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, {z":hmt
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, |$$gj[+^
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? z&a%_
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那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? M\%LB}4M
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function z = zernfun(n,m,r,theta,nflag) K6vF}A|
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Kk?]z7s-4
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Z0fl]3p
% and angular frequency M, evaluated at positions (R,THETA) on the M$|r8%z1
% unit circle. N is a vector of positive integers (including 0), and ql
Z()
% M is a vector with the same number of elements as N. Each element a'sa{>
% k of M must be a positive integer, with possible values M(k) = -N(k) nveHLHvC7
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, #2N']VP
% and THETA is a vector of angles. R and THETA must have the same mFL"h
% length. The output Z is a matrix with one column for every (N,M) ('SA9JG
% pair, and one row for every (R,THETA) pair. f7y a0%N
% :o!bz>T
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike '|v??`o#
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), >Ln/ )j
% with delta(m,0) the Kronecker delta, is chosen so that the integral VBHDI{HzRv
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, B,`B!rU
% and theta=0 to theta=2*pi) is unity. For the non-normalized g>])O
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. FlWgTn>
% RbexsBq
% The Zernike functions are an orthogonal basis on the unit circle. 5C03)Go3Z
% They are used in disciplines such as astronomy, optics, and :n1^Xw0q
% optometry to describe functions on a circular domain. LyEM^d]
% q7itznQSKc
% The following table lists the first 15 Zernike functions. zF+NS]XK
% ]p,svevo
% n m Zernike function Normalization C26vH#C
% -------------------------------------------------- <"Ox)XG3]W
% 0 0 1 1 }Mh@%2$
% 1 1 r * cos(theta) 2 mM6g-)cV
% 1 -1 r * sin(theta) 2 (}$pf6s
% 2 -2 r^2 * cos(2*theta) sqrt(6) *2K/)(
% 2 0 (2*r^2 - 1) sqrt(3) 7=$@bHEF#*
% 2 2 r^2 * sin(2*theta) sqrt(6) ~Ibq,9i
% 3 -3 r^3 * cos(3*theta) sqrt(8) RyI(6TZl
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) s7x&x;-
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) hJuR,NP
% 3 3 r^3 * sin(3*theta) sqrt(8) i{#5=np H
% 4 -4 r^4 * cos(4*theta) sqrt(10) u@(z(P
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) i_ha^mq3
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) =dVPx<l5
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6 WD(
% 4 4 r^4 * sin(4*theta) sqrt(10) 7~gIOu
% -------------------------------------------------- zv1#PfO@)
% '}\#bMeObg
% Example 1: Z*9Qeu-N:
% "OIra2O
% % Display the Zernike function Z(n=5,m=1) yvPcD5s5
% x = -1:0.01:1;
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% [X,Y] = meshgrid(x,x); h92'~X36
% [theta,r] = cart2pol(X,Y); C\~!2cy
% idx = r<=1; YQ\c0XG
% z = nan(size(X)); v/W\k.?q/
% z(idx) = zernfun(5,1,r(idx),theta(idx)); L7~9u|7a#
% figure [8- . T4
% pcolor(x,x,z), shading interp
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% axis square, colorbar #FAy
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% title('Zernike function Z_5^1(r,\theta)') xWZ87
% <Cbah%X
% Example 2: Hr?_`:
% Dz<"eyB\
% % Display the first 10 Zernike functions bO)voJ<
% x = -1:0.01:1; K,ccM[hu|
% [X,Y] = meshgrid(x,x); j_3X
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% [theta,r] = cart2pol(X,Y); y:C=Ni&,"
% idx = r<=1; {LwV&u(
% z = nan(size(X));
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% n = [0 1 1 2 2 2 3 3 3 3]; NuL.l__W
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 3RwDIk?>%
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ,*y\b|<j
% y = zernfun(n,m,r(idx),theta(idx)); 676r0`
% figure('Units','normalized') RDX$Wy$@L
% for k = 1:10 td}%reH
% z(idx) = y(:,k); _LVi}mM
% subplot(4,7,Nplot(k)) TzPG(f
% pcolor(x,x,z), shading interp NCid`a$
% set(gca,'XTick',[],'YTick',[]) Ng1{NI+S
% axis square @X$~{Vp__
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) !foiGZ3g
% end Hp#IOsP~
% +>w %j&B
% See also ZERNPOL, ZERNFUN2. i4Ps#R_wx
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% Paul Fricker 11/13/2006 ji8)/
