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， j _d}?jh
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？ a& 0g0n6
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ Sed 8Q-m
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function z = zernfun(n,m,r,theta,nflag)
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 8H-yT1
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N | J4sQ!%K
% and angular frequency M, evaluated at positions (R,THETA) on the QuEX|h,F
% unit circle. N is a vector of positive integers (including 0), and OD7^*j(p`
% M is a vector with the same number of elements as N. Each element Y=|p}>.}
% k of M must be a positive integer, with possible values M(k) = -N(k) ;`^_9
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% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, /ojx$Um
% and THETA is a vector of angles. R and THETA must have the same Q>Klkd5(
% length. The output Z is a matrix with one column for every (N,M) ;6 W[%{
% pair, and one row for every (R,THETA) pair. XYR
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% 9QX!HQ|5y8
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike m-$}'mEO
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), %\-E
R !b
% with delta(m,0) the Kronecker delta, is chosen so that the integral m8PS84."]M
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, FRR05%K
% and theta=0 to theta=2*pi) is unity. For the non-normalized 5.ab/uk;M
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. f.$[?Fi
% 7b08Lo7b
% The Zernike functions are an orthogonal basis on the unit circle. m5
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% They are used in disciplines such as astronomy, optics, and R~iv%+
% optometry to describe functions on a circular domain. cH*")oD
% %\,9S`0
% The following table lists the first 15 Zernike functions. Z[w}PN,xV
% Q*I8RAfd
% n m Zernike function Normalization 9#7 W+9
% -------------------------------------------------- i$%Bo/Y
% 0 0 1 1 u;
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% 1 1 r * cos(theta) 2 WPsfl8@D
% 1 -1 r * sin(theta) 2 .xwskzJ3
% 2 -2 r^2 * cos(2*theta) sqrt(6) T0dD:s N
% 2 0 (2*r^2 - 1) sqrt(3) L ,.~VNy-
% 2 2 r^2 * sin(2*theta) sqrt(6) n_; s2,2r
% 3 -3 r^3 * cos(3*theta) sqrt(8) D|Q7dIZm
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) q=->) &D%
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Y!oLNGY
% 3 3 r^3 * sin(3*theta) sqrt(8) vE^tdzAG
% 4 -4 r^4 * cos(4*theta) sqrt(10) LA_{[VWYp>
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) E"VF BKB
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) \8$~ i
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *GoTN
% 4 4 r^4 * sin(4*theta) sqrt(10) $m#^0%
% -------------------------------------------------- XX /s@ C
% :,JjN&
% Example 1: pD({"A.x9z
% NW5OLa")J<
% % Display the Zernike function Z(n=5,m=1) o$</At
% x = -1:0.01:1; ? -:2f#bC
% [X,Y] = meshgrid(x,x); 2Q %7J3I
% [theta,r] = cart2pol(X,Y); 4 j=K3m
% idx = r<=1; V: L%GWU
% z = nan(size(X)); . ,z6a
% z(idx) = zernfun(5,1,r(idx),theta(idx)); %gO/mj3*
% figure S=-$:65
% pcolor(x,x,z), shading interp 5z 0VMt
% axis square, colorbar PlH~um[J
% title('Zernike function Z_5^1(r,\theta)') h-1?c\Qq:
% T4wk$R
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% Example 2: Z[IM\# "
% 1Zn8CmE V
% % Display the first 10 Zernike functions \Aro Sy9
% x = -1:0.01:1; bD ,X.
% [X,Y] = meshgrid(x,x); u*Xp%vNe
% [theta,r] = cart2pol(X,Y); 2H4vK]]Nl
% idx = r<=1; sq`X z 8u
% z = nan(size(X)); \t=0r FV)t
% n = [0 1 1 2 2 2 3 3 3 3]; v5' `iO0o
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; seEo)m`d
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ) %Fwfb
% y = zernfun(n,m,r(idx),theta(idx)); 7xeqs
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% figure('Units','normalized') r ~)fAb?
