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，
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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， gaU1A"S}
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？ ^C70b)68
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ 8<PQ31
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function z = zernfun(n,m,r,theta,nflag) *IIuGtS
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. `{ ` W-C
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ((T6z$:hA
% and angular frequency M, evaluated at positions (R,THETA) on the )| 0(#R
% unit circle. N is a vector of positive integers (including 0), and !<= ^&\A
% M is a vector with the same number of elements as N. Each element aqKrf(Rv
% k of M must be a positive integer, with possible values M(k) = -N(k) O[W/=j[
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, wH]Y1 m
% and THETA is a vector of angles. R and THETA must have the same lc\%7-%:5
% length. The output Z is a matrix with one column for every (N,M) LjPpn jU
% pair, and one row for every (R,THETA) pair. r;SOAucX
% s(cC ;
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *s$:"g-
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), &FY7
D<
% with delta(m,0) the Kronecker delta, is chosen so that the integral 5;X3{$y
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, OEhDRU%k
% and theta=0 to theta=2*pi) is unity. For the non-normalized )Ag{S[yZ
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 8RjFp2) W
% "J>8ZUP
% The Zernike functions are an orthogonal basis on the unit circle. H' %#71
% They are used in disciplines such as astronomy, optics, and `Tc"a_p9t
% optometry to describe functions on a circular domain. 9" f
% DT3koci(
% The following table lists the first 15 Zernike functions. #D
.hZ=!
% F&$~]R=&
% n m Zernike function Normalization Cp^`-=r+
% -------------------------------------------------- q*7: L
% 0 0 1 1 g<^-[w4/
% 1 1 r * cos(theta) 2 rn RWL4
% 1 -1 r * sin(theta) 2 lX/ :e=
% 2 -2 r^2 * cos(2*theta) sqrt(6) %6E:SI 4
% 2 0 (2*r^2 - 1) sqrt(3) 8XD_p);Oy
% 2 2 r^2 * sin(2*theta) sqrt(6) Huf; A1.
% 3 -3 r^3 * cos(3*theta) sqrt(8) aPm2\Sq$
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) PZk"! I<oN
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) cHX~-:KOr
% 3 3 r^3 * sin(3*theta) sqrt(8) +k\cmDcb
% 4 -4 r^4 * cos(4*theta) sqrt(10) Y InPmR
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 2-beq <I
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) KEo?Cy?%ff
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^ b6yN\,S
% 4 4 r^4 * sin(4*theta) sqrt(10) =O>E>Q
% -------------------------------------------------- Ti $_V_
% x,UP7=6
% Example 1: kerBy\^
% %a|m[6+O
% % Display the Zernike function Z(n=5,m=1) Ue(\-b\)
% x = -1:0.01:1; S3ZI C\2
% [X,Y] = meshgrid(x,x); t)hi j&wzu
% [theta,r] = cart2pol(X,Y); !#dp [,nk
% idx = r<=1; VF:95F;@
% z = nan(size(X)); MS;^@>|wj
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 91M5F $
% figure SHRn $<
% pcolor(x,x,z), shading interp oa6&?4K?F
% axis square, colorbar (lt{$0
% title('Zernike function Z_5^1(r,\theta)') 4rUOk"li
% }NKnV3G/Z
% Example 2: ~2[mZias
% b)7v-1N
% % Display the first 10 Zernike functions tg C)vZ&a
% x = -1:0.01:1; 2X6L'!=
% [X,Y] = meshgrid(x,x); mT,#"k8
% [theta,r] = cart2pol(X,Y); <ToRPx&E
% idx = r<=1; oW3| b2D
% z = nan(size(X)); }s:~E2?In
% n = [0 1 1 2 2 2 3 3 3 3]; > *soc!# Y
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; R<;;Ph
% Nplot = [4 10 12 16 18 20 22 24 26 28]; $y,tR.5.)[
% y = zernfun(n,m,r(idx),theta(idx)); bp>M&1^KY
% figure('Units','normalized') naV bcY
% for k = 1:10 ?<1~KLPMhY
% z(idx) = y(:,k); o8fY!C)
% subplot(4,7,Nplot(k)) ,AwX7gx22
% pcolor(x,x,z), shading interp ^w z 2e
% set(gca,'XTick',[],'YTick',[]) G{gc]7\=Cd
% axis square f0+vk'Z
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]|_ +lik#
% end +!$]a^3l
% 2*a5pFkb
% See also ZERNPOL, ZERNFUN2. <aQ5chf7
1t }
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% Paul Fricker 11/13/2006 f7:}t+d
gl 27&'?E*
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% Check and prepare the inputs: W~/{ct$Y
% ----------------------------- ;e$YM;;d
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) A+hA'0isF@
error('zernfun:NMvectors','N and M must be vectors.') {'yr)(:2M
end +aN"*//i
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if length(n)~=length(m) /6:qmh2
error('zernfun:NMlength','N and M must be the same length.') 8w MwS6s:
end j+("4b'
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n = n(:); .JV y }^Q\
m = m(:); EkoT U#w5
if any(mod(n-m,2)) [F
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error('zernfun:NMmultiplesof2', ... Mb-AzGsV
'All N and M must differ by multiples of 2 (including 0).') ~>Xq R/v
end ydMSL25<+
.$ o
A~
%:Z_~7ZR
if any(m>n) X n02p,,
error('zernfun:MlessthanN', ... u{S"NEc
'Each M must be less than or equal to its corresponding N.') 7m8(8$-6
end 6[-[6%o#z
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if any( r>1 | r<0 ) B7PdavO#
error('zernfun:Rlessthan1','All R must be between 0 and 1.') +v<
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end d<[L^s9
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. f!dH
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) rTqGtmulG
error('zernfun:RTHvector','R and THETA must be vectors.') %DM0Z8P$B-
end "O~kIT?/v
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r = r(:); SGu`vN]
theta = theta(:); /!fJ`pu!
