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, _y .]3JNm
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, 2q}..
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? bzi|s5!'<
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? @U -$dw'4
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function z = zernfun(n,m,r,theta,nflag) e~v(eK_
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. g<\z= H
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N + E"[
% and angular frequency M, evaluated at positions (R,THETA) on the ;HOPABWz)
% unit circle. N is a vector of positive integers (including 0), and jw6Tj;c
% M is a vector with the same number of elements as N. Each element y|_Eu:
% k of M must be a positive integer, with possible values M(k) = -N(k) *@ED}Mj+
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 1@XgTL4
% and THETA is a vector of angles. R and THETA must have the same
+f4W"t
% length. The output Z is a matrix with one column for every (N,M) pJ,@Y>
% pair, and one row for every (R,THETA) pair. \Btk;ivg
% 6Gn4asoA
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike V:bV ?lt
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), TOI4?D]
% with delta(m,0) the Kronecker delta, is chosen so that the integral P"7ow-
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, g dj^df+2F
% and theta=0 to theta=2*pi) is unity. For the non-normalized e<gx~N9l'
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 'PdmI<eXQ
% GO5 ~!g
% The Zernike functions are an orthogonal basis on the unit circle. m(sXk}e;1
% They are used in disciplines such as astronomy, optics, and BQ05`nkF
% optometry to describe functions on a circular domain. ,yLw$-
% O2-M1sd$
% The following table lists the first 15 Zernike functions. )WR_
ug
% EY>8O+
% n m Zernike function Normalization 9-jO,l
% -------------------------------------------------- 'b:Ne,<
% 0 0 1 1 igDyp0t
% 1 1 r * cos(theta) 2 p*;Qz
% 1 -1 r * sin(theta) 2 %6 =\5>
% 2 -2 r^2 * cos(2*theta) sqrt(6) Gg0#H^s( (
% 2 0 (2*r^2 - 1) sqrt(3) `hB1b["(
% 2 2 r^2 * sin(2*theta) sqrt(6) >R,?hWT
% 3 -3 r^3 * cos(3*theta) sqrt(8) YT2'!R
1
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) VTe.M[:
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) _py2kjA6
% 3 3 r^3 * sin(3*theta) sqrt(8) heD,&OX
% 4 -4 r^4 * cos(4*theta) sqrt(10) 0|)19LR
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) DOm-)zl{|x
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) r!/0 j)
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) WO%h"'iJ
% 4 4 r^4 * sin(4*theta) sqrt(10) !eD+GDgE]
% -------------------------------------------------- Nh)[rx
% w;`m- 9<Y
% Example 1: hH+bt!aH
% q/6UK =
% % Display the Zernike function Z(n=5,m=1) @Y'I,e
% x = -1:0.01:1; m7 XjP2
% [X,Y] = meshgrid(x,x); = hX[
% [theta,r] = cart2pol(X,Y); ~mILA->F
% idx = r<=1; L]zNf71RD
% z = nan(size(X)); oK-!(1A-
% z(idx) = zernfun(5,1,r(idx),theta(idx)); j|'R$|
% figure q9}2
% pcolor(x,x,z), shading interp -gKpL\
% axis square, colorbar B7"Fp
% title('Zernike function Z_5^1(r,\theta)') \K`jCsT
% l`rC0kJ]
% Example 2: 8&a_A:h
% *PB/iVH%6
% % Display the first 10 Zernike functions =l|>.\-
% x = -1:0.01:1; R+.
N n
% [X,Y] = meshgrid(x,x); 5t'Fv<g
% [theta,r] = cart2pol(X,Y); <%,'$^'DS
% idx = r<=1; lYQtv=q
% z = nan(size(X)); x1DVD!0 ~{
% n = [0 1 1 2 2 2 3 3 3 3]; ~u/@rqF
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; H%.zXQ4}n
% Nplot = [4 10 12 16 18 20 22 24 26 28]; TU%"jb5
% y = zernfun(n,m,r(idx),theta(idx)); @P70W<<
% figure('Units','normalized') (UW6F4:$
% for k = 1:10 U1^l+G^,~
% z(idx) = y(:,k); w#{l4{X|
% subplot(4,7,Nplot(k)) :,C%01bH|l
% pcolor(x,x,z), shading interp ze"~Ird
% set(gca,'XTick',[],'YTick',[]) i]M"Cu*
% axis square -lp"#^ ;
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 5^|"_Q#:
% end U?6yke
% g3a/;wl
% See also ZERNPOL, ZERNFUN2. !"(u_dFw
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% Paul Fricker 11/13/2006 J8h7e}n?
