下面这个函数大家都不会陌生，计算zernike函数值的，并根据此可以还原出图像来， |'U,/
我输入10阶的n、m，r,theta为38025*1向量，最后得到的z是29525*10阶的矩阵， qa
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这个，跟我们用zygo干涉仪直接拟合出的36项zernike系数，有何关系呢？ r~Y>+ln.
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ k
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function z = zernfun(n,m,r,theta,nflag) u;+%Qh
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ee&nU(pK
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N zQL!(2
% and angular frequency M, evaluated at positions (R,THETA) on the y\F`B0#$
% unit circle. N is a vector of positive integers (including 0), and Po Yr:=S?
% M is a vector with the same number of elements as N. Each element CDQJ bvx
% k of M must be a positive integer, with possible values M(k) = -N(k) "C:rTIH
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ^H5w41
% and THETA is a vector of angles. R and THETA must have the same _-@ZOhw&
% length. The output Z is a matrix with one column for every (N,M) C+/E qq^(
% pair, and one row for every (R,THETA) pair. 9USrgY6_
% ,pDp>-vI%
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike yD "] {
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), qxf+#
% with delta(m,0) the Kronecker delta, is chosen so that the integral ,3VG.u;U
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, X!U]`Q h
% and theta=0 to theta=2*pi) is unity. For the non-normalized /yx=7<
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2-8YSHlh
% a<f;\$h]
% The Zernike functions are an orthogonal basis on the unit circle. gXq!a|eH
% They are used in disciplines such as astronomy, optics, and Y[f,ia
% optometry to describe functions on a circular domain. m3U+ du
% Xy[}G p
% The following table lists the first 15 Zernike functions. ?D1x;i9<
% `[X6#` <
% n m Zernike function Normalization c *.G]nRc
% -------------------------------------------------- sEoZ1E
% 0 0 1 1 :0nK`$'
% 1 1 r * cos(theta) 2 G+ :bL S#:
% 1 -1 r * sin(theta) 2 NOF?LV
% 2 -2 r^2 * cos(2*theta) sqrt(6) |tG05 +M
% 2 0 (2*r^2 - 1) sqrt(3) I") H~
% 2 2 r^2 * sin(2*theta) sqrt(6) B1y<.1k
% 3 -3 r^3 * cos(3*theta) sqrt(8) lN);~|IOv7
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) U^ B"|lc:[
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) '/ Cg*o/
% 3 3 r^3 * sin(3*theta) sqrt(8)
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% 4 -4 r^4 * cos(4*theta) sqrt(10) / p O{2[
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ov1Wr#s
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) NV:>a
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) '!pAnsXfO
% 4 4 r^4 * sin(4*theta) sqrt(10) =ZG<BG_
% -------------------------------------------------- ah 4kA LO
% buRhQ"
% Example 1: A)OdQFet(
% u06t DJ[
% % Display the Zernike function Z(n=5,m=1) !)N YW4"
% x = -1:0.01:1; h{\t*U 54'
% [X,Y] = meshgrid(x,x); /C Ix$G
% [theta,r] = cart2pol(X,Y); : @s8?eg
% idx = r<=1; )y6QAp
% z = nan(size(X)); NI^{$QMj
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Z#CxQ D%\
% figure {" :c@I
% pcolor(x,x,z), shading interp R'Sa?6xS4
% axis square, colorbar >+L7k^[,0
% title('Zernike function Z_5^1(r,\theta)') &xgZF Sq
% }(m1ql
% Example 2: Cm^Yl p
% Xc{ZN1 4n
% % Display the first 10 Zernike functions 9`&?hi49nK
% x = -1:0.01:1; B
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% [X,Y] = meshgrid(x,x); yX kt:O,i
% [theta,r] = cart2pol(X,Y); P%iP:16
% idx = r<=1; 5;}2[3}[
% z = nan(size(X)); hMv2"V-X
% n = [0 1 1 2 2 2 3 3 3 3]; {JX f*IJ
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; `4_c0 q)N4
% Nplot = [4 10 12 16 18 20 22 24 26 28]; qbH %Hx
% y = zernfun(n,m,r(idx),theta(idx)); SBC~QD>L+
% figure('Units','normalized') Xj%,xm>}!u
% for k = 1:10 +.=1^+a
% z(idx) = y(:,k); XWJ SLN(O
% subplot(4,7,Nplot(k)) s} s|~
% pcolor(x,x,z), shading interp >8%M*-= p
% set(gca,'XTick',[],'YTick',[]) lbd(j {h>4
% axis square \/n+j!
