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 7m:, -xp
function z = zernfun(n,m,r,theta,nflag) ],|B4\b ;
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. QeoDq
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N m6D4J=59
% and angular frequency M, evaluated at positions (R,THETA) on the *Y~64FM
% unit circle. N is a vector of positive integers (including 0), and 9~yuyv4$
% M is a vector with the same number of elements as N. Each element KMcP !N.I
% k of M must be a positive integer, with possible values M(k) = -N(k) :Ib\v88WIv
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 0b'R5I.M
% and THETA is a vector of angles. R and THETA must have the same ":ycyN@g
% length. The output Z is a matrix with one column for every (N,M) EK_^#b
% pair, and one row for every (R,THETA) pair. J;dFmZOk
% #4>F%_
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike dGe
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), yk/XfwQ5
% with delta(m,0) the Kronecker delta, is chosen so that the integral "5K:"m
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, r^)<Jy0|r
% and theta=0 to theta=2*pi) is unity. For the non-normalized v},sWjv
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 9`AQsZ2
% 1YxI q565
% The Zernike functions are an orthogonal basis on the unit circle. M;R>]wP"V
% They are used in disciplines such as astronomy, optics, and 2_Z60]
% optometry to describe functions on a circular domain. _NFJm(X.
% Z/x~:u_
% The following table lists the first 15 Zernike functions. 0'uj*Y{L
% FceT'
% n m Zernike function Normalization &0raa
% -------------------------------------------------- ,U}8(D~:
% 0 0 1 1 C'ZU .Y
% 1 1 r * cos(theta) 2 Yi`.zm
% 1 -1 r * sin(theta) 2 [Wc 73-
% 2 -2 r^2 * cos(2*theta) sqrt(6) \N30SG?o
% 2 0 (2*r^2 - 1) sqrt(3) 4~4Hst#^
% 2 2 r^2 * sin(2*theta) sqrt(6) *O~D lf
% 3 -3 r^3 * cos(3*theta) sqrt(8) uY,FugWbl
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) mwxJ#
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) vq}V0-
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% 3 3 r^3 * sin(3*theta) sqrt(8) ]CjODa
% 4 -4 r^4 * cos(4*theta) sqrt(10) SW7%SX,xM
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ad `IgZ
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) .;'xm_Gw<
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) tR3hbL$W
% 4 4 r^4 * sin(4*theta) sqrt(10) Zh@\+1]
% -------------------------------------------------- b~}}{fm&f
% 9YKEME+:
% Example 1: "<n{/x(
% {<@~;iq
% % Display the Zernike function Z(n=5,m=1) IF=rD-x
% x = -1:0.01:1; IWkBq]Y
% [X,Y] = meshgrid(x,x); ;
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% [theta,r] = cart2pol(X,Y); \iFE,z
% idx = r<=1; J0IK=Y
% z = nan(size(X)); hY!G>d{J
% z(idx) = zernfun(5,1,r(idx),theta(idx)); dn Xc- <
% figure aozk,{9-
% pcolor(x,x,z), shading interp (&S v$L@
% axis square, colorbar kQ +
% title('Zernike function Z_5^1(r,\theta)') -}%zus5
% wZ^/-
% Example 2: 0*q:p`OLw*
% UxW~yk
% % Display the first 10 Zernike functions xK/`XY
% x = -1:0.01:1; k(MQ:9'|
% [X,Y] = meshgrid(x,x); m5S/T\,X
% [theta,r] = cart2pol(X,Y); 2}NfR8
N
% idx = r<=1;
#xmUND`@
% z = nan(size(X));
m ]\L1&
% n = [0 1 1 2 2 2 3 3 3 3]; bnlL-]]9z
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; `F)Iv:;y,
% Nplot = [4 10 12 16 18 20 22 24 26 28]; IAfYlS#<yD
% y = zernfun(n,m,r(idx),theta(idx)); Th8xh=F[
% figure('Units','normalized') &5B+8>
% for k = 1:10 7
ir T6O<.
