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， ] kdU]}z
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， 0u}+n+\ g
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？ 1EMrXnv,
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ o%E-K=a
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function z = zernfun(n,m,r,theta,nflag) O:j=L{,d^
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. \*mKctpz]6
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Bo4iX,zu
% and angular frequency M, evaluated at positions (R,THETA) on the .A F94OlE/
% unit circle. N is a vector of positive integers (including 0), and 6m0- he~
% M is a vector with the same number of elements as N. Each element Wgm{
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% k of M must be a positive integer, with possible values M(k) = -N(k) (V>/[Ev
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, nNhb,J
% and THETA is a vector of angles. R and THETA must have the same G&q@B`I
% length. The output Z is a matrix with one column for every (N,M) +{\b&q_
% pair, and one row for every (R,THETA) pair. Wb#ON|.2
% Mk/ZEy q^
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /6FPiASbS
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), .*Axr\x3
% with delta(m,0) the Kronecker delta, is chosen so that the integral P E MuIYm$
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +7Qj%x\
% and theta=0 to theta=2*pi) is unity. For the non-normalized _ ]M :
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ;HmQRiCg
% 8Yw V"+Fu/
% The Zernike functions are an orthogonal basis on the unit circle. l (3\ek U!
% They are used in disciplines such as astronomy, optics, and \ eCQL(_
% optometry to describe functions on a circular domain. %(~8a
% *40Z }1ng
% The following table lists the first 15 Zernike functions. -v~XS-F
% R 5Cy%
% n m Zernike function Normalization +pjU4>)
% -------------------------------------------------- R ]y9>5 'U
% 0 0 1 1 { \5-b:#_
% 1 1 r * cos(theta) 2 <xAlp;8m5
% 1 -1 r * sin(theta) 2 ZIJTGa}B
q
% 2 -2 r^2 * cos(2*theta) sqrt(6) h<FEe~
% 2 0 (2*r^2 - 1) sqrt(3) D,SL _*r{
% 2 2 r^2 * sin(2*theta) sqrt(6) l?zWi[Zf
% 3 -3 r^3 * cos(3*theta) sqrt(8) y:dwx *Q9I
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) @v=A )L
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) MKr:a]-'f~
% 3 3 r^3 * sin(3*theta) sqrt(8) 6N'HXL UlQ
% 4 -4 r^4 * cos(4*theta) sqrt(10) {-,^3PI\
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !'
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% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) m NAp FwZ
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) L z;E/ a}s
% 4 4 r^4 * sin(4*theta) sqrt(10) {y 7,n
% -------------------------------------------------- +ywd(Tuzm
% !d=Q@oy5
% Example 1: !FR1yO'd >
% 7x :j4
% % Display the Zernike function Z(n=5,m=1) da53XEF&
% x = -1:0.01:1; a3oSSkT
% [X,Y] = meshgrid(x,x); dM3V2 TT
% [theta,r] = cart2pol(X,Y); ! YEU<9
% idx = r<=1; %]@ K}!)2
% z = nan(size(X)); i*%2 e)
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Y&_1U/}h
% figure 7&9'=G
% pcolor(x,x,z), shading interp vH^^QI:em
% axis square, colorbar )SYZ*=ezl.
% title('Zernike function Z_5^1(r,\theta)') yD!V;?EnK
% 5-pz/%,
% Example 2: ]]Da/^K=Z
% 8M;G@ Q80
% % Display the first 10 Zernike functions DqHVc)9
% x = -1:0.01:1;
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% [X,Y] = meshgrid(x,x); kM\O2 ay
% [theta,r] = cart2pol(X,Y); >AW=N
% idx = r<=1; ~f2zMTI|
% z = nan(size(X)); :1wMGk
% n = [0 1 1 2 2 2 3 3 3 3]; +xlxhF
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; a 4}2^K
% Nplot = [4 10 12 16 18 20 22 24 26 28]; :/IcFU~)M
% y = zernfun(n,m,r(idx),theta(idx)); ft[g1
% figure('Units','normalized') +mT}}; -TS
% for k = 1:10 VBssn]w
% z(idx) = y(:,k); w%k )J{\
% subplot(4,7,Nplot(k)) Ga%]$4u
% pcolor(x,x,z), shading interp ${ ~U A 6
% set(gca,'XTick',[],'YTick',[]) 05MtQB
% axis square v|Tg %
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) G1~|$X@@
% end v[6 BESu
% HJ5m5':a
% See also ZERNPOL, ZERNFUN2. =D4EPfQn1
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% Paul Fricker 11/13/2006 E!VAA=
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% Check and prepare the inputs: 6EK+] 0
% ----------------------------- `CK;,>i
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Ai->,<I g]
error('zernfun:NMvectors','N and M must be vectors.') !nj%n
end e|Sg?ocR
hDlk! #*
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if length(n)~=length(m) /"CKVQ
error('zernfun:NMlength','N and M must be the same length.') L''0`a. +S
end : 6>H\
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D hZtiqL#_
n = n(:); HqXo;`Yy}
m = m(:); @IiT8B
if any(mod(n-m,2)) Msd!4 TrBJ
error('zernfun:NMmultiplesof2', ... m]-8?B1`Y
'All N and M must differ by multiples of 2 (including 0).') ( $S{td;
end CY>NU
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if any(m>n) AVyZ#`,
error('zernfun:MlessthanN', ... K%pmE?%,8
'Each M must be less than or equal to its corresponding N.') <,E*,&0W
end Q?a"uei[
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if any( r>1 | r<0 ) bf7 4 "
error('zernfun:Rlessthan1','All R must be between 0 and 1.') \4 j+pU
end D+SpSO7yg
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ,{Ab=xV
error('zernfun:RTHvector','R and THETA must be vectors.') \W}EyA
end +uLo~GdbE
i 52R, hz
ob a*w;
r = r(:); 6fV)8,F3
theta = theta(:); r/4]b ]n
length_r = length(r); GBphab|
if length_r~=length(theta) Z>,X$ Y6<
error('zernfun:RTHlength', ... z;/'OJ[.