% for k = 1:10 . + u
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% z(idx) = y(:,k); V2}\]x'1
% subplot(4,7,Nplot(k)) 9r]|P}yuS
% pcolor(x,x,z), shading interp 8 -x-?7
% set(gca,'XTick',[],'YTick',[]) \wA:58 -j
% axis square Kb(11$ U
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) b*?u+tWP_
% end =D$ ED^W
% t([}a ~1}
% See also ZERNPOL, ZERNFUN2. 1`7zYW&L
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% Paul Fricker 11/13/2006 klJ21j0Bb2
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% Check and prepare the inputs: 7m-%
% ----------------------------- O<cP1T F
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) )
.fl r
error('zernfun:NMvectors','N and M must be vectors.') @ M"gEeI9
end t 6nRg
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if length(n)~=length(m) D8Fi{?A#FV
error('zernfun:NMlength','N and M must be the same length.') y+ze`pL?
end 2HFn\kjj.s
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n = n(:); 7_i8'(``
m = m(:); mtv8Bm=<
if any(mod(n-m,2)) Lg7A[\c
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error('zernfun:NMmultiplesof2', ... GjhTF|
'All N and M must differ by multiples of 2 (including 0).') d5m - f/
end 3^y(@XFt
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if any(m>n) f( 5(V
%
error('zernfun:MlessthanN', ... 6^Wep- $
'Each M must be less than or equal to its corresponding N.') O{X~,Em=q
end Tzex\]fw
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if any( r>1 | r<0 ) O{LCHtN
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Ki;SONSV~|
end E]`7_dG+T
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) G>V6{g2 Q
error('zernfun:RTHvector','R and THETA must be vectors.') {.:$F 3T
end p
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r = r(:); x z _sejKB
theta = theta(:); xR 1G
length_r = length(r); A;TP~xq\
if length_r~=length(theta) /\8I l+0
error('zernfun:RTHlength', ... "313eeIt%i
'The number of R- and THETA-values must be equal.') FO2e7p^Q
end o
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% Check normalization: j !u)V1,
% -------------------- kTvM,<
if nargin==5 && ischar(nflag) ~Bzzu % S
isnorm = strcmpi(nflag,'norm'); I P62|~Ap
if ~isnorm ShB]U5b:k
error('zernfun:normalization','Unrecognized normalization flag.') EA& 3rI>U)
end C%X O| sP
else s*izhjjX
isnorm = false; ~K;QdV=YX
end n<ZPWlJ
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =c[tHf
% Compute the Zernike Polynomials =hPXLCeC
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "%-Vrb =:Y
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% Determine the required powers of r: 8`q7Yss6F
% ----------------------------------- 5'lPXKn+L
m_abs = abs(m); EbC!tR
rpowers = []; xVm-4gB
for j = 1:length(n) X ,QsE{
rpowers = [rpowers m_abs(j):2:n(j)]; &R94xh%@(
end -pu5O 9
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rpowers = unique(rpowers); cr1x
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% Pre-compute the values of r raised to the required powers, $]2)r[eA)
% and compile them in a matrix: f`9Mcli !
% ----------------------------- wcGK *sWG-
if rpowers(1)==0 *pKTJP
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ++0)KSvw
rpowern = cat(2,rpowern{:}); F-yY(b]$
rpowern = [ones(length_r,1) rpowern]; D|;O9iks#
else r"7n2
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); #.Rn6|V/4
rpowern = cat(2,rpowern{:}); sXIYl% d
end </h ^%mnd
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% Compute the values of the polynomials: ht3.e[%'b
% -------------------------------------- ~4~`bT 9
y = zeros(length_r,length(n)); ]?Ef0?4 4
for j = 1:length(n) }Z!D?(
s = 0:(n(j)-m_abs(j))/2; tq3Wga!5
pows = n(j):-2:m_abs(j); *r7v Dc
for k = length(s):-1:1 7},A. q
p = (1-2*mod(s(k),2))* ... kx"1 0Vw
prod(2:(n(j)-s(k)))/ ... YDt+1Kw}D
prod(2:s(k))/ ... )#=J<OpG
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ?e7]U*jEU
prod(2:((n(j)+m_abs(j))/2-s(k))); ^t;z;. g
idx = (pows(k)==rpowers); r~4uIUE{
y(:,j) = y(:,j) + p*rpowern(:,idx); J$dwy$n
end P15
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if isnorm p9(y b
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 4f EDg{ T
end %,$ n^{v
end KpLm pK1
% END: Compute the Zernike Polynomials +X}i%F'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {zdMmpQF
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% Compute the Zernike functions: mgs(n5V5
% ------------------------------ V~J5x >O
idx_pos = m>0; & d# R'Z
idx_neg = m<0; :+rGBkw1m
#(8|9
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z = y; ]Bw 0Qq F#
if any(idx_pos) 1>!LK_
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); G0cG%sIl
end J=4>zQLW
if any(idx_neg) EY} : aur
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); eI
#Gx_mg
end =YO ]m<
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% EOF zernfun *GP2>oEM
