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, Cq;t;qN,nQ
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, "2(lgxhj
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? #ebT$hf30
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? KB <n-'
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function z = zernfun(n,m,r,theta,nflag) p}_bu@;.Z
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. /=>z|?z3
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N "12.Bi.O"[
% and angular frequency M, evaluated at positions (R,THETA) on the S*Un$ngAh
% unit circle. N is a vector of positive integers (including 0), and tKrr5SRb
% M is a vector with the same number of elements as N. Each element ,,S5 8\x
% k of M must be a positive integer, with possible values M(k) = -N(k) K2>(C$Z
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, B5*{85p(u
% and THETA is a vector of angles. R and THETA must have the same FYR%>Em
% length. The output Z is a matrix with one column for every (N,M) KG6ki_
% pair, and one row for every (R,THETA) pair. B2:6=8<
% X+;Ivx
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 8:0QI kqk
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ~b/lr
% with delta(m,0) the Kronecker delta, is chosen so that the integral 3&_O\nD
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Mz#
&"WjF
% and theta=0 to theta=2*pi) is unity. For the non-normalized A
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5t#+UR
% ))cL+r
% The Zernike functions are an orthogonal basis on the unit circle. ~V[pu
% They are used in disciplines such as astronomy, optics, and $r *7)/
% optometry to describe functions on a circular domain. FFb`4.
% YpoO:
% The following table lists the first 15 Zernike functions. 6 /gh_'&
% eWS[|'dl
% n m Zernike function Normalization gN1b?_g
% -------------------------------------------------- )a.Y$![
% 0 0 1 1 _Sy-&}c+
+
% 1 1 r * cos(theta) 2 Z0g3> iItM
% 1 -1 r * sin(theta) 2 W_9-JM(r
% 2 -2 r^2 * cos(2*theta) sqrt(6) \~d|MP}"F:
% 2 0 (2*r^2 - 1) sqrt(3) v~e@:7d i
% 2 2 r^2 * sin(2*theta) sqrt(6) 5:/
zbt\C
% 3 -3 r^3 * cos(3*theta) sqrt(8) s$css{(ek
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) r-v;A
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) I-oI,c%+
% 3 3 r^3 * sin(3*theta) sqrt(8) rlk0t159
% 4 -4 r^4 * cos(4*theta) sqrt(10) Wk"4mq
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 2s>dlz
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) -;&aU;k
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }GJIM|7^
% 4 4 r^4 * sin(4*theta) sqrt(10) Ls )y.u
% -------------------------------------------------- Q(
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% .^j#gE&B
% Example 1: *gfx'$
% <DP_`[+C
% % Display the Zernike function Z(n=5,m=1) kmPK |R
% x = -1:0.01:1; >B/ jTn5=
% [X,Y] = meshgrid(x,x); }UJS*mR
% [theta,r] = cart2pol(X,Y); /uS(Z-@
% idx = r<=1; \.y|=Ql_u
% z = nan(size(X)); 2%U)y;$m2
% z(idx) = zernfun(5,1,r(idx),theta(idx)); )QEvV:\
% figure F%@(
$f
% pcolor(x,x,z), shading interp hX?L/yf
% axis square, colorbar Q1 ?O~ao
% title('Zernike function Z_5^1(r,\theta)') y}*rRm.:
% S453oG"
% Example 2: l:mC'aR
% x*&
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% % Display the first 10 Zernike functions =8O057y
% x = -1:0.01:1; &54fFyJF
% [X,Y] = meshgrid(x,x); lMz5))Rr
% [theta,r] = cart2pol(X,Y); i*B@#;;F
% idx = r<=1; 5_Yl!=
% z = nan(size(X)); __r]@hY
% n = [0 1 1 2 2 2 3 3 3 3]; H((!
BRl
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; [` ~YPUR*
% Nplot = [4 10 12 16 18 20 22 24 26 28]; rStfluPL
% y = zernfun(n,m,r(idx),theta(idx)); zM{'GB+en
% figure('Units','normalized') 3&'ll51t
% for k = 1:10 ss63/
% z(idx) = y(:,k); V{@
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% subplot(4,7,Nplot(k)) $~vy,^
% pcolor(x,x,z), shading interp Ug02G
% set(gca,'XTick',[],'YTick',[]) c=]qUhnH
% axis square uqwB`<>KJ
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ',j'Hf
% end =<w6yeko
% $s<,xY 9
% See also ZERNPOL, ZERNFUN2. ;ZZ%(P=-
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% Paul Fricker 11/13/2006 CY;ML6c@
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% Check and prepare the inputs: OrP-+eg
% ----------------------------- n^P=a'+
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) BE. v+'c"
error('zernfun:NMvectors','N and M must be vectors.') )R$+dPu>
end %^s;{aN*!
