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 uj@<_|7
function z = zernfun(n,m,r,theta,nflag) }X?*o`sW
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. LNb![Rq
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N P:TpB6.=q
% and angular frequency M, evaluated at positions (R,THETA) on the ]3,0
8JW=
% unit circle. N is a vector of positive integers (including 0), and +g[B &A!d+
% M is a vector with the same number of elements as N. Each element w;(gi
% k of M must be a positive integer, with possible values M(k) = -N(k) :&%;s*-9
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 9|`@czw
% and THETA is a vector of angles. R and THETA must have the same yM2&cMHH~
% length. The output Z is a matrix with one column for every (N,M) ChGM7uu2
% pair, and one row for every (R,THETA) pair. m [g}vwS
% ""d>f4,S
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike v\eBL&WK
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), SDwSlwf
% with delta(m,0) the Kronecker delta, is chosen so that the integral "=(;l3-o
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, E-D5iiF
% and theta=0 to theta=2*pi) is unity. For the non-normalized _ XZ=4s
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. B`aAvD`7
% NjxW A&[ng
% The Zernike functions are an orthogonal basis on the unit circle. SS~Q ;9o
% They are used in disciplines such as astronomy, optics, and sdWl5 "
% optometry to describe functions on a circular domain. xNkY'4%
% "BRE0Ir:
% The following table lists the first 15 Zernike functions. Z]f2&
% >B
% n m Zernike function Normalization OpLSjr
% -------------------------------------------------- nS4S[|w"
% 0 0 1 1 8tMte!E
% 1 1 r * cos(theta) 2 02[II_< 1
% 1 -1 r * sin(theta) 2 )mdNvb[*n
% 2 -2 r^2 * cos(2*theta) sqrt(6) s>\g03=
% 2 0 (2*r^2 - 1) sqrt(3) pG6-.F;
% 2 2 r^2 * sin(2*theta) sqrt(6) BT3O_X`u
% 3 -3 r^3 * cos(3*theta) sqrt(8) ntV>m*^
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) =fG8YZ(
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) LDeVNVM
% 3 3 r^3 * sin(3*theta) sqrt(8) 63S1ed[
% 4 -4 r^4 * cos(4*theta) sqrt(10) :$aW@?zAY
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) L@r.R_*H?s
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 17)M.(qmuP
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) HW726K*
% 4 4 r^4 * sin(4*theta) sqrt(10) M[u3]dN
% -------------------------------------------------- , ~xU>L^
% ]ECZU
% Example 1: ;;!{m(;LS}
% Rk%M~ D*-
% % Display the Zernike function Z(n=5,m=1) o$VH,2 QF
% x = -1:0.01:1; 3gy;$}Lq T
% [X,Y] = meshgrid(x,x); *^6xt7
% [theta,r] = cart2pol(X,Y); +c`C9RXk
% idx = r<=1; "NH+qQhs
% z = nan(size(X)); [!?,TGM}^
% z(idx) = zernfun(5,1,r(idx),theta(idx)); [9om"'
% figure ZHlin#"
% pcolor(x,x,z), shading interp Z(mn
U;9{v
% axis square, colorbar .oj" ru
% title('Zernike function Z_5^1(r,\theta)') KHz838C]
% g/Jj]X#r
% Example 2: D{c>i`\G
% Z'dI!8(Nf
% % Display the first 10 Zernike functions 8M+F!1-#
% x = -1:0.01:1; :TYzzl43
% [X,Y] = meshgrid(x,x); zl
0^EltiU
% [theta,r] = cart2pol(X,Y); up3<=u{>
% idx = r<=1;
MVP)rugU
% z = nan(size(X)); \Ntdl:fSw
% n = [0 1 1 2 2 2 3 3 3 3]; ({kGK0
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ?>jArzI
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 50bP&dj&
% y = zernfun(n,m,r(idx),theta(idx)); efkie}
% figure('Units','normalized') [pgkY!R?)
% for k = 1:10 sk6|_
% z(idx) = y(:,k); yn":!4U1
% subplot(4,7,Nplot(k))
"rDzrz
% pcolor(x,x,z), shading interp [I<'E
LX
% set(gca,'XTick',[],'YTick',[]) q\y#
% axis square T>Rf?%o
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ajW$d!