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% Check and prepare the inputs: ^]{)gk8P~2
% ----------------------------- sQIzcnKB
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) \&KfIh8
error('zernfun:NMvectors','N and M must be vectors.') bhqV2y*'
end \$Jz26
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if length(n)~=length(m) B1u.aa$
error('zernfun:NMlength','N and M must be the same length.') SU8vz/\%y
end ?y1G,0,
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n = n(:); @xqjAcfg
m = m(:); `A\|qH5`W
if any(mod(n-m,2)) t XbMP
error('zernfun:NMmultiplesof2', ... 7uI~Xo?N
'All N and M must differ by multiples of 2 (including 0).') gq:2`W&5
end ^U5g7Emf
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if any(m>n) V\2&?#GZ
error('zernfun:MlessthanN', ... 3|Vh[iAa\
'Each M must be less than or equal to its corresponding N.') }O7!>T
end P
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if any( r>1 | r<0 ) ,cF
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error('zernfun:Rlessthan1','All R must be between 0 and 1.') gf]k@-)
end Z~s"=kF,
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) m})EYs1
error('zernfun:RTHvector','R and THETA must be vectors.') 1dE|q{
end k~|5TO
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r = r(:); 7hq$vI%0
theta = theta(:); iN]#XIQ%
length_r = length(r); $I$ B8
if length_r~=length(theta) '|jN!y^2p
error('zernfun:RTHlength', ... :'+- %xUM
'The number of R- and THETA-values must be equal.') =0" Zse,
end Y`tv"v2
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% Check normalization: XW2{I.:in>
% -------------------- ~d)2>A2:
if nargin==5 && ischar(nflag) 9NPOdt:@
isnorm = strcmpi(nflag,'norm'); m0$~O5|4
if ~isnorm g"P!KPrf1p
error('zernfun:normalization','Unrecognized normalization flag.') V9SkB3-'
end zF-M9f$_PY
else F8T.}qI
isnorm = false; qz]g4hS
end e ab_"W
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~4t7Q
% Compute the Zernike Polynomials Rv^
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #1#?k
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% Determine the required powers of r: A]WR-0Z7
% ----------------------------------- u&7c2|Q
m_abs = abs(m); KgCQ4w9
rpowers = []; {Bd 0
for j = 1:length(n) PRpW*#"EI
rpowers = [rpowers m_abs(j):2:n(j)]; m~xO;_m
end ]u(EEsG/
rpowers = unique(rpowers); y G{;kJ P
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% Pre-compute the values of r raised to the required powers, sm1(I7y
% and compile them in a matrix: J-3%.fX,
% ----------------------------- >kN%R8*Sx
if rpowers(1)==0 Qjl.O HO
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); _w\A=6=q|
rpowern = cat(2,rpowern{:}); k5X& |L/
rpowern = [ones(length_r,1) rpowern]; D)my@W0,
else { :~D
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 5[\LQtM
rpowern = cat(2,rpowern{:}); h,u?3}Knnb
end {:!CA/0Jx
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% Compute the values of the polynomials: Ki_8g
% -------------------------------------- 6k%Lc4W
y = zeros(length_r,length(n)); re-;s
for j = 1:length(n) pk&;5|cCD
s = 0:(n(j)-m_abs(j))/2; 1p%75VW
pows = n(j):-2:m_abs(j); &!=[.1H<
for k = length(s):-1:1 /GQN34RD
p = (1-2*mod(s(k),2))* ... &|8R4l C|
prod(2:(n(j)-s(k)))/ ... 6_#:LFke
prod(2:s(k))/ ... pMy];9SvW
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... (uk-c~T!u
prod(2:((n(j)+m_abs(j))/2-s(k))); @|hn@!YK
idx = (pows(k)==rpowers); '9R.$,N
y(:,j) = y(:,j) + p*rpowern(:,idx); k9|8@3(h
end K:3u/C`
K>a+-QWK3
if isnorm 1[&V6=n
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); {*jo,<4ee
end 0qPbmLMK
end zP(UaSXz/
% END: Compute the Zernike Polynomials j4fv-{=$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% w$2Z7S
[G=+f6 a
;
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% Compute the Zernike functions: ys#V_ysb
% ------------------------------ rCTH 5"
idx_pos = m>0; &LD=Zp%
idx_neg = m<0; *sPG,6>
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#F~^m
z = y; u#c3T'E
if any(idx_pos) i4XE26B;e
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); \j$q';9p
end s?g`ufF.t
if any(idx_neg) )PNeJf|@
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); jZ5 mpYUO
end Q-qM"8I
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% EOF zernfun 2
u{"R