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 ?Tuh22J{Q
function z = zernfun(n,m,r,theta,nflag) s+zb[3}
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. KH pxWq
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 3kR- WgVF,
% and angular frequency M, evaluated at positions (R,THETA) on the x?CjRvT$
% unit circle. N is a vector of positive integers (including 0), and VPN@q<BV
% M is a vector with the same number of elements as N. Each element 9}}D -&Mc
% k of M must be a positive integer, with possible values M(k) = -N(k) u"T5m
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, @TgCI`E
% and THETA is a vector of angles. R and THETA must have the same 5\|[)~b
% length. The output Z is a matrix with one column for every (N,M) }QJE9;<e
% pair, and one row for every (R,THETA) pair. S\0"G*
% U^pe/11)H
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike
IR/0gP
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), u x:,io
% with delta(m,0) the Kronecker delta, is chosen so that the integral ; UrwK
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ,?&hqM\
% and theta=0 to theta=2*pi) is unity. For the non-normalized 8(3vNuyP
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Q|5wz]!5Y(
% tm]75*?
% The Zernike functions are an orthogonal basis on the unit circle. [*}[W6
3v
% They are used in disciplines such as astronomy, optics, and FNc[2sI
% optometry to describe functions on a circular domain. I">z#@CT
% Kq{s^G
% The following table lists the first 15 Zernike functions. { +C>^b
% 6Ir
?@O1'!
% n m Zernike function Normalization 0A 4(RLGg
% -------------------------------------------------- VKN^gz
% 0 0 1 1 h)y"?Jj
% 1 1 r * cos(theta) 2 ORtl~V'
% 1 -1 r * sin(theta) 2 1GEE ^Eu
% 2 -2 r^2 * cos(2*theta) sqrt(6) W;KHLHp-
% 2 0 (2*r^2 - 1) sqrt(3) _&F6As
!{
% 2 2 r^2 * sin(2*theta) sqrt(6) sp_(j!]jX
% 3 -3 r^3 * cos(3*theta) sqrt(8) a'T|p)N.;T
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ; $y.+5 q
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) sgCIY:8
% 3 3 r^3 * sin(3*theta) sqrt(8) 8:t-I]dzk
% 4 -4 r^4 * cos(4*theta) sqrt(10) -CL7^
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) aaaC8;.
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 8<]> q
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) iicrRGp3
% 4 4 r^4 * sin(4*theta) sqrt(10) ,YkQJ$
% -------------------------------------------------- 8|d[45*q
% vxqMo9T
% Example 1: Nju7!yVM_
% }*R"yp
% % Display the Zernike function Z(n=5,m=1) Hfc^<q4a.
% x = -1:0.01:1; |_w*:NCV5
% [X,Y] = meshgrid(x,x); 3}h&/KN{
% [theta,r] = cart2pol(X,Y); "V9!srIC
% idx = r<=1; 5T
% z = nan(size(X)); S>"dUM
% z(idx) = zernfun(5,1,r(idx),theta(idx)); *g=*}2
% figure x65e,'
% pcolor(x,x,z), shading interp 2<988F
% axis square, colorbar +- .BF"}
% title('Zernike function Z_5^1(r,\theta)') rBny*! n
% hG
]j m
% Example 2: QY^ y(I49
% lZ.,"F@
% % Display the first 10 Zernike functions X,OxvmDm
% x = -1:0.01:1; Yb?#vp I
% [X,Y] = meshgrid(x,x); IvO3*{k,
% [theta,r] = cart2pol(X,Y); ED$gnFa3I
% idx = r<=1; 2XXEg>CU
% z = nan(size(X)); u{3KV6MS
% n = [0 1 1 2 2 2 3 3 3 3]; /S;?M\
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; gq0gr?
% Nplot = [4 10 12 16 18 20 22 24 26 28]; GB
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% y = zernfun(n,m,r(idx),theta(idx)); NKB!_R+
% figure('Units','normalized') I+<`}
% for k = 1:10 B[$SA-ZHi
% z(idx) = y(:,k); [2FXs52
% subplot(4,7,Nplot(k)) 6g.@I!j E
% pcolor(x,x,z), shading interp >)5vsqGZaK
% set(gca,'XTick',[],'YTick',[]) ~z'0~3
% axis square Tl1?5
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ,%jJ
,G,
% end Qy$QOtrv
% keWgbj
% See also ZERNPOL, ZERNFUN2. ,2cw9?<
/0\pPc*kA{
% Paul Fricker 11/13/2006 @+2Zt%
ZHF@k'vm/9
Mr1pRIYMd
% Check and prepare the inputs: >N3{*W
% ----------------------------- AVJF[t ,
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) pTN_6=Y"
error('zernfun:NMvectors','N and M must be vectors.') ^{Vm,nAQqs
end r;'!qwr
Th8Q~*v
if length(n)~=length(m) -5qO}^i$a
error('zernfun:NMlength','N and M must be the same length.') y%GV9
end 2`},;i~[
y~d W=zO
n = n(:); *PI3L/*
m = m(:); tT
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if any(mod(n-m,2)) \.{JS>!
