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, lfeWtzOf
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, B{(l5B6
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? Y[?Wt/O;
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? iB`]Z@ZC
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function z = zernfun(n,m,r,theta,nflag) ; 2-kQK9
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ;-^9j)31+F
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N gdY/RDxn:
% and angular frequency M, evaluated at positions (R,THETA) on the !Qa7-
% unit circle. N is a vector of positive integers (including 0), and \9zC?Cw
% M is a vector with the same number of elements as N. Each element E9-'!I !
% k of M must be a positive integer, with possible values M(k) = -N(k) 3g:+p
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, e-=PT1T`
% and THETA is a vector of angles. R and THETA must have the same ulo7d1OVkJ
% length. The output Z is a matrix with one column for every (N,M) 31Mc<4zI8
% pair, and one row for every (R,THETA) pair. 6dp_R2zH~o
% CoXL;\
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike XQ;dew+
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), K):sq{
% with delta(m,0) the Kronecker delta, is chosen so that the integral l #z`4<
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, )!-'S H
% and theta=0 to theta=2*pi) is unity. For the non-normalized
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 1*b%C"C
% YKd?)$J
% The Zernike functions are an orthogonal basis on the unit circle. Bd[Gsns
% They are used in disciplines such as astronomy, optics, and %y+j~]^:
% optometry to describe functions on a circular domain. $Ws2g*i
% (OJ9@_fgG[
% The following table lists the first 15 Zernike functions. )E2Lf]
% M'7x:Uw;
% n m Zernike function Normalization P~Owvs/=
% -------------------------------------------------- boovCW
% 0 0 1 1 zZiVBUmE<
% 1 1 r * cos(theta) 2 `2
% 1 -1 r * sin(theta) 2 Av]N.HB$
% 2 -2 r^2 * cos(2*theta) sqrt(6) x^BBK'
% 2 0 (2*r^2 - 1) sqrt(3) I!'(>VlP7
% 2 2 r^2 * sin(2*theta) sqrt(6) SX;IUvVE5
% 3 -3 r^3 * cos(3*theta) sqrt(8) Ooy96M~_G
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) x%&V!L
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) -v@^6bQVp
% 3 3 r^3 * sin(3*theta) sqrt(8) j,jUg}b
% 4 -4 r^4 * cos(4*theta) sqrt(10) n//a;m
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) O v6=|]cW
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ~zRd||qv
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) SoIMf tX
% 4 4 r^4 * sin(4*theta) sqrt(10) D40VJ3TUc
% -------------------------------------------------- ;\.&FMi
% j<?4N*S
% Example 1: hp}8
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% sOmYQ{R
% % Display the Zernike function Z(n=5,m=1) ep|u_|sB/r
% x = -1:0.01:1;
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% [X,Y] = meshgrid(x,x); XWV ~6"
% [theta,r] = cart2pol(X,Y); omP7|
% idx = r<=1; H5)WxsZ R
% z = nan(size(X)); r; !us~
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 4R6 .GO
% figure r$zXb9a|<
% pcolor(x,x,z), shading interp ]A[~2]
% axis square, colorbar +.St"f/1
% title('Zernike function Z_5^1(r,\theta)') ,0xN#&?Ohh
% G>"[nXmcu
% Example 2: u e~1144
% Jo]g{GX[
% % Display the first 10 Zernike functions [$X(i|6
% x = -1:0.01:1; F!8425oAw
% [X,Y] = meshgrid(x,x); )DMbO"7
% [theta,r] = cart2pol(X,Y); (aLnbJeJ
% idx = r<=1; 2e&Zs%u
% z = nan(size(X)); =6:Iv"<
% n = [0 1 1 2 2 2 3 3 3 3]; d1N&J`R\1
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; _G`aI*rKsy
% Nplot = [4 10 12 16 18 20 22 24 26 28]; WxdYvmp6z[
% y = zernfun(n,m,r(idx),theta(idx));
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% figure('Units','normalized') 6 ?cV1:jh
% for k = 1:10 S7R^%Wck/6
% z(idx) = y(:,k); FS[CUoA
% subplot(4,7,Nplot(k)) UF4QPPH4
% pcolor(x,x,z), shading interp @VFg XN
% set(gca,'XTick',[],'YTick',[]) N]~q@x;<)3
% axis square xhv)rhu@
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) WD]dt!V%
% end 6}0#({s:R
% h 9/68Gc?6
% See also ZERNPOL, ZERNFUN2. 3? "GH1e
@ M-bE=
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% Paul Fricker 11/13/2006 -.y3:^){^
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% Check and prepare the inputs: XOoND
% ----------------------------- M II]sF
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) @:
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error('zernfun:NMvectors','N and M must be vectors.')
