下面这个函数大家都不会陌生,计算zernike函数值的,并根据此可以还原出图像来, .N2nJ/
我输入10阶的n、m,r,theta为38025*1向量,最后得到的z是29525*10阶的矩阵, ,H#qgnp
这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? ?{ 8sT-Z-L
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? >#"jfjDuR
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function z = zernfun(n,m,r,theta,nflag) g%fJyk'
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. rw=UK`
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N %>I?'y^
% and angular frequency M, evaluated at positions (R,THETA) on the \>aa8LOe
% unit circle. N is a vector of positive integers (including 0), and WIH4Aw
% M is a vector with the same number of elements as N. Each element Xn
ZX *Y]"
% k of M must be a positive integer, with possible values M(k) = -N(k) a@S4IoBg%
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, )\I? EU8
% and THETA is a vector of angles. R and THETA must have the same }jyS\drJ
% length. The output Z is a matrix with one column for every (N,M) Z CQt1;
% pair, and one row for every (R,THETA) pair. VFO&)E/-
% ]U^d 1&k
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 1K*f4BnDr~
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), "M5ro$qZ}
% with delta(m,0) the Kronecker delta, is chosen so that the integral 6ljRV)
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, -UD~>s
% and theta=0 to theta=2*pi) is unity. For the non-normalized cV=_GE
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ai;gca_P#
% wCC~tuTpr
% The Zernike functions are an orthogonal basis on the unit circle. wE8a4.
% They are used in disciplines such as astronomy, optics, and z7.C\l
% optometry to describe functions on a circular domain. ;SlS!6.W-
% ^b|Nw:
% The following table lists the first 15 Zernike functions. {KpH|i
% .^N#|hp^
% n m Zernike function Normalization (61twutC
% -------------------------------------------------- ]\9B?W(#
% 0 0 1 1 I$6
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% 1 1 r * cos(theta) 2 He71h(BHm
% 1 -1 r * sin(theta) 2 x}8T[
% 2 -2 r^2 * cos(2*theta) sqrt(6) f'i8Mm4IL
% 2 0 (2*r^2 - 1) sqrt(3) >y06s{[
% 2 2 r^2 * sin(2*theta) sqrt(6) {, *Y
% 3 -3 r^3 * cos(3*theta) sqrt(8) \,cKt_{ u
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 3W0E6H"
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) m|cWX"#g
% 3 3 r^3 * sin(3*theta) sqrt(8) * /Ry6Yu
% 4 -4 r^4 * cos(4*theta) sqrt(10) lTOM/^L
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #+ lq7HJ1
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) J&U0y
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) [|;Zxb:
% 4 4 r^4 * sin(4*theta) sqrt(10) ?D^,K`wY=B
% -------------------------------------------------- a^}P_hg}-
% /%q9hI
% Example 1: qxcBj
% ]{6yS9_tuI
% % Display the Zernike function Z(n=5,m=1) 53+rpU_
% x = -1:0.01:1; (R*jt,x
% [X,Y] = meshgrid(x,x); F?,&y)ri
% [theta,r] = cart2pol(X,Y); ):\{n8~
% idx = r<=1; dV=5_wXZ$
% z = nan(size(X)); >8fz ?A
% z(idx) = zernfun(5,1,r(idx),theta(idx)); @G$<6CG\
% figure M^JZ]W(
% pcolor(x,x,z), shading interp Q]g 4gj
% axis square, colorbar A%w]~ chC9
% title('Zernike function Z_5^1(r,\theta)') a*8.^SdzR
% aE cg_es
% Example 2: '>mb@m
% 6.7Kp
% % Display the first 10 Zernike functions irw 7
% x = -1:0.01:1; -hR\Y2?
% [X,Y] = meshgrid(x,x); l5OV!<7~X
% [theta,r] = cart2pol(X,Y); =
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% idx = r<=1; #!&R7/
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% z = nan(size(X)); v*fc5"3eO
% n = [0 1 1 2 2 2 3 3 3 3];
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% m = [0 -1 1 -2 0 2 -3 -1 1 3]; %s :
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ;AB ,:*
% y = zernfun(n,m,r(idx),theta(idx)); z==}~|5
% figure('Units','normalized') FRQ("6(
% for k = 1:10 lnSE+YJ>
% z(idx) = y(:,k); `b`52b\6S
% subplot(4,7,Nplot(k)) 78J.~v/
% pcolor(x,x,z), shading interp |b~g^4
% set(gca,'XTick',[],'YTick',[]) 4x?u5L
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% axis square `/ReJj&~
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) V+~{a:8[pq
% end Wy>\KrA1
% NWwtq&pz2
% See also ZERNPOL, ZERNFUN2. !enz05VW6.
