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， C,+
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， $(D>v!dp
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？ F62 uDyY
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ fhN\AjB6Td
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function z = zernfun(n,m,r,theta,nflag) H-pf8
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. "yQBHYP
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N
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% and angular frequency M, evaluated at positions (R,THETA) on the SN@>m pcJS
% unit circle. N is a vector of positive integers (including 0), and K[iAN;QCe%
% M is a vector with the same number of elements as N. Each element .;7V]B1o
% k of M must be a positive integer, with possible values M(k) = -N(k) q!Ek
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% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 7<WUj K|
% and THETA is a vector of angles. R and THETA must have the same 8:& ! F`o
% length. The output Z is a matrix with one column for every (N,M) $CMye; yL
% pair, and one row for every (R,THETA) pair. i_N8)Z;r
% Kfb(wW
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike (UkDww_!
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), eQuw uT
% with delta(m,0) the Kronecker delta, is chosen so that the integral T9$~tv,5F
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, *l`yxz@U
% and theta=0 to theta=2*pi) is unity. For the non-normalized %"r9;^bj&<
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. c"tlNf ?
% RI8*'~ix]
% The Zernike functions are an orthogonal basis on the unit circle. \r:* `Z*y
% They are used in disciplines such as astronomy, optics, and >;9g`d
% optometry to describe functions on a circular domain. 'sI ne>
% 3T.V*&
% The following table lists the first 15 Zernike functions. `WH$rx!
% 9BZ B1o X
% n m Zernike function Normalization 1,=:an
% -------------------------------------------------- b/[X8w'VP
% 0 0 1 1 p+~Imf-Jk
% 1 1 r * cos(theta) 2 ^ ^}htg
% 1 -1 r * sin(theta) 2 1P"7.{
% 2 -2 r^2 * cos(2*theta) sqrt(6) AsE77AUA
% 2 0 (2*r^2 - 1) sqrt(3) /#T {0GBXe
% 2 2 r^2 * sin(2*theta) sqrt(6) qZ! kVrmg&
% 3 -3 r^3 * cos(3*theta) sqrt(8) ng+sK
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) > 8{w0hh;
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) xKE=$SV(
% 3 3 r^3 * sin(3*theta) sqrt(8) BC!) g+8
% 4 -4 r^4 * cos(4*theta) sqrt(10) VB90 5 %
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) jo&j<3i
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) TY% c`Q5
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) s/@uGC0>
% 4 4 r^4 * sin(4*theta) sqrt(10) ~/A2 :}Cp=
% -------------------------------------------------- %' WC7s
% mRAt5a#is
% Example 1: ? <.a>"!
% qn yacI
% % Display the Zernike function Z(n=5,m=1) +)yoQRekX
% x = -1:0.01:1; EXeV @kg
% [X,Y] = meshgrid(x,x); >dK0&+A
% [theta,r] = cart2pol(X,Y); xkFa
% idx = r<=1; O-7)"
% z = nan(size(X)); uq[5 om"
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ">=E p+ix
% figure c*\i%I#f2
% pcolor(x,x,z), shading interp 9j^rFG!n
% axis square, colorbar #m{(aa9;
% title('Zernike function Z_5^1(r,\theta)') ^`#7(S)a/
% &iu]M=Y b
% Example 2: '2Zs15)V
% .B x QF
% % Display the first 10 Zernike functions $hC S-9%&
% x = -1:0.01:1; tt-ci,X+
% [X,Y] = meshgrid(x,x); Da)p%E>Q
% [theta,r] = cart2pol(X,Y); 0.+Eo.AX4M
% idx = r<=1; &;?+ ^L>
% z = nan(size(X)); :4[>]&:u3
% n = [0 1 1 2 2 2 3 3 3 3]; xKBi".wA
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Kn$t_7AF^
% Nplot = [4 10 12 16 18 20 22 24 26 28];
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% y = zernfun(n,m,r(idx),theta(idx)); v(Kj 6 '
% figure('Units','normalized') M^\`~{*T
% for k = 1:10 Q1*_l
% z(idx) = y(:,k); ~rI2 RJ
% subplot(4,7,Nplot(k)) 8h)7K/!\
% pcolor(x,x,z), shading interp cg^~P-i@*
% set(gca,'XTick',[],'YTick',[]) 4xT /8>v2|
% axis square <WWZb\"{
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])
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% end @?J7=}bzz
% FT>>X P8
% See also ZERNPOL, ZERNFUN2. 3%r/w7Fc
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% Paul Fricker 11/13/2006 =?4[:#Rh
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% Check and prepare the inputs: \H>Psv{
% ----------------------------- QsPg4y3?D
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) x(Uv>k~i}
error('zernfun:NMvectors','N and M must be vectors.') s+_8U}R
end 8 [,R4@
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if length(n)~=length(m) ( bsywM
error('zernfun:NMlength','N and M must be the same length.') GMZ6 dK
end W\0u[IV.x
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n = n(:); cGo_qR/B(>
m = m(:); P()n=&XO6
if any(mod(n-m,2)) _I EbRVpb
error('zernfun:NMmultiplesof2', ... y+$vHnS/jC
'All N and M must differ by multiples of 2 (including 0).') @\gE{;a8
end pUmT?N!
