Reconstruction algorithm - From Adaptive Optics systems to Point Spread Function Reconstruction

48 CHAPTER 4. A CUMULATIVE RECONSTRUCTOR ON FINITE ELEMENT BASIS

guarantee continuity over the whole domain Ω, instead of η_y a different constant has to be subtracted in every subdomain. We also want to have mean zero again over Ω.

In the following, we present the adjusted constants for a regionΩbeing symmetric with respect to the x- and y-axis and four regions of the same size. If additionally the split is done alongx“0 andy “0, as indicated by the bold lines in Figure 4.2, the correct constants are defined as

c^ll :“ 1

4pη_y^ll`η^ul_y `η_y^lr`η_y^ur`κ₁`κ₂`2κ₃q, c^ul :“c^ll´κ₁, c^lr :“c^ll´κ₃, c^ur :“c^lr´κ₂,

κ1 :“

w^´_yxp0q `

k´1

n“1

fy,n

ffll

κ₂ :“

w^´_yxp0q `

k´1

n“1

f_y,n fflr

κ₃ :“

l_xp0, yq ´w^´_xxpyq `w^´_yxpyq `

jpyq

n“1

f_y,n ff^ll

´w_xx^´pyq `w_yx^´pyq `

jpyq

n“1

f_y,n fflr

, where y in κ₃ can be any element in rcp0q, dp0qs.

It is also possible to define κ₃ as an average over such values for several y-elements, yielding analytically the same value. However, the discretized version will give different values. Setting y“cp0q “C, the result is simply

κ3 “ rlxp0, Cq ´wxxpCqs^ll` rwxxpCqs^lr.

In formula (4.3), also some adjustments are necessary. At a first glance they look similar with the roles of x and y interchanged, but in fact they are not. We will come back to this in the next section.

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BASIS 49

still is different compared to existing algorithms, as we treat general sets Ω and our basis functions for the reconstructionRsare linear finite elements (known as Courant elements) on the triangulation shown in Figure 4.3. Thus, we can also use five inter-mediate nodes in addition to the four corners of Ω_ij. We define px_p, y_qq as the nodes of the grid. Then,

x_p :“A`ph

2 and y_q :“C`qh 2,

withpp, qq PM. The set of admissible indices Mis obtained using Figure 4.3 and the definition of Min Section 4.1, as

M:“ tpp, qq PN0 ˆN0 : 0ďq ď2N_y, pP K_qu, where

Kq :“

’’

t2pσn, . . . ,2p_σ_nu, 2σ_n´1 ďqă2σ_n,1ďn ďk, t2pσ1, . . . ,2p_σ₁u, q“0,

t2 mintp

σn, p

σn`1u, . . . ,2 maxtp_σ_n, p_σ_n`1uu, q“2σ_n,1ďn ďk, t2p

σk, . . . ,2p_σ

ku, q“2N_y.

Of course, the roles ofp and q can be interchanged, leading to a similar definition.

Figure 4.3: The triangulation on a part of a telescope aperture with WFS subapertures (bold lines). The dots indicate the nodes which are used for constructing the finite element reprensentation via piecewise linear functions. Corners of subapertures of the WFS are filled dots, intermediate points are empty dots. Source: [117]

The aim of the presented algorithm is to choose the approximate wavefront ¯ϕ such that the sensor measurements s are reproduced exactly, i.e.,

Γ ¯ϕ“ΓRs“s,

50 CHAPTER 4. A CUMULATIVE RECONSTRUCTOR ON FINITE ELEMENT BASIS

where R is the reconstruction operator.

We define the reconstruction of the wavefront via a finite element representation as

Rspx, yq “ϕpx, yq “¯ ÿ

pp,qqPM

c_p,qϕ_p,qpx, yq, (4.4)

where the functions ϕp,q are piecewise linear functions satisfying

ϕ_p,qpx¯_p_˜,y¯_q_˜q “δ_p,˜_pδ_q,˜_q,

where δ_r,t denotes the usual Kronecker symbol. Due to this special choice, it holds that

ϕp¯x_p,y¯_qq “c_p,q.

As a Shack-Hartmann WFS cannot measure constant functions, known as piston mode, the algorithm should not introduce a constant shift to the reconstructed wavefront.

