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, XM3N>OR.
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, BVxg=7%St
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? CjM+%l0MW
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? PIo/|1
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function z = zernfun(n,m,r,theta,nflag) ^_JD
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ks7g*; 3{@
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N {*H&NI
% and angular frequency M, evaluated at positions (R,THETA) on the ;HM&
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% unit circle. N is a vector of positive integers (including 0), and B:5(sK
% M is a vector with the same number of elements as N. Each element g^(wZ$NH
% k of M must be a positive integer, with possible values M(k) = -N(k) !*.mcIQT
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, c&2ZjM
% and THETA is a vector of angles. R and THETA must have the same <CJua1l\
% length. The output Z is a matrix with one column for every (N,M) >z6(fM`i
% pair, and one row for every (R,THETA) pair. OA2<jrGB!
% m8H|cQ@Uu
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike `CW8Wj
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), PJN TIa
% with delta(m,0) the Kronecker delta, is chosen so that the integral bp2l%A;
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 9@Yk8
% and theta=0 to theta=2*pi) is unity. For the non-normalized XJsHy_6
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. -;'8#"{`^
% A1@tp/L=o
% The Zernike functions are an orthogonal basis on the unit circle. 9 )u*IGj
% They are used in disciplines such as astronomy, optics, and =?T\zLN=
% optometry to describe functions on a circular domain. vrdlI^
% .&.j?kb
% The following table lists the first 15 Zernike functions. ?hvPPEJf
% KDgJ~T
% n m Zernike function Normalization /j./
% -------------------------------------------------- Gvv~P3Dm
% 0 0 1 1 npg.*I/>
% 1 1 r * cos(theta) 2 0 V*Di2
% 1 -1 r * sin(theta) 2 ?8. $A2(Xw
% 2 -2 r^2 * cos(2*theta) sqrt(6) Adgh:'h
% 2 0 (2*r^2 - 1) sqrt(3) ,Cj1S7GFR
% 2 2 r^2 * sin(2*theta) sqrt(6) XodA(73`i
% 3 -3 r^3 * cos(3*theta) sqrt(8) (=0W[@k
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) R6]/g
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) =v=a:e
% 3 3 r^3 * sin(3*theta) sqrt(8) ;oVdkp
% 4 -4 r^4 * cos(4*theta) sqrt(10) Ojq>4=Z\
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) WM NcPHcj
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) DCM,|FE
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) EsXCi2]1
% 4 4 r^4 * sin(4*theta) sqrt(10) .MMFN}1O
% -------------------------------------------------- !Sfy'v.
% x)l}d3
% Example 1: 5b3Wt7
% _KC)f'Cx
% % Display the Zernike function Z(n=5,m=1) qI\qpWS\
% x = -1:0.01:1; $[5ihV$u
% [X,Y] = meshgrid(x,x); Q.#@xaX'{`
% [theta,r] = cart2pol(X,Y); {NXc<0a(
% idx = r<=1; w-};\]I
% z = nan(size(X)); y$7Fq'
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ;$l!mv7
% figure X|t?{.p
% pcolor(x,x,z), shading interp e~=fo#*2?@
% axis square, colorbar G+
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% title('Zernike function Z_5^1(r,\theta)') {o<
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% ~F^=7oq
% Example 2: -}@3,G
% 048BQ
% % Display the first 10 Zernike functions [>::@[
% x = -1:0.01:1; d_gm'
% [X,Y] = meshgrid(x,x); pa Uh+"y>
% [theta,r] = cart2pol(X,Y); 9d^o2Yo
% idx = r<=1; kM|akG
% z = nan(size(X)); DtG><g}[]
% n = [0 1 1 2 2 2 3 3 3 3]; T!eeMsI
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; rc1EJ(c
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 0*YLFqN
% y = zernfun(n,m,r(idx),theta(idx)); :q S=_!1
% figure('Units','normalized') *5 ]fjh{
% for k = 1:10 J/8aDr(+
% z(idx) = y(:,k); S*Un$ngAh
% subplot(4,7,Nplot(k)) q PuxYU
% pcolor(x,x,z), shading interp ,,S5 8\x
% set(gca,'XTick',[],'YTick',[]) K2>(C$Z
% axis square sguE{!BO
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) FYR%>Em
% end j!GJ$yd=-6
% RzQ1Wq
% See also ZERNPOL, ZERNFUN2. YW9 [^
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% Paul Fricker 11/13/2006 Y*;Z(W.V#
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% Check and prepare the inputs: 'A
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% ----------------------------- Q
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7kx)/Rw\B
error('zernfun:NMvectors','N and M must be vectors.') Enm#\(j
end EWNh:<F?