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if any(m>n) n(LO`{
error('zernfun:MlessthanN', ... dtV*CX.D.7
'Each M must be less than or equal to its corresponding N.') G3!O@j!7w$
end }jce5E
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if any( r>1 | r<0 ) .L^j:2(L
error('zernfun:Rlessthan1','All R must be between 0 and 1.') N0$
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end =^ Ws/k
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) )
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error('zernfun:RTHvector','R and THETA must be vectors.') ('uUf!h?\
end $z)egh(z
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r = r(:); <
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theta = theta(:); dR>$vbjh1Z
length_r = length(r); 5>e<|@2
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if length_r~=length(theta) 6
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error('zernfun:RTHlength', ... J^DyhCs
'The number of R- and THETA-values must be equal.') n/BoK6g
end bx6=LK
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% Check normalization: )eSQce7H
% -------------------- DH$ Nz
if nargin==5 && ischar(nflag) Ln+ .$ C
isnorm = strcmpi(nflag,'norm'); I_?R(V[9
if ~isnorm #jxPh! %9
error('zernfun:normalization','Unrecognized normalization flag.') l.;^ w
end Je^ ;[^
else Mw+
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isnorm = false; Ps 0<CUyI
end x}` )'a[
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Compute the Zernike Polynomials 9&kPcFX B
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% XdlA)0S)
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% Determine the required powers of r: 8OS^3JS3"
% ----------------------------------- 2} .~
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m_abs = abs(m); Jfv'M<I
rpowers = []; 6>&(OV
for j = 1:length(n) 6" [,
rpowers = [rpowers m_abs(j):2:n(j)]; ?%Q=l;W.
end u,=?|M\
rpowers = unique(rpowers); v$;URF%^
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% Pre-compute the values of r raised to the required powers, |Z o36@s
% and compile them in a matrix: I&^hG\D
% ----------------------------- ]gA2.,)}D
if rpowers(1)==0 D~Q -:G$x
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); EuVA"~PA
rpowern = cat(2,rpowern{:}); '['x'G50
rpowern = [ones(length_r,1) rpowern]; ]_!NmB_3
else w&hCt c
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); d?/g5[
rpowern = cat(2,rpowern{:}); x A*6Z)Y
end )T
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% Compute the values of the polynomials: MQ' =qR
% -------------------------------------- 7#N= G N
y = zeros(length_r,length(n)); ~xJr|_,gp
for j = 1:length(n) ;D(6Gy9~
s = 0:(n(j)-m_abs(j))/2; cxPO O #
pows = n(j):-2:m_abs(j); @6;ZP1
for k = length(s):-1:1 #z*,-EV|
p = (1-2*mod(s(k),2))* ... k
$# ,^)T
prod(2:(n(j)-s(k)))/ ... 02:`Joy2D
prod(2:s(k))/ ... 4 4WyfpTJ*
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... !b$~Sm)
prod(2:((n(j)+m_abs(j))/2-s(k))); t`eIkq|NxI
idx = (pows(k)==rpowers); O zTR#`oey
y(:,j) = y(:,j) + p*rpowern(:,idx); F+D
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end L?Ih;
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if isnorm r. rzU
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Wrm3U/>e
end
z' 5
end zCS }i_ p
% END: Compute the Zernike Polynomials <)L[V
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5RF*c,cNq
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% Compute the Zernike functions: p~-)6)We?
% ------------------------------ szOa yAS
idx_pos = m>0; :o:/RR p[
idx_neg = m<0; }n,LvA@[0
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z = y; 3CSw cD
if any(idx_pos) ,s,AkH
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Pn ?gB}l
end +.u
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if any(idx_neg) 530Kk<%^}8
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); sr<\fW
end \M Av's4b@
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% EOF zernfun sei%QE]!/
