非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 s(.-bjR
function z = zernfun(n,m,r,theta,nflag) |+~2sbM
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~2}ICU5
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ~MQf($]
% and angular frequency M, evaluated at positions (R,THETA) on the 7Ej#7\TB]
% unit circle. N is a vector of positive integers (including 0), and WA5kX SdIb
% M is a vector with the same number of elements as N. Each element 3'e 4{
% k of M must be a positive integer, with possible values M(k) = -N(k) =xet+;~ji
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, &Q+V I/p
% and THETA is a vector of angles. R and THETA must have the same %9Fg1LH42r
% length. The output Z is a matrix with one column for every (N,M) 1AV1W_"
% pair, and one row for every (R,THETA) pair. 6lAo`S\)eX
%
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% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 1H
6Wrik
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), s9bP6N!,
% with delta(m,0) the Kronecker delta, is chosen so that the integral HKw:fGt/o^
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, k $&A
% and theta=0 to theta=2*pi) is unity. For the non-normalized "a{f?
.X.
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. v>!}cB/6
% K3D $
hb
% The Zernike functions are an orthogonal basis on the unit circle. G_mu7w
% They are used in disciplines such as astronomy, optics, and P`9A?aG.Z
% optometry to describe functions on a circular domain. KptLeb:Om
% i~L7h=__
% The following table lists the first 15 Zernike functions. to=##&ld<
% s7}
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% n m Zernike function Normalization 5c7a\J9>
% -------------------------------------------------- n7uD(cL
% 0 0 1 1 p'} %pAY
% 1 1 r * cos(theta) 2 M?u)H&kEl
% 1 -1 r * sin(theta) 2 w! 7/;VJ3d
% 2 -2 r^2 * cos(2*theta) sqrt(6) 3U$fMLx]k
% 2 0 (2*r^2 - 1) sqrt(3) e,UgTxZ
% 2 2 r^2 * sin(2*theta) sqrt(6) =ApT#*D)o
% 3 -3 r^3 * cos(3*theta) sqrt(8) ,SwaDWNO
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) e'&{KD,-T
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) W%cPX0
% 3 3 r^3 * sin(3*theta) sqrt(8) hDMp^^$
% 4 -4 r^4 * cos(4*theta) sqrt(10) j=S"KVp9NF
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9<mj@bI$
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) .&.CbE8K[
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u;g}N'"
% 4 4 r^4 * sin(4*theta) sqrt(10) @R{&>Q:.
% -------------------------------------------------- 0O4mA&&!oK
% ~A4WuA
% Example 1: X5[sw;rk
% z\
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% % Display the Zernike function Z(n=5,m=1) 0u\@-np
% x = -1:0.01:1; Bx>@HU
% [X,Y] = meshgrid(x,x); a$8?0`(
% [theta,r] = cart2pol(X,Y); =^v Ub
% idx = r<=1; ;A!i V|
% z = nan(size(X)); ek!N eu>
% z(idx) = zernfun(5,1,r(idx),theta(idx)); nQ~L.V
% figure U$bM:d
% pcolor(x,x,z), shading interp :tG5~sK
% axis square, colorbar .X1niguXH
% title('Zernike function Z_5^1(r,\theta)') =x>k:l~s
% 0in6z
% Example 2: |D:0BATRP
% w2[R&hJ
% % Display the first 10 Zernike functions xpwzz O*U
% x = -1:0.01:1; iX p8u**
% [X,Y] = meshgrid(x,x); {*9i}w|2
% [theta,r] = cart2pol(X,Y); v^G5
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% idx = r<=1; b\Ub<pE
% z = nan(size(X)); yl%F<5
% n = [0 1 1 2 2 2 3 3 3 3]; 5Ncd1
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; m(Ynl=c
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ^5}3FvW
% y = zernfun(n,m,r(idx),theta(idx)); g*M3;G
% figure('Units','normalized') ^(:Rbsl
% for k = 1:10 i,T{SV
% z(idx) = y(:,k); Rw`s O:eZ
% subplot(4,7,Nplot(k)) H l@rS
% pcolor(x,x,z), shading interp M(f'qFY=K
% set(gca,'XTick',[],'YTick',[]) _P:P5H8
% axis square 9qA_5x%"%u
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) V-3]h
ba,
% end dX=^>9hN/
% 9nE%r\H
% See also ZERNPOL, ZERNFUN2. 04t_
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% Paul Fricker 11/13/2006 iy9]Y5b
/([aD~.
