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， +:jT=V"X
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， &217l2X
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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？ x ;BbTBc >
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ ^%oUmwP<$
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function z = zernfun(n,m,r,theta,nflag) +zOOdSFk.
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. w5~i^x
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ? S=W&
% and angular frequency M, evaluated at positions (R,THETA) on the D>T],3U(H
% unit circle. N is a vector of positive integers (including 0), and ySN V^+
% M is a vector with the same number of elements as N. Each element =)<3pG O
% k of M must be a positive integer, with possible values M(k) = -N(k) {M~lbU
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 2C+(":=}
% and THETA is a vector of angles. R and THETA must have the same ;\)=f6N
% length. The output Z is a matrix with one column for every (N,M) %I4zQiJ%
% pair, and one row for every (R,THETA) pair. <ZPZk'53<f
% J 0<p4%Cf
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike \x\.
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), :pKG\A
% with delta(m,0) the Kronecker delta, is chosen so that the integral m24v@?*
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]gd/}m)1
% and theta=0 to theta=2*pi) is unity. For the non-normalized DR+,Y2!_GT
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ~\_T5/I%
% 2 g`[u|
% The Zernike functions are an orthogonal basis on the unit circle. )BV=|,j
% They are used in disciplines such as astronomy, optics, and $@[)nvV\
% optometry to describe functions on a circular domain. r:l96^xs
% pz}mF D&[
% The following table lists the first 15 Zernike functions. ,a(O`##Bn
% JAb$M{t
% n m Zernike function Normalization nX
x=1 *X
% -------------------------------------------------- ;lfWu U%R
% 0 0 1 1 *=nO
% 1 1 r * cos(theta) 2 NtZ6$o<Y
% 1 -1 r * sin(theta) 2 t3F?>G#y
% 2 -2 r^2 * cos(2*theta) sqrt(6) fNhT;Bux
% 2 0 (2*r^2 - 1) sqrt(3) (.^8^uc 7X
% 2 2 r^2 * sin(2*theta) sqrt(6) @!H
'+ c
% 3 -3 r^3 * cos(3*theta) sqrt(8) C!UEXj`l9
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) !]DuZ=
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) {OxWcK\2@h
% 3 3 r^3 * sin(3*theta) sqrt(8) 23E 0~O
% 4 -4 r^4 * cos(4*theta) sqrt(10) H @ !#;w
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ]tVl{" .{
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Va9q`XbyO
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #MM &BC
% 4 4 r^4 * sin(4*theta) sqrt(10) 4]BJ0+|mT
% -------------------------------------------------- l BiovT
% cF.mb*$K
% Example 1: ,olwwv_8G
% d^aNR
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% % Display the Zernike function Z(n=5,m=1) {[3YJkrM
% x = -1:0.01:1; @ M[Q $:
% [X,Y] = meshgrid(x,x); NW ISS
% [theta,r] = cart2pol(X,Y); m`9^.>]P
% idx = r<=1; |3@=CE7G
% z = nan(size(X)); &:8T$U V
% z(idx) = zernfun(5,1,r(idx),theta(idx)); m3?e]nL4W
% figure XtW _
% pcolor(x,x,z), shading interp _7 `E[&v
% axis square, colorbar F E6C6dW{
% title('Zernike function Z_5^1(r,\theta)') R~c1)[[E
% qc-C>Ra
% Example 2: Y\8+}g;KR
%
^@q# $/z
% % Display the first 10 Zernike functions QN #)F
% x = -1:0.01:1; cdp{W
% [X,Y] = meshgrid(x,x); SQIdJG^:
% [theta,r] = cart2pol(X,Y); 44Qk;8*
% idx = r<=1; 5&%fkZ0
% z = nan(size(X)); @U7Dunu*f
% n = [0 1 1 2 2 2 3 3 3 3]; syMm`/*/G-
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; }bgo )<i
% Nplot = [4 10 12 16 18 20 22 24 26 28]; |8;?
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% y = zernfun(n,m,r(idx),theta(idx)); >Fh#DmQ
% figure('Units','normalized') |UZOAGiBg
% for k = 1:10 ^je5 28%H
% z(idx) = y(:,k); >W~=]&7{s4
% subplot(4,7,Nplot(k)) &?}1AQAYg
% pcolor(x,x,z), shading interp @Y NGxg~*g
% set(gca,'XTick',[],'YTick',[]) W^|J/Y48
% axis square ReqE?CeV
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) G|V\^.f<
% end m9b(3
% i0i`k^bA
% See also ZERNPOL, ZERNFUN2. 7uA\ &/
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% Paul Fricker 11/13/2006 >gGi l|I
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% Check and prepare the inputs: Xiw@
% ----------------------------- jRwa0Px(
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ytob/tc
error('zernfun:NMvectors','N and M must be vectors.') F b2p(.
