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, kX8C'D4 gX
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, Sf?;j{?G
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? /2p*uv}IP
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? zj7ta[<tr
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function z = zernfun(n,m,r,theta,nflag) %jL^sA2;c+
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ,ua1sTgQ
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N D, ")n75
% and angular frequency M, evaluated at positions (R,THETA) on the n\+c3
% unit circle. N is a vector of positive integers (including 0), and 5f*_K6 ,v
% M is a vector with the same number of elements as N. Each element R/=rNUe
% k of M must be a positive integer, with possible values M(k) = -N(k) gH//@`6
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, iVFOOsJ@
% and THETA is a vector of angles. R and THETA must have the same >ai,6!
% length. The output Z is a matrix with one column for every (N,M) {;{U@Z
% pair, and one row for every (R,THETA) pair. VM$n|[C~
% t'U=K>7
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike kyHli~Nr"
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ji ?Hw
% with delta(m,0) the Kronecker delta, is chosen so that the integral qHk{5O3
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, <Z^by;d|z
% and theta=0 to theta=2*pi) is unity. For the non-normalized PK+sGV
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Uj5-x%~
%
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% The Zernike functions are an orthogonal basis on the unit circle. 3X gJZ
% They are used in disciplines such as astronomy, optics, and x0# Bc7y
% optometry to describe functions on a circular domain. 19$A!kH\
% Xl4}S"a
% The following table lists the first 15 Zernike functions. rg^\gE6_
% c!\Gj|
% n m Zernike function Normalization ]?}>D?5
% -------------------------------------------------- @_do<'a
% 0 0 1 1 c5^HGIe1
% 1 1 r * cos(theta) 2 Jj=qC{]
% 1 -1 r * sin(theta) 2 6 - 3?&+
% 2 -2 r^2 * cos(2*theta) sqrt(6) f./K/
% 2 0 (2*r^2 - 1) sqrt(3) 8"&!3_
% 2 2 r^2 * sin(2*theta) sqrt(6) m}l);P^
% 3 -3 r^3 * cos(3*theta) sqrt(8) Wep^He\:
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 'maX
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ~uhW~bT
% 3 3 r^3 * sin(3*theta) sqrt(8) ]W3_]N 3
% 4 -4 r^4 * cos(4*theta) sqrt(10) %M96m
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) pB:XNkxL
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) $9YQ aN%
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9Jwd *gevV
% 4 4 r^4 * sin(4*theta) sqrt(10) 7Yg1z%%U
% -------------------------------------------------- Bc8&-eZ,
% 7n5gXiI"
% Example 1: cM%?Ot,mK"
% /5sn*,
% % Display the Zernike function Z(n=5,m=1) $UzSPhv[
% x = -1:0.01:1; Gi)Vr\Q.
% [X,Y] = meshgrid(x,x); We y*\@
% [theta,r] = cart2pol(X,Y); as@8L|i*
% idx = r<=1; 1WtE ]
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% z = nan(size(X)); Q^ W,)%
% z(idx) = zernfun(5,1,r(idx),theta(idx)); <,:{Q75
% figure +QN4hJK
% pcolor(x,x,z), shading interp [l-o*@
% axis square, colorbar :aOR@])>o
% title('Zernike function Z_5^1(r,\theta)') >*EZZ\eU!
% DQ8/]Z{H
% Example 2: d}O\:\}y
% b|_e):V|
% % Display the first 10 Zernike functions uUjjAGZ
% x = -1:0.01:1; `dm*vd
% [X,Y] = meshgrid(x,x); i`+w.zJOH8
% [theta,r] = cart2pol(X,Y); J=-z~\f56
% idx = r<=1; x{;{fMN1
% z = nan(size(X)); 2{j$1EdI@-
% n = [0 1 1 2 2 2 3 3 3 3]; ir6aV|ea!
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; W/UA%We3+L
% Nplot = [4 10 12 16 18 20 22 24 26 28]; R([zlw~B5
% y = zernfun(n,m,r(idx),theta(idx)); bkdXBCBx?
% figure('Units','normalized') "" UyfC[
% for k = 1:10 rfonM~3?'
