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, {h7 vJ^
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, \X:e9~
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? p35=CX`T.
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 7;fC%Fq
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function z = zernfun(n,m,r,theta,nflag) P3&s<mh
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6M*z`B{hV
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 6iyl8uL0J
% and angular frequency M, evaluated at positions (R,THETA) on the i D IY|
% unit circle. N is a vector of positive integers (including 0), and 1@}F8&EZ
% M is a vector with the same number of elements as N. Each element M?eP1v:<+G
% k of M must be a positive integer, with possible values M(k) = -N(k) v'@gUgC
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, T+}|$/Tv
% and THETA is a vector of angles. R and THETA must have the same mvEhP{w
% length. The output Z is a matrix with one column for every (N,M) WMf /
S"=
% pair, and one row for every (R,THETA) pair. S} m=|3%y
% tb+gCs'D
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike @kFZN 6
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), #:gd9os :
% with delta(m,0) the Kronecker delta, is chosen so that the integral DWt|lO
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ltNCti{Q
% and theta=0 to theta=2*pi) is unity. For the non-normalized JX=rL6Y@:;
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. f=F:Af!
% .n]"vpWm[
% The Zernike functions are an orthogonal basis on the unit circle. *OG<+#*\_?
% They are used in disciplines such as astronomy, optics, and V/ G1C^'/
% optometry to describe functions on a circular domain. ?% 24M\
% MeEa| .
% The following table lists the first 15 Zernike functions. i<^X z
% u?Ffqt9'
% n m Zernike function Normalization 6VGY4j}:(
% -------------------------------------------------- cAW}a
% 0 0 1 1 h+Co:pr
% 1 1 r * cos(theta) 2 2 ?t@<M]
% 1 -1 r * sin(theta) 2 oe|#!SM(
% 2 -2 r^2 * cos(2*theta) sqrt(6) >&PM'k
% 2 0 (2*r^2 - 1) sqrt(3) 2LtDS?)@
% 2 2 r^2 * sin(2*theta) sqrt(6) _nMd
% 3 -3 r^3 * cos(3*theta) sqrt(8) \)~d,M}kK
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) PXMd=,}
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) =0g!Q
% 3 3 r^3 * sin(3*theta) sqrt(8) |2j,
% 4 -4 r^4 * cos(4*theta) sqrt(10) p3=Py7iz
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) gXdMGO>
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Tz @=N] D
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "]S
% 4 4 r^4 * sin(4*theta) sqrt(10) @|b-X? `
% -------------------------------------------------- W@T\i2r$z
% Jl~ *@0(
% Example 1: 5qz,FKx5
% xnZnbgO+
% % Display the Zernike function Z(n=5,m=1) *:n~j9V-
% x = -1:0.01:1; [Yt{h9
% [X,Y] = meshgrid(x,x); >O-KJZ'GV
% [theta,r] = cart2pol(X,Y); u<ed O+
% idx = r<=1; V"YeF:I
% z = nan(size(X)); [:y:_ECs6
% z(idx) = zernfun(5,1,r(idx),theta(idx)); #f2Ot<#-
% figure !O_G%+>5W
% pcolor(x,x,z), shading interp Ul}RT xJ
% axis square, colorbar }=-0DSLVj
% title('Zernike function Z_5^1(r,\theta)') o}rG:rhIh
% ybBmg'198
% Example 2: |.N[NY
% XGl2rX&
% % Display the first 10 Zernike functions (P|[<Sd
% x = -1:0.01:1; ?})A-$f ~
% [X,Y] = meshgrid(x,x); r2=@1=?8
% [theta,r] = cart2pol(X,Y); PQ&*(G
% idx = r<=1; I&1Lm)W&
% z = nan(size(X)); ix!xLm9\
% n = [0 1 1 2 2 2 3 3 3 3]; Hl$W+e|tj
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; <V&0GAZ
% Nplot = [4 10 12 16 18 20 22 24 26 28]; AP&//b,^M
% y = zernfun(n,m,r(idx),theta(idx)); #t
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% figure('Units','normalized') d0(zB5'}
% for k = 1:10 E5ce=$o
% z(idx) = y(:,k); uM2@&)u
% subplot(4,7,Nplot(k)) k =! Q
% pcolor(x,x,z), shading interp 'ITq\1z
% set(gca,'XTick',[],'YTick',[]) $mQ0w~:@
% axis square =]7o+L4
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) t8^1wA@@V
% end >d + }$dB
% w(oK
% See also ZERNPOL, ZERNFUN2. 5XKTb
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% Paul Fricker 11/13/2006 ~5KcbGD~
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% Check and prepare the inputs: Q-<]'E#\(
% ----------------------------- l2
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) T\l`Y-vu