error('zernfun:NMmultiplesof2', ... IR>Kka(B
'All N and M must differ by multiples of 2 (including 0).') iKK=A.g
end ydFY<Mb(o
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if any(m>n) XmnqZWB
error('zernfun:MlessthanN', ... 0hv[Ff
'Each M must be less than or equal to its corresponding N.') ?y
'.sQ
end jsG9{/Ov3
%z2nas$$g
if any( r>1 | r<0 ) ^g0 Ig2'
error('zernfun:Rlessthan1','All R must be between 0 and 1.') j[HKC0C6
end !of7]s
f*I5m=
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) J\%:jg( m
error('zernfun:RTHvector','R and THETA must be vectors.') e,x@?L*
end 0N}5sF
;klDt|%3j
r = r(:); 2ReulL8j
theta = theta(:); q[#\qT&QU
length_r = length(r); ]@f6O*&=
if length_r~=length(theta) jTqba:q@
error('zernfun:RTHlength', ... w iq{Jo#
'The number of R- and THETA-values must be equal.') Q?>#sN,
end i~.[iZf|
V?"^Ff3m!
% Check normalization: vW_A.iI"e
% -------------------- 4EpzCaEZ
if nargin==5 && ischar(nflag) U%4g:s
isnorm = strcmpi(nflag,'norm'); NqWHR~&
if ~isnorm 5A)w.i&V
error('zernfun:normalization','Unrecognized normalization flag.') Z 0:2x(x9
end 8x{vgx @M
else l=oVC6C
isnorm = false; EK'&S=]
end e%P;Jj476
7mjj%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% WNx^Rg"
>'
% Compute the Zernike Polynomials bXfOZFzq)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1Q%.-vs
^Pf&C0xXv
% Determine the required powers of r: z>{KeX:
% ----------------------------------- :}-u`K*
m_abs = abs(m); xsvs3y |
rpowers = []; v Y\O=TZT
for j = 1:length(n) WU4i-@Bm8
rpowers = [rpowers m_abs(j):2:n(j)]; !$?@;}=
end <7 rK
rpowers = unique(rpowers); JA}'d7yEa
vP4Ij
% Pre-compute the values of r raised to the required powers, cg.e(@(
% and compile them in a matrix: ^ZlV1G;/W@
% ----------------------------- 9e&#;6l
if rpowers(1)==0 n96gDH*
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Psa8OJan
rpowern = cat(2,rpowern{:}); :6/OU9f/R
rpowern = [ones(length_r,1) rpowern]; U{_s1
else 5:d2q<x:{
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8?YW i
rpowern = cat(2,rpowern{:}); S=g-&lK
end gB(W`:[
J){\h-4
% Compute the values of the polynomials: QT$1D[>
% -------------------------------------- ."X~?Nk
y = zeros(length_r,length(n)); |BkY"F7m9
for j = 1:length(n) ?>8zU;Aj
s = 0:(n(j)-m_abs(j))/2; T/;hIX:R
pows = n(j):-2:m_abs(j); <`3(i\-X
for k = length(s):-1:1 C6M/$_l&a
p = (1-2*mod(s(k),2))* ... }Yl=lcvw
prod(2:(n(j)-s(k)))/ ... +E~`H^
prod(2:s(k))/ ... ] )L'Rk#4
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 5)7mjyo%
prod(2:((n(j)+m_abs(j))/2-s(k))); ewG21 q$
idx = (pows(k)==rpowers); a~7osRmp0
y(:,j) = y(:,j) + p*rpowern(:,idx); ws?s
end ?v8k& q^q
*`-29eR"8
if isnorm sri#L+I
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ->&VbR)
end J\ +gd%
end bJF/daC5
% END: Compute the Zernike Polynomials P i!r}m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @%nUfG7TQ
u^$Md WP
% Compute the Zernike functions: n E0~Y2
% ------------------------------ >Ro n+
oe
idx_pos = m>0; Q|KD/s??
idx_neg = m<0; t/WnDR/fM
idS
RWa
z = y; T\!SA
if any(idx_pos) llfiNEK5;
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); .0E4c8R\X
end 51opP8
if any(idx_neg) <lmJa#
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); K_|~3g
end (E}cA&{
'xZPIj+
% EOF zernfun