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，
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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， ? R282l
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？ hg4J2m
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ u]0!|Jd0
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function z = zernfun(n,m,r,theta,nflag) %]!?{U\* k
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. H(?e&Qkg
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N %;
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% and angular frequency M, evaluated at positions (R,THETA) on the soDfi-2o3
% unit circle. N is a vector of positive integers (including 0), and kR _E6Fl
% M is a vector with the same number of elements as N. Each element &*jixqzvn
% k of M must be a positive integer, with possible values M(k) = -N(k) 6# R;HbkO
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, v|>BDN@,6
% and THETA is a vector of angles. R and THETA must have the same f@L \E>t
% length. The output Z is a matrix with one column for every (N,M) L PMb0F}"5
% pair, and one row for every (R,THETA) pair. `!_? uT
% eiOi3q
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike \wTW?>o Z
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), yG4 MUf6
% with delta(m,0) the Kronecker delta, is chosen so that the integral WFXx 70n
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, q4 $sc_0i
% and theta=0 to theta=2*pi) is unity. For the non-normalized I'P!,Y/>
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. |N XFla
% m8p4U-*j
% The Zernike functions are an orthogonal basis on the unit circle. 4@gl4&<h
% They are used in disciplines such as astronomy, optics, and CO7CNN
% optometry to describe functions on a circular domain. uQ-WTz|*
% X=\x&Wt
% The following table lists the first 15 Zernike functions. oUCVd}wH
% } cR i
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% n m Zernike function Normalization ga,A'Z
% -------------------------------------------------- L-SdQTx_
% 0 0 1 1 E|\3f(aF
% 1 1 r * cos(theta) 2 WGluZhRuT3
% 1 -1 r * sin(theta) 2 Xp.|.)Od
% 2 -2 r^2 * cos(2*theta) sqrt(6) []!tT-Gzy
% 2 0 (2*r^2 - 1) sqrt(3) gZ=) qT]Pj
% 2 2 r^2 * sin(2*theta) sqrt(6) 2zwuvgiZ
% 3 -3 r^3 * cos(3*theta) sqrt(8) v#w4{ .8)
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ?! !;X W
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) MV7}
% 3 3 r^3 * sin(3*theta) sqrt(8) 0GF%~6
% 4 -4 r^4 * cos(4*theta) sqrt(10) 3KbUHSx
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) N IO;
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) UqY J#&MqY
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) P"U>tsHK:
% 4 4 r^4 * sin(4*theta) sqrt(10) A5`#Ot* 3
% -------------------------------------------------- Gt?!E6^ !
% _$~ex ~v
% Example 1: 1V#B]x:
% X*(gT1"t
% % Display the Zernike function Z(n=5,m=1) 'Wd3`4V$
% x = -1:0.01:1; 9(V=Ubj
% [X,Y] = meshgrid(x,x); }Z<D^Z~w
% [theta,r] = cart2pol(X,Y); (8+.# 1!*
% idx = r<=1; 'cWlY3%t
% z = nan(size(X)); 8s \8 `2=
% z(idx) = zernfun(5,1,r(idx),theta(idx)); PL9zNCr-[
% figure 9N`+ O
% pcolor(x,x,z), shading interp Fa Qu$q
% axis square, colorbar _gis+f/8h
% title('Zernike function Z_5^1(r,\theta)') Z:W')N d(
% g9RzzE!
% Example 2: s qgD?:@J
% 9Cg X c5
% % Display the first 10 Zernike functions =P@M&Yy'
% x = -1:0.01:1; ayB=|*Q"
% [X,Y] = meshgrid(x,x);
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% [theta,r] = cart2pol(X,Y); WMt&8W5
% idx = r<=1; ]0 at2
% z = nan(size(X)); &6=TtTp"9
% n = [0 1 1 2 2 2 3 3 3 3]; XY&]T'A
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; (Q*2dd>
% Nplot = [4 10 12 16 18 20 22 24 26 28]; yHV^a0e7EH
% y = zernfun(n,m,r(idx),theta(idx)); /1s 9;'I
% figure('Units','normalized') $_%2D3-;D
% for k = 1:10 eP-R""uPw
% z(idx) = y(:,k); |:J*>"sq
% subplot(4,7,Nplot(k)) ~)oWSo5ll
% pcolor(x,x,z), shading interp b7F3]W<`&
% set(gca,'XTick',[],'YTick',[]) ;@ h'Mb
% axis square 8Iqk%n~(
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _"Fbj Q"
% end n]kQ tjJ
% q329z>
% See also ZERNPOL, ZERNFUN2. tIg CF?