error('zernfun:NMvectors','N and M must be vectors.') _uIS[%4g
end eEZgG=s
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if length(n)~=length(m) 8b/$Qp4d
error('zernfun:NMlength','N and M must be the same length.') @DysM~I
end xC`!uPk/pL
0 +=sBk (
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n = n(:); `ASDUgx Mq
m = m(:); ',EI[
]+
if any(mod(n-m,2)) kdNo<x1o
error('zernfun:NMmultiplesof2', ... T[\1=h]
'All N and M must differ by multiples of 2 (including 0).') t1 3V>9to
end L `1 ITz
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if any(m>n) 82LE9<4A
error('zernfun:MlessthanN', ... VF?H0}YSHb
'Each M must be less than or equal to its corresponding N.') b| L;*<KU
end $)M3fZ$#
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P $`1}
if any( r>1 | r<0 ) Q|_F
P:
error('zernfun:Rlessthan1','All R must be between 0 and 1.') {$frR "K
end 2-4N)q
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) tYjG8P#
error('zernfun:RTHvector','R and THETA must be vectors.') u&zY>'}zm
end !^arWH[od
Y%
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r = r(:); u zZ|0
theta = theta(:); X$kLBG[o_
length_r = length(r); a{8a[z
if length_r~=length(theta) Hx#YN*\.M
error('zernfun:RTHlength', ... -@N-i$!;J
'The number of R- and THETA-values must be equal.') <pX?x3-'
end 5%,3)H{;t
u]*7",R
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yT^2;/Z
% Check normalization: un "I
% -------------------- ^+(5[z
if nargin==5 && ischar(nflag) Z ]A
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isnorm = strcmpi(nflag,'norm'); 45yP {+/-Q
if ~isnorm rNN>tpZ}
error('zernfun:normalization','Unrecognized normalization flag.') iK}p#"si
end tD8fSV
else ifn=De3+
isnorm = false; Cv#aBH'N
end !u7KgB<=/F
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 88?bUA3]
% Compute the Zernike Polynomials O,%UNjx9K
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N#u'SGTG
i c{I
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% Determine the required powers of r: r5j$FwY
% ----------------------------------- \,;glY=M!
m_abs = abs(m); J jAxNviG
rpowers = []; 9^*RK6
for j = 1:length(n) 4?pb!@l
rpowers = [rpowers m_abs(j):2:n(j)]; 1H-Wk
end $yO B-
rpowers = unique(rpowers); &4%pPL\f
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% Pre-compute the values of r raised to the required powers, Fyvo;1a
% and compile them in a matrix: D`XXR}8V
% ----------------------------- nlv,j&
if rpowers(1)==0 ;+75"=[YT
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); n@pwOHQn<|
rpowern = cat(2,rpowern{:}); _9BL7W $;
rpowern = [ones(length_r,1) rpowern]; y[McdlH m
else m=}h7&5 p
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); EZ!! V~
rpowern = cat(2,rpowern{:}); k8 #8)d
end $:u*)&"t|
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% Compute the values of the polynomials: aR2N,<Cp5
% -------------------------------------- }8#olZ/(q
y = zeros(length_r,length(n)); x(c+~4:_M
for j = 1:length(n) (MXy\b<
s = 0:(n(j)-m_abs(j))/2; M7BpOmK'
pows = n(j):-2:m_abs(j); s_ZPo6p
for k = length(s):-1:1 f`4=Bl&"{
p = (1-2*mod(s(k),2))* ... nf
pO
prod(2:(n(j)-s(k)))/ ... yu_PZ"l
prod(2:s(k))/ ... HQ+{9Z8
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prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 7~2_'YX>:
prod(2:((n(j)+m_abs(j))/2-s(k))); %Z6Q/+#fn
idx = (pows(k)==rpowers); yl$Ko
y(:,j) = y(:,j) + p*rpowern(:,idx); bg~CV&]M
end X1w11Z7o
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if isnorm 1XiA
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1e\cJ{B
end %Za}q]?
end [60y.qE
% END: Compute the Zernike Polynomials :uYZ1O
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |ts0j/A]Pi
ltOS()[X
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% Compute the Zernike functions: =*fq5v
% ------------------------------ La6
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idx_pos = m>0; xR-;,=J
idx_neg = m<0; X!h>13fW
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z = y; &/m^}x/_W
if any(idx_pos) j~_iv~[
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Fepsa;\sU
end 3^KR{N p
if any(idx_neg) XrUI[ryE
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); bR3Crz(9G
end +8<$vzB
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% EOF zernfun 8i$|j~M a