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if length(n)~=length(m) 'U
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error('zernfun:NMlength','N and M must be the same length.') nM:e<`r
end YSwAu,$jf
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n = n(:); xc dy/J&
m = m(:); =g4^tIYq
if any(mod(n-m,2)) RG/M-
error('zernfun:NMmultiplesof2', ... d%_v
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'All N and M must differ by multiples of 2 (including 0).') 2|]$hjs
end *KNj5>6=
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if any(m>n) =H6"\`W
error('zernfun:MlessthanN', ... jqq96hP,
'Each M must be less than or equal to its corresponding N.') z-fP#.
end )\!_`ob
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if any( r>1 | r<0 )
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error('zernfun:Rlessthan1','All R must be between 0 and 1.') Dyv 6K_,
end ?dMyhU}
: l>&5w;
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Jydz2
zt!
error('zernfun:RTHvector','R and THETA must be vectors.') 7=C$*)x
end 2RXU75VY
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r = r(:); (|%YyRaX
theta = theta(:); 3YT>3f!\
length_r = length(r); 0S8v41i6
if length_r~=length(theta) _mVq9nBEf
error('zernfun:RTHlength', ... =9$hZ c
'The number of R- and THETA-values must be equal.') $g&,$7}O_
end S <~"\<ED
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% Check normalization: {n9]ej^
% -------------------- &}}c>]m
if nargin==5 && ischar(nflag) sYnf
# '
isnorm = strcmpi(nflag,'norm'); \|
qr&(PG
if ~isnorm 8a{S*
error('zernfun:normalization','Unrecognized normalization flag.') ?K,xxH
end &K2J$(.t
else :m*r(i3
isnorm = false; USF&; M3
end %oVoE2T{@
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <=WSX{_D
% Compute the Zernike Polynomials eytd@-7uX
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {pJ{UJKv?
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% Determine the required powers of r: 1 .\|,$
% ----------------------------------- =xwA'D9]
m_abs = abs(m); ;/gH6Z?
rpowers = []; 9W{`$30
for j = 1:length(n) I4]|r k9
rpowers = [rpowers m_abs(j):2:n(j)]; H}m%=?y@
end I|jGu9G
rpowers = unique(rpowers); hAx#5@*5
t(3<w)r2
/)I:Cz/f
% Pre-compute the values of r raised to the required powers, E?%SOU<
% and compile them in a matrix: ygt7;};!
% ----------------------------- [@ExR*
if rpowers(1)==0 -*q:B[d
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);
N7%iz+
rpowern = cat(2,rpowern{:}); E]eVoC
rpowern = [ones(length_r,1) rpowern]; MbY?4i00%h
else E`vCYhf{
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); vLQ!kB^\W
rpowern = cat(2,rpowern{:}); ho*44=j
end Glz)-hjJ:n
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% Compute the values of the polynomials: OsS5WY0H
% -------------------------------------- !uaV6K
y = zeros(length_r,length(n)); T\ ;7'
for j = 1:length(n) _86pbr9
s = 0:(n(j)-m_abs(j))/2; 9qyA{
|3
pows = n(j):-2:m_abs(j); r$3{1HXc
for k = length(s):-1:1 o$dnp`E
p = (1-2*mod(s(k),2))* ... CX](^yU_
prod(2:(n(j)-s(k)))/ ... z"4UObVs
prod(2:s(k))/ ... W)WL1@!Z
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... s)_Xj`Q#
prod(2:((n(j)+m_abs(j))/2-s(k))); 32DT]{-N!
idx = (pows(k)==rpowers); 29:1crzx~
y(:,j) = y(:,j) + p*rpowern(:,idx); _`6fGu& W
end /?<tjK' "H
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if isnorm X\X*-.]{
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); S $wx>715
end 5sbMp;ZM
end #$!(8>YJ
% END: Compute the Zernike Polynomials B8Ob~?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]Z/<HP$#
wc#+Yh6
#vk-zx*v7=
% Compute the Zernike functions: B> kx$_~
% ------------------------------ eWjLP{W
idx_pos = m>0; /S}0u}jID?
idx_neg = m<0; CiE
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\@yx;}bdI
z = y; sT|$@$bN
if any(idx_pos) INca
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); #=)(t${7'
end t*<@>] k
if any(idx_neg) ,TrrqCw>
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); o*5<Cxg
end *E"QFirk0
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yKJKQ9
% EOF zernfun j$%KKl8j