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end ^H>vJT
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if length(n)~=length(m) Ty5\zxC|
error('zernfun:NMlength','N and M must be the same length.') #t\Oq9}^
end zuOIos
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}
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n = n(:); lVtgg?
m = m(:); L/shF}<
if any(mod(n-m,2)) /lUb9&yV
error('zernfun:NMmultiplesof2', ... [Gu]p&
'All N and M must differ by multiples of 2 (including 0).') 0&Qn7L
end ) ":~`Z*@
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if any(m>n) 2>mDT
error('zernfun:MlessthanN', ... I".r`$XZ
'Each M must be less than or equal to its corresponding N.') 5 p750`n
end @$aCUJ/mE
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if any( r>1 | r<0 ) tNtP+v-{
error('zernfun:Rlessthan1','All R must be between 0 and 1.') =|6IyL_N
end ?x:\RNB/
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 3Z`oI#-x
error('zernfun:RTHvector','R and THETA must be vectors.') 4aGHks8Z,\
end ~-,<`VY
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r = r(:); h><;TAp
theta = theta(:); \KG{
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length_r = length(r); Qf"gH<vT
if length_r~=length(theta) R+5x:mpHy
error('zernfun:RTHlength', ... X(/W|RY{@
'The number of R- and THETA-values must be equal.') Hkpn/,D5
end %H:!/'45
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' e-FJ')|
% Check normalization: >Z/,DIn,I
% -------------------- M6?* \9E
if nargin==5 && ischar(nflag) XI
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isnorm = strcmpi(nflag,'norm'); (fq>P1-
if ~isnorm ~6R|
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error('zernfun:normalization','Unrecognized normalization flag.') $g*|h G/{
end Pb!kl #
else 8c#u"qF
isnorm = false; {>Zc#U'
end $ U<xrN>O
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% op[5]tjL
% Compute the Zernike Polynomials 5gi`&t`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% XjWoUnz
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% Determine the required powers of r: R5qC;_0cV
% ----------------------------------- +DksWbD
m_abs = abs(m); ;A1pqHr
rpowers = []; TR]~r2z
for j = 1:length(n) eEXer>Rm
rpowers = [rpowers m_abs(j):2:n(j)]; p1CY?K
end nKch_Jb
rpowers = unique(rpowers); Q4C28-#
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% Pre-compute the values of r raised to the required powers, (`xhh
% and compile them in a matrix: Lylw('zZ
% ----------------------------- kpcIU7|e
if rpowers(1)==0 Rm{S,
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); N^B
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rpowern = cat(2,rpowern{:}); Uk5jZ|
rpowern = [ones(length_r,1) rpowern]; UV$v:>K#
else $#1i@dI
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); h0L*8P`t
rpowern = cat(2,rpowern{:}); [P407Sa"
end 7$k[cL1
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% Compute the values of the polynomials: * 3WK`9q
% -------------------------------------- >#<o7]
y = zeros(length_r,length(n)); #O*
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for j = 1:length(n) L@XhgQ
s = 0:(n(j)-m_abs(j))/2; Jn-iIl
pows = n(j):-2:m_abs(j); hU@9vU<U
for k = length(s):-1:1 Z[s{
p = (1-2*mod(s(k),2))* ... Oe5=2~4O
prod(2:(n(j)-s(k)))/ ... a=T_I1
prod(2:s(k))/ ... :VX?j3qW
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... YD 1u
prod(2:((n(j)+m_abs(j))/2-s(k))); +v{<<
idx = (pows(k)==rpowers); aHvTbpJ
y(:,j) = y(:,j) + p*rpowern(:,idx); tgKmCI
end 43^%f-J5
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if isnorm ^
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y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); FE,&_J"
end ]^uO3!+
end l
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% END: Compute the Zernike Polynomials 1$]4g/":o
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4Bsx[~ u&
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% Compute the Zernike functions: -y;SR+
% ------------------------------ WgF
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idx_pos = m>0; l1fP@|
idx_neg = m<0; :)_Ap{9J
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z = y; ?kMG!stgp}
if any(idx_pos)
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z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); <nOK#;O)
end ~&8ag`
if any(idx_neg) RoFy2A=_
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); TL lR"L5
end r~N0P|Tq
hosw :%
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% EOF zernfun gpB3\