computationally simple method isthen to interpolate these data points. This providesa result that is both close to the true gradient andalso provides low pass filtering to alleviate any pre-sent measurement noise. This approach is computa-tionally simpler than the one in [9], which is based onfinding systems of linear equations to force the curlfor a closed path to become zero.
3. Flat Mirror Reflection
Reflection of the graduate measurement data using aflat mirror is a very simple operation, and it can beeasily shown that the reflected gradient data will sa-tisfy the zero curl condition. When the mirror surfaceis chosen to be parallel to the principal axes of therectangular measurement grid, reflection involvesonly a positive or negative copying of the gradientdata and does not involve any computations. If themirror surface is chosen in any of the diagonal direc-
tions, reflection of the gradient data implies a rota-tion by 90°. Rotating Cartesian gradient data by 90°is simply a swap of the horizontal and vertical direc-tions with the appropriate signs depending on thediagonal direction. Clearly, flat mirror reflectionrequires less computation than the curved mirror re-flection discussed in the previous subsection. It istherefore from the computational point of view con-venient to use the curvedmirror extrapolation (or thediagonally flat from Subsection 3.A) to fill up thepoints from the circular aperture to an octagonalaperture and then continue with flat mirror reflec-tions. For an octagonal aperture, the flat mirror sur-faces are defined such that reflected outer pixels arepositioned on the extension of the same rectangulargrid as the inner pixels shown in Fig. 10.
The simple relationship between the inner gradi-ent measurements and the outer gradient extrapola-tions can be given by
� ∂Φðxout;youtÞ∂x
∂Φðxout;youtÞ∂y
�¼
�− cos 2θ − sin 2θ− sin 2θ cos 2θ
�� ∂Φðxin;yinÞ∂x
∂Φðxin;yinÞ∂y
�; ð25Þ
xin ¼ xout − 2Δd × cos θ; ð26Þ
yin ¼ yout − 2Δd × sin θ; ð27Þ
where θ is the angle from the positive x axis to thenormal of the flat mirror, shown in Fig. 10. In thiscase θ ¼ 25πk, where k is integer k ¼ 0;…; 7. For keven, this is reduced to a mirror surface that is par-allel to the Cartesian axes, and therefore the extra-polation of the gradient measurements does notinvolve any interpolation. For k odd, the matrix inEq. (25) has entries that are �1 for the sine terms
Fig. 8. Relationship between extrapolated grid points (x) and thecorresponding required measurement positions (o). Measurementdata (·) do not coincide with (o), so the process requires interpola-tion of the actual measurement points by matrix A.
Fig. 9. Example of relationship between coordinate systems.Fig. 10. Lower right quadrant of the octagon. The octagon is sym-metric about the x and y axes.
and 0 for the cosine terms, which models the rotationdue to the diagonal mirror.The case Δd ¼ 0 is a special case that occurs for
points on the simulated diagonal mirror surface.For such points the extrapolated gradient cannotbe computed using reflection and is computed usingthe diagonally flat extrapolation approach discussedin Subsection 3.A.Remark: It is interesting to note that, although
gradients in AO systems are considered to be mea-sured using a circular aperture, the difference be-tween the circular and an octagonal pupil is onlyfew points for low and medium resolutions. As an ex-ample, Fig. 11 shows that using a WFS with 15 × 15resolution, there are only 2 points per quadrant thatare fully within the octagon but outside the circularaperture.
4. Extrapolation Process
With the tools developed in the previous sections, theprocess of converting a circular aperture to a squareone can be summarized in the block diagram ofFig. 12.The computational requirements for the extrapola-
tion can be estimated as follows. The diagonally flatextrapolation requires computation only for thepoints on the boundary of the circular aperture.The computation required to evaluate both x and ygradients using Eq. (6) is 6 flops per data location.Every other extrapolated point using the diagonallyflat method beyond the points on the boundary ismerely a data copy. It is obvious that the numberof data points on the circular boundary is the circum-ference measured in pixels. So, if there are N pixelswithin the circular aperture, than the number of pix-els is approximately D ¼ 2ðN=πÞ1=2 on the diameterand πD on the boundary. Therefore the computationcost of the diagonally flat extrapolation is approxi-mately 21N1=2.
The extrapolation using the circular mirror equa-tions in Eq.(17) has a cost of 21 flops per extrapola-tion point. Thus the total number of operationsdepends on whether the extrapolation ends at the oc-tagon, the smallest square, or a square of any size.The area of the smallest square that contains the cir-cular pupil isD2, and the area of the smallest octagonthat contains the circular pupil is ð81=2 − 2ÞD2. Thedifference between the area of the smallest squareand the circle is about 0:273N pixels, and the areabetween the octagon and the circle is 0:055N. There-fore the cost of using the circular mirror approach is5:74N to extrapolate to the smallest square and only1:16N if this process is stopped at the octagon.
