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 /W0E(8:C)
function z = zernfun(n,m,r,theta,nflag) [,GU5,o
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. /ISLVp%H
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N shNE~TA
% and angular frequency M, evaluated at positions (R,THETA) on the f,JX"
% unit circle. N is a vector of positive integers (including 0), and T*R{L
% M is a vector with the same number of elements as N. Each element ?DRR+n _
% k of M must be a positive integer, with possible values M(k) = -N(k) !pl_Ao~(
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, `{<JC{yc?
% and THETA is a vector of angles. R and THETA must have the same KD=bkZ&
% length. The output Z is a matrix with one column for every (N,M) $NdH*
% pair, and one row for every (R,THETA) pair. 0:#7M}U
% 5v+L';wx[T
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /vy?L\`)#
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), %b9fW
% with delta(m,0) the Kronecker delta, is chosen so that the integral xRB7lV*
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, EzUPah
% and theta=0 to theta=2*pi) is unity. For the non-normalized ^F&A6{9f/h
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 7~D`b1||
% ?jFc@t*\:
% The Zernike functions are an orthogonal basis on the unit circle. 2$3kKY6$e
% They are used in disciplines such as astronomy, optics, and _\!0t
% optometry to describe functions on a circular domain. *.xZfi_|
% g&XhQ.aa
% The following table lists the first 15 Zernike functions. mgxz1d
% a 1NCVZ
% n m Zernike function Normalization &jFKc0\i@
% -------------------------------------------------- *n,UOHlO
% 0 0 1 1 Ir^ BC!<2>
% 1 1 r * cos(theta) 2 T6;>O`B.r
% 1 -1 r * sin(theta) 2 EL"4E',
% 2 -2 r^2 * cos(2*theta) sqrt(6) 0T=jR{j!o
% 2 0 (2*r^2 - 1) sqrt(3) ea>[BB3#
% 2 2 r^2 * sin(2*theta) sqrt(6) BJ"Ay@D*
% 3 -3 r^3 * cos(3*theta) sqrt(8) "AV1..mu
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Bg5;Q)
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) S7Qen6lm
% 3 3 r^3 * sin(3*theta) sqrt(8) ?F9hDLX
% 4 -4 r^4 * cos(4*theta) sqrt(10) UQSX<6"
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |HNQ|r_5S
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) JE/l#Q!
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #DrZ`Aq
% 4 4 r^4 * sin(4*theta) sqrt(10) {7/ A
% -------------------------------------------------- zV6AuUIt
% !'Gb$l!
% Example 1: anpJAB:1
% )H.ubM1
% % Display the Zernike function Z(n=5,m=1) \\y}DNh
% x = -1:0.01:1; _!|=AIX
% [X,Y] = meshgrid(x,x); @"jmI&hYn
% [theta,r] = cart2pol(X,Y); k\Yu5)
% idx = r<=1; yY-FL`-
% z = nan(size(X)); yp( ?1
% z(idx) = zernfun(5,1,r(idx),theta(idx)); e?_c[`sg
% figure 8}ii3P y
% pcolor(x,x,z), shading interp D!81(}p
% axis square, colorbar l2z`<2mp
% title('Zernike function Z_5^1(r,\theta)') ,?P8m"
% %ZJ),9+
% Example 2: 2<p5_4"-U*
% K7)j
% % Display the first 10 Zernike functions -='8_B/75
% x = -1:0.01:1; oHYD_8'f
% [X,Y] = meshgrid(x,x); %4QoF
% [theta,r] = cart2pol(X,Y); !7kAJG g
% idx = r<=1; Dx p>
% z = nan(size(X)); AH"g^ gw~T
% n = [0 1 1 2 2 2 3 3 3 3]; HV#?6,U}
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; l5":[C$
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 8=ukS_?Vy
% y = zernfun(n,m,r(idx),theta(idx)); b/a?\0^
% figure('Units','normalized') hY4)W
% for k = 1:10 n.;5P {V1
% z(idx) = y(:,k); ;]l{D}
% subplot(4,7,Nplot(k)) *il]$i
% pcolor(x,x,z), shading interp \N'hbT=
% set(gca,'XTick',[],'YTick',[]) *SMoodFBS
% axis square te! ]9rR
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) IPr*pQ{;c
% end %^Q@*+{:f
% ~/]\iOL
% See also ZERNPOL, ZERNFUN2. )-TeDIfm
b3CspBgC
% Paul Fricker 11/13/2006 '6dD^0dZ
`-9*@_-=M
#J<`p
% Check and prepare the inputs: s)`1Rf
% -----------------------------
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) J*^,l`C/
error('zernfun:NMvectors','N and M must be vectors.') SSA%1l2!
