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 R2Fjv@Egk
function z = zernfun(n,m,r,theta,nflag) h<Aq|*
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~Ba=nn8Cq
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N K"0IW A
% and angular frequency M, evaluated at positions (R,THETA) on the \x}\)m_7M<
% unit circle. N is a vector of positive integers (including 0), and 2]5{Xmmo9
% M is a vector with the same number of elements as N. Each element h= sNj
% k of M must be a positive integer, with possible values M(k) = -N(k) E&P2E3P
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Oo|PZ_P
% and THETA is a vector of angles. R and THETA must have the same \EySKQ=
% length. The output Z is a matrix with one column for every (N,M) PW5]+ |#
% pair, and one row for every (R,THETA) pair. -^xbd_'
% QJVbt
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike i7Up AHd/
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), DW. w=L|5R
% with delta(m,0) the Kronecker delta, is chosen so that the integral u=.8M`FxP
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, f82%nT
% and theta=0 to theta=2*pi) is unity. For the non-normalized *5%vU|9b
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 4 O!2nP
% C!VhVOy>d
% The Zernike functions are an orthogonal basis on the unit circle. lvO6&sF1
% They are used in disciplines such as astronomy, optics, and #J"xByQKK
% optometry to describe functions on a circular domain. 0X=F(,>9
% ec&/a2M
% The following table lists the first 15 Zernike functions. LjI`$r.B
% \Oeo"|
% n m Zernike function Normalization
QrYF Lh
% -------------------------------------------------- Wo1xZZ
% 0 0 1 1 M^o_='\bE
% 1 1 r * cos(theta) 2 f+h\RE=BGt
% 1 -1 r * sin(theta) 2 q>$MqKWM
% 2 -2 r^2 * cos(2*theta) sqrt(6) %F;BL8d
% 2 0 (2*r^2 - 1) sqrt(3) bv[#|^/
% 2 2 r^2 * sin(2*theta) sqrt(6) s@F&N9oh
% 3 -3 r^3 * cos(3*theta) sqrt(8) ]vvYPRV76
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) }/cReX,so
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) =,6H2ew
% 3 3 r^3 * sin(3*theta) sqrt(8) &lQ%;)'
% 4 -4 r^4 * cos(4*theta) sqrt(10) X Q#K1Z
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) D.K""*ula
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) : ky`)F`
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) MCKN.f%lP
% 4 4 r^4 * sin(4*theta) sqrt(10) 1<YoGm&
% -------------------------------------------------- 'hpOpIsHa
% YB 38K(
% Example 1: tbFAVGcAM
% ZL(
j5E
% % Display the Zernike function Z(n=5,m=1) oac)na:O#
% x = -1:0.01:1; 'Gy`e-yB
% [X,Y] = meshgrid(x,x); ,;$OaJFT
% [theta,r] = cart2pol(X,Y); F]aoTy
% idx = r<=1; xXe3E&
% z = nan(size(X)); uX_H;,n
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 5Gz!Bf@!!
% figure M/N8bIC! Q
% pcolor(x,x,z), shading interp v:t;Uk^Y
% axis square, colorbar 0*gvHVd/l
% title('Zernike function Z_5^1(r,\theta)') D:z'`v0j
% ^A$=6=CX
% Example 2: {^N,=m\
% YuK+N
% % Display the first 10 Zernike functions ?I}RX~Tgg
% x = -1:0.01:1; 'ygKP6M
% [X,Y] = meshgrid(x,x); Q{[@n
% [theta,r] = cart2pol(X,Y); ingG
% idx = r<=1; Ykxk`SJ
% z = nan(size(X)); cQ8[XNa
% n = [0 1 1 2 2 2 3 3 3 3]; (95|DCL
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Cv**iW
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Gv-VDRS
% y = zernfun(n,m,r(idx),theta(idx)); 7(Fas(j3
% figure('Units','normalized') ,aP6ct
% for k = 1:10 B7%K}|Qg
% z(idx) = y(:,k); h^Wb<O`S
% subplot(4,7,Nplot(k)) Sdu\4;(
% pcolor(x,x,z), shading interp 8y
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% set(gca,'XTick',[],'YTick',[]) d_9 Cm@
% axis square _Mw3>GNl
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) )w7vE\n3
% end q$:1Xkl
% TM)INo^
% See also ZERNPOL, ZERNFUN2. CMj =4e
;UQGi}?CD
% Paul Fricker 11/13/2006 ? i{?Q,
W A/dt2D|
) /raTD
% Check and prepare the inputs: AdDX_\V,*
% ----------------------------- oD2:19M@p
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /Hr|u
error('zernfun:NMvectors','N and M must be vectors.')
