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, Yn#8uaU
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, XMdc n,
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? a2 SQ:d
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? .( J/*H
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function z = zernfun(n,m,r,theta,nflag) LBE".+
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. $"i690
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N BNy"YK$
% and angular frequency M, evaluated at positions (R,THETA) on the saT9%?4-
% unit circle. N is a vector of positive integers (including 0), and n=&c5!
% M is a vector with the same number of elements as N. Each element r#Mx~Zg~
% k of M must be a positive integer, with possible values M(k) = -N(k) .$k"+E
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, l+6\U6_)B
% and THETA is a vector of angles. R and THETA must have the same ]/bE${W*]
% length. The output Z is a matrix with one column for every (N,M) 'l:2R,cP
% pair, and one row for every (R,THETA) pair. y#0w\/<
% g@2.A;N0
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike #SYWAcTkO}
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), lP
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% with delta(m,0) the Kronecker delta, is chosen so that the integral -1:Z^&e/
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, HFr3(gNj@
% and theta=0 to theta=2*pi) is unity. For the non-normalized [z~Nw#
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. V\ "5<>+O
% !.9vW&t
% The Zernike functions are an orthogonal basis on the unit circle.
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% They are used in disciplines such as astronomy, optics, and wb[(_@eZ
% optometry to describe functions on a circular domain. mc'p-orAf
% _Pkh`}W:
% The following table lists the first 15 Zernike functions. TJpv"V
%
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% n m Zernike function Normalization s]c$]&IGG
% -------------------------------------------------- @"8QG^q8de
% 0 0 1 1 50&F#v%YB
% 1 1 r * cos(theta) 2 { 9 ".o,
% 1 -1 r * sin(theta) 2 )0mDN.
% 2 -2 r^2 * cos(2*theta) sqrt(6) _w;+Jh
% 2 0 (2*r^2 - 1) sqrt(3) %B*dj9n^q
% 2 2 r^2 * sin(2*theta) sqrt(6) kDq%Y[6Z
% 3 -3 r^3 * cos(3*theta) sqrt(8) Aa>gN
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) K]8wW;N4
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) !h!9SE
% 3 3 r^3 * sin(3*theta) sqrt(8) 3.X0!M;x
% 4 -4 r^4 * cos(4*theta) sqrt(10) =on!&M
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Jt6J'MOq
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) LFyceFbm
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~ fEs!hl
% 4 4 r^4 * sin(4*theta) sqrt(10) h&bV!M
% -------------------------------------------------- V^I/nuy
% t3$gwO$
% Example 1: n-3j$x1Ne
% ATM:As:<@
% % Display the Zernike function Z(n=5,m=1) k_<{j0z.
% x = -1:0.01:1; [IFRwQ^%_O
% [X,Y] = meshgrid(x,x); HFuaoS+b*
% [theta,r] = cart2pol(X,Y); b',bi.FH
% idx = r<=1; vQmackY
% z = nan(size(X)); @z)tC@
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Tki/d\!+
% figure 1lyOp
% pcolor(x,x,z), shading interp :ZS8Zm"
% axis square, colorbar 7&V^BW
% title('Zernike function Z_5^1(r,\theta)') ^:DhHqvK
% Dh No +"!z
% Example 2: arS'th:j
% C'/M/|=Q#
% % Display the first 10 Zernike functions xg,]M/J
% x = -1:0.01:1; 6BU0hV
% [X,Y] = meshgrid(x,x); @:+n6
% [theta,r] = cart2pol(X,Y); 8UT%:DlxQ
% idx = r<=1; Xm:=jQn
% z = nan(size(X)); |sqo+E
% n = [0 1 1 2 2 2 3 3 3 3]; c48J!,jCd'
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; _$\5ZVe
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 8V|jL?a~
% y = zernfun(n,m,r(idx),theta(idx)); BX(d"z b<
% figure('Units','normalized') 8o7]XZE=)
% for k = 1:10 e=o{Zo?H=
% z(idx) = y(:,k); >'-w%H/
% subplot(4,7,Nplot(k)) >Ug?O~-
% pcolor(x,x,z), shading interp j%Z{.>mJ
% set(gca,'XTick',[],'YTick',[]) _8b]o~[Z+
% axis square XDdcq ]*|
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) PR@4' r|a
% end x)VIA]
% `)=A!x y
% See also ZERNPOL, ZERNFUN2. ?3,64[
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--W+=
% Paul Fricker 11/13/2006 r` `iC5Ii
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nN=:#4
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% Check and prepare the inputs: `}9 1S
% ----------------------------- >[XOMKgQ](
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) B}q
error('zernfun:NMvectors','N and M must be vectors.') +# RlX3P
end N=Uc=I7C
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if length(n)~=length(m) Jn[q<e"
error('zernfun:NMlength','N and M must be the same length.') Lk`k>Nn)
end !