end ],fwZd[t
r(?'Y y
if length(n)~=length(m) Fw_bY/WN{
error('zernfun:NMlength','N and M must be the same length.') V5(tf'
end &t9XK 8S
l1iF}>F2
n = n(:); {Vt^Xc
m = m(:); /pSUn"3
if any(mod(n-m,2)) dwf #~7h_
error('zernfun:NMmultiplesof2', ... 8KGv?^M
6W
'All N and M must differ by multiples of 2 (including 0).') 1o5Y9#7
end c9cphZ(z
]C!Y~
if any(m>n) hq&
error('zernfun:MlessthanN', ... -G^t-I
'Each M must be less than or equal to its corresponding N.') ;nAg4ll8Q
end .9[8H:Fe
X T)hPwg.
if any( r>1 | r<0 ) X{9JSq
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 'nj&}A'
end kVG6\<c]
f@xfb
ie!
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^S;RX*
error('zernfun:RTHvector','R and THETA must be vectors.') _sf0{/< )
end ]%Q]C
8[C
kgbr+Yw2X
r = r(:); HLyFyv\
theta = theta(:); ;5JIY7t
length_r = length(r); L]L~TA<D9i
if length_r~=length(theta) +(h6{e%)
error('zernfun:RTHlength', ... wEHrer
'The number of R- and THETA-values must be equal.') O(
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end ]cGz~TN~
Z+h70,|
% Check normalization: 65`'Upu
% -------------------- n[cyK$"
if nargin==5 && ischar(nflag) PE6u8ZAb"
isnorm = strcmpi(nflag,'norm'); V~uA(3\U
if ~isnorm p?`|CE@h7
error('zernfun:normalization','Unrecognized normalization flag.') >-tH&X^
end wor'=byh\
else KiRt'
isnorm = false; Rcx'a:k
end GYb2m"a)
>.nt'BQ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Rp%\`'+Xz
% Compute the Zernike Polynomials %OfDTs
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e5/DCz
Mbi+Vv-
% Determine the required powers of r: >"$-V Y6 i
% ----------------------------------- /CQQ^/
m_abs = abs(m); x8rFMR#S=
rpowers = []; 4Z
T
for j = 1:length(n) (+Nmio
rpowers = [rpowers m_abs(j):2:n(j)]; ;x0 KaFk
end aXid;v,
rpowers = unique(rpowers); \$\(9!=
'/qe#S
% Pre-compute the values of r raised to the required powers, "a`0w9Mm}
% and compile them in a matrix: *,*:6^t
% ----------------------------- d# ?*62
if rpowers(1)==0 Vx4pP$S
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); bHH}x"d[x
rpowern = cat(2,rpowern{:}); PG~m-W+
rpowern = [ones(length_r,1) rpowern]; fjZveH0
else JU2' ~chh
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); aFc'_FrQ
rpowern = cat(2,rpowern{:}); /a/uS3&
end S?z j&XY3
*[5#g3
% Compute the values of the polynomials: /z- C
:k\
% -------------------------------------- n,'AFb4AF
y = zeros(length_r,length(n)); &I'F-F;
for j = 1:length(n) #?d>S;)+
s = 0:(n(j)-m_abs(j))/2; P9cI{RI
pows = n(j):-2:m_abs(j); ;\&bvGj8V
for k = length(s):-1:1 %fSk
"%u%<
p = (1-2*mod(s(k),2))* ... o!dkS/u-m
prod(2:(n(j)-s(k)))/ ... 1bAp{u&
prod(2:s(k))/ ... b({b5z.A
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... g$+O<a@ n
prod(2:((n(j)+m_abs(j))/2-s(k))); ?*5l}y=
idx = (pows(k)==rpowers); 4a-F4j'
y(:,j) = y(:,j) + p*rpowern(:,idx); }sNZQ89V*v
end W)P_t"'@L
|;1:$E"
if isnorm c+M@{EbuN
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ]mU*Y:<
end )Zr0_b"V:e
end K<9MK>T
% END: Compute the Zernike Polynomials ]CJ>iS!V
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r
($t.iS
iQR})=Q
% Compute the Zernike functions: 'eXw`kw(
% ------------------------------ O9IjU10:
idx_pos = m>0; x};g!FYfkB
idx_neg = m<0; wDTV /"Y
QO^X7A"?X
z = y; .Zz7LG{
if any(idx_pos) _)H+..=
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Xg#([}b
end U"G+su->e
if any(idx_neg) g}j>;T
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); *)SgdC/f
end o|im
]
:#IZ0#
% EOF zernfun