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, _r>kR7A\{
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, @km4qJZ
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? &4ndi=.#rg
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? q1v7(`O
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function z = zernfun(n,m,r,theta,nflag) BS(jC
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. cg_ " }]Y1
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N `@ny!S|1/
% and angular frequency M, evaluated at positions (R,THETA) on the oW^>J-
% unit circle. N is a vector of positive integers (including 0), and X ]W)D
S
% M is a vector with the same number of elements as N. Each element g#`}HuPoE
% k of M must be a positive integer, with possible values M(k) = -N(k) AN3oh1xe:
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, +*,!q7Gt
% and THETA is a vector of angles. R and THETA must have the same bg|dV
% length. The output Z is a matrix with one column for every (N,M) 41P0)o
% pair, and one row for every (R,THETA) pair. >'4$g7o,
% 6lT< l zT
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Jg)( F|>o
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), $0vWC#.A]
% with delta(m,0) the Kronecker delta, is chosen so that the integral %!eRR
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, g/}d> 6
% and theta=0 to theta=2*pi) is unity. For the non-normalized v|KIVBkbT
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. mG$N%`aG
% .)=*Yr M
% The Zernike functions are an orthogonal basis on the unit circle. \GQRpJ#h1
% They are used in disciplines such as astronomy, optics, and p3Ozfk
% optometry to describe functions on a circular domain. QUaV;6
4
% EV-sEl8ki
% The following table lists the first 15 Zernike functions. D+BiclJ
% w]nt_xj
% n m Zernike function Normalization }a#T\6rY
% -------------------------------------------------- 8:)[.
% 0 0 1 1 9HEqB0|ZRu
% 1 1 r * cos(theta) 2 _`gkYu3R+
% 1 -1 r * sin(theta) 2 bRrSd:e
% 2 -2 r^2 * cos(2*theta) sqrt(6) ({@"{
% 2 0 (2*r^2 - 1) sqrt(3) JZ+6)R
% 2 2 r^2 * sin(2*theta) sqrt(6) w>8kBQ?b
% 3 -3 r^3 * cos(3*theta) sqrt(8) v9FR
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 1zCu1'Wv
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 'n>44_7 L
% 3 3 r^3 * sin(3*theta) sqrt(8) 4f~sRubK
% 4 -4 r^4 * cos(4*theta) sqrt(10) EZ:?
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% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) pJs`/
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 8EMBqhl
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) IZm6.F
% 4 4 r^4 * sin(4*theta) sqrt(10) $_;rqTk]g
% -------------------------------------------------- U;IGV~oT
% l1|*(%p?X
% Example 1: *xmC`oP
% rk4KAX_[
% % Display the Zernike function Z(n=5,m=1) SvQ|SKE':
% x = -1:0.01:1; +H?g9v40
% [X,Y] = meshgrid(x,x); Z,SV9
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% [theta,r] = cart2pol(X,Y); !.^x^OK%y
% idx = r<=1; j`q>YPp
% z = nan(size(X)); 2wnk~URj
% z(idx) = zernfun(5,1,r(idx),theta(idx)); #d3_7rI0V
% figure QH4m7M@ni
% pcolor(x,x,z), shading interp *0Z6H-Do,
% axis square, colorbar SXYwhID=
% title('Zernike function Z_5^1(r,\theta)') 1LSJy*yY
% jnbR}a=fJ
% Example 2:
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% XR9kxTuk
% % Display the first 10 Zernike functions `?.6}*4@_A
% x = -1:0.01:1; X Db% -
% [X,Y] = meshgrid(x,x); -,YI>!
% [theta,r] = cart2pol(X,Y); 0TA8#c
% idx = r<=1; 1Az&BZU[
% z = nan(size(X)); & wtE"w
% n = [0 1 1 2 2 2 3 3 3 3]; m1jEky(
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; :RukW.MR
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 2;*G!rE&*`
% y = zernfun(n,m,r(idx),theta(idx)); re/u3\S
% figure('Units','normalized') A'7Y{oPHX
% for k = 1:10 p>\[[Md
% z(idx) = y(:,k); <*z'sUh+}
% subplot(4,7,Nplot(k)) -zMvpe-am&
% pcolor(x,x,z), shading interp u/wX7s
% set(gca,'XTick',[],'YTick',[]) a@&qdp
% axis square }Hg\
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% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) h$$JXf
% end x9\{a
% xi.?@Lff
% See also ZERNPOL, ZERNFUN2. U&:-Vf~&
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% Paul Fricker 11/13/2006 j DEym&-
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% Check and prepare the inputs: BR?DW~7J j
% ----------------------------- )'g4Ty
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +h/OQ]`/m
error('zernfun:NMvectors','N and M must be vectors.') p=eSJ*
end RrrlfF ms
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%
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if length(n)~=length(m) )BI%cD
error('zernfun:NMlength','N and M must be the same length.') IcQpbF0
end *P7n YjG
n} !')r
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n = n(:); !,Gavt7f
m = m(:); 2Hx*kh2
if any(mod(n-m,2)) QD^= ;!
