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, qRGb3l
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, /[IQ:':^
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? EH "g`r
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? xA5$!Oq7
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function z = zernfun(n,m,r,theta,nflag) 3ElpS^2W
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Mqtp}<*@-
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N jgo@~,5R
% and angular frequency M, evaluated at positions (R,THETA) on the 1;gSf.naG
% unit circle. N is a vector of positive integers (including 0), and #Fd([Zx#.
% M is a vector with the same number of elements as N. Each element Z=c@Gd
% k of M must be a positive integer, with possible values M(k) = -N(k) QPcB_wUqu
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, td&l T(7
% and THETA is a vector of angles. R and THETA must have the same D)sEAfvX
% length. The output Z is a matrix with one column for every (N,M)
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% pair, and one row for every (R,THETA) pair. _z 5CplO
% ed*AU,^@v
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike e,*[5xQ
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), /a|NGh%
% with delta(m,0) the Kronecker delta, is chosen so that the integral c6m,oS^
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Xh/av[Q
% and theta=0 to theta=2*pi) is unity. For the non-normalized fx-*')
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ">9CN$]J
% `j![
% The Zernike functions are an orthogonal basis on the unit circle. MX0B$yc$
% They are used in disciplines such as astronomy, optics, and 7:<Ed"rdE
% optometry to describe functions on a circular domain. k9xKaJ%1
% @#tSx
% The following table lists the first 15 Zernike functions. 6 {Z\cwP)c
% !gf3%!%
% n m Zernike function Normalization //@=Q!MW
% -------------------------------------------------- ,AM-cwwT:u
% 0 0 1 1 0cUt"(]
% 1 1 r * cos(theta) 2 xH[yIfHkG@
% 1 -1 r * sin(theta) 2 OJ 5 !+#>
% 2 -2 r^2 * cos(2*theta) sqrt(6) ?$c
% 2 0 (2*r^2 - 1) sqrt(3) <Y2!c,"
% 2 2 r^2 * sin(2*theta) sqrt(6) *~uuCLv_
% 3 -3 r^3 * cos(3*theta) sqrt(8) z0[ZO1Fo(
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Z5[:Zf?h7J
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) [;AcV73
% 3 3 r^3 * sin(3*theta) sqrt(8) F8Wq&X#r
% 4 -4 r^4 * cos(4*theta) sqrt(10) Ohag%<1#
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ig KAD#2a
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) }2,#[mM
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?|GxVOl
% 4 4 r^4 * sin(4*theta) sqrt(10) [9_ (+E[}
% -------------------------------------------------- 8|NJ(D-$
%
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% Example 1: -(}1o9e\7
% G9inNz*Cx
% % Display the Zernike function Z(n=5,m=1) p'k+0=
% x = -1:0.01:1; V9_HC f
% [X,Y] = meshgrid(x,x); A_~5|
% [theta,r] = cart2pol(X,Y); o~&!M_ED
% idx = r<=1; am+mXb
% z = nan(size(X)); XSjelA?
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ~J{[]wi
% figure a3O_#l-Z
% pcolor(x,x,z), shading interp ja- ~`
% axis square, colorbar AuipK*&g
% title('Zernike function Z_5^1(r,\theta)') zxUj1
% y?#J`o-
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% Example 2: {dXBXC/Ju
% GPLt<K!<#
% % Display the first 10 Zernike functions ~"2@A
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% x = -1:0.01:1; !o':\hex6
% [X,Y] = meshgrid(x,x); zn1Rou]6
% [theta,r] = cart2pol(X,Y); (<ZkmIXN
% idx = r<=1; r Ob"S*
% z = nan(size(X)); s[@>uP
% n = [0 1 1 2 2 2 3 3 3 3]; .4FcZJvy
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; rE `}?d
% Nplot = [4 10 12 16 18 20 22 24 26 28]; )2Ru!l#
% y = zernfun(n,m,r(idx),theta(idx)); l)*,18n
% figure('Units','normalized') qK
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% for k = 1:10 2RUR=%C
% z(idx) = y(:,k); yUmsE-W
% subplot(4,7,Nplot(k)) {V%O4/
% pcolor(x,x,z), shading interp ZWRRh^
% set(gca,'XTick',[],'YTick',[]) D#Yx,`Ui
% axis square EQ63VF
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) "Lq|66
% end )]c3bMVE-
% ]_: TrH
% See also ZERNPOL, ZERNFUN2. _<RR`
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% Paul Fricker 11/13/2006 J7wQ=!g
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% Check and prepare the inputs: -jtC>_/
% ----------------------------- wYO"znd
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) m_!vIUOz
error('zernfun:NMvectors','N and M must be vectors.') k3>ur>aW
end v<3o[m q
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if length(n)~=length(m) zP(=,)d
error('zernfun:NMlength','N and M must be the same length.') LX\*4[0%K
end s'aV q B
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n = n(:); 0YpiHoM
m = m(:); nz(q)"A
if any(mod(n-m,2)) ^/C$L8#
error('zernfun:NMmultiplesof2', ... CI!Eq&D,
'All N and M must differ by multiples of 2 (including 0).') v=.z|QD^1
end }x?H ~QQT
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if any(m>n) MO~~=]Y'
error('zernfun:MlessthanN', ... 12tJrS*Z
'Each M must be less than or equal to its corresponding N.') ewAH'H]o
end JU'WiR
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if any( r>1 | r<0 ) i,'Ka[6
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ^s2m\Q(
end t$H':l0
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) m+Q5vkW
error('zernfun:RTHvector','R and THETA must be vectors.') Foe>}6~{?