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0 0,9azs
% Check and prepare the inputs: 5vGioO
% ----------------------------- =L16hDk o
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) foyB{6q8
error('zernfun:NMvectors','N and M must be vectors.') A5+5J_)*
end DrFu r(=T
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if length(n)~=length(m) afy/K'~
error('zernfun:NMlength','N and M must be the same length.') E.#6;HHzN
end ^+a
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u7(];
n = n(:); Okoo(dfM
m = m(:); tWRf'n[+]
if any(mod(n-m,2)) ioWJj.%
error('zernfun:NMmultiplesof2', ... #'g^Za
'All N and M must differ by multiples of 2 (including 0).') Z*h ;e;
end .S6ji~;r
y;,y"W
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if any(m>n) aYBTrOd z
error('zernfun:MlessthanN', ... skK*OO2-
'Each M must be less than or equal to its corresponding N.') NJ>,'s
end CnQg *+
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if any( r>1 | r<0 ) Dx0O'uwR
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 9IOGc}
end =1Ri]b
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) y,^";7U
error('zernfun:RTHvector','R and THETA must be vectors.') /+N|X
end BMY>a
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r = r(:); pIvfmIm
theta = theta(:); {Wa~}1`Kl
length_r = length(r); L2d:.&5
if length_r~=length(theta) 6#O#T;f)
error('zernfun:RTHlength', ... )ib7K1GJ
'The number of R- and THETA-values must be equal.') O%prD}x
end
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=g >.X9lr
% Check normalization: 5^b i
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% -------------------- e& p_f<
if nargin==5 && ischar(nflag) CJm.K
isnorm = strcmpi(nflag,'norm'); / =-6:L
if ~isnorm wLpkUa
error('zernfun:normalization','Unrecognized normalization flag.') p %L1uwLG
end .<HC[ls
else #n=A)#'my
isnorm = false; pFEZDf}:
end YsZ{1W
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3;y_qwA
% Compute the Zernike Polynomials TR~|c|B
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% z;[gEA+I
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% Determine the required powers of r: is9}ePC7Xu
% ----------------------------------- =l_rAj~I|
m_abs = abs(m); Z^{+,$H@
rpowers = []; IKGTsA;
for j = 1:length(n) "/Om}*VhD
rpowers = [rpowers m_abs(j):2:n(j)]; AfUZO^<
end &{ DR6
rpowers = unique(rpowers); \Bt=bu>Z
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.N/GfR`0/<
% Pre-compute the values of r raised to the required powers, ax4*xxU
% and compile them in a matrix: sfyBw
% ----------------------------- 3R'.}^RN
if rpowers(1)==0 l6V%"Lo/)
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ] xb]8]
rpowern = cat(2,rpowern{:}); vc )9Re$
rpowern = [ones(length_r,1) rpowern]; &S<?07Z
else qC\]"Z`m
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 2H[=lY
rpowern = cat(2,rpowern{:}); +mivqR~{{
end ^eT@!N
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% Compute the values of the polynomials: r|l53I5
% -------------------------------------- tp#Z@5=
y = zeros(length_r,length(n)); RV(
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for j = 1:length(n) ]Mn&76fu
s = 0:(n(j)-m_abs(j))/2; y*}AX%8`e~
pows = n(j):-2:m_abs(j); cT_uJbP+
for k = length(s):-1:1 mr@_%U
p = (1-2*mod(s(k),2))* ... {-o7w0d_
prod(2:(n(j)-s(k)))/ ... TG4\%S$w
prod(2:s(k))/ ... >sn"
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... #D=
tX
prod(2:((n(j)+m_abs(j))/2-s(k))); hK:#+hg,
idx = (pows(k)==rpowers); +xn&K"]:3
y(:,j) = y(:,j) + p*rpowern(:,idx); Jz=;mrW
end Y=5!QLV4
BO8%:/37[4
if isnorm M_qP!+Y
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); =]!8:I?C<
end xR0~S
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end }/_('q@s\
% END: Compute the Zernike Polynomials {'h)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6].yRNy"
%|#P&`
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% Compute the Zernike functions: PZKbnu
% ------------------------------ <dq,y>
idx_pos = m>0; WA<H
idx_neg = m<0; +A'}PXm*tu
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bs:C1j\&
z = y; <FXQxM5"
if any(idx_pos) FI3sLA
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); }W - K
end Z|]l"W*w
if any(idx_neg) F;cI0kP=>
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Iu)L3_+
end (jp1; #P!
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% EOF zernfun Hi9 G^Q