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， 9TIyY`2!
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， .{1G"(z
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？ 1XSA3;ZEc
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ 9z$]hl
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function z = zernfun(n,m,r,theta,nflag) g{.>nE^Sc5
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. K kP}z
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N u_;*Ay
% and angular frequency M, evaluated at positions (R,THETA) on the +FfT)8@W
% unit circle. N is a vector of positive integers (including 0), and nm'sub
% M is a vector with the same number of elements as N. Each element o@>{kzCx
% k of M must be a positive integer, with possible values M(k) = -N(k) ;5:g%Dt
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 0 #pjfc `:
% and THETA is a vector of angles. R and THETA must have the same }zY)H9J~
% length. The output Z is a matrix with one column for every (N,M) |5 _bFB+&
% pair, and one row for every (R,THETA) pair. ;2Db/"`t
% R7;SZo
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike nd3=\. (P
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), uSLO"\zysX
% with delta(m,0) the Kronecker delta, is chosen so that the integral )xX(Et6+`
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, >J_{ mU
% and theta=0 to theta=2*pi) is unity. For the non-normalized 5cO}Jp%PA
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ^m;dEe&@F
% J]UlCg
% The Zernike functions are an orthogonal basis on the unit circle. d)1)/Emyj
% They are used in disciplines such as astronomy, optics, and >!s = f
% optometry to describe functions on a circular domain. WMnR+?q
% \HLI
y
% The following table lists the first 15 Zernike functions.
F' s($n
% SweaE Rl
% n m Zernike function Normalization ?BT\)@ h
% -------------------------------------------------- ^.5 L\
% 0 0 1 1 /+l3
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% 1 1 r * cos(theta) 2 pJr c\`D
% 1 -1 r * sin(theta) 2 _QbLg"O
% 2 -2 r^2 * cos(2*theta) sqrt(6) ;
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% 2 0 (2*r^2 - 1) sqrt(3) !8yw!hA
% 2 2 r^2 * sin(2*theta) sqrt(6) | :$D[=
% 3 -3 r^3 * cos(3*theta) sqrt(8) vpcHJ ^19
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) %;yDiQ !+
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) #DApdD9M
% 3 3 r^3 * sin(3*theta) sqrt(8) #t:]a<3Y2
% 4 -4 r^4 * cos(4*theta) sqrt(10) Pk9s~}X
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ePdM9%
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ZKzXSI4
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) iwWy]V m7
% 4 4 r^4 * sin(4*theta) sqrt(10) jY
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% -------------------------------------------------- Xt~`EN
% zvf:*N a")
% Example 1: @P#uH5U
% qIcQPJn!}
% % Display the Zernike function Z(n=5,m=1) eZWN9#p2
% x = -1:0.01:1; V#.;OtF]
% [X,Y] = meshgrid(x,x); sUN>uroi !
% [theta,r] = cart2pol(X,Y); ^8$CpAK]M
% idx = r<=1; Y^m 2ealC
% z = nan(size(X)); jXvG L
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Y$b4Ga9j
% figure 6NH.!}"G9
% pcolor(x,x,z), shading interp lS]<~
% axis square, colorbar <8Ek-aNNt
% title('Zernike function Z_5^1(r,\theta)') 1{4d)z UB
% kKVd4B[#*
% Example 2: =Xh^@ OR
% _/ bF t6
% % Display the first 10 Zernike functions F+,X%$A#?
% x = -1:0.01:1; ljV tFm<
% [X,Y] = meshgrid(x,x); []:;8fY
% [theta,r] = cart2pol(X,Y); !|_b}/
% idx = r<=1; .w/#S-at
% z = nan(size(X)); fL.;-
% n = [0 1 1 2 2 2 3 3 3 3]; r ?Jxl<
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 9)0AwLlv
% Nplot = [4 10 12 16 18 20 22 24 26 28]; s^ rO I~
% y = zernfun(n,m,r(idx),theta(idx)); <$wh@$PK
% figure('Units','normalized') ,Q4 U<`ds!