% end FJ,\?ooGf
% S%s|P=u
% See also ZERNPOL, ZERNFUN2. 'A(-MTd%
m\Fb ,
% Paul Fricker 11/13/2006 Ldj^O9p(
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d=
us,,W(q
% Check and prepare the inputs: ~K#_'Ldrd
% ----------------------------- \3(|c#c
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?CW^*So
error('zernfun:NMvectors','N and M must be vectors.') qMdtJ(gq
end t2%@py*bU
_KhEwd
if length(n)~=length(m) 'j<:FUDJ
error('zernfun:NMlength','N and M must be the same length.') 0/00W6r0
end [xs)u3b
m>-^K
n = n(:); ^AjYe<RU}
m = m(:); (=tF2YBV
if any(mod(n-m,2)) aU]O$Pg{
error('zernfun:NMmultiplesof2', ... g yH7((#i
'All N and M must differ by multiples of 2 (including 0).') a0/n13c?G
end t"bPKFRy9E
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if any(m>n) -f;j1bQ
error('zernfun:MlessthanN', ... CbH T #
'Each M must be less than or equal to its corresponding N.') {{[jC"4AY
end k1Mxsd
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if any( r>1 | r<0 ) B/9<b{6
error('zernfun:Rlessthan1','All R must be between 0 and 1.') JXRf4QmG
end 0@e}hv;
>[%.h(h/%
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 6[3Ioh
error('zernfun:RTHvector','R and THETA must be vectors.') Rr;LV<q+
end qfP"UAc{/
D.&eM4MZ
r = r(:); 5IE+M
theta = theta(:); mLk6!&zN
length_r = length(r); z1SMQLk
if length_r~=length(theta) )<x;ra^
error('zernfun:RTHlength', ... kSDa\l!W]
'The number of R- and THETA-values must be equal.') NtA|#"^
end eYD9#y
ZaUcP6[h
% Check normalization: Yr"!&\[oz
% -------------------- J.e8UQ@=5
if nargin==5 && ischar(nflag) j#nO6\&o
isnorm = strcmpi(nflag,'norm'); x+*L5$;h
if ~isnorm "U5Ln2X{J
error('zernfun:normalization','Unrecognized normalization flag.') 0q>NE<L
end K@j^gF/0B
else mb~=Xyk&
isnorm = false; zmL~]!~&
end DvRA2(M
S `m-5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y5AXL5
% Compute the Zernike Polynomials =6Kv`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4<3?al&
Z*vpQBbu
% Determine the required powers of r: d[>N6?JA/
% ----------------------------------- O2'bNR
m_abs = abs(m); ll<9f)
rpowers = []; `3sy>GU?
for j = 1:length(n) B=Zukg1G
rpowers = [rpowers m_abs(j):2:n(j)]; 9OQ0Yc!3
end UP~WP@0F
rpowers = unique(rpowers); 7k`*u) Q
-M>K4*%K
% Pre-compute the values of r raised to the required powers, S4{ Mu(^xT
% and compile them in a matrix: K5)yM @cq
% ----------------------------- g@k#J"Q'[
if rpowers(1)==0 4*D fI
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Dc&9emKI
rpowern = cat(2,rpowern{:}); M]4 =(Vv+5
rpowern = [ones(length_r,1) rpowern]; 7{Ki;1B[w
else V$-~%7@>;9
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ].k+Nzf_
rpowern = cat(2,rpowern{:}); J2oWssw"
end *b6I%MZn
^ =/?<C4
% Compute the values of the polynomials: >TlW]st
% -------------------------------------- O7'<I|aD
y = zeros(length_r,length(n)); ;Oi[:Ck
for j = 1:length(n) |Uy e>%*}4
s = 0:(n(j)-m_abs(j))/2; Ha=_u+@
pows = n(j):-2:m_abs(j); _\4`
for k = length(s):-1:1 n ,&/D
p = (1-2*mod(s(k),2))* ... Uxk[O
prod(2:(n(j)-s(k)))/ ... hr_9;,EPh
prod(2:s(k))/ ... :~ZqB\>i
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ]< s\V-y
prod(2:((n(j)+m_abs(j))/2-s(k))); j]!7B HC
idx = (pows(k)==rpowers); 9k=U0]!ch
y(:,j) = y(:,j) + p*rpowern(:,idx); n2xLgK=
end (<bm4MPf
xb+RRTgj
if isnorm `x{.z=xC
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); (b/A|hl
end wQD0vsD
end MG7 ?N #
% END: Compute the Zernike Polynomials Q)LXL.0h
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d}{LM!s
Ci7P%]9
% Compute the Zernike functions: O6m.t%*
% ------------------------------ {)
:%WnM9
idx_pos = m>0; %]a
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idx_neg = m<0; ;~Q
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z = y; /~3N@J
if any(idx_pos) b 0LGH.
z4
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); &v5G92
end v`#j
if any(idx_neg) "3{#d9Gs
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); @uI?
end w(76H^e
Vs#"SpH{'
% EOF zernfun