end ip674'bq7R
s%bUgO%&
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if length(n)~=length(m) ,aYU$~ o#
error('zernfun:NMlength','N and M must be the same length.') [DL|Ht>
end `M6YblnJZ
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n = n(:); `_ 0)kdu
m = m(:); "p`o]$Wv
if any(mod(n-m,2)) Djyp3uUA/
error('zernfun:NMmultiplesof2', ... m"q/,}DR
'All N and M must differ by multiples of 2 (including 0).') |thad!?
end a6P!Wzb
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if any(m>n) BZ}`4W'
error('zernfun:MlessthanN', ... . 2/,XwIr
'Each M must be less than or equal to its corresponding N.') ?|)rv
end )L|C'dJ<k`
h 9U+ %=^O
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if any( r>1 | r<0 ) 9 Dy)nm^
error('zernfun:Rlessthan1','All R must be between 0 and 1.') >Rr!rtc'x
end {dDq*sLf
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) h^`{ . TlN
error('zernfun:RTHvector','R and THETA must be vectors.') cu :-MpE
end #*+;B93 )
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r = r(:); p AMo
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theta = theta(:); U>bP}[&S
length_r = length(r); jm4)gmC
if length_r~=length(theta) \I:UC
%
error('zernfun:RTHlength', ... /%-o.hT
'The number of R- and THETA-values must be equal.') IC\E , m
end +J% 6bn)U
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% Check normalization: ys/ `{:w8p
% -------------------- LPb]mC6#
if nargin==5 && ischar(nflag) ,!jR:nApE
isnorm = strcmpi(nflag,'norm'); JThk Wx
if ~isnorm \f6lT3"VN
error('zernfun:normalization','Unrecognized normalization flag.') <\+Po<)3 j
end 3e#x)H/dr
else zI1(F67d`
isnorm = false; /7.wQeL9
end :fl*w""V@
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% D` 1I;Tb#
% Compute the Zernike Polynomials GOUY _&}tL
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZCj >MA
^ b=5 6~[
[^h/(a`
% Determine the required powers of r: 6-D%)Z(
% ----------------------------------- D WsCYo
m_abs = abs(m); >+zAWK9
rpowers = []; J 11 dqj
for j = 1:length(n) 8'' 9 @xz
rpowers = [rpowers m_abs(j):2:n(j)]; ^H
f+du
end 1!K !o Y
rpowers = unique(rpowers); FEge+`{,
W ]a7&S
Y+|L 3'H
% Pre-compute the values of r raised to the required powers, -HO6K) ur
% and compile them in a matrix: ?, .HA@T%
% ----------------------------- 40`9t Xn
if rpowers(1)==0 #-l!`\@
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); V5hp
Y ]
rpowern = cat(2,rpowern{:}); pE9aT5
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rpowern = [ones(length_r,1) rpowern]; Fcu Eeca
else ,e}mR>i=e
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); J R 8 Z6
rpowern = cat(2,rpowern{:}); " 8~f
end 8 /:X&
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% Compute the values of the polynomials: Dx9$H++6$X
% -------------------------------------- ^EnNbFI
y = zeros(length_r,length(n)); p{\q SPK
for j = 1:length(n) sDz )_;;%
s = 0:(n(j)-m_abs(j))/2; l4R<`b\Jt
pows = n(j):-2:m_abs(j); |H3?ox*
for k = length(s):-1:1 <z~2d
p = (1-2*mod(s(k),2))* ... RZcx4fL }x
prod(2:(n(j)-s(k)))/ ... m-~V+JU;x
prod(2:s(k))/ ... r"Hbr Qn
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ]%vGC^
prod(2:((n(j)+m_abs(j))/2-s(k))); EhmUX@k],
idx = (pows(k)==rpowers); ogkz(wZ
y(:,j) = y(:,j) + p*rpowern(:,idx); 6KBzlj0T+
end GN~[xX JU
x"z jN'|
if isnorm S'v V"
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); .=et{\
end WF3DGqs_]
end KoxGxHz^Y3
% END: Compute the Zernike Polynomials yhJA;&}>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4{Yy05PFS
oF 1W}DtA
UIm[DYMS
% Compute the Zernike functions: xPn'yo
% ------------------------------ U_ N5~#9
idx_pos = m>0; hPEp0("
idx_neg = m<0; -Ib + #pX
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z = y; ci{9ODN
if any(idx_pos) 6x (L&>F
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); priT 7!
end b}} 1TnS)
if any(idx_neg) [EW$7 se ~
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Tvksf!ba
end 1b
%T_a
&?5{z\;1"
jU3 ;jm.)
% EOF zernfun Ok~W@sYST