length_r = length(r); 8vQR'<,
if length_r~=length(theta) A=wG};%_
error('zernfun:RTHlength', ... g} pD%
'The number of R- and THETA-values must be equal.') &0]5zQ
end 6FY.kN\
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`1uGU[{ x
% Check normalization: ; ,s9jw
% -------------------- &@=W+A=c~
if nargin==5 && ischar(nflag) =M T'e,T
isnorm = strcmpi(nflag,'norm'); ,c&gw tdl
if ~isnorm L3A2A
error('zernfun:normalization','Unrecognized normalization flag.') ,&L}^ Up
end dWdD^>8Ef
else "28zLo3
isnorm = false; ;=WwJ Np~
end -A zOujSS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *%*B o9a/
% Compute the Zernike Polynomials |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [s{[
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% Determine the required powers of r: RT${7=
% ----------------------------------- Wb[k2V
m_abs = abs(m); L|B! ]}
rpowers = []; a ,"
for j = 1:length(n) S&Q Xf<v
rpowers = [rpowers m_abs(j):2:n(j)]; zRbY]dW
end _3. rPS,s
rpowers = unique(rpowers); cICf V,j
}9&dY!h +
)sNPWn8<Uy
% Pre-compute the values of r raised to the required powers, I?^(j;QpS
% and compile them in a matrix: ci/qm\JI<<
% ----------------------------- O<E8,MCA[a
if rpowers(1)==0 u:mndTpB6x
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4c[/%e:\-
rpowern = cat(2,rpowern{:}); $x,EPRNs
rpowern = [ones(length_r,1) rpowern]; SPXv i0Jg
else "YW Z&_n**
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); _3< P(w{
rpowern = cat(2,rpowern{:}); :wG
)
end ]"~
x
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% Compute the values of the polynomials: Hp>L}5 y[
% -------------------------------------- C!c h
!E#
y = zeros(length_r,length(n)); 'GT^araz
for j = 1:length(n) :Z x|=
s = 0:(n(j)-m_abs(j))/2; J_;*@mW
pows = n(j):-2:m_abs(j); EB *C;ms
for k = length(s):-1:1 lRNm
&3:-
p = (1-2*mod(s(k),2))* ... +vxOCN4}v
prod(2:(n(j)-s(k)))/ ... *C<; yPVc
prod(2:s(k))/ ... lfre-pS+
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... /sj*@HF=
prod(2:((n(j)+m_abs(j))/2-s(k))); Ow.DBL)x'>
idx = (pows(k)==rpowers); O6vxp?:^
y(:,j) = y(:,j) + p*rpowern(:,idx); '5LdiSk
end 5[4nFa}R:5
6 q>}M
if isnorm CWocb=E
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); -
jCj_@n
end L#uU. U=
end vhAgX0k
% END: Compute the Zernike Polynomials AI|+*amTd
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O5Z9`_9<
YB<nz<;JR
tfZ@4%'
% Compute the Zernike functions: M=lU`Sm
% ------------------------------ :8hI3]9
idx_pos = m>0; GZ,MC?W
idx_neg = m<0; _> x }MW+
vSC1n8 /
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z = y; I:G8B5{J
if any(idx_pos) d;]m wLB0
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); p6K ~ b
end &)gc{(4$
if any(idx_neg) 6 /5,n0
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); T^(W _S
end JJ%@m; ~
0<a|=kZ
BV:Ca 34&
% EOF zernfun `[g$EXX