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n = n(:); ,dZ#,<
m = m(:); nI*(a:
if any(mod(n-m,2)) n=G>y7b
error('zernfun:NMmultiplesof2', ... RUS7Z~5
'All N and M must differ by multiples of 2 (including 0).') 9xK4!~5V
end mI7rx`4H
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if any(m>n) ayH%
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error('zernfun:MlessthanN', ... mo|PrLV
'Each M must be less than or equal to its corresponding N.') EtR@sJ<
end xxLgC;>[
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if any( r>1 | r<0 ) ?GUz?'d
error('zernfun:Rlessthan1','All R must be between 0 and 1.') }RA3$%3
end Bbl)3$`,
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) HzD> -f
error('zernfun:RTHvector','R and THETA must be vectors.') `R=a@DQ
end 23}BW_m
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r = r(:); z"6ZDC6
theta = theta(:); {t844La"
length_r = length(r); RwAbIXG{0
if length_r~=length(theta) aCU7w5
error('zernfun:RTHlength', ... PPwxk;
'The number of R- and THETA-values must be equal.') y wW-p.
end 3x"@**(Q
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r@*=|0(OrK
% Check normalization: c&7Do}
% -------------------- `a9k!3_L
if nargin==5 && ischar(nflag) _`bS[%CJ
isnorm = strcmpi(nflag,'norm'); 1DEO3p
if ~isnorm `e'G.@
error('zernfun:normalization','Unrecognized normalization flag.') .sd B3x
end zW"~YaO%C
else I@3Q=14k%
isnorm = false; $ZQlIJZ
end OW+ e_im}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TK;\_yN
% Compute the Zernike Polynomials `pP9z;/Xq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _MM
r?64!VS;
Ey r5jXt%;
% Determine the required powers of r: d^KBIz8$5l
% ----------------------------------- !(kX~S
m_abs = abs(m); zc6Ho
rpowers = []; =ud`6{R
for j = 1:length(n) jA4PDH f+
rpowers = [rpowers m_abs(j):2:n(j)]; 7<h.KZPc
end u$WBc\j
rpowers = unique(rpowers); r#LnDseW
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x Z|&/Ci
% Pre-compute the values of r raised to the required powers, @4;HC=~
% and compile them in a matrix: 'n~fR]h}
% ----------------------------- |.1qy,|!X
if rpowers(1)==0 E9^(0\Z
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rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); e Wc_ N
rpowern = cat(2,rpowern{:}); E;9Z\?P
rpowern = [ones(length_r,1) rpowern]; jMK3T
else Hab!qWK`
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); hZ!oRWIU%G
rpowern = cat(2,rpowern{:}); ?sV[MsOsC
end S*4f%!
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% Compute the values of the polynomials: BRe{1i 6
% -------------------------------------- GA.BI"l
y = zeros(length_r,length(n)); T'hml
for j = 1:length(n) 5!<o-{J[(=
s = 0:(n(j)-m_abs(j))/2;
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pows = n(j):-2:m_abs(j); O'fk&&l
for k = length(s):-1:1 ;U
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p = (1-2*mod(s(k),2))* ... rxQ<4
prod(2:(n(j)-s(k)))/ ... ;vMn/
prod(2:s(k))/ ... 8GY.){d!l
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Ru:n~77{
prod(2:((n(j)+m_abs(j))/2-s(k))); (/'h4KS@
idx = (pows(k)==rpowers); :JR<SFjm
y(:,j) = y(:,j) + p*rpowern(:,idx); ~u!gUJ:
end &(g|="T
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if isnorm 7k `_#
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 3:UA<&=s
end UIn^_}jF`
end d^tVD`Fm
% END: Compute the Zernike Polynomials VQ2Fnb4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% oB4#J*
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% Compute the Zernike functions: _D-5}a"
% ------------------------------ D%A@lMru
idx_pos = m>0; d4J<,
idx_neg = m<0; zHV|-R
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z = y; E[UO5X
if any(idx_pos) mk\i}U>`
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); l2(.>-#
end _e* c
if any(idx_neg) *E}Oh
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 2hy NVG&$
end Yc
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% EOF zernfun Mm.Ql