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 ri8=u$!
function z = zernfun(n,m,r,theta,nflag) hDB(y4/
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. $%DoLpE>
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 2?q>yL! Gz
% and angular frequency M, evaluated at positions (R,THETA) on the TaYl[I
% unit circle. N is a vector of positive integers (including 0), and 2yn"K|
% M is a vector with the same number of elements as N. Each element {v]L|e%{
% k of M must be a positive integer, with possible values M(k) = -N(k) B<r0y
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ef '?O
% and THETA is a vector of angles. R and THETA must have the same NO[A00m|OL
% length. The output Z is a matrix with one column for every (N,M) Ro9:kEG$
% pair, and one row for every (R,THETA) pair. Ot-P
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% duEXp]f!
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike |=YK2};
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), T~/>U&k}J
% with delta(m,0) the Kronecker delta, is chosen so that the integral ohKoX$|p~
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, o5&b'WUJ=
% and theta=0 to theta=2*pi) is unity. For the non-normalized ZYWGP:Y
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. VNT?
% t{iRCj
% The Zernike functions are an orthogonal basis on the unit circle. 2@Yu:|d4U
% They are used in disciplines such as astronomy, optics, and $%bd`d*S
% optometry to describe functions on a circular domain. &t8,326;
% Yl&[_
l
% The following table lists the first 15 Zernike functions. 5\h 6"/6Df
% G) KI{D
% n m Zernike function Normalization }FS_"0
% -------------------------------------------------- 59 g//;35@
% 0 0 1 1 bi+M28m
% 1 1 r * cos(theta) 2 ]vf0 f,F
% 1 -1 r * sin(theta) 2 t27UlFX
% 2 -2 r^2 * cos(2*theta) sqrt(6) ,i}EGW,9q
% 2 0 (2*r^2 - 1) sqrt(3) 2#5Q~
% 2 2 r^2 * sin(2*theta) sqrt(6) QObVJg,GD
% 3 -3 r^3 * cos(3*theta) sqrt(8) c]x-mj =
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Z ;rM@x
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) \K\eq>@6
% 3 3 r^3 * sin(3*theta) sqrt(8) :n13v@q
% 4 -4 r^4 * cos(4*theta) sqrt(10) kZ@UQ{>`
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) D6@ c|O{Q
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Ey:?!
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `=hCS0F
% 4 4 r^4 * sin(4*theta) sqrt(10) iYT?6Y|+
% -------------------------------------------------- 0'F/z%SMj
% PQlA(v+S
% Example 1: s) s9Z,HY
% YFu,<8"swe
% % Display the Zernike function Z(n=5,m=1) In?+
% x = -1:0.01:1; [>dDRsZ
% [X,Y] = meshgrid(x,x); 7P3/Ky@6
% [theta,r] = cart2pol(X,Y); g`'!Vgd?M[
% idx = r<=1; ,}W|cm>
% z = nan(size(X)); <& PU%^Ha
% z(idx) = zernfun(5,1,r(idx),theta(idx)); x f{`uHa8
% figure B<BS^waU
% pcolor(x,x,z), shading interp &@@PJ!&
% axis square, colorbar 6BA$v-VVU
% title('Zernike function Z_5^1(r,\theta)') g#74c'+
% VOr:G85*s
% Example 2: 30WOH
'n
% (=u!E+N
% % Display the first 10 Zernike functions &8i$`6wY
% x = -1:0.01:1; t=}]4&Yp
% [X,Y] = meshgrid(x,x); *ilVkV"U
% [theta,r] = cart2pol(X,Y); _/Ve~(
"
% idx = r<=1; 3HuocwWbz
% z = nan(size(X)); L7<30"7
% n = [0 1 1 2 2 2 3 3 3 3]; o9|
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% m = [0 -1 1 -2 0 2 -3 -1 1 3]; u=L Dfn
% Nplot = [4 10 12 16 18 20 22 24 26 28]; {_(R?V]w,
% y = zernfun(n,m,r(idx),theta(idx)); TDk[,4
% figure('Units','normalized') P-T@'}lW
% for k = 1:10 ;&|I/MVm
% z(idx) = y(:,k); cz/E
% subplot(4,7,Nplot(k)) z0\
$#r^I
% pcolor(x,x,z), shading interp 2jhJXM=~
% set(gca,'XTick',[],'YTick',[]) dr"$@
% axis square ?;UR9f|!