'The number of R- and THETA-values must be equal.') .u*].As=
end zl:D|h77
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% Check normalization: o(` `7A@7a
% -------------------- @}?D<O8#"#
if nargin==5 && ischar(nflag) V^ {!d}
isnorm = strcmpi(nflag,'norm'); {6n \532@
if ~isnorm `e9uSF:9C
error('zernfun:normalization','Unrecognized normalization flag.') *h4m<\^U
end dI !/:x
else Qwa"AY 5pW
isnorm = false; [;=ky<K 0E
end -*lP1Nbp
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Yy)a,clZ*$
% Compute the Zernike Polynomials "? {yVu~9
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PbPP1G')
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9Bk}g50$#
% Determine the required powers of r: )A0&16<
% ----------------------------------- [~<',,tA0|
m_abs = abs(m); D%idlL2%J
rpowers = []; 9-Qtj49
for j = 1:length(n) _;q-+"6L;
rpowers = [rpowers m_abs(j):2:n(j)]; O|RO
j
end lDU:EJ&DHE
rpowers = unique(rpowers); 8 -5 jr_*
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% Pre-compute the values of r raised to the required powers, nh=Us^xD
% and compile them in a matrix: 'q'Y:A?,
% ----------------------------- pt v 4v[gQ
if rpowers(1)==0 'Xl>,\'6
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &{/>Sv!6#
rpowern = cat(2,rpowern{:}); H27Oq8
rpowern = [ones(length_r,1) rpowern]; OZ;E&IL
else Zax] i,Bx
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); =+h!JgY/L
rpowern = cat(2,rpowern{:}); S.)7u6/_!
end NoAb}1uae
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% Compute the values of the polynomials: ePRM v
% -------------------------------------- ba9<(0`
y = zeros(length_r,length(n)); &E &iaw!
for j = 1:length(n) U9o*6`"o
s = 0:(n(j)-m_abs(j))/2; m9 0R8 V
pows = n(j):-2:m_abs(j); eH!|MHe
for k = length(s):-1:1 6&QTVdK'O
p = (1-2*mod(s(k),2))* ... m=.7f9
prod(2:(n(j)-s(k)))/ ... q7 oR9
prod(2:s(k))/ ... .&x? `pER
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... :ZfUjqRE
prod(2:((n(j)+m_abs(j))/2-s(k))); cNr][AzU@
idx = (pows(k)==rpowers); pt cLJ]+)
y(:,j) = y(:,j) + p*rpowern(:,idx); :/[YY?pg-
end q uGPk)c
Z)O>h^0
if isnorm /-YlC (kL
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); <SM&VOiaOz
end uP =_-ZUW
end 9;Pu9s[q2
% END: Compute the Zernike Polynomials HjK<)q8b
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3:8nwt
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% Compute the Zernike functions: khN:+V|
% ------------------------------ ]6%%X+$7
idx_pos = m>0; `{|}LFS >
idx_neg = m<0; @oqi@&L'C
h NOYFH
x\bR j>%(
z = y; YTjuSV
if any(idx_pos) 9poEUjBI
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); v8vh~^X%P
end k *;{n8o?)
if any(idx_neg) h,'mN\6t
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); mf;^b.mKh
end ilRm}lU|x
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rI^ ~9Rz
% EOF zernfun N]s 7/s