HF[%/Tu
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% Paul Fricker 11/13/2006 9,> Y
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% Check and prepare the inputs: }M@Jrq+7
% ----------------------------- J$*["y`+
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) G*p.JsZP
error('zernfun:NMvectors','N and M must be vectors.') rB|:r\Z(jG
end ~0GX~{;r
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if length(n)~=length(m) Dohe(\C@
error('zernfun:NMlength','N and M must be the same length.') s(Bi&C\
end `z`;eR2oX
%-po6Vf
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n = n(:); >8t[EsW/
m = m(:); )?y"NVc*
if any(mod(n-m,2)) GhA~Pj ZS
error('zernfun:NMmultiplesof2', ... cty#@?"e
'All N and M must differ by multiples of 2 (including 0).') 7B"aFnK;[J
end 4>`w9
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z ,/d
if any(m>n) PP|xIAc
error('zernfun:MlessthanN', ... SYLkC
[0k
'Each M must be less than or equal to its corresponding N.') o%Q2.
end v3#47F)
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if any( r>1 | r<0 ) YP97D n
error('zernfun:Rlessthan1','All R must be between 0 and 1.') o:ob1G[p%
end Py<vN!
.AS,]*?Zn%
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ]xs\,}I%
error('zernfun:RTHvector','R and THETA must be vectors.') 5OE?;PJ(
end H[U*'
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r = r(:); {+}Lc$O#C
theta = theta(:); FiL
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length_r = length(r); 8yl/!O,v
if length_r~=length(theta) qpCi61lTDJ
error('zernfun:RTHlength', ... FA,CBn5%
'The number of R- and THETA-values must be equal.') vS<e/e+
end >48Y-w
r?*?iw2g
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% Check normalization: X_2I4Jz]6
% -------------------- `dhK$jYD
if nargin==5 && ischar(nflag) <u\G&cd_tA
isnorm = strcmpi(nflag,'norm'); -B!pg7>'##
if ~isnorm >[U$n.
error('zernfun:normalization','Unrecognized normalization flag.') G#>X~qk()
end iV=#'yY
else -Zh+5;8g
isnorm = false; !)]3@$#
end 3 -FNd~%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2}AV_]]
% Compute the Zernike Polynomials zb (u?U
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /k,p]/e
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% Determine the required powers of r: <LE>WfmC
% ----------------------------------- xXtDGP
m_abs = abs(m); n3w2&
rpowers = []; P\R3/g
for j = 1:length(n) ~zx-'sc?
rpowers = [rpowers m_abs(j):2:n(j)]; `i-&Z`
end &'R]oeag
rpowers = unique(rpowers); k&2I(2S
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% Pre-compute the values of r raised to the required powers, Hd:ZE::Q'#
% and compile them in a matrix: cX2b:
% ----------------------------- BB-`=X~:m
if rpowers(1)==0
`@p*1
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); C5m*pGImG
rpowern = cat(2,rpowern{:}); n\QG-?%Pi
rpowern = [ones(length_r,1) rpowern]; h1"#DnK7
else OG.`\G|
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Wrlmo'31
rpowern = cat(2,rpowern{:}); Eb*DP_
end Z^sO`C
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? in&/ZrB
% Compute the values of the polynomials: N%0Z>
G
% -------------------------------------- `VHm,g2
y = zeros(length_r,length(n)); =U:iR
for j = 1:length(n) Yz,*Q<t
s = 0:(n(j)-m_abs(j))/2; pDu~84!])
pows = n(j):-2:m_abs(j); '?QZ7A
for k = length(s):-1:1 {#7t(:x
p = (1-2*mod(s(k),2))* ... <#c2Hg%jh
prod(2:(n(j)-s(k)))/ ... /q]WV^H
prod(2:s(k))/ ... REHfk6YE
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... |/q *Fg[f
prod(2:((n(j)+m_abs(j))/2-s(k))); (A1 !)c
idx = (pows(k)==rpowers); HzWZQ6o
y(:,j) = y(:,j) + p*rpowern(:,idx); _yU
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end ~!UxmYgO
m'%F,c)
if isnorm {D7!'Rq,
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); /H\ZCIu/7
end )]v vp{
end 7^S &g.A
% END: Compute the Zernike Polynomials 'I;pS)sb
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]Dx5t&
%uQ^mK
u'}DG#@ -
% Compute the Zernike functions: eGZIdv1
% ------------------------------ j'~xe3j
idx_pos = m>0; a}MOhM6T
idx_neg = m<0; E-l>z%
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z = y; F7a\Luae
if any(idx_pos) nAg|m,gA
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); g(|p/%H
end @eR>?.:&
if any(idx_neg) z;1yZ4[G
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); vfmKY iLp
end r*y4Vx7
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% EOF zernfun DL<r2h