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if length(n)~=length(m) )dgXS//Y
error('zernfun:NMlength','N and M must be the same length.') KRQKL`}}
end ^_#0\f
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n = n(:); \~d|MP}"F:
m = m(:); v~e@:7d i
if any(mod(n-m,2)) 5:/
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error('zernfun:NMmultiplesof2', ... s$css{(ek
'All N and M must differ by multiples of 2 (including 0).') z(d@!Cd
end &$t BD@7
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if any(m>n) mv|eEz)r
error('zernfun:MlessthanN', ... /~NsHStn
'Each M must be less than or equal to its corresponding N.') rCi7q]_
end _fha9`
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if any( r>1 | r<0 ) >Lp^QP1gU
error('zernfun:Rlessthan1','All R must be between 0 and 1.') zQM3n =y
end dqO!p6
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5n!
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) p0~=
error('zernfun:RTHvector','R and THETA must be vectors.') NH$%g\GPs
end 0H,1"~,w]
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r = r(:); n#t{3qzpD
theta = theta(:); MEMD8:['
length_r = length(r); U.is:&]E
if length_r~=length(theta) ]C_g:|q
error('zernfun:RTHlength', ... @-nCK Yj
'The number of R- and THETA-values must be equal.') ['ol]ZJ
end B?9K! c
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% Check normalization: &54fFyJF
% -------------------- lMz5))Rr
if nargin==5 && ischar(nflag) i*B@#;;F
isnorm = strcmpi(nflag,'norm'); 5_Yl!=
if ~isnorm __r]@hY
error('zernfun:normalization','Unrecognized normalization flag.') H((!
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end }ozlED`E
else vKN"o* q
isnorm = false; .}>d[},F
end . [DCL
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9 X}F{!p~1
% Compute the Zernike Polynomials qYv/"
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s2A3.SN
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% Determine the required powers of r: <Z5ak4P
% ----------------------------------- yL/EIN
m_abs = abs(m); }YJ(|z""
rpowers = []; d2lOx|jt
for j = 1:length(n) M,@\*qlEJ
rpowers = [rpowers m_abs(j):2:n(j)]; WF\
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end n B4)%
rpowers = unique(rpowers);
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% Pre-compute the values of r raised to the required powers, Jcf'Zw"\
% and compile them in a matrix: 7uG@hL36
% ----------------------------- %^s;{aN*!
if rpowers(1)==0 It'hmwu#
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); "+3p??h%Rq
rpowern = cat(2,rpowern{:}); 'U
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rpowern = [ones(length_r,1) rpowern]; nM:e<`r
else YSwAu,$jf
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); A5-y+
rpowern = cat(2,rpowern{:}); fy04/_,q
end xc dy/J&
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% Compute the values of the polynomials: d%_v
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% -------------------------------------- 2|]$hjs
y = zeros(length_r,length(n)); *KNj5>6=
for j = 1:length(n) >m='#x0>Y
s = 0:(n(j)-m_abs(j))/2; Sx)b~ *
pows = n(j):-2:m_abs(j); =H6"\`W
for k = length(s):-1:1 jqq96hP,
p = (1-2*mod(s(k),2))* ... tWR>I$O8F
prod(2:(n(j)-s(k)))/ ... )\!_`ob
prod(2:s(k))/ ... 'Lu7cb^
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... $z,lq#zzl
prod(2:((n(j)+m_abs(j))/2-s(k))); .Tr!/mf_
idx = (pows(k)==rpowers); 'qcLK>E
y(:,j) = y(:,j) + p*rpowern(:,idx); gTWl];xja
end ceBu i8a
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if isnorm $;NxO0$
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); #<df!)
end Yqz(@( %
end KdU!wsKfG
% END: Compute the Zernike Polynomials K{)N:|y%!$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .),ql_sXr
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% Compute the Zernike functions: MmH[ 7R
% ------------------------------ m<L.H33'
idx_pos = m>0; 4mR{\
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idx_neg = m<0; ufF$7@(+
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z = y; D?;8bI%"
if any(idx_pos) S*;8z}5<\
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); )]x/MC:9r
end z5G<h
if any(idx_neg) l`c&nf6
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); t.wB\Kmt\
end sLiKcR8^
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% EOF zernfun RbB
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