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， A+69_?B
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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， IO/2iSbW
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？ 12~zS
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ x4pl#~Su
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function z = zernfun(n,m,r,theta,nflag) ]tanvJG}'
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. s{2BG9s
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 5tX| @Z:
z
% and angular frequency M, evaluated at positions (R,THETA) on the /RT3 r
% unit circle. N is a vector of positive integers (including 0), and iKu[j)F
% M is a vector with the same number of elements as N. Each element 68kxw1xY
% k of M must be a positive integer, with possible values M(k) = -N(k) 5^t68
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% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, /T v=BXL-
% and THETA is a vector of angles. R and THETA must have the same <=/v%VXPm
% length. The output Z is a matrix with one column for every (N,M) &$.Vi&{.
% pair, and one row for every (R,THETA) pair. 3o%JJ In&
% jW}n6w5
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike p)(mF"\8=
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), KN'l/9.
% with delta(m,0) the Kronecker delta, is chosen so that the integral `Yn^ -W
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, )Mx[;IwE
% and theta=0 to theta=2*pi) is unity. For the non-normalized n6ETWjP
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. HIcx "y
% >&f .^p
% The Zernike functions are an orthogonal basis on the unit circle. \O/EY&
% They are used in disciplines such as astronomy, optics, and L ~cswG'K
% optometry to describe functions on a circular domain. pv~XZ(J.1
% NDm@\<MIzB
% The following table lists the first 15 Zernike functions. SXSH9;j
% /t ikLJ
% n m Zernike function Normalization OY*BVJ^
% -------------------------------------------------- @] 1E ~
% 0 0 1 1 Is` S
% 1 1 r * cos(theta) 2 i,NN"
% 1 -1 r * sin(theta) 2 %np b.C|+
% 2 -2 r^2 * cos(2*theta) sqrt(6) jJg9M'@2!
% 2 0 (2*r^2 - 1) sqrt(3) 0NK]u~T<
% 2 2 r^2 * sin(2*theta) sqrt(6) : c"J$wT/
% 3 -3 r^3 * cos(3*theta) sqrt(8) c=<d99Cu!
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) J*F-tRuEw
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) m0t 5oO
% 3 3 r^3 * sin(3*theta) sqrt(8) yb1A(~
% 4 -4 r^4 * cos(4*theta) sqrt(10) R XkE"H{
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /7De .O~H
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) %;(+s7
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) F=kD/GCB
% 4 4 r^4 * sin(4*theta) sqrt(10) !!E_WDZ#9
% -------------------------------------------------- f(=yC} si
% M@UkXA}
% Example 1: ^QTl (L
% 'D#}ce)s#
% % Display the Zernike function Z(n=5,m=1) 0I* ^VGZ
% x = -1:0.01:1; #.?DsK_:@
% [X,Y] = meshgrid(x,x); H6 ( ~6Bp5
% [theta,r] = cart2pol(X,Y); '\ H {Y[
% idx = r<=1; ?u` ?_us
% z = nan(size(X)); lb2m Wsg"
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ,G S8Gu
% figure Ne ,7[ k
% pcolor(x,x,z), shading interp _j-k*:
% axis square, colorbar }UMg ph:2:
% title('Zernike function Z_5^1(r,\theta)') J\b,rOI f
% 7qt<C LJ
% Example 2:
%1 <No/
% ?q1&(g]qO
% % Display the first 10 Zernike functions HuBG?4Qd
% x = -1:0.01:1; Na=9 ju
% [X,Y] = meshgrid(x,x); L.$9ernVY
% [theta,r] = cart2pol(X,Y); {g@Wd2-J}
% idx = r<=1; 8Y3c,p/gS>
% z = nan(size(X)); EC&t+"= R
% n = [0 1 1 2 2 2 3 3 3 3]; fu}NH \{
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; a8rsF
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Bs = V-0
% y = zernfun(n,m,r(idx),theta(idx)); 1e l?f>
% figure('Units','normalized') LTG#nM0
% for k = 1:10 GeWB "(t
% z(idx) = y(:,k); > ~ _y\
% subplot(4,7,Nplot(k)) 9E)*X
% pcolor(x,x,z), shading interp N{46D S
% set(gca,'XTick',[],'YTick',[]) } >b4s!k,
% axis square d%Jl9!u
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ^LaI{UDw%h
% end 'Sppm;?