5. Experiments
The performance of the proposed extrapolation tech-niques is evaluated based on a series of experimentsusing a simulated phase screen that represents aber-rations caused by the Earth’s atmosphere. Thisphase screen was generated using the same turbu-lence parameters as the ones used in [15,16] andare briefly described here. Atmospheric phasescreens are generated using a Kolmogorov turbu-lence model with r0 of 0:25m, wavelength of500nm, pupil diameter of 32m, and outer scale of22m. These are the same parameters as in [15,16]with the exception of the outer scale, which wasnot given in [15] and was chosen [16] based on mea-surements at the Paranal European Southern Obser-vatory in Chile [19]. The phase screen generationsoftware was developed according to the discussionin [20]. The gradient data are obtained from the gen-erated phase screen using a simulated sensor thatuses the Fried aligned sensor model of Eqs. (2)and (3). The gradient measurements are contami-nated with additive zero mean Gaussian white noiseas in [14–16] with various signal to noise ratio (SNR)
Fig. 11. Circular and octagonal apertures on a square 15 × 15grid. For this resolution there are only 8 total points (2 per quad-rant) that are fully within the octagon but outside the circularaperture. Points on the pupil edge are considered outside the pupil.
Fig. 12. Decision flow chart for presented extrapolation options.
values. The generated phase screen is 1024 × 1024and the resulting square gradient measurement gridis 1023 × 1023. Phase screens smaller than 1024 ×1024 resolution are obtained via simple 2 × 2 aver-aging filters and downsampling by 2 operations asis the result in Fig. 13.Two simulation experiments will be discussed in
the rest of this section. The first uses a 64 × 64 phasescreen, while the second deals with phase screens ofsizes varying from 8 × 8 to 1024 × 1024.
A. Modern Adaptive Optics Resolution
For the first set of simulation experiments the reso-lution of 64 × 64 pixels is chosen to conform with thecurrent highest resolution deformable mirrors avail-able such as the Boston Micromachines 64 × 64microelectromechanical system (MEMS) [21]. A 64×64 phase screen is generated, and the noisy gradientmeasurement data are obtained from the phase datacontained in a circular aperture of 63 pixels dia-meter. The missing gradient measurements are ex-trapolated from the noisy gradient measurementsinside the circular aperture using the techniques pro-posed in the previous section, and the extrapolated63 × 63 gradient measurements are used for phasereconstruction. For the gradient extrapolation, fourdifferent options have been evaluated. They are gi-ven by all possible combinations of the flow chartin Fig. 12. For comparison of what the error is whenthe phase screen is reconstructed without any extra-polation, in Figs. 14–17 the “no extrapolation” case isincluded.The results from the first set of simulation experi-
ments using the wavelet based reconstruction meth-od by Hampton et al.[16] are shown in Fig. 14.Figure 14 shows the normalized residual rms errorfor these experiments as a function of the gradientmeasurement SNR and also compares the quality
of the reconstruction using extrapolation to the idealcase, i.e., the quality of reconstruction using gradientmeasurements over the full square. Each plot in thissection is obtained as the mean of 10 separate recon-structions that are each contaminated by a differentnoise profile. It can be seen from Fig. 14 that usingthe diagonally flat extrapolation leads to nearly the
Fig. 13. Atmospheric turbulence phase screen downsampled to512 × 512. Amplitude normalized to 1 unit rms. r0 ¼ 0:25m,L0 ¼ 22m, 32m width, 500nm wavelength, and 25m=s windspeed.
100
101
102
103
10−4
10−3
10−2
10−1
100
101
Pupil masked 64 x 64Hampton−Agathoklis Full Reconstruction
Fig. 14. Residual error from using the proposed extrapolationmethods and the Hampton–Agathoklis reconstruction method[16]. For each pair of similar lines, the black line is superior at highgradient SNR, and the gray line is inferior. Figure 15 shows per-formance at low gradient SNR.
10−0.01
100
10−0.31
10−0.3
10−0.29
10−0.28
Pupil masked 64 x 64Hampton−Agathoklis Full Reconstruction
Fig. 15. Zoomed representation of Fig. 14 showing residual errorat low gradient SNR. The curved mirror extrapolation was theworst method at high gradient SNR and is shown here to providesuperior reconstruction for low gradient SNR.