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% Paul Fricker 11/13/2006 $A5 B{2
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% Check and prepare the inputs: \6nQ-S_
% ----------------------------- "OlI-^y
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) &D&5UdN
x
error('zernfun:NMvectors','N and M must be vectors.') sk %:Sp
end umHs " d
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if length(n)~=length(m) }(vOaD|k =
error('zernfun:NMlength','N and M must be the same length.') `
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end sbq44L)
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n = n(:); T{ v <
m = m(:); D{Jc+Q$
if any(mod(n-m,2)) o#KPrW`XJ/
error('zernfun:NMmultiplesof2', ... Kr+Bt y
'All N and M must differ by multiples of 2 (including 0).') Xbsj:Ko]]U
end \e5,`
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if any(m>n) ^QHMN 7r/
error('zernfun:MlessthanN', ... [ XY:MU e
'Each M must be less than or equal to its corresponding N.') j_YZ(: =
end L{o >D"
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if any( r>1 | r<0 ) /i]=ndAk
error('zernfun:Rlessthan1','All R must be between 0 and 1.') e4NX\tCpw
end j{Qbzczy,
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%]= 'Uv^x
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 3vvFF]D5k
error('zernfun:RTHvector','R and THETA must be vectors.') +XaO?F[c
end q5K/+N ^2?
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r = r(:); :8GxcqvCWq
theta = theta(:); E )Zd{9A5)
length_r = length(r); e^l+ #^fR
if length_r~=length(theta) YQ[&h
error('zernfun:RTHlength', ... bU g 2Bm!y
'The number of R- and THETA-values must be equal.') :N'[d e
end 6 [Pr<4J
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% Check normalization: Wn9Mr2r!*,
% -------------------- iRr& 'k
if nargin==5 && ischar(nflag) {TN@KB
isnorm = strcmpi(nflag,'norm'); .qU%SmQ^
if ~isnorm pa> 2JF*
error('zernfun:normalization','Unrecognized normalization flag.') 4>ce,*B1
end !E. l yz
else fB ,!|u
isnorm = false; #L*@~M^]
end |(8Hk@\CT>
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4sD:J-c
% Compute the Zernike Polynomials t;~`Lm@hY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /
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% Determine the required powers of r: LyPBFo[?
% ----------------------------------- #d i_V"
m_abs = abs(m); ~ X(xa
rpowers = []; kAF}*&Kzd~
for j = 1:length(n) ke6cZV5w
rpowers = [rpowers m_abs(j):2:n(j)]; l$~bkVNL
end Q1&dB{L
rpowers = unique(rpowers); L}S4Zz18
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% Pre-compute the values of r raised to the required powers, g4qdm{BL
% and compile them in a matrix: u#k6v\/
% ----------------------------- GpQF * x
if rpowers(1)==0 9TN5|x
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); CH+&
rpowern = cat(2,rpowern{:}); 7wEG<,D
rpowern = [ones(length_r,1) rpowern]; V4i%|vV
else T-8nUo}i
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); I\6<)2j/L
rpowern = cat(2,rpowern{:}); G+^$JN=
end KIl. ?_61O
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% Compute the values of the polynomials: T[=cKYp8\
% -------------------------------------- cT x/ Y&\9
y = zeros(length_r,length(n)); o[ W3/
for j = 1:length(n) +\$|L +@Z
s = 0:(n(j)-m_abs(j))/2; l%5%oN`4
pows = n(j):-2:m_abs(j); ]@}BdMlHp
for k = length(s):-1:1 ?v~3zHK
p = (1-2*mod(s(k),2))* ... q;~ >h
prod(2:(n(j)-s(k)))/ ... R'HA>?D
prod(2:s(k))/ ... 0BD((oNg
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ;<R_ j%*
prod(2:((n(j)+m_abs(j))/2-s(k))); O}!@28|3"
idx = (pows(k)==rpowers); To?
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y(:,j) = y(:,j) + p*rpowern(:,idx); j;'Wf[V
end 5&Kn #
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if isnorm O +}EE^*a
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ryLNMh
end &V{,D))6[
end ZTC1t_
% END: Compute the Zernike Polynomials RteTz _ z{
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B;je|M!d
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% Compute the Zernike functions: ,w>?N\w!}
% ------------------------------ Dx)XC?'xO
idx_pos = m>0; ,]qX_`qF
idx_neg = m<0; {# _ C
%-|q3 ^s
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z = y; &J M;jS z
if any(idx_pos) o^6 j(~
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ,lM2BXz%
end =nZd"t'p|
if any(idx_neg) =)5a=^
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z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 6u;(R 0n
end J :(\o=5 5
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% EOF zernfun .LHe*J C