% I: U /%cr,
% See also ZERNPOL, ZERNFUN2. HAEgR
x=Qy{eIe
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% Paul Fricker 11/13/2006 e'[T 5H I
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% Check and prepare the inputs: #HWz.Wb
% ----------------------------- W:O<9ZbQ_
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) QG ?7L_I
error('zernfun:NMvectors','N and M must be vectors.') DalQ.
end Jy@cMq2
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if length(n)~=length(m) /|t
vGC.#
error('zernfun:NMlength','N and M must be the same length.') >"jV8%!sM
end au9r)]p-
y T`[9u ,
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n = n(:); 4A^hP![c#]
m = m(:); T~cq= i|O
if any(mod(n-m,2)) z@> z.d4
error('zernfun:NMmultiplesof2', ... 7J~6J .m
'All N and M must differ by multiples of 2 (including 0).') .{k(4_ Q?I
end UBOCd [
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s=(~/p #M
if any(m>n) %}%D8-d}G
error('zernfun:MlessthanN', ... 33J}AK^FE
'Each M must be less than or equal to its corresponding N.') Fe.Y4\xz
end > C+0LF`U
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if any( r>1 | r<0 ) Qc;`n ck
error('zernfun:Rlessthan1','All R must be between 0 and 1.') _DMj )enH"
end P{)H7B>
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ,r{*o6
error('zernfun:RTHvector','R and THETA must be vectors.') r=n| MT^O
end %2^C
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r = r(:); g,Kb9['
theta = theta(:); ?*u)T%S
length_r = length(r); EhEn|%S
if length_r~=length(theta) ~53E)ilB
error('zernfun:RTHlength', ... WEqHL,Uh]
'The number of R- and THETA-values must be equal.') #I%< 1c%XA
end (6 u<w#u
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.!KlN% As
% Check normalization: }E]`ly<Z
% -------------------- $Bz |[=
if nargin==5 && ischar(nflag) nuw90=qj!]
isnorm = strcmpi(nflag,'norm'); (Ew o
if ~isnorm rr3NY$W
error('zernfun:normalization','Unrecognized normalization flag.') -}{\C]%
end \9Itu(<f
else -2v|d]3qG
isnorm = false; ij r *_=
end 4@5rR~DQq
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vMzR3@4e
% Compute the Zernike Polynomials fB1JU1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% on*?O O'
TmKO/N@}
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% Determine the required powers of r: dLD"Cx
% ----------------------------------- 4dMwJ"V
m_abs = abs(m); @ MtF^y
rpowers = [];
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for j = 1:length(n) 5Ag]1 k{
rpowers = [rpowers m_abs(j):2:n(j)]; H4k`wWOk
end uP |AP
rpowers = unique(rpowers); VOG DD@
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% Pre-compute the values of r raised to the required powers, )#ic"UtR
% and compile them in a matrix: G8QJM0VpS
% ----------------------------- L$ ]D&f8:
if rpowers(1)==0 /Ia=/Jj7N
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @ )< uQ S
rpowern = cat(2,rpowern{:}); s]L`&fY]O
rpowern = [ones(length_r,1) rpowern]; 5tP0dQYd
else xw%?R=&L
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); rM [Ps=5
rpowern = cat(2,rpowern{:}); *2 MUG
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end \5s!lv*&
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% Compute the values of the polynomials: Lrq+0dI 65
% -------------------------------------- 8k_,Hni
y = zeros(length_r,length(n)); 4DuZF
-y
for j = 1:length(n) "kP.Kx!
s = 0:(n(j)-m_abs(j))/2; e6s L N
pows = n(j):-2:m_abs(j); YvBUx#\
for k = length(s):-1:1 Ma-^o<{
p = (1-2*mod(s(k),2))* ... 'G-VhvM v
prod(2:(n(j)-s(k)))/ ... )KXLL;]
prod(2:s(k))/ ... Pl1:d{"d
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 1)u= &t,
prod(2:((n(j)+m_abs(j))/2-s(k))); {:6VJ0s\
idx = (pows(k)==rpowers); .4_ ~ku
y(:,j) = y(:,j) + p*rpowern(:,idx); VrF]X# \)
end jq.@<<j|$
]d$)G4X 1
if isnorm YLPiK
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); $23="Jcl
end c0Q`S"o+
end ucoBeNsHx
% END: Compute the Zernike Polynomials C,tl p
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A ba6/
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#\`6ZHW
% Compute the Zernike functions: Yv"uIj+']
% ------------------------------ Lb2Bu >
idx_pos = m>0; Z]9
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idx_neg = m<0; v]VIUVd
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z = y; OEB_LI'
if any(idx_pos) %}j/G l5
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); i]Kq
end s Gdt)
if any(idx_neg) Lg Bs<2
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 3kKXzIh
end oWXvkDN
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{_k 6 t
% EOF zernfun \BJnJk!%
