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， t_^X$pL
我输入10阶的n、m，r,theta为38025*1向量，最后得到的z是29525*10阶的矩阵， nfSbM3D]h
这个，跟我们用zygo干涉仪直接拟合出的36项zernike系数，有何关系呢？ ^zzP.
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ %
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function z = zernfun(n,m,r,theta,nflag) e}/Lk5q!
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. J]l rS
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N jp8@vdRg
% and angular frequency M, evaluated at positions (R,THETA) on the RqEH| EUZ
% unit circle. N is a vector of positive integers (including 0), and gI^o U 4mq
% M is a vector with the same number of elements as N. Each element X+L) -d
% k of M must be a positive integer, with possible values M(k) = -N(k) DI +]D~N
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 3$.deYa$R
% and THETA is a vector of angles. R and THETA must have the same ^k5ll=}
% length. The output Z is a matrix with one column for every (N,M) |F ,R&<2
% pair, and one row for every (R,THETA) pair. "k*PA\ U
% 3.22"U\1:
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ;c~cet4
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), u H/w\v_I
% with delta(m,0) the Kronecker delta, is chosen so that the integral @1.QE yX G
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, B~o\+n
% and theta=0 to theta=2*pi) is unity. For the non-normalized j
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. - p3Re9
% *.L81er5~
% The Zernike functions are an orthogonal basis on the unit circle. kB?al#`
% They are used in disciplines such as astronomy, optics, and w0w G-R ?
% optometry to describe functions on a circular domain. Y<1QY?1sd
% i1H\#;`$
% The following table lists the first 15 Zernike functions. Eskb9^A
% M@ed>.
% n m Zernike function Normalization G!K ]W:m
% -------------------------------------------------- IDnC< MO>
% 0 0 1 1 6qcO?U
% 1 1 r * cos(theta) 2 @d Jr/6Yx
% 1 -1 r * sin(theta) 2 5 >BK%`
% 2 -2 r^2 * cos(2*theta) sqrt(6) ]|`C uc
% 2 0 (2*r^2 - 1) sqrt(3) qM#R0ZUIe\
% 2 2 r^2 * sin(2*theta) sqrt(6) j'9"cE5_
% 3 -3 r^3 * cos(3*theta) sqrt(8) n 1!?"m!
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) !r # ?C9Sq
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) nX$XL=6mJ&
% 3 3 r^3 * sin(3*theta) sqrt(8) 0a-: <zm
% 4 -4 r^4 * cos(4*theta) sqrt(10) 9U$EJ N_G
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /~ x "wo
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5)
=-_B:d;
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >3z5 ww
% 4 4 r^4 * sin(4*theta) sqrt(10) 6iCrRjY*
% -------------------------------------------------- K| dso]b/
% C@th O
% Example 1: Q}-~O1
% } CeCc0M
% % Display the Zernike function Z(n=5,m=1) cA% %IL$R
% x = -1:0.01:1; e{ m2l2Tx:
% [X,Y] = meshgrid(x,x); v4C{<8:X
% [theta,r] = cart2pol(X,Y); JV;OGh>
% idx = r<=1; u m9_ru~
% z = nan(size(X)); FQ=@mjh
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ]Dw]p! @
% figure rETRTp0HT
% pcolor(x,x,z), shading interp Htti X /2~
% axis square, colorbar zbq@pj)Qu
% title('Zernike function Z_5^1(r,\theta)') $ @UN4B?y
% 7)s ^8+
% Example 2: D1__n6g[
% &197P7&o
% % Display the first 10 Zernike functions N!g9 *Z
% x = -1:0.01:1; ;[fw]P n
% [X,Y] = meshgrid(x,x); K_d Oq68_
% [theta,r] = cart2pol(X,Y); O%FPS=
% idx = r<=1; J>/w5$h5
% z = nan(size(X)); M)U{7c$c7
% n = [0 1 1 2 2 2 3 3 3 3]; hiQha5
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; j ? MAED
% Nplot = [4 10 12 16 18 20 22 24 26 28]; .k,kTr$ S
% y = zernfun(n,m,r(idx),theta(idx)); gG/!,Q.Qh
% figure('Units','normalized') !Y-98<|b
M
% for k = 1:10 TYy.jFT-
% z(idx) = y(:,k); fl\ly `_
% subplot(4,7,Nplot(k)) z<yU-m2h
% pcolor(x,x,z), shading interp R) c'#St
% set(gca,'XTick',[],'YTick',[]) ~Q\3pI. |
% axis square *hw\35%P`?