% for k = 1:10 | qtdmm
% z(idx) = y(:,k); "}Kvx{L8
% subplot(4,7,Nplot(k)) A`<#}~A
% pcolor(x,x,z), shading interp ;8/w'oe *j
% set(gca,'XTick',[],'YTick',[]) #P *%FgROl
% axis square *@ o@>
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 26JP<&%L
% end /]"&E"X"
% VTk6.5!8
% See also ZERNPOL, ZERNFUN2. H+vONg
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% Paul Fricker 11/13/2006 _Z+tb ]
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l.}=
% Check and prepare the inputs: ;tTM3W-h
% ----------------------------- EJ{Z0R{{
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) IK5FSN]s/
error('zernfun:NMvectors','N and M must be vectors.') UgDa i?b1
end &[,g `S0
IF~i*
L;/#D> U (
if length(n)~=length(m) 6cz%>@
error('zernfun:NMlength','N and M must be the same length.') ;KJJK#j
end c nvx TI<
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rt$ z&#M
n = n(:); n4\6\0jq6
m = m(:); W7.O(s,32
if any(mod(n-m,2)) mE)65@3%
error('zernfun:NMmultiplesof2', ... 2uFaAAT
'All N and M must differ by multiples of 2 (including 0).') v'i"Q
end h
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if any(m>n) ,Hh7' `
error('zernfun:MlessthanN', ... nL!h hseH
'Each M must be less than or equal to its corresponding N.') nR4L4tdS
end XSt5s06TM
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if any( r>1 | r<0 ) '7o' u]
error('zernfun:Rlessthan1','All R must be between 0 and 1.') hu~02v5
end FQNhn+A
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) zk5sAHQ
error('zernfun:RTHvector','R and THETA must be vectors.') U g^C}".&
end W>
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r = r(:); {YZ)Ia qZ
theta = theta(:); Q>7#</i\.
length_r = length(r); ,e+.Q#r*Y
if length_r~=length(theta) 1 6;l,@
error('zernfun:RTHlength', ... :Q2\3
'The number of R- and THETA-values must be equal.') Z)'jn8?P
end _P tf^+
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]_I<-}?;
% Check normalization: /b6Y~YbgU
% -------------------- L` FsK64@
if nargin==5 && ischar(nflag) Hf+A5 2lrf
isnorm = strcmpi(nflag,'norm');
/Z! ,1
if ~isnorm gXI_S 9 z
error('zernfun:normalization','Unrecognized normalization flag.') Djx9TBZ5
end /eDah3%d
else @dX0gHU[c
isnorm = false; asP> (Li
end RyD2LAf)J
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7}Z.g9<
% Compute the Zernike Polynomials C yC<{D+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mm Y~V:,Kd
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% Determine the required powers of r: !? ?Cxs'
% ----------------------------------- Zm%}AzM
m_abs = abs(m); 1mOZ\L!m*
rpowers = []; OTtSMO
for j = 1:length(n) c} Jy'F7&f
rpowers = [rpowers m_abs(j):2:n(j)]; dDW],d}B;
end hw_7N)}
rpowers = unique(rpowers); 0LoA-c<Ay
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Wr`=P ,
% Pre-compute the values of r raised to the required powers, l,h#RTfry
% and compile them in a matrix: Bp^> R`,
% ----------------------------- d(, -13
if rpowers(1)==0 Q9K
Gf;
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 8 /b_4!5c
rpowern = cat(2,rpowern{:}); 9L%&4V}BIS
rpowern = [ones(length_r,1) rpowern]; }n=Tw92g
else \ :}) R{
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); kSU*d/}*u
rpowern = cat(2,rpowern{:}); %P_\7YBC>
end =%U t&6}sQ
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% Compute the values of the polynomials: AEPgQ9#E
% -------------------------------------- 4}580mBc
y = zeros(length_r,length(n)); 7`X"B*`~b
for j = 1:length(n) )t&|oQ3sVG
s = 0:(n(j)-m_abs(j))/2; cWFv YF
pows = n(j):-2:m_abs(j); ;oh88,*'
for k = length(s):-1:1 QI= SR
p = (1-2*mod(s(k),2))* ... G D [~4G
prod(2:(n(j)-s(k)))/ ... CvQ LF9 |
prod(2:s(k))/ ... z &<Rx[
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... m>:%[vm
prod(2:((n(j)+m_abs(j))/2-s(k))); i=o>Bl@f
idx = (pows(k)==rpowers); 2_r}4)z
y(:,j) = y(:,j) + p*rpowern(:,idx); b% $S6.
end 6J%SkuxR
b@1 QE
if isnorm dUb(C1h
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 6ap,XFRMh
end Z|8f7@k{|+
end \vQ_ :-A
% END: Compute the Zernike Polynomials Q2 rZMK
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% = 1}-]ctVn
/f%u_ 8pV%
`R:<(:
% Compute the Zernike functions: &~E=T3
% ------------------------------ ~d{E> J77j
idx_pos = m>0; /cI]Z^&
idx_neg = m<0; !+>yCy$~_
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z = y; y]?$zbB
if any(idx_pos) 9s*Lzi[}
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); /E] 4N =T
end tD4IwX
if any(idx_neg) cK-!Evv
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ,{oP`4\Lm
end (O`=$e
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nosEo? {
% EOF zernfun l }O` cC