% z(idx) = y(:,k); )M<+?R$];
% subplot(4,7,Nplot(k)) \~8W0q.4M
% pcolor(x,x,z), shading interp W_@ b. 1
% set(gca,'XTick',[],'YTick',[]) /rpr_Xw}
% axis square ,6]ID1o:y
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #;8)UNc)}
% end ,Mw93Kp
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% VKPEoy8H
% See also ZERNPOL, ZERNFUN2. 9"^ib9M
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% Paul Fricker 11/13/2006 0y|1@CS
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% Check and prepare the inputs: rd&d~R6
% ----------------------------- ;>2-
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) K\ Wzh;
error('zernfun:NMvectors','N and M must be vectors.') 5
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end N?m)u,6-l
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if length(n)~=length(m) %( OP
[
error('zernfun:NMlength','N and M must be the same length.') #jBmWaP.
end s YTJ^K d
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n = n(:); D,=~7/g
m = m(:); 9wCgJ$te
if any(mod(n-m,2)) p[&Jl
error('zernfun:NMmultiplesof2', ... =ttD5p
'All N and M must differ by multiples of 2 (including 0).') t8Pf~v
end s:'>G;p
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if any(m>n) Oa:C'M
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error('zernfun:MlessthanN', ... &wU"6E
'Each M must be less than or equal to its corresponding N.') nZ=[6?
end 28v^j*=*
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if any( r>1 | r<0 ) "{@[06|1
error('zernfun:Rlessthan1','All R must be between 0 and 1.') rbOJ;CK
end 4w|t|?
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 9q0s
error('zernfun:RTHvector','R and THETA must be vectors.') |}^u<S8X
end YCP D+
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r = r(:); i<m)
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theta = theta(:); q;R&valn
length_r = length(r); w)J-e gc
if length_r~=length(theta) RCa1S^.
error('zernfun:RTHlength', ... gWjYS#D
'The number of R- and THETA-values must be equal.') fqbWD)L]
end W`LG.`JW
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% Check normalization: "&XhMw4
% -------------------- 7]So=%q
if nargin==5 && ischar(nflag) z z]~IxQ
isnorm = strcmpi(nflag,'norm'); 8=bn
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if ~isnorm ?$)a[UnqX
error('zernfun:normalization','Unrecognized normalization flag.') cb'Ya_
end q9x@Pc29d
else :?EZ\WM7
isnorm = false; ~:,}?9
end ga KZ4#
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8MW-JZ
% Compute the Zernike Polynomials 4D5Wse
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GYy8kp84
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% Determine the required powers of r: :xmj42w>^
% ----------------------------------- m{>"
m_abs = abs(m); x]Nx,tt
rpowers = []; g_PP9S_?
for j = 1:length(n) .mwW`D
rpowers = [rpowers m_abs(j):2:n(j)]; (MqQ3ys
end |j/Y#.k;{0
rpowers = unique(rpowers); $EIKi'!8
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i.K!;E>
% Pre-compute the values of r raised to the required powers, [+5g 9tBJ
% and compile them in a matrix: X:f5t` ;
% ----------------------------- 'rXf
if rpowers(1)==0 w&}<b%l
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); \ eba9i^
rpowern = cat(2,rpowern{:}); 5`}za-
rpowern = [ones(length_r,1) rpowern]; DdISJWc'`5
else ADxje%!1O
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); e7n0=U0
rpowern = cat(2,rpowern{:}); F W2x
end ])v61B
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% Compute the values of the polynomials: LB.B w
% -------------------------------------- k!z.6di
y = zeros(length_r,length(n)); 2_bEo
for j = 1:length(n) @ZYJY
s = 0:(n(j)-m_abs(j))/2; 1W.oRD&8j/
pows = n(j):-2:m_abs(j); >sAaLR4
for k = length(s):-1:1 8t< X
p = (1-2*mod(s(k),2))* ... M4;M.zxJv
prod(2:(n(j)-s(k)))/ ... ( ,mV6U%
prod(2:s(k))/ ... q b=%W
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... s7jNRY V
prod(2:((n(j)+m_abs(j))/2-s(k))); iVVR$uzhH
idx = (pows(k)==rpowers); ?|NsaW
y(:,j) = y(:,j) + p*rpowern(:,idx); LH`$<p2''r
end ETX>wZ
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if isnorm $zhvI*0
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); jpXbFWgN
end ;X+.Ag
end ME)='~E
% END: Compute the Zernike Polynomials 4S+E%b|)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |"b|Q
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% Compute the Zernike functions: B1I{@\z0G
% ------------------------------ PxWH)4
idx_pos = m>0; k^KpQ&n
idx_neg = m<0; p.MLKp-'
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z = y; kE+fdr\ T
if any(idx_pos) qv2J0'd'.
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); {w,^Z[<
end 9J_vvq`%`
if any(idx_neg) S<*1b 6%D
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); V'za,.d-
end a~!7A
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% EOF zernfun C=K{;.