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 phwBil-vUU
function z = zernfun(n,m,r,theta,nflag) #eF
k
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Y^XZ.R
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N N Ff`V
% and angular frequency M, evaluated at positions (R,THETA) on the tg9{(_t/W
% unit circle. N is a vector of positive integers (including 0), and ):n'B` f}z
% M is a vector with the same number of elements as N. Each element _,f7D/dq
% k of M must be a positive integer, with possible values M(k) = -N(k) nB}eJD|
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, b=SCyGxlZ5
% and THETA is a vector of angles. R and THETA must have the same ~K
('t9|
% length. The output Z is a matrix with one column for every (N,M) `1#Z9&bO
% pair, and one row for every (R,THETA) pair. ']Z%6_WF
% 7Jpq7;
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 6zGeGW
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), R'oGsaPB2
% with delta(m,0) the Kronecker delta, is chosen so that the integral q#"lnc<S
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, >x
]{cb/m
% and theta=0 to theta=2*pi) is unity. For the non-normalized sWi4+PAM0
% polynomials, max(Znm(r=1,theta))=1 for all [n,m].
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% M}
+s_h9
% The Zernike functions are an orthogonal basis on the unit circle. `9A`pC
% They are used in disciplines such as astronomy, optics, and r&~]6
U
% optometry to describe functions on a circular domain. <<-BQ
l~
% 6p.y/LMO
% The following table lists the first 15 Zernike functions. ^KV:.up6
% b{
tp
qNm~
% n m Zernike function Normalization ?/(*cA
% -------------------------------------------------- Fw^^sB
% 0 0 1 1 FS*J8)
% 1 1 r * cos(theta) 2 +6L.a3&(b
% 1 -1 r * sin(theta) 2 KbH|'/w
% 2 -2 r^2 * cos(2*theta) sqrt(6) ziv+*Qn_b4
% 2 0 (2*r^2 - 1) sqrt(3) _*xY>?Aq
% 2 2 r^2 * sin(2*theta) sqrt(6) -oY8]HrXfK
% 3 -3 r^3 * cos(3*theta) sqrt(8) V|<'o<h8
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) mt[ #=Yba
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) IiY%y:!g
% 3 3 r^3 * sin(3*theta) sqrt(8) $-
Y8@bw
% 4 -4 r^4 * cos(4*theta) sqrt(10) 5JG`FRW!
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) th5UzpB4
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) !P6?nS
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7_eV.'h
% 4 4 r^4 * sin(4*theta) sqrt(10) j|b$b,rF\
% -------------------------------------------------- _P%PjFQ)
% ZbH_h]1$D
% Example 1: `6PBV+]Vm3
% LCb0Kq}*/(
% % Display the Zernike function Z(n=5,m=1) QJI]@3
Y
% x = -1:0.01:1; %~0]o@LW7
% [X,Y] = meshgrid(x,x); {#&D=7LP
% [theta,r] = cart2pol(X,Y); sGa "
% idx = r<=1; _j~y;R)
% z = nan(size(X));
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% z(idx) = zernfun(5,1,r(idx),theta(idx)); xX8c>p
% figure MYVb !
% pcolor(x,x,z), shading interp zv^+8h7k
% axis square, colorbar =r&i`L{]
% title('Zernike function Z_5^1(r,\theta)') yz)Nco]
% [lzH%0
V
% Example 2: "Q{7X[$$^
% bvT$/(7
% % Display the first 10 Zernike functions 8SCXA9}
% x = -1:0.01:1; .mxc~
% [X,Y] = meshgrid(x,x); \t? ;p-+ta
% [theta,r] = cart2pol(X,Y); e|^.N[W
% idx = r<=1; oMNBK/X_
% z = nan(size(X)); cq/@ng*o
% n = [0 1 1 2 2 2 3 3 3 3]; dx.Jv/Mb
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; tn |H~iF{
% Nplot = [4 10 12 16 18 20 22 24 26 28]; _9S"rH[
% y = zernfun(n,m,r(idx),theta(idx)); C k/DV
% figure('Units','normalized') 'a~F'FN$
% for k = 1:10 9|K:\!7
% z(idx) = y(:,k); m,F4N$
% subplot(4,7,Nplot(k)) p.,`3"C1
% pcolor(x,x,z), shading interp $M1;d1e6'
% set(gca,'XTick',[],'YTick',[]) #=Whh
9-d
% axis square *#GX~3A
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]=Wq&~
% end V! .I>
% !)`m mr
% See also ZERNPOL, ZERNFUN2. W>[TFdH?
