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(S!xr_
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, \SkCsE#H
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? 8sOM%y9M
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ]d&6 ?7 !>
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function z = zernfun(n,m,r,theta,nflag) B8NOPbT
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. yk5-@qo
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Xhe2 5
% and angular frequency M, evaluated at positions (R,THETA) on the UxzZr%>s
% unit circle. N is a vector of positive integers (including 0), and 7z9gsi
% M is a vector with the same number of elements as N. Each element ^EdY:6NJ=A
% k of M must be a positive integer, with possible values M(k) = -N(k) -8X*(7
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, {aqceg
% and THETA is a vector of angles. R and THETA must have the same o
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% length. The output Z is a matrix with one column for every (N,M) 44B)=p7
% pair, and one row for every (R,THETA) pair. V7.xKmB
% / Li?;H
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike }A'QXtI/G
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Y-hGHnh]'
% with delta(m,0) the Kronecker delta, is chosen so that the integral '9>z4G*Td
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, f7mP4[+dS
% and theta=0 to theta=2*pi) is unity. For the non-normalized sNZ{OD+
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. @5K/z<p%
% js/N qf2>
% The Zernike functions are an orthogonal basis on the unit circle. W2J"W=:z
% They are used in disciplines such as astronomy, optics, and BY.'0,H=k
% optometry to describe functions on a circular domain. yeqZPzn
% rIR~YMv!
% The following table lists the first 15 Zernike functions. 7 [N1Vr(1
% \74+ cN
% n m Zernike function Normalization /\"=egB9
% -------------------------------------------------- _"6{Rb53v=
% 0 0 1 1 6":=p:PT.
% 1 1 r * cos(theta) 2 );$_|]#
% 1 -1 r * sin(theta) 2 SsiAyQ|Ma
% 2 -2 r^2 * cos(2*theta) sqrt(6) BFc=GiPnQ
% 2 0 (2*r^2 - 1) sqrt(3) c7.%Bn,
% 2 2 r^2 * sin(2*theta) sqrt(6) _ #288`bU
% 3 -3 r^3 * cos(3*theta) sqrt(8) D'2&'7-sm\
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Rm`_0}5
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) WDNuR#J?
% 3 3 r^3 * sin(3*theta) sqrt(8) `6koQZm
% 4 -4 r^4 * cos(4*theta) sqrt(10) %:yJ/&-Q,Z
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ZNNgi@6>
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) /MKcS%/H/
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) avrf]raM|
% 4 4 r^4 * sin(4*theta) sqrt(10) QL%&b\K
% -------------------------------------------------- 3Z;`n,g
% 7'uuc]\5>
% Example 1: yPL1(i;
% |fkz=*rn
% % Display the Zernike function Z(n=5,m=1) ?(UeWLC#
% x = -1:0.01:1; eD5.*O
% [X,Y] = meshgrid(x,x); me"}1REa
% [theta,r] = cart2pol(X,Y); Z_Ffiw(p
% idx = r<=1; Sa7bl~p\
% z = nan(size(X)); YYwFjA@
% z(idx) = zernfun(5,1,r(idx),theta(idx)); T!u&r
% figure :^]rjy/|+
% pcolor(x,x,z), shading interp ~fbFA?g3
% axis square, colorbar XgE\q
% title('Zernike function Z_5^1(r,\theta)') {3cT\u
% YMx]i,u'+
% Example 2: ~{lSc/SP|
% IIcG+zwx
% % Display the first 10 Zernike functions :23w[vt=
% x = -1:0.01:1; -,+zA.{+W
% [X,Y] = meshgrid(x,x); sw
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% [theta,r] = cart2pol(X,Y); #m[R1G#
% idx = r<=1; yXyL,R
% z = nan(size(X)); NN\>(
=
% n = [0 1 1 2 2 2 3 3 3 3];
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% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 1'ts>6b
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 3BHPD;U
% y = zernfun(n,m,r(idx),theta(idx)); OOJg%y*H
% figure('Units','normalized') y}Ji( q~
% for k = 1:10 8>Az<EF^=#
% z(idx) = y(:,k); "@uKe8r|y
% subplot(4,7,Nplot(k)) KG7 ~)g
% pcolor(x,x,z), shading interp ObJgJr
% set(gca,'XTick',[],'YTick',[]) r$<-2lW
% axis square &p|+K
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% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) L[;U
Z)V@
% end gD`|N@W$5
% OI:G~Wg
% See also ZERNPOL, ZERNFUN2. #pDWwnP[rt
IL*Ghq{/
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% Paul Fricker 11/13/2006 2:b3+{\f
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lsJl+%&8
Z',Z7QW7
/Wos{}Z0
% Check and prepare the inputs: dQW=k^X 'U
% ----------------------------- C{Y0}ZrmlF
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) E<6Fjy
error('zernfun:NMvectors','N and M must be vectors.') v0psth?qV
end ktE~)G
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c;
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if length(n)~=length(m) YO&=fd*
error('zernfun:NMlength','N and M must be the same length.') l;F\s&^
end Fl8*dXG&
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n = n(:); jd>ug=~x
m = m(:); ,v<GSiO
if any(mod(n-m,2)) ,_+Gb
error('zernfun:NMmultiplesof2', ... ~O|g~H5;
'All N and M must differ by multiples of 2 (including 0).') jTSN`R9@
end P_7QZ0k/
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if any(m>n) 'c]Fhe fb
error('zernfun:MlessthanN', ... 4\?z^^
'Each M must be less than or equal to its corresponding N.')
