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, V7[6jWgH
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, :hJHjh
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,w'Op
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? IppzQ0'=y1
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function z = zernfun(n,m,r,theta,nflag) dw#K!,g
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. `%IzW2v6
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 4[m})X2(
% and angular frequency M, evaluated at positions (R,THETA) on the tS!FnQg4
% unit circle. N is a vector of positive integers (including 0), and m5m}RWZ#
% M is a vector with the same number of elements as N. Each element Aslh}'$}-
% k of M must be a positive integer, with possible values M(k) = -N(k) %sxLxx_x!
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, sU! h^N$
% and THETA is a vector of angles. R and THETA must have the same }(k#,&Fv`
% length. The output Z is a matrix with one column for every (N,M) "O{j}QwY
% pair, and one row for every (R,THETA) pair. ^0)Mc"&{
% =r"-Pm{
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ,cZhkXd
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), C))5,aX
% with delta(m,0) the Kronecker delta, is chosen so that the integral ,5!&}
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, cq lA"Eof
% and theta=0 to theta=2*pi) is unity. For the non-normalized K.) ionb
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. f++MH]I;
% /kV3[Rw+
% The Zernike functions are an orthogonal basis on the unit circle. x\PZ.o
% They are used in disciplines such as astronomy, optics, and JjA3G`m=
% optometry to describe functions on a circular domain. R=&9M4
% |osu4=s|
% The following table lists the first 15 Zernike functions. wpgO09
% MDV<[${
% n m Zernike function Normalization EQe !&;
% -------------------------------------------------- 9{[I|
% 0 0 1 1 \9"
% 1 1 r * cos(theta) 2 9
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% 1 -1 r * sin(theta) 2 ?*DM|hzOi
% 2 -2 r^2 * cos(2*theta) sqrt(6) paKur%2u
% 2 0 (2*r^2 - 1) sqrt(3) x}\x3U
% 2 2 r^2 * sin(2*theta) sqrt(6) f>*T0"\c
% 3 -3 r^3 * cos(3*theta) sqrt(8) 7egE."
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) w`BY>Xft0
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) SeuC7!q{
% 3 3 r^3 * sin(3*theta) sqrt(8) xgDd5`W
% 4 -4 r^4 * cos(4*theta) sqrt(10) +85#`{ D
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (~>uFH
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 44Dytpvg
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) YcQ$nZAU
% 4 4 r^4 * sin(4*theta) sqrt(10) 6/-]
% -------------------------------------------------- A]0A,A0
% 9NF2a)&~
% Example 1: F/pq9
% ')R+Z/hG.
% % Display the Zernike function Z(n=5,m=1) C@x\ZG5rA
% x = -1:0.01:1; )6+Z9 9w
% [X,Y] = meshgrid(x,x); f^JiaU4 [
% [theta,r] = cart2pol(X,Y); PP*6nW8
% idx = r<=1; CzMCd
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% z = nan(size(X)); 8y:/!rRN
% z(idx) = zernfun(5,1,r(idx),theta(idx)); B":9C'tip
% figure 5D]3I=kj
% pcolor(x,x,z), shading interp 1G}f83yR
% axis square, colorbar 1`hmD1d
% title('Zernike function Z_5^1(r,\theta)') } 6 ,m2u
% IRhi1{K$"
% Example 2: @}, |i*H/
% Q};n%&n&
% % Display the first 10 Zernike functions #ovausK[7
% x = -1:0.01:1; kM6i{{Q
% [X,Y] = meshgrid(x,x); dU}Cb?]7s
% [theta,r] = cart2pol(X,Y); p9>{X\eT:
% idx = r<=1; ^VC/tJ
% z = nan(size(X)); _0cCTQE
% n = [0 1 1 2 2 2 3 3 3 3]; C/$bgK[ev
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 18^#:=Z
% Nplot = [4 10 12 16 18 20 22 24 26 28]; --fRh N>
% y = zernfun(n,m,r(idx),theta(idx)); SND@#?hiO
% figure('Units','normalized') D`|8Og
% for k = 1:10 ^ps6\>=0cW
% z(idx) = y(:,k); kzE<Y
% subplot(4,7,Nplot(k)) M)F_$
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% pcolor(x,x,z), shading interp %p48=|+
% set(gca,'XTick',[],'YTick',[]) >jU25"XI[
% axis square Y/x>wNW
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @T"-%L8PL
% end m?<^b_a}
% `uKsFXM
% See also ZERNPOL, ZERNFUN2. /!#A'#Z
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% Paul Fricker 11/13/2006 YpOcLxFL
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% Check and prepare the inputs: ,i jB3J
% ----------------------------- &SG5f[