r h*F
end htBA.eQ
Y~"tL(WfJl
if length(n)~=length(m) 69c4bT:b"
error('zernfun:NMlength','N and M must be the same length.') Z/Rp?Jz\j/
end IiPX`V>RC
y``\^F
n = n(:); UqK.b}s
m = m(:); `<7\Zl
if any(mod(n-m,2)) ;s+/'(*
error('zernfun:NMmultiplesof2', ... Y{}
ub]i
'All N and M must differ by multiples of 2 (including 0).') (?z?/4>7<
end csP4Oq\g[
K~L&Z?~|E
if any(m>n) 7`|'Om?'
error('zernfun:MlessthanN', ... u
r$
'Each M must be less than or equal to its corresponding N.') \{h_i
FU!
end "wcaJ;Os
+( LH!\{^
if any( r>1 | r<0 ) h FU8iB`Q
error('zernfun:Rlessthan1','All R must be between 0 and 1.') l.}PxZ
end +7.|1x;C
j.=:S;
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 6n9/`D!
error('zernfun:RTHvector','R and THETA must be vectors.') ,rB(WKU
end 2yfU]`qN
Fb,*;M1'
r = r(:); Ao K9=F}
theta = theta(:); v5[gFY(?
length_r = length(r); AiHU*dp6
if length_r~=length(theta) "r^RfZ;
error('zernfun:RTHlength', ... wB)y@w4k
'The number of R- and THETA-values must be equal.') IdmP!(u
end g QBS#NY
EQyX!
% Check normalization: #2]*qgA4
% -------------------- KI9Pw]]{-
if nargin==5 && ischar(nflag) G&oD;NY@/
isnorm = strcmpi(nflag,'norm'); 7Z>vQ f B
if ~isnorm >Na. C(DZ
error('zernfun:normalization','Unrecognized normalization flag.') 8m0*89HEu
end Snkb^Kt
else xp|1yud
isnorm = false; utck{]P
end bB<S4@jF8z
JD*HG]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )Xdq+$w.
% Compute the Zernike Polynomials %R GZu\p
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% , Q0Y} )
}83
8F&
% Determine the required powers of r: 8g-u
% ----------------------------------- (wu'FFJp#
m_abs = abs(m); d(^8#4
rpowers = []; qc(e3x
for j = 1:length(n) YP,,vcut
rpowers = [rpowers m_abs(j):2:n(j)]; k|OM?\
end ';R]`vWFe
rpowers = unique(rpowers); B Ewa QvQ!
Ou[`)|>
% Pre-compute the values of r raised to the required powers, S(.J
% and compile them in a matrix: 1uw1(iL+
% ----------------------------- pCt2-aam
if rpowers(1)==0 z}-CU GS
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); _|e&zr
rpowern = cat(2,rpowern{:}); "|JbdI]%P
rpowern = [ones(length_r,1) rpowern]; $~5H-wJ
else dNR/|
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); <lzC|>BG
rpowern = cat(2,rpowern{:}); SY
Bp-o
end 0[UI'2
h[dJNawL
% Compute the values of the polynomials: sqhMnDn[
% -------------------------------------- "E+;O,N-
y = zeros(length_r,length(n)); 4A+g-{d
for j = 1:length(n) h] ho? K
s = 0:(n(j)-m_abs(j))/2; L9) gN.#
pows = n(j):-2:m_abs(j); P[fy
for k = length(s):-1:1 0_qr7Ui8(
p = (1-2*mod(s(k),2))* ... :?~)P!/xl5
prod(2:(n(j)-s(k)))/ ... e!J5h<:
prod(2:s(k))/ ... YWU@e[
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... '=nmdqP
prod(2:((n(j)+m_abs(j))/2-s(k))); Xc[ym
idx = (pows(k)==rpowers); +C\79,r
y(:,j) = y(:,j) + p*rpowern(:,idx); 9Qszr=C0
end A@o7
G+#bO5
if isnorm |6^a[x3/U
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); xDeM7L'
end 6n/=n%US
end RF*>U a
% END: Compute the Zernike Polynomials ?5't1219
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% od#Lad@p
v8F{qT50
% Compute the Zernike functions: &n,v@
gt
% ------------------------------ wdj?T`4
idx_pos = m>0; yl?LXc[)
idx_neg = m<0; -W6@[5 c
6<@mBZ
z = y; X8v)yDtw
if any(idx_pos) (}F@0WYT^O
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); K
'I6iCrD
end $m
;p@#n
if any(idx_neg) AAfhh5i
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); [;hkT
end Z42q}Fhm*R
Pg.JI:>2Ku
% EOF zernfun