same reconstruction quality as having gradient mea-surements over the full square. The curved mirrorbased extrapolations always have an error above aminimum amount, which is indicated by the horizon-tal asymptotes as the SNR becomes high, i.e., thereare reconstruction errors even when there is no mea-surement noise. This is due to interpolation. Thediagonally flat extrapolations, on the other hand,do not have a measured horizontal asymptote, whichimplies that the reconstruction would be losslesswhen there is no noise.Figure 15 shows a close-up of the low SNR region
in Fig. 14, an area where all the methods converge tosimilar performance. This plot shows that the curvedmirror extrapolation outperforms the others in termsof reconstruction error. It even surpasses the ideal
square aperture case, likely due to the noise suppres-sing properties of the interpolation (low pass) filters.A computationally faster method is the curved-octa-gonal extrapolation, which only uses the curved mir-ror equations to extrapolate gradients to the smallestoctagon that surrounds the circle. The performanceof this method is very close to that of the ideal squareaperture case. This plot also shows that there is aperformance improvement to reflecting the diagonalextrapolation with a simulated octagonal mirror.There is no additional computation cost of this deci-sion since the diagonally flat process properly definesgradient on the octagonal mirror edge automatically.
The results presented in Figs. 14 and 15 show thatthere are performance trade-offs between the variousextrapolation approaches that are dependent on theSNR of the gradient measurements. The circularmirror reflections become superior in the low SNRregion for the Hampton–Agathoklis reconstructionmethod, and the circular–octagonal approach isindistinguishable from the ideal square aperturecase in this region.
One interesting question is, what is the effect ofthe proposed extrapolation and reconstruction ap-proach to the waffle modes? It is well known [17] thatit is desirable to suppress the waffle modes asso-ciated with an AO correction. One of the advantagesof the wavelet based reconstruction approach givenin [16] is that, in the wavelet decomposition of thephase obtained directly from the gradient sensormeasurements, the waffle modes are organized intoa well defined quadrant. All waffle modes can there-fore be easily removed from the reconstruction byignoring this section of the decomposition. This pro-vides the additional benefit of reducing the computa-tion cost of the reconstruction by about 40% [16]. Toevaluate the effect of removing the waffle modes, thesame approach as the one presented in Figs. 14 and15 was used with the wavelet reconstruction [16],with the waffle modes ignored. The results are shownin Figs. 16 and 17.
Figure 16 indicates that the reconstruction basedon gradient data extrapolated using curved mirror isabout the same as in Fig. 14. For the other extrapola-tion techniques, the removal of all waffle modes leadsto a minimal error (of about 5% in this case) that can-not be overcome. Further, it can be seen that thediagonally flat based extrapolations are very closeto the ideal case of having measurements in a fullsquare. This indicates that when the waffle modesare neglected, the diagonally flat extrapolation isas good as having gradient measurements for the fullsquare. Figure 17 shows a close-up view of Fig. 16 forlow gradient SNR. It can be seen that for low SNR,the curved mirror extrapolations provide the bestreconstruction.
The above results indicate that the proposedapproach for wavefront reconstruction on a circularaperture is promising for use in correcting aberra-tions using deformable mirrors (DMs) with resolu-tions of 64 × 64 actuators such as the one in [21].
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Fig. 16. Residual error from using the proposed extrapolationmethods and the Hampton–Agathoklis reconstruction methodwith all waffle modes suppressed. For each pair of similar lines,the black line is superior at high gradient SNR, and the gray lineis inferior. Figure 17 shows performance at low gradient SNR.
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The model of the interaction between a SH-WFS,DM, control computer, and intervening electronicsmay be developed by the methods presented byHampton et al. [22], and evaluation of the perfor-mance of the DM can be determined by the methodspresented by Wallace et al. [23].In the next set of simulations the performance of
the previously discussed extrapolation and recon-struction methods using noisy gradient measure-ments in a circular aperture are compared. Theresults are presented in Fig. 18. They indicate thatthe full reconstruction method performs better forgradient SNR greater than 10, while suppressingwaffle modes performs better when the gradientSNR is 10 or less. These results are for gradient datadefined as in the Fried model.This performance of the Hampton–Agathoklis
method indicates that this method could be a goodchoice for open loop AO control, such as the VictoriaOpen Loop Testbed (VOLT) experiment by Andersonand Fischer [24]. Further, the Hampton–Agathoklismethod could also be used for a fast and accurate es-timation of the phase screen as an intermediate stepapplied to tomographic reconstruction techniquessuch as the ones presented by Tokovinin and Viard[25] or Gavel [26].Remark: It was determined that the gradient ex-
trapolation methods do not perform well for Fouriertransform based reconstruction algorithms. The rea-son for this is that the methods presented in thiswork do not explicitly satisfy the strict requirementof the Fourier transform methods that the curl mustbe zero across the edge when the data are tiled. Thisis so because it is not a criterion of the wavelet basedapproach. For a method that handles this require-
ment for Fourier transforms, the authors recommendthe works of Poyneer et al. [9,27].