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) xfF;u9$;
error('zernfun:NMvectors','N and M must be vectors.') 2tb+3K1
end T@Bu Fr`]<
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if length(n)~=length(m) i]#"@xQ
error('zernfun:NMlength','N and M must be the same length.') Dm@h'*
end zfD@/kU
6b7c9n Z
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n = n(:); bUsX~R-
m = m(:); ECyG$j0
if any(mod(n-m,2)) Pn,>eD*g
error('zernfun:NMmultiplesof2', ... )Q 5 x%
'All N and M must differ by multiples of 2 (including 0).') ~<.{z]*O
end J-|&[-Z
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if any(m>n) ?I8r2M]
error('zernfun:MlessthanN', ... cL<,]%SkE
'Each M must be less than or equal to its corresponding N.') bv;.6C(T<
end nC%<BatQ
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if any( r>1 | r<0 ) =r)LG,w212
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ? 'qyI^m@
end W}y)vrL
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) a/v]E]=qI
error('zernfun:RTHvector','R and THETA must be vectors.') +I\54PBws
end ]h#QA;
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r = r(:); !PA ><F
theta = theta(:); !>"fDz<w`
length_r = length(r); k*u6'IKi.4
if length_r~=length(theta) o[1#)&
error('zernfun:RTHlength', ... Q 5hOVD%
'The number of R- and THETA-values must be equal.') =it @U/
end GKbbwT0T|
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% Check normalization: 5A$,'%d
% -------------------- mr2Mu
if nargin==5 && ischar(nflag) !MrQ-B (
isnorm = strcmpi(nflag,'norm'); lX-i <0`
if ~isnorm RH:vd|q+
error('zernfun:normalization','Unrecognized normalization flag.') 1{5t.
end eh%{BXW[p
else &qK:LHhj
isnorm = false; jU kxA7 }}
end ::+;PRy_E
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% W"*2,R[}%
% Compute the Zernike Polynomials $hHV Ie]+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >gs_Bzy]
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% Determine the required powers of r: |d z2Drc
% ----------------------------------- BG8/
m_abs = abs(m); 98)C
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rpowers = []; 2X[oge0@
for j = 1:length(n) L,.AY?)+7
rpowers = [rpowers m_abs(j):2:n(j)]; |V4<eF-0S
end &XdTY +
rpowers = unique(rpowers); Kj "X!-
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TpwN2 =
% Pre-compute the values of r raised to the required powers, 9R2"(.U
% and compile them in a matrix: *Wvk~
% ----------------------------- dA (n,@{
if rpowers(1)==0 ?;_>BX|Zjl
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); K3<A<&W_-
rpowern = cat(2,rpowern{:}); t,dm3+R
rpowern = [ones(length_r,1) rpowern]; u#rbc"
else >MKj~Ud
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); w3"L5;oH
rpowern = cat(2,rpowern{:}); \{]y(GT
end }3 _b%{
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% Compute the values of the polynomials: 83I 5n&)
% -------------------------------------- t$~'$kM)<
y = zeros(length_r,length(n)); TTFs|T6`q
for j = 1:length(n) 5y 5Dn!`
s = 0:(n(j)-m_abs(j))/2; 8!cHRtqK
pows = n(j):-2:m_abs(j); UgK
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for k = length(s):-1:1 iF Mf[qBg
p = (1-2*mod(s(k),2))* ... T&MhSJf#
prod(2:(n(j)-s(k)))/ ... 0MroHFh9`
prod(2:s(k))/ ... @&EIH,c
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... xp'Q>%v
prod(2:((n(j)+m_abs(j))/2-s(k))); m2"e ]I
idx = (pows(k)==rpowers); @M B)B5
y(:,j) = y(:,j) + p*rpowern(:,idx); +-(,'slov
end Z)$@1Q4P?1
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if isnorm cV0CI&
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); OA=~i/n~
end :UP8nq
end r8eJ&-Yi{Z
% END: Compute the Zernike Polynomials s2NBYDi$?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %7}j|eS)G
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% Compute the Zernike functions: TEY~E*=}$
% ------------------------------ _K!.TM+9
idx_pos = m>0; ~gW^9nWYU
idx_neg = m<0; kyvl>I0q@
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z = y; xQDQgvwa
if any(idx_pos) \.O&-oi
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); jq*`| m;Q
end ;s{'cN[.
if any(idx_neg) dd<l;4(
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Y
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end w-#
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% EOF zernfun ^}p##7t[