wid
% Paul Fricker 11/13/2006 |WgFLF~k
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1
GMpg+rK
% Check and prepare the inputs: s|R`$+'{
% ----------------------------- k7 Ne(4P
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 8]4W@~c
error('zernfun:NMvectors','N and M must be vectors.') ?O9|
end 41+@!`z7
HrRw
if length(n)~=length(m) LfvRH?<W
error('zernfun:NMlength','N and M must be the same length.') g c<Y?a-
end j(c;r>
hKems3
n = n(:); A|m0.'/
m = m(:); /5?tXH"
if any(mod(n-m,2)) u\f QaQV
error('zernfun:NMmultiplesof2', ... _A=i2?g
'All N and M must differ by multiples of 2 (including 0).') R l)g[s
end "}0)~,{xB
- P4X@s_;
if any(m>n) B!J&=*=e
error('zernfun:MlessthanN', ... JRDIGS_~
'Each M must be less than or equal to its corresponding N.') 3+!G9T!
end z%$M
IC
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if any( r>1 | r<0 ) GwmYhG<{
error('zernfun:Rlessthan1','All R must be between 0 and 1.') P[H 4Yp
end ^KQZ;[B
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Z|d+1i
error('zernfun:RTHvector','R and THETA must be vectors.') Qn@[{%),4
end L;
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eqyUI|e
r = r(:); '.I0n
theta = theta(:); | 8akp
length_r = length(r); zOis}$GR
if length_r~=length(theta) @680.+Kw
error('zernfun:RTHlength', ... &p55Cg@e)
'The number of R- and THETA-values must be equal.') VrJf g
end M4t:)!dji?
)c.!3n/pb
% Check normalization: ~{t<g;F
% -------------------- 3.Jk-:u %m
if nargin==5 && ischar(nflag) S]gV! Q4%
isnorm = strcmpi(nflag,'norm'); ",S146Y+
if ~isnorm w%%*3[--X
error('zernfun:normalization','Unrecognized normalization flag.') z#d*Odc
end
$qiM_06
else .F0Q<s9
isnorm = false; Q|7m9~
end w[u>*I
SjvSnb_3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +O7GgySx
% Compute the Zernike Polynomials $]J<^{v
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i`!>zl+D
$IJ"fs
% Determine the required powers of r: )vGxF}I3
% ----------------------------------- lXutZ<S[
m_abs = abs(m); ~b6c:db3
rpowers = []; W A#y&
for j = 1:length(n) w$jSlgUHy)
rpowers = [rpowers m_abs(j):2:n(j)]; tSVS ogGd
end C-^8;xd
rpowers = unique(rpowers); c7]0>nU;
<lRjh7
% Pre-compute the values of r raised to the required powers, @={
qy}
% and compile them in a matrix: r>6FJ:Tx
% ----------------------------- e1ExB#
if rpowers(1)==0 }|],UXk{xB
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); jEL"Q?#
rpowern = cat(2,rpowern{:}); ##s:Ww
rpowern = [ones(length_r,1) rpowern]; 8>|<m'e^\r
else mJsU7bD`
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); {O4&HW%
rpowern = cat(2,rpowern{:}); D@m3bsMwe
end b{.Y?.U
8(}cbW
% Compute the values of the polynomials: D4T(Dce
% -------------------------------------- m:cWnG
y = zeros(length_r,length(n)); E*L5D4Kw
for j = 1:length(n) \cHFV
s = 0:(n(j)-m_abs(j))/2; OUy}1%HY
pows = n(j):-2:m_abs(j); hcR^?
for k = length(s):-1:1 *`t3z-L
p = (1-2*mod(s(k),2))* ... -gv[u,R
prod(2:(n(j)-s(k)))/ ... .i1|U8" X
prod(2:s(k))/ ... 5YXMnYt9
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Sd\oL*lN
prod(2:((n(j)+m_abs(j))/2-s(k))); A$l
idx = (pows(k)==rpowers); Rpxg
5
y(:,j) = y(:,j) + p*rpowern(:,idx); Mz;KXP
end 72= 4#
QiNLE'19^
if isnorm zT'(I6S:)
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); :#=BwdC
end 03!#99
end |A2o$H
% END: Compute the Zernike Polynomials {&nDm$KTD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ESuP ZB
C-/+n5J
% Compute the Zernike functions: H:mcex
% ------------------------------ [+qB^6I+P%
idx_pos = m>0; )00jRuF
idx_neg = m<0; xj JoWB
G~/*!?&z
z = y; [>lQiX
if any(idx_pos) d,o|>e$
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); jV#1d8qm
end }S}%4c>
if any(idx_neg) ?_.
SV g
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); iAXF;'|W
end eH%i8a
|wuN`;gc"
% EOF zernfun