error('zernfun:NMmultiplesof2', ... 5>CeFy
'All N and M must differ by multiples of 2 (including 0).') RT'5i$q[
end v,N!cp1
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if any(m>n) !@Vj&>mH$
error('zernfun:MlessthanN', ... ak3WER|f#
'Each M must be less than or equal to its corresponding N.') qkc,93B3
end S\sy^Kt~4:
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if any( r>1 | r<0 ) K)`R?CZ:s
error('zernfun:Rlessthan1','All R must be between 0 and 1.') .3Smqwm=Y
end Gv 8Z
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) I 6<LKI/
error('zernfun:RTHvector','R and THETA must be vectors.') #3?"#),q
end L:lnm9<
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r = r(:); 9S'\&mRl
theta = theta(:); Ly, ];
length_r = length(r); 4U)%JK.ta
if length_r~=length(theta) }c4F}Cy
error('zernfun:RTHlength', ... C5Fq%y{$.
'The number of R- and THETA-values must be equal.') 93w$ck},?G
end 4T&Jlu?:
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% Check normalization: sZqi)lo-s
% -------------------- \[+':o`LH
if nargin==5 && ischar(nflag) G8^b9xoA+.
isnorm = strcmpi(nflag,'norm'); :t+LuH g
if ~isnorm )0;O<G] d
error('zernfun:normalization','Unrecognized normalization flag.') flBJO.2
end lu1T+@t
else Ja\B%f
isnorm = false; {=R
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end X/Fip0i
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !a5e{QG0
% Compute the Zernike Polynomials #]} G{
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =`gFwH<
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% Determine the required powers of r: _1$+S0G;
% ----------------------------------- Qej<(:J5
m_abs = abs(m); OW> >6zM
rpowers = []; {`L,F
for j = 1:length(n) jJ_6_8#
rpowers = [rpowers m_abs(j):2:n(j)]; \nV oBW(
end .8|5;!`WB
rpowers = unique(rpowers); <("P5@cExU
,?GAFgK:
.M\0+,%/
% Pre-compute the values of r raised to the required powers, ,}Ic($To
% and compile them in a matrix: IifH=%2Y
% ----------------------------- R*O6Z"h
if rpowers(1)==0 <jVk}gi)Jp
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); o`ODz[04
rpowern = cat(2,rpowern{:}); JlH5 <:#PN
rpowern = [ones(length_r,1) rpowern]; -f(<2i
else jin?;v
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `jDmbD
+=
rpowern = cat(2,rpowern{:}); -32.g\]
end :4238J8
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% Compute the values of the polynomials: gi\UNT9x
% -------------------------------------- EmcwX4|
y = zeros(length_r,length(n)); zhwajc
for j = 1:length(n) X@B,w_b
s = 0:(n(j)-m_abs(j))/2; MWc{7,
pows = n(j):-2:m_abs(j); FEg&EYI
for k = length(s):-1:1 UC9w T
p = (1-2*mod(s(k),2))* ... 0`e- ;
prod(2:(n(j)-s(k)))/ ... ';x5 $5k'
prod(2:s(k))/ ... g\,HiKBXd
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... W3i X;-Z
prod(2:((n(j)+m_abs(j))/2-s(k))); \RTX fe-`
idx = (pows(k)==rpowers); N
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y(:,j) = y(:,j) + p*rpowern(:,idx); y
`w5u.'
end
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if isnorm gLv";"4S
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 3sGe#s%
end 4,R1}.?BzJ
end ^S`c-N
% END: Compute the Zernike Polynomials C[(Exe
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %OsV(7
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% Compute the Zernike functions: E|^~R}z)
% ------------------------------ I#hzU8Cc
idx_pos = m>0; ~4~>;e
idx_neg = m<0; mh`VZQ@
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z = y; =u`^QE
if any(idx_pos) Y3I+TI>x
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); -T-h~5
end ;zvg] %
if any(idx_neg) WAcQRa~C
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); M3dNG]3E
end G@QZmuj&KH
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% EOF zernfun R|AGN*.