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 ,}15Cse
function z = zernfun(n,m,r,theta,nflag) 5y7rY!]Bf
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. /\L|F?+@
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N V5y8VT=I
% and angular frequency M, evaluated at positions (R,THETA) on the 3w9j~s
% unit circle. N is a vector of positive integers (including 0), and 'P{0K?{H-4
% M is a vector with the same number of elements as N. Each element }Z
T{
% k of M must be a positive integer, with possible values M(k) = -N(k) `IQ01FuP
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, I`"8}d@Jm
% and THETA is a vector of angles. R and THETA must have the same /0Q=}:d
% length. The output Z is a matrix with one column for every (N,M) YUo{e=m|
% pair, and one row for every (R,THETA) pair. J(*qOGBD
% rj[2XIO
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike m1x7f%_
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), sS 5 ]d8
% with delta(m,0) the Kronecker delta, is chosen so that the integral {@Y|"qIN
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, WPVur{?<
% and theta=0 to theta=2*pi) is unity. For the non-normalized V{17iRflf
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. F&US-ce:M
% :TU;%@7
% The Zernike functions are an orthogonal basis on the unit circle. ,]?Xf>
% They are used in disciplines such as astronomy, optics, and \\F^uM7,
% optometry to describe functions on a circular domain. c"BFkw
% 3V:{_~~
% The following table lists the first 15 Zernike functions. ~_WsjD0O
% GOJ*>GpS
% n m Zernike function Normalization [r'PGx
% -------------------------------------------------- sg"J00
% 0 0 1 1 L3:dANG
% 1 1 r * cos(theta) 2 yM$@*od
% 1 -1 r * sin(theta) 2 }
DY{> D>
% 2 -2 r^2 * cos(2*theta) sqrt(6) m&/{iCwp
% 2 0 (2*r^2 - 1) sqrt(3) S,Q!Xb@
% 2 2 r^2 * sin(2*theta) sqrt(6) "&jA
CI
% 3 -3 r^3 * cos(3*theta) sqrt(8) mG4myQ?$
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) QC7Ceeh]4
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) R;,&s!\<
% 3 3 r^3 * sin(3*theta) sqrt(8) Uc,D&Og
% 4 -4 r^4 * cos(4*theta) sqrt(10) H..g2;D
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) / fBi9=}+
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) P7GuFn/p~2
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) UhuEE
% 4 4 r^4 * sin(4*theta) sqrt(10) YXE?b@W"
% -------------------------------------------------- j^ L"l;m
% #m_3ls}W$
% Example 1: ]v=*WK
% qzk/P1{-
% % Display the Zernike function Z(n=5,m=1) Q 6djfEN>
% x = -1:0.01:1; 0TA{E-A
% [X,Y] = meshgrid(x,x); Kx.'^y
% [theta,r] = cart2pol(X,Y); hE>ux"_2/
% idx = r<=1; j)4:*R.Z]
% z = nan(size(X)); xWk:7 ,/
% z(idx) = zernfun(5,1,r(idx),theta(idx)); z3!j>X_w
% figure +a$'<GvP
% pcolor(x,x,z), shading interp 5\RTy}w3x
% axis square, colorbar $hexJzX
% title('Zernike function Z_5^1(r,\theta)') kO:|?}Koc
% RlH|G
% Example 2: 0* Ox>O>
% eQh@.U*S)
% % Display the first 10 Zernike functions *^j'G^n
% x = -1:0.01:1; hdky:2^3
% [X,Y] = meshgrid(x,x); -#0(Jm'
% [theta,r] = cart2pol(X,Y); V~j:!=b%v
% idx = r<=1; P{YUW~
% z = nan(size(X)); rQ~7BlE
% n = [0 1 1 2 2 2 3 3 3 3]; D$C >ZF
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 3vx5dUgl,
% Nplot = [4 10 12 16 18 20 22 24 26 28]; \Eq,4-q
% y = zernfun(n,m,r(idx),theta(idx)); [ kI|Thx
% figure('Units','normalized') f681i(q"
% for k = 1:10 &L3OP@;
% z(idx) = y(:,k); X}T/6zk
% subplot(4,7,Nplot(k)) YyOPgF] M
% pcolor(x,x,z), shading interp +O`3eP`u
% set(gca,'XTick',[],'YTick',[]) 2aQR#lcv
% axis square =l6aSr
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) }j=UO*|
% end 12
y=Eh
% ${(v
Er#}k
% See also ZERNPOL, ZERNFUN2. #-76E
^PwZP;On