end 5"kx}f2$
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r = r(:); `&u<aLA
theta = theta(:); ,l$NJt
length_r = length(r); lk[G;=K:.
if length_r~=length(theta) !_U37Uj<m
error('zernfun:RTHlength', ... :T7?
'The number of R- and THETA-values must be equal.') >,>;)B@J
end Gpdv]SON{
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% Check normalization: C{UF~
% -------------------- 0~+NB-L}
if nargin==5 && ischar(nflag) ShWHHU(QQ
isnorm = strcmpi(nflag,'norm'); selP=Q!
if ~isnorm 8ji^d1G,
error('zernfun:normalization','Unrecognized normalization flag.') 8"km_[JE e
end (ve+,H6w\
else y Y>-MoF/t
isnorm = false; 83KfM!w
end a[1sA12
w0 Fwd
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %bIsrQ~B
% Compute the Zernike Polynomials Y&vHOA
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y)3~]h\a
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% Determine the required powers of r: L!;^#g
% ----------------------------------- R9tckRG#
m_abs = abs(m); 0LWdJ($?
rpowers = []; ycgfZ 3K
for j = 1:length(n) 1@A7h$1P
rpowers = [rpowers m_abs(j):2:n(j)]; gB]C&Q
end ==]Z \jk
rpowers = unique(rpowers); 'FShNY5
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% Pre-compute the values of r raised to the required powers, {UB%(E[Mr
% and compile them in a matrix: a(8>n
Z,V
% ----------------------------- C
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if rpowers(1)==0 EfKM*;A
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); IWAj Mwo
rpowern = cat(2,rpowern{:}); 89zuL18V
rpowern = [ones(length_r,1) rpowern]; ^DBD63N"
else q}>M& *
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); |/@0~O(6
rpowern = cat(2,rpowern{:}); sf Dg/ a
end C@%iQ]=
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% Compute the values of the polynomials: Q> kiVvc
% -------------------------------------- qh%i5Mu
y = zeros(length_r,length(n)); hzaU8kb
for j = 1:length(n) F?7u~b|@{
s = 0:(n(j)-m_abs(j))/2; P,(9cyS{
pows = n(j):-2:m_abs(j); %fHH{60
for k = length(s):-1:1 !0`lu_ZN
p = (1-2*mod(s(k),2))* ... GF&_~48GD
prod(2:(n(j)-s(k)))/ ... SijtTY#r
prod(2:s(k))/ ... mv{<'
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... $h,d?
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prod(2:((n(j)+m_abs(j))/2-s(k))); neF8V"-u&
idx = (pows(k)==rpowers); c8T/4hU
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y(:,j) = y(:,j) + p*rpowern(:,idx); $u:<x
end O{~KR/
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if isnorm pg+b[7
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); \H^;'agA
end $Jc q7E~
end \fTTkpM
% END: Compute the Zernike Polynomials 6VC-KY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /w0sj`;"
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% Compute the Zernike functions: }7%9}2}Iw
% ------------------------------ >E,Q
idx_pos = m>0; f_rp<R>Uu
idx_neg = m<0; Hoj8okP
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z = y; M;qBDT~)
if any(idx_pos) K!p,x;YX
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ^_sQG
end NddO*`8+)
if any(idx_neg) Y17hOKc`
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); "TfI+QgLF
end _C20 +PMO
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% EOF zernfun e gI&epN