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) J> \B`E
% end Z,=7Tu bR#
% f "<O0Qw
% See also ZERNPOL, ZERNFUN2. ~$~5qwl
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% Paul Fricker 11/13/2006 3g >B"t
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% Check and prepare the inputs: YGrg
% ----------------------------- %HL*c =
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) q}]XYys
error('zernfun:NMvectors','N and M must be vectors.') !P -^O
end +t?3T-@Ks
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if length(n)~=length(m) hC\6-
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error('zernfun:NMlength','N and M must be the same length.') 6Aku1h
end {|'E
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n = n(:); (Tbw3ENz
m = m(:); QnJZr:4b
if any(mod(n-m,2)) A T%u%cE-
error('zernfun:NMmultiplesof2', ... svq<)hAf<
'All N and M must differ by multiples of 2 (including 0).') /bi}'H+#
end *" {lMZ +
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#H :7@
if any(m>n) F~P%AjAx'
error('zernfun:MlessthanN', ... ;S>])5<
'Each M must be less than or equal to its corresponding N.') wbst8 *$
end jJ5W>Q1mK$
%;rHrDP(>
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if any( r>1 | r<0 ) u4 /kR
error('zernfun:Rlessthan1','All R must be between 0 and 1.') $GTU$4u
end D`$hPYK|_
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) +w+} b^4
error('zernfun:RTHvector','R and THETA must be vectors.') BYMi6w ts
end i ~{Ufi
|%'
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r = r(:); #N`~xZ|$
theta = theta(:); |563D#?c R
length_r = length(r); E/%9jDTQ
if length_r~=length(theta) L{8xlx`
error('zernfun:RTHlength', ... 28 UU60
'The number of R- and THETA-values must be equal.') BBtzs^C|
end <=>= .kmGt
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% Check normalization: Ro9tZ'N!S
% -------------------- k @z,Iq8
if nargin==5 && ischar(nflag) 70eb]\ %
isnorm = strcmpi(nflag,'norm'); SN1}xR$
if ~isnorm G1o3l ~x
error('zernfun:normalization','Unrecognized normalization flag.') \Ol kM<
end R
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else "$2 y-|
isnorm = false; ;e1ku|>$
end $d_|NssvU
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[O?z@)dx
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% T Kg aV;92
% Compute the Zernike Polynomials ,}K7Dg^1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3ZXQoC '
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SUU !7Yd|
% Determine the required powers of r: sXD1C2o
% ----------------------------------- {/
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m_abs = abs(m); 5
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rpowers = []; !& z(:d
for j = 1:length(n) O v k_\On
rpowers = [rpowers m_abs(j):2:n(j)]; fmb} 2h
end @T'i/}nl
rpowers = unique(rpowers); Q|D @Yd\
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% Pre-compute the values of r raised to the required powers, G}ob<`o|"
% and compile them in a matrix: VB, ?Mo}R
% ----------------------------- `s8{C
b=}1
if rpowers(1)==0 ,#L=v]
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Yd#/1!A7u
rpowern = cat(2,rpowern{:}); Jc|6&
rpowern = [ones(length_r,1) rpowern]; A^y|J ` k|
else /Z@tv .f
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); no9;<]4
rpowern = cat(2,rpowern{:}); 5*,f
Fib
end 4<lRPsvg c
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% Compute the values of the polynomials: V;:j Zp G
% -------------------------------------- L_wk~ z
y = zeros(length_r,length(n)); >JC
for j = 1:length(n) SU/BQ3
s = 0:(n(j)-m_abs(j))/2; DUC#NZgw
pows = n(j):-2:m_abs(j); C'o6 4+W^
for k = length(s):-1:1 vM*($qpAy
p = (1-2*mod(s(k),2))* ... O slL~ <
prod(2:(n(j)-s(k)))/ ... / ;,Md,p
prod(2:s(k))/ ... \ ytJ=0r
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... @jsDq
Ln
prod(2:((n(j)+m_abs(j))/2-s(k))); VBcy9|lD
idx = (pows(k)==rpowers); :$m}UA-9
y(:,j) = y(:,j) + p*rpowern(:,idx); =1 Oj*x@*4
end C
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if isnorm R}4So1
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ,u`YT%&L
end 2E)wpgUc?e
end JAQb{KefdO
% END: Compute the Zernike Polynomials S/ODq L|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9B w|(J
uoX:^'q
r? Wk<> %>
% Compute the Zernike functions: (<} &DE
% ------------------------------ ZRg;/sX]
idx_pos = m>0; GJtZ&H
idx_neg = m<0; m6[0Kws&
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z = y; b#hDHSdZ,
if any(idx_pos) fi$-;Gz
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 7lBAxqr2
end }A7j/uy}s
if any(idx_neg) _PlKhv}
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); t-0a7
1#e
end f'VX Y-
f=mZu1(FZ
qp E&go=k'
% EOF zernfun fH > NJK;