% z(idx) = y(:,k); 1xw},y6T2
% subplot(4,7,Nplot(k)) 1yhx)m;f
% pcolor(x,x,z), shading interp o` e~1
% set(gca,'XTick',[],'YTick',[]) m'pihFR:f
% axis square &rn,[w_F[
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) q+K`+& @\
% end 5+U~ZW0|+
% IflpM ]
% See also ZERNPOL, ZERNFUN2. `]%{0 Rx
dWI\VS 9
% Paul Fricker 11/13/2006 {S|uQgs6j
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m+o>`1>a
% Check and prepare the inputs: lB-Njr
% ----------------------------- {vaq,2_w
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 69_c,(M0
error('zernfun:NMvectors','N and M must be vectors.') MFC= oKD
end s#4
"f
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if length(n)~=length(m) b&LhydaJ
error('zernfun:NMlength','N and M must be the same length.') Va1|XQ<CL
end "MyYu}AD
4-m}W;igu
n = n(:); `aCcTs7~]p
m = m(:); QPBf++|
if any(mod(n-m,2)) C4b3ZcD2
error('zernfun:NMmultiplesof2', ... 1f}Dza9
'All N and M must differ by multiples of 2 (including 0).') V482V#BP
end er 97&5
0py0zE6,,
if any(m>n) Cu0 /TeEM
error('zernfun:MlessthanN', ... 9{+B lNZ
'Each M must be less than or equal to its corresponding N.') d@C93VYp
end RNm/&F1C$
/ZAEvdO*P
if any( r>1 | r<0 ) OrzDr
error('zernfun:Rlessthan1','All R must be between 0 and 1.') \wTWhr0
end ~V (WD;Mk
r=s7be
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) zhFGMF1
error('zernfun:RTHvector','R and THETA must be vectors.') ll6~8PN
end A6-JV8^
IuRKj8J)o
r = r(:); e\\ I,
theta = theta(:); dD#A.C,Rz
length_r = length(r); w@hm>6j
if length_r~=length(theta) M*%iMz
error('zernfun:RTHlength', ... qF>}"m
'The number of R- and THETA-values must be equal.') Cfa?LgSz
end {AJspLcG
*ozeoX'5D
% Check normalization: ujHqwRh
% -------------------- ~]}7|VN.}
if nargin==5 && ischar(nflag) ptX;-'j(
isnorm = strcmpi(nflag,'norm'); `^RpT]S
if ~isnorm 75gE>:f
error('zernfun:normalization','Unrecognized normalization flag.') M,NYF`;a
end ao Y"uT+
else 0&Zm3(}
isnorm = false; ]Rz]"JZ\S
end $n!saPpxS
=p$1v{L8
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )fv0H&g
% Compute the Zernike Polynomials FhW\23OC
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7n
{uxE#U)
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% Determine the required powers of r: @99@do|C
% ----------------------------------- {i3]3V"Xp
m_abs = abs(m); nT9B?P>
rpowers = []; ;ZB[g78%R%
for j = 1:length(n) 0zetOlFbO
rpowers = [rpowers m_abs(j):2:n(j)]; m%l\EE
end `9%@{Ryo
rpowers = unique(rpowers); zaa>]~g .
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% Pre-compute the values of r raised to the required powers, c:M~!CXO
% and compile them in a matrix: o[0Cv*
% ----------------------------- Fd9ypZs
if rpowers(1)==0 Y0
Ta&TYZ0
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?(4=:o
rpowern = cat(2,rpowern{:}); #D&eov?
rpowern = [ones(length_r,1) rpowern]; NO8)XJ3s
else l>Z"y\l=
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); c&J,O1){\
rpowern = cat(2,rpowern{:}); NvQN
end +.:- :
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% Compute the values of the polynomials: 5_9mA4gs@
% -------------------------------------- +\2{{~_z
y = zeros(length_r,length(n)); Wyd,7]'z)Z
for j = 1:length(n) 4DaLmQ2O
s = 0:(n(j)-m_abs(j))/2; f9?\Q'v8
pows = n(j):-2:m_abs(j); a^>0XXr}Y
for k = length(s):-1:1 1!~=8FTv
p = (1-2*mod(s(k),2))* ... |1uyJ?%B
prod(2:(n(j)-s(k)))/ ... ?zM]p"M
prod(2:s(k))/ ... B;@7
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... k|'{$/n
prod(2:((n(j)+m_abs(j))/2-s(k))); M3!A?!BU
idx = (pows(k)==rpowers); by (xv0v;
y(:,j) = y(:,j) + p*rpowern(:,idx); v=R=K
end #41~`vq3
S=@.<gS
if isnorm 8m\*~IX=
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 8GgZAu'X
end h(l4\)
end 5ro^<P0f**
% END: Compute the Zernike Polynomials W_8N?coM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FCgr
qwM71B!r
% Compute the Zernike functions: JTA65T{3
% ------------------------------ Nk*d=vj
idx_pos = m>0; -|YG**i/
idx_neg = m<0; L3/m}AH,
Fuq ;4UcbL
z = y; )O*\}6:S
if any(idx_pos) 4+"2K-]
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); GH[ATL
end eg!s[1[_
if any(idx_neg) lA>^k;+>
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); &c>%E%!"
end G<:_O-cPSv
K%iWUl;
% EOF zernfun