B. Reconstruction for Varying Resolutions
The second set of experiments deals with phasescreens that are 2m × 2m, where m is an integer from3 to 10, in order to validate that the proposed ap-proach can be used with a wide range of data sizes.In this set of experiments, noisy gradient measure-ments over circular apertures with diameter up to1023 pixels were generated for a constant pupil sizeof 32m, as discussed at the beginning of this section.These measurements were extrapolated using thediagonally flat extrapolation method and recon-structed using the Hampton–Agathoklis [16] full re-construction method. The results presented inFig. 19 demonstrate that the diagonally flat recon-struction process does not have a lower limit forthe residual error when the gradient SNR is in therange of SNR ¼ 0:1 to SNR ¼ 1000. This indicatesthat, for current AO systems with pixel diametersfrom 7 to 63 as well as future AO systems with higherresolutions up to 1023 pixel diameter, there is nolimit on the quality of reconstruction due to extrapo-lation errors.
Figure 19 allows comparison of reconstructionerror among resolutions for a given gradient SNR.In AO applications it is unlikely that the gradientSNR would remain equal with varying resolution,since the noise level may remain unchanged whilethe gradient signal strength would vary. A methodproposed in [9] is to determine the amount thatthe mean square noise is amplified when converted
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Comparison of Full Reconstruction and Waffle Suppression. Diagonal−Octagonal Extrapolation is Used for Both Cases
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Fig. 18. Comparison between the full reconstruction and wafflesuppression. Both are combined with the diagonal-octagonal ex-trapolation method.
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Residual Solution Error of Various Data Set Sizes for the Diagonally Flat Extrapolated Hampton−Agathoklis Method
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Fig. 19. The residual reconstruction error of various sizes of datasets. The extrapolation method is diagonally flat. The reconstruc-tion method is the Hampton–Agathoklis method [16]. The unit ofdiameter, D, is pixels. The physical diameter of the pupil is a con-stant 32m.
to mean square reconstruction error. For noiseless re-construction error of zero, this can be expressed as
σ2e ¼ ασ2n; ð28Þwhere σ2e is the mean square reconstruction error, σ2nis themean square noise, and α is the amplification ofthe mean square noise. Figure 20 shows how muchthe mean square noise is amplified when extra-polated by the diagonal-octagonal method followedby full reconstruction by the Hampton–Agathoklismethod. This shows that, when the mean squareof noise is maintained constant for differentresolutions, the mean square error increasesnearly linearly with respect to the logarithm of thenumber of sensors. The quadratic fit of data pointsin Fig. 20 is
αðNÞ ¼ 0:0261 ln2N þ 0:2405 lnN þ 0:6833; ð29Þwhere lnN is the natural logarithm of the number ofsensors on the square,N. The rms error is the squareroot of the mean square error, so the amplification ofa noise signal into rms error would increase logarith-mically with the number of sensors.
6. Conclusions
Techniques for extrapolating gradient measure-ments obtained over a circular aperture to a squareaperture have been presented. The gradient data ona fictitious square aperture are used to evaluate theperformance of the proposed extrapolation processeswith respect to computation time and reconstructionerror inside the circular aperture. Results indicatethat the diagonally flat extrapolation method wasvirtually indistinguishable from the unrealistic idealcase of having a square aperture when applied to theHampton–Agathoklis reconstruction method. The
extrapolation using the diagonally flat techniquesdoes not cause errors when there is no noise, as isevident from the lack of a knee point in the residualerror plots. Further, the computation cost of this ex-trapolation technique is OðN1=2Þ and is expected tobe insignificant compared to theOðNÞ reconstructionmethod.
The curved mirror extrapolation techniques areshown to be superior even to the ideal square aper-ture case when the gradient SNR is very low. This isdue to the noise filtering effect of interpolating datain the interior of the pupil to obtain the extrapolateddata. This approach comes with a cost that is OðNÞflops. The octagonal mirror reflection reduces the er-ror of the diagonal extrapolation approach with noadditional computation cost. This combination ofmethods performs best for high gradient SNR. Theoctagonal mirror reflection combined with the curvedmirror extrapolation leads to a reduction in compu-tation cost by at least a factor of 5 and results inreduced error for high SNR gradient data and ampli-fied error in the low SNR gradient data. It was foundthat when the gradient SNR is constant as resolutionincreases, the amount of reconstruction error de-creases. If only the noise is kept constant instead,the noise amplification into rms error increaseslogarithmically with the number of sensors. Theexamples indicate that the Hampton–Agathoklis re-construction method [16] can be used for reconstruc-tion of gradient data from circular apertures withresults as good as if data from a square aperturewere available.
The authors acknowledge financial support fromthe Natural Sciences and Engineering ResearchCouncil of Canada (NSERC), the Canada Foundationfor Innovation (CFI), and the British ColumbiaKnowledge Development Fund (Canada) (BCKDF)for this research.
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