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) JT}.F!q6E
% end uN8/Q2
% :Pc(DfkS
% See also ZERNPOL, ZERNFUN2. 36nyu_h:R
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% Paul Fricker 11/13/2006 B&
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% Check and prepare the inputs: w~]T<^fW~
% ----------------------------- as(; ]
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 6s5yyy=L%~
error('zernfun:NMvectors','N and M must be vectors.') w E?CvL
end g@Ld"5$^2
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if length(n)~=length(m) qIz}$%!A
error('zernfun:NMlength','N and M must be the same length.') 7_KXD#
end q~j)W $k
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oXdel
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n = n(:); W+K.r?G<j
m = m(:); 07FT)QTE
if any(mod(n-m,2)) v$;@0t:;#
error('zernfun:NMmultiplesof2', ... h
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'All N and M must differ by multiples of 2 (including 0).') Ch t%uzb,
end Y([d;_#P
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if any(m>n) &a tyDFJ'
error('zernfun:MlessthanN', ... m<3w^mww
'Each M must be less than or equal to its corresponding N.') Kr]z]4.d@
end eVx~n(m!}
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if any( r>1 | r<0 ) lov%V*tL
error('zernfun:Rlessthan1','All R must be between 0 and 1.') SB/3jH
end z0
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) W~yLl%
error('zernfun:RTHvector','R and THETA must be vectors.') zq f[Z3
end !b63ik15O~
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r = r(:); !D;c,{Oz
theta = theta(:); VX!hv`E
length_r = length(r); GyK(Vb" h6
if length_r~=length(theta) bcn7,ht
error('zernfun:RTHlength', ... @$c!/
'The number of R- and THETA-values must be equal.') K{2h9 ]VF
end 3ev -Iqz
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% Check normalization: MMd.0Ju aO
% -------------------- )~dOmfw%|
if nargin==5 && ischar(nflag) |IN[u Q
isnorm = strcmpi(nflag,'norm'); 8k H< $9
if ~isnorm `[Sl1saZ$S
error('zernfun:normalization','Unrecognized normalization flag.') TF2KZL#A|
end I .P6l*$
else ISBF\ wQY
isnorm = false; *)D1!R<\,R
end >f@ G>H )+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Sc_5FX\Yx
% Compute the Zernike Polynomials `tVy_/3(9
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QNpu TZn#Q
l!@ 1u^v2
#U"1 9@|}
% Determine the required powers of r: I_>`hTiR
% ----------------------------------- gr+Pl>C{
m_abs = abs(m); ]r959+\$
rpowers = []; x.UaQ |F
for j = 1:length(n) F0.z i>5
rpowers = [rpowers m_abs(j):2:n(j)]; oY.\)eJ~>
end
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rpowers = unique(rpowers); ) rpq+~b
b# ='^W3
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% Pre-compute the values of r raised to the required powers, MI*Sq\-i
% and compile them in a matrix: taDQ65
% ----------------------------- .iT4-
if rpowers(1)==0 Hi8Y6|y$D
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); C:j]43`
rpowern = cat(2,rpowern{:}); &*gbK6 JB
rpowern = [ones(length_r,1) rpowern]; !_x*m@/
else J\A8qh8
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); HeozJ^u \?
rpowern = cat(2,rpowern{:}); mb{q(WEPP
end @GeHWv
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% Compute the values of the polynomials: ri=+(NKo-
% -------------------------------------- {y-`Q S
y = zeros(length_r,length(n)); h<N RE0-
for j = 1:length(n) ,YB1 y)x
s = 0:(n(j)-m_abs(j))/2; A 3q*$.[
pows = n(j):-2:m_abs(j); (B}+h
for k = length(s):-1:1 j^ EbO3
p = (1-2*mod(s(k),2))* ... 28 UV DG1?
prod(2:(n(j)-s(k)))/ ... s
MZ[d\
prod(2:s(k))/ ... ^y Vl"/
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... zP nC=h|g
prod(2:((n(j)+m_abs(j))/2-s(k))); ChE_unw
idx = (pows(k)==rpowers); ? ,XC =}
y(:,j) = y(:,j) + p*rpowern(:,idx); ti9}*8
end P
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if isnorm hW'
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y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); i0ybJOa4
end a.,_4;'UE1
end % rc FT_
% END: Compute the Zernike Polynomials {ERje uDm]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m
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% Compute the Zernike functions: "&-C$J5
Id
% ------------------------------ 7>,rvW:]
idx_pos = m>0; TB#N k5
idx_neg = m<0; D^$OCj\
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z = y; X4Xf2aXI
if any(idx_pos) o5 WW{)Q
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Funj!x'uE
end 3_zSp.E\l
if any(idx_neg) 2 ~-( A
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ' ^a!`"Bc
end 8*Zvr&B,G
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% EOF zernfun [Vp\$;\nT