6"(&lK\^
% Check and prepare the inputs: )Be;Zw.|
% ----------------------------- oL;/Qan
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) @gOgs
error('zernfun:NMvectors','N and M must be vectors.') dmO|PswW
end ZHJzh\?
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if length(n)~=length(m) /2@@v|QL
error('zernfun:NMlength','N and M must be the same length.') =[&Jxy>Y
end p\K5B,
i747( ^
n = n(:); yrX]w3kr%
m = m(:); p
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if any(mod(n-m,2)) -E1}mL}I`
error('zernfun:NMmultiplesof2', ... a=R-F!P)
'All N and M must differ by multiples of 2 (including 0).') M*N8p]3Cq
end #z.x3D@^r6
RZZB?vx
if any(m>n) q'q{M-U<
error('zernfun:MlessthanN', ... I
f(_$>
'Each M must be less than or equal to its corresponding N.') By9/tB
end Sy_M!`B
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if any( r>1 | r<0 ) ?kSs7e>
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ]{hfM
end xjYFTb}!
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) )
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error('zernfun:RTHvector','R and THETA must be vectors.') w'!gLta
end I(.XK ucU
yT4|eHl
r = r(:); !`gg$9
theta = theta(:); ! [X<>
length_r = length(r); y[cAU:P?
if length_r~=length(theta) `W9_LROD
error('zernfun:RTHlength', ... /[OMpP
'The number of R- and THETA-values must be equal.') =ZQIpc
end n!p&.Mt
s5.2gu|"%
% Check normalization: \0$?r4A
% -------------------- Vk"QcW
if nargin==5 && ischar(nflag) VYBl0!t
isnorm = strcmpi(nflag,'norm'); >\'yj|
U,
if ~isnorm >Ry4Cc
error('zernfun:normalization','Unrecognized normalization flag.') ]WG\+1x9
end ^6`U0|5mRX
else h5JXKR.1]c
isnorm = false; n;U|7it7
end 6=
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7;XdTx
% Compute the Zernike Polynomials D|xSO~M5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yVL~SH|
AXyuXB
% Determine the required powers of r: Y9WH%
% ----------------------------------- >g?,BK@
m_abs = abs(m); eg3{sDv,
rpowers = []; Abl=Ev
for j = 1:length(n) 5XhV+t
g.
rpowers = [rpowers m_abs(j):2:n(j)]; <ANKoPNie
end ,FTF@h-Cs
rpowers = unique(rpowers); gC 4w&yL
>4Lb+]
% Pre-compute the values of r raised to the required powers, 6jn<YR
E-
% and compile them in a matrix: 43eGfp'
% ----------------------------- yS?1JWUC>
if rpowers(1)==0 cX*^PSM
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ~&pk</Dl
rpowern = cat(2,rpowern{:}); -x7L8Wj
rpowern = [ones(length_r,1) rpowern]; W46sKD;\^W
else %>f:m!.
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Rk'Dd4"m,
rpowern = cat(2,rpowern{:}); ''Hq-Ng
end yCz?V[49
th]9@7UE,
% Compute the values of the polynomials: 3y@'p(}Az
% -------------------------------------- 8Hhe&B
y = zeros(length_r,length(n)); eq"~by[Uq
for j = 1:length(n) 4U((dx*m
s = 0:(n(j)-m_abs(j))/2; x*YJ:t
pows = n(j):-2:m_abs(j); d.{RZq2cp
for k = length(s):-1:1 (Yx rZ_F'b
p = (1-2*mod(s(k),2))* ... tDi<n}
prod(2:(n(j)-s(k)))/ ... ?znSA
>
prod(2:s(k))/ ... NE(6`Wq`
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 43/|[
prod(2:((n(j)+m_abs(j))/2-s(k))); Jzr(A^vwo
idx = (pows(k)==rpowers); w}'E]y2.
y(:,j) = y(:,j) + p*rpowern(:,idx); W4Eo1 E
end _h5@3>b3r
Abj`0\
if isnorm 40Du*5M
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ~2pctqMA
end "xh]>_;&'
end Tj.;\a|d
% END: Compute the Zernike Polynomials r`"
? K]rI
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yXDf;`J
tn p]wZ
% Compute the Zernike functions: 7Npz
{C{I
% ------------------------------ 1{DHlyA6g
idx_pos = m>0; vHao
y
idx_neg = m<0; N^)L@6
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z = y; =6? 3c\
if any(idx_pos) 5:O"T
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); +('jqbV
end {4#'`Eejj
if any(idx_neg) 4).q+{#k
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)');
pO"V9[p]
end ?+51 B-
p#3P`I>ZrT
% EOF zernfun