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end >]/RlW[
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if any( r>1 | r<0 ) Z|t`}lK
error('zernfun:Rlessthan1','All R must be between 0 and 1.') @la/sd4`
end ,1|Qm8O
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$%:=;1Jl
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ab-z 7g
error('zernfun:RTHvector','R and THETA must be vectors.') %?sPKOh3N}
end ;*J_V/&?
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:ebu8H9f%
r = r(:); !4Oj^yy%
theta = theta(:); r(qwzUI
length_r = length(r); qpt},yn)C
if length_r~=length(theta) ;#)vw;XR
error('zernfun:RTHlength', ... ":I@>t{H*
'The number of R- and THETA-values must be equal.') s@$SM,tnn
end %tK^&rw%
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% Check normalization: )(/Bw&$
% -------------------- /s~(? =qYH
if nargin==5 && ischar(nflag) +<})`(8
isnorm = strcmpi(nflag,'norm'); ._X|Ye9/
if ~isnorm !_P-?u
error('zernfun:normalization','Unrecognized normalization flag.') >?L)+*^
end 7QXp\<7
else Zws[C
isnorm = false; IE*5p6IM~
end l_lK,=cLj+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Y2!P!u+Q
% Compute the Zernike Polynomials \D5_g8m:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `Q1;Y
%E\ pd@
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% Determine the required powers of r: j>I.d+
% ----------------------------------- p|`[8uY?
m_abs = abs(m); Io*mFa?
rpowers = []; =XhxD<kI
for j = 1:length(n) S-7ryHH*0
rpowers = [rpowers m_abs(j):2:n(j)]; ETQL,t9m
end .L=C7 w1
rpowers = unique(rpowers); {P7 I<^,
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CBu$8]9=
% Pre-compute the values of r raised to the required powers, CubBD+hl*
% and compile them in a matrix: .a_xQ]eQ
% ----------------------------- p5V.O20
if rpowers(1)==0 6DxT(VU}
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); IAFj_VWC0
rpowern = cat(2,rpowern{:}); +01bjM6F_1
rpowern = [ones(length_r,1) rpowern]; 2tMa4L%@C
else W5U;{5
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); f1wwx|b%.
rpowern = cat(2,rpowern{:}); V }wh
end @"vTz8oY@
m^%Xl@V:c-
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% Compute the values of the polynomials: q +*>T=k
% -------------------------------------- rXF=/
y = zeros(length_r,length(n)); cS;O]>/5
for j = 1:length(n) Dy|DQ> ?}
s = 0:(n(j)-m_abs(j))/2; ZK?:w^Z
pows = n(j):-2:m_abs(j); <=gf|(
for k = length(s):-1:1 ]%<0V,G
q
p = (1-2*mod(s(k),2))* ... FX&)~)
prod(2:(n(j)-s(k)))/ ... E[8i$
prod(2:s(k))/ ... qYbPF|Y=Z
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &?0hj@kd~
prod(2:((n(j)+m_abs(j))/2-s(k))); c]3^2Ag,
idx = (pows(k)==rpowers); f'&
y(:,j) = y(:,j) + p*rpowern(:,idx); rT!9{uK
end 8
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0$I!\y\
if isnorm D]zpG
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); W<OO:B.ty
end c
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end tRzo}_+N
% END: Compute the Zernike Polynomials 5imqZw
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a4D4*=!G0
^#,cWG}z
:}[[G2|9
% Compute the Zernike functions: ~\~XD+jy"
% ------------------------------ %q5iy0~P
idx_pos = m>0; S$%Y{
idx_neg = m<0; 5:x .<
t.]c44RY
90]{4 ]y;
z = y; 7).zed^
if any(idx_pos) !#Hca
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); R:FyCT_,
end n$YCIW)0
if any(idx_neg) J6*B=PX=(
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); _.ELN/$-
end ]J6+nA6)
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a FrVP
% EOF zernfun C@q&0\HN