In order to fulfill condition (2.20), we shift ¯ϕto zero mean. Furthermore, ¯ϕshould be close to the exact incoming wavefront ϕ.

For the implementation, the formulae of [117, Section 3] have to be discretized. We need to compute sums over each line in x- and y-direction and shifts to connect the restored parts together. We recall the discretized versions of all needed quantities from [117]. For 1 ď j ď N_y, 1 ď i ď N_x, 1 ď i₁ ă i₂ ď N_x, 1 ď j₁ ă j₂ ď N_y, hą0, the subaperture size in meter, and sxri, js and syri, js the WFS measurements at subaperture ri, js, we have

lxri, js “h

p“pj`1

sxrp, js, (4.5)

lyri, js “h

q“qi`1

syri, qs. (4.6)

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BASIS 51

The integrals overl_x and l_y are approximated by the trapeziodal rule, leading to:

wxxrjsri1, i2s “ 1 2

p“i1`1

plxrp´1, js `lxrp, jsq, (4.7) wxxrjsri1, i2s “ pi2´i1q^´1w˜xxrjsri1, i2s, (4.8)

w_yxrjsri₁, i₂s “

p“i1`1

l_yrp, js, (4.9)

wyxrjsri1, i2s “ pi2´i1q^´1w˜yxrjsri1, i2s, (4.10)

w_yyrisrj₁, j₂s “ 1 2

q“j1`1

pl_yri, q´1s `l_yri, qsq, (4.11) w_yyrisrj₁, j₂s “ pj₂´j₁q^´1w˜_yyrisrj₁, j₂s, (4.12)

w_xyrisrj₁, j₂s “

q“j1`1

l_xri, qs, (4.13)

wxyrisrj1, j2s “ pj2´j1q^´1w˜xyrisrj1, j2s. (4.14) For 1ďn ďk, define:

f_yrns “ pw_yxrσ_nsrp

σn, p_σ_ns ´w_yxrσ_nsrp

σn`1, p_σ_n_`1sq (4.15)

` pw_xxrσ_n`1srp

σn`1, p_σ_n_`1s ´w_xxrσ_n`1srα_n, β_nsq

` pw_xxrσ_nsrα_n, β_ns ´w_xxrσ_nsrp

σn, p_σ_ns, with

α_n “maxtp

σn, p

σn`1u, β_n “mintp

σn, p

σn`1u.

Moreover, we introduce gyrjs “

k´1

n“1 σnăj

fyrns, 1ďj ďNy, (4.16)

ηy “N^´1

n“2

gyrσnspp_σ_n´p

σnqpσn´σn´1q, (4.17)

ηy “N^´1

i“1

wyyrisrq_i, q_is. (4.18) Analogously, we define for 1ďn ďl,

fxrns “ pwxyrτnsrq

τn, q_τ_ns ´wxyrτnsrq

τn`1, q_τ_n_`1sq (4.19)

` pw_yyrτ_n`1srq

τn`1, q_τ_n_`1s ´w_yyrτ_n`1srγ_n, δ_nsq

` pwyyrτnsrγn, δns ´wyyrτnsrq_τ

n, q_τ_ns,

52 CHAPTER 4. A CUMULATIVE RECONSTRUCTOR ON FINITE ELEMENT BASIS

with

γn“maxtq

τn, q

τn`1u, δ_n“mintq

τn, q

τn`1u, and

g_xris “

l´1

n“1 τnăi

f_xrns, 1ďiďN_x, (4.20)

η_x “N^´1

n“2

g_xrσ_nspq_τ_n´q

τnqpτ_n´τ_n´1q, (4.21)

η_x “N^´1

i“1

˜ w_yyrjsrp

j, p_js. (4.22)

From (4.7)–(4.22),we get all needed quantities and can compute the following approxi-mations of ˜ϕas one-sided limits from all four possible directions, i.e., ˜ϕri, j^`s, ˜ϕri, j^´s,

ϕri^`, js, ˜ϕri^´, js for all admissible pairs pi, jq, as follows

ϕri, j´1^`s “l_xri, js ´w_xxrjsrp

j, p_js `w_yxrj´1srp

j, p_js `g_yrjs ´η˜_y ´η¯_y, (4.23)