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])
"[]oWPOj
% end ]Zh$9YK
% aO}hE2]
% See also ZERNPOL, ZERNFUN2. '")'h
`'iO+/;GY
% Paul Fricker 11/13/2006 J?#vL\8
I__b$
0OG
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% Check and prepare the inputs: j|[$P4w}U
% ----------------------------- R73@!5N%
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Yg5o!A
error('zernfun:NMvectors','N and M must be vectors.') 9 9:.j=
end V!. Y M)B
E71H=C 4
if length(n)~=length(m) m#[c]v{
error('zernfun:NMlength','N and M must be the same length.') 6:}n}q,V
end _0u=}tc
T}?b,hNl$
n = n(:); <f}:YDY'
m = m(:); }@
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if any(mod(n-m,2)) ,ko#z}Z4r,
error('zernfun:NMmultiplesof2', ... $;=^|I4E
'All N and M must differ by multiples of 2 (including 0).') 1Z_w2D*
end C%<Dq0j
{I0!q"sF
if any(m>n) _-{=Z=?6}
error('zernfun:MlessthanN', ... ]QY-LO(
'Each M must be less than or equal to its corresponding N.') _?felxG[
end WRbdv{1E
80%"2kG
if any( r>1 | r<0 ) 7~1Fy{tc
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 9-{.W Z
end 4@F8-V3q4
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) N@Ap|`Ei
error('zernfun:RTHvector','R and THETA must be vectors.') $aT '~|?
end "UY34a^I
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r = r(:); # ^,8JRA
theta = theta(:); =s:kC`O
length_r = length(r); r&v!2A]:
if length_r~=length(theta) P^Og(F8;
error('zernfun:RTHlength', ... s H'FqV,)
'The number of R- and THETA-values must be equal.') Zd-QZ<c";t
end H 9BqE+
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40g1}
% Check normalization: @$p6w
% -------------------- h0 %M+g
if nargin==5 && ischar(nflag) &l`_D?{<#
isnorm = strcmpi(nflag,'norm'); V$$9Rh
if ~isnorm Xe`$SNM
error('zernfun:normalization','Unrecognized normalization flag.') _a$5"
end VJ&-Z |
else g=v'[JPd
isnorm = false; uJ1oo| sn
end *<{hLf
K",Xe>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ESIeZhXVH
% Compute the Zernike Polynomials =b )!l9TX
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d{WOO)j
Y nTx)uW
% Determine the required powers of r: -c0*
% ----------------------------------- *fy aAv
m_abs = abs(m); 6PWw^Cd
rpowers = []; .hf%L1N%F
for j = 1:length(n) ]-heG'y]{
rpowers = [rpowers m_abs(j):2:n(j)]; 8c%N+E]
end K-.%1d@$y
rpowers = unique(rpowers); 8 f~M6
h6`VU`pPI
% Pre-compute the values of r raised to the required powers, |a\,([aU
% and compile them in a matrix: 1!+0]_8K
% ----------------------------- #w^Ot*{!N
if rpowers(1)==0 RWDPsZC
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 3-0jxx(
rpowern = cat(2,rpowern{:}); Z]Z&PbP
rpowern = [ones(length_r,1) rpowern]; YWANBM(v+
else X2np.9hie
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); }LWrtmc
rpowern = cat(2,rpowern{:}); Vd)
%qw
end "x:-#2+h
@@!]Raj=
% Compute the values of the polynomials: h^{aG ])
% -------------------------------------- o/RGz PR
y = zeros(length_r,length(n)); PI*Z>VE?
for j = 1:length(n) OMjx,@9
s = 0:(n(j)-m_abs(j))/2; g'-hSV/@}@
pows = n(j):-2:m_abs(j); !.q#X^@>L
for k = length(s):-1:1 xTZJ5iZ17
p = (1-2*mod(s(k),2))* ... `Y '-2Fv
prod(2:(n(j)-s(k)))/ ... ']X0g{%
prod(2:s(k))/ ... PIsXX#`7;
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... !0X"^VB
prod(2:((n(j)+m_abs(j))/2-s(k))); !83 N#Y_Mz
idx = (pows(k)==rpowers); Us>n`Lj@
y(:,j) = y(:,j) + p*rpowern(:,idx); Sn;q:e3i{A
end 2:[G4
`;Fs
if isnorm z5f3T D6,
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); )Z:maz
end `V[ hE
r|
end [Fd[(
% END: Compute the Zernike Polynomials U!lWP#m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q6
4bP4K
?/Aql_?3
% Compute the Zernike functions: .MxMBrM
% ------------------------------ @]],H0
idx_pos = m>0; fAT
M?
idx_neg = m<0; E3_ 5~>
DeN$YE#*
z = y; 1!ijRr
if any(idx_pos) vb\R~%@T,
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); +ldgT"
end Xu{S4#1
if any(idx_neg) "[ >ql1t{b
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); OZl0I#@A
end W%#LHluP
0 n)UvJ
% EOF zernfun