% Paul Fricker 11/13/2006 {Ju
&PY~m<F
P2y`d9,Q
% Check and prepare the inputs: K9{3,!1
% ----------------------------- e/+_tC$@p@
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |wF_CZ*1
error('zernfun:NMvectors','N and M must be vectors.') bf1Tky=/
end 0,~f"Dyqy
9a\H+Y~
if length(n)~=length(m) \o-9~C\c*
error('zernfun:NMlength','N and M must be the same length.') a%\6L
end m]C|8b7Y
WiDl[l"{9
n = n(:); C\%T|ZDE
m = m(:); s98Jh(~
if any(mod(n-m,2)) %6A."sePO
error('zernfun:NMmultiplesof2', ... .3xpDVW^e
'All N and M must differ by multiples of 2 (including 0).') x`7Ch3`4}
end 3y&N}'R(F
6"3-8orj
if any(m>n) t:MeSO
error('zernfun:MlessthanN', ... I,[njlO:
'Each M must be less than or equal to its corresponding N.') 'gBns
end hw2'.}B"(
-PfBL8
if any( r>1 | r<0 ) tX'`4!{@+
error('zernfun:Rlessthan1','All R must be between 0 and 1.') @#HB6B
end ;Fo%R$y
G
=`-w
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) xIt' o(jQH
error('zernfun:RTHvector','R and THETA must be vectors.') O}#Ic$38
end b/#SkxW#S
_*&I[%I5
r = r(:); p\;\hHai
theta = theta(:); hc"l^a!7ic
length_r = length(r); TJYup%q
if length_r~=length(theta) )FLDCer
error('zernfun:RTHlength', ... MP/@Mf\<E
'The number of R- and THETA-values must be equal.') 3H^0v$S
end ^)J2tpr;]=
RIC\f_Dv
% Check normalization: 'SW%EVB
% -------------------- }-Ds%L
if nargin==5 && ischar(nflag) Uu_g_b:z
isnorm = strcmpi(nflag,'norm'); I |PEC-(
if ~isnorm tLH:'"{zx
error('zernfun:normalization','Unrecognized normalization flag.') t`M4@1S"'
end ppm=o4`s[
else b]0]*<~y
isnorm = false; )2z<5 `
end z6IOVQ*r
+N6IdDN3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I45 kPfu
% Compute the Zernike Polynomials D=+md
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /"+CH\)
E
^_4e^D]P"
% Determine the required powers of r: <mrvuWg0
% ----------------------------------- 0Cg}yy Oz
m_abs = abs(m); }4uHT.)
rpowers = []; C33BP}c]
for j = 1:length(n) x5w5xw
rpowers = [rpowers m_abs(j):2:n(j)]; x/fhlf}a}=
end vU,V[1^a
rpowers = unique(rpowers); ~mF^t7n]
F_U9;*f]
% Pre-compute the values of r raised to the required powers, ^l:~r2
% and compile them in a matrix: [X9T$7q#
% ----------------------------- {})d}dEC
if rpowers(1)==0 9T\uOaC"
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); d/8p?Km
rpowern = cat(2,rpowern{:}); 'iM#iA8
rpowern = [ones(length_r,1) rpowern]; r* q
else Z bW!c1s{
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); @Ojbu@A
rpowern = cat(2,rpowern{:}); {gC?kp
end ybC0Ee@
~|lEi1|
% Compute the values of the polynomials: <~ Dq8If
% -------------------------------------- l`bl^~xRo
y = zeros(length_r,length(n)); ;tJ}*!z
W
for j = 1:length(n) pqCp>BO?O
s = 0:(n(j)-m_abs(j))/2; sck.2-f"
pows = n(j):-2:m_abs(j); HUFm@?
for k = length(s):-1:1 :[:*kbWN-
p = (1-2*mod(s(k),2))* ... 2M+}o"g
prod(2:(n(j)-s(k)))/ ... dO1h1yJJ
prod(2:s(k))/ ... &gg Om
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... *@VS^JB
prod(2:((n(j)+m_abs(j))/2-s(k))); 1gA^Qv~?
idx = (pows(k)==rpowers); .GSK!1{@
y(:,j) = y(:,j) + p*rpowern(:,idx); 3v91 yMx
end Zv0'OX~8i
j].=,M<dxE
if isnorm MpVZL29)
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); %p(X*mVX
end @CtnV|
end uv&4
A,h
% END: Compute the Zernike Polynomials SIZ&0V
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ez/>3:;
zNO,vR[\
% Compute the Zernike functions: )Z*nm<=
% ------------------------------ cCx_tGR"
idx_pos = m>0; *`_2uBz
idx_neg = m<0; S l`F`
~<Z7\yS)
z = y; aKFY&zN?
if any(idx_pos) tZ.hSDH
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); h+!@`c>)Y
end >g;995tG
if any(idx_neg) #Q1
|]
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); <^w4+5sT/
end FfC\uuRe
Eb7GiRT#
% EOF zernfun