ϕri, j^´s “l_xri, js ´w_xxrjsrp

j, p_js `w_yxrjsrp

j, p_js `g_yrjs ´η˜_y´η¯_y, (4.24) with 1ďj ďN_y, p

j ďiďp_j, and,

ϕri´1^`, js “l_xri, js ´w_yyrisrq

i, q_is `w_xyri´1srq

i, q_is `g_xris ´η˜_x´η¯_x, (4.25)

ϕri^´, js “l_xri, js ´w_yyrisrq

i, q_is `w_xyrisrq

i, q_is `g_xris ´η˜_x´η¯_x, (4.26) with 1 ď i ď N_x, q

i ď j ď q_i. The numbers p

j, p_j and q

i, q_i, respectively, stem from the underlying geometry of the WFS, see Section 4.1.

From these approximations we compute ¯ϕat the corners of Ω_ij as c_2i,2j “ ϕri, j˜ ^´sµpi, jq `ϕri, j˜ ^`sµpi, j`1q

2pµpi, jq `µpi, j`1qq ` ϕri˜ ^´, jsνpi, jq `ϕri˜ ^`, jsνpi`1, jq 2pνpi, jq `νpi`1, jq

(4.27) for all admissible values i, j and where

µpi, jq “

’&

’% 2, p

j ăiăp_j, 1, iP tp

j, p_ju, 0, else,

νpi, jq “

’&

’% 2, q

i ăj ăq_i, 1, j P tq

i, q_iu, 0, else.

Note that for boundary nodes, some values are not well-defined, but the appropriate weights are 0 anyways.

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BASIS 53

The values of ¯ϕat intermediate points can be calculated by

c_2i,2j´1 “ϕri, j˜ ´1^`s `ϕri, j˜ ^´s ´ 1

2pc_2i,2j´2`c_2i,2jq, (4.28)

c_2i´1,2j “ϕri˜ ´1^`, js `ϕri˜ ^´, js ´ 1

2pc_2i´2,2j `c_2i,2jq, (4.29)

c_2i´1,2j´1 “ 1

4pϕri˜ ´1, j´1^`s `ϕri, j˜ ^´s `ϕri˜ ´1^`, j´1s `ϕri˜ ^´, jsq. (4.30) These steps are summarized in Algorithm 4.1.

Algorithm 4.1 FinECuRe (from [117])

Lets“ ps_x, s_yq PR^N ˆR^N. ThenRs“ϕ is defined as in (4.4). For the computation of the appropriate coefficientsc_p,q the following steps have to be performed:

(i) Calculate the chains l_x, l_y according to (4.5), (4.6), respectively.

(ii) Calculate wxx, wyx, wyy, wxy, the jump numbers fx, fy, the sums gx, gy, and the constants ηr_x,rη_y, η_x, η_y according to their definitions in (4.7) - (4.22).

(iii) Calculate the approximations ˜ϕpi, j^`q,ϕpi, j˜ ^´q,ϕpi˜ ^`, jq,ϕpi˜ ^´, jq from (4.23) -(4.26).

(iv) Compute the corner nodes c_2i,2j, given by (4.27).

(v) Compute the intermediate nodes at the edges c_2i,2j´1, c_2i´1,2j, from (4.28) and (4.29).

(vi) Finally, compute the subaperture mid point nodes c_2i´1,2j´1, from (4.30).

(vii) Compute ¯ϕ from (4.4).

We emphasize that the reconstruction algorithm is linear using only 14N operations for the first four steps and the first three steps can be performed independently on each subdomain, if we have to split the domain for geometrical reasons. The last two steps require another 12N operations.

In the control system of a big telescope, the reconstructed wavefront has to be mapped into actuator commands for the deformable mirror. Depending on the actual geometry of the deformable mirror, it might be possible to omit the last two steps, e.g., if the actuators of the deformable mirror are perfectly aligned with the corners of the subapertures.

54 CHAPTER 4. A CUMULATIVE RECONSTRUCTOR ON FINITE ELEMENT BASIS

Im Dokument From Adaptive Optics systems to Point Spread Function Reconstruction and Blind Deconvolution for Extremely Large Telescopes / eingereicht von Dipl.-Ing. Roland Wagner, Bakk.techn. (Seite 60-66)