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, CJh,-w{wJ"
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, T`pDjT
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? $m~&| s
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? *59|
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function z = zernfun(n,m,r,theta,nflag) mA$86 X_
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. l53Q"ajG
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 94et ]u%7
% and angular frequency M, evaluated at positions (R,THETA) on the \2=I//YF
% unit circle. N is a vector of positive integers (including 0), and DA iS|x
% M is a vector with the same number of elements as N. Each element sV-PR]
% k of M must be a positive integer, with possible values M(k) = -N(k) ? %8%1d
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, M9o/6
% and THETA is a vector of angles. R and THETA must have the same ]cv|dc=
% length. The output Z is a matrix with one column for every (N,M) F-b]>3r
% pair, and one row for every (R,THETA) pair. nSh~mP
% 9_d#F'#F
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike _68vSYr
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), lyFlJm i,r
% with delta(m,0) the Kronecker delta, is chosen so that the integral :!Dm,PP%
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, LC##em=Y
% and theta=0 to theta=2*pi) is unity. For the non-normalized ,52Lm=n
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. U}DE9e{/!
% &zB>
% The Zernike functions are an orthogonal basis on the unit circle. ]LZ#[xnM7
% They are used in disciplines such as astronomy, optics, and Wu<;QY($5
% optometry to describe functions on a circular domain. J=78p#XUg
% JNXzZ4U
% The following table lists the first 15 Zernike functions. t:V._@
% 4h_YVG]ur
% n m Zernike function Normalization 9B;WjXSe
% -------------------------------------------------- [zm@hxym
% 0 0 1 1 /n(0w`
% 1 1 r * cos(theta) 2 wu
eDedz\
% 1 -1 r * sin(theta) 2 *k_<|{>j(
% 2 -2 r^2 * cos(2*theta) sqrt(6) 4i{Xs5zk
% 2 0 (2*r^2 - 1) sqrt(3) )`{m |\b
% 2 2 r^2 * sin(2*theta) sqrt(6) QW,:'\G
% 3 -3 r^3 * cos(3*theta) sqrt(8) |a"]@W$>
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Jn d_cJ ]a
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) pZeOdh
% 3 3 r^3 * sin(3*theta) sqrt(8) -`{W~yz
% 4 -4 r^4 * cos(4*theta) sqrt(10) uq-`1m}
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &y1iLk h ^
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) spm)X-[1
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?!jJxhK<h
% 4 4 r^4 * sin(4*theta) sqrt(10) 6{^\7`
% -------------------------------------------------- T?f{.a)
% &+@`Si=
% Example 1: zj]
g^c;
% Q:Pp'[ RK
% % Display the Zernike function Z(n=5,m=1) %z1^
% x = -1:0.01:1; xRgdU+,Mj
% [X,Y] = meshgrid(x,x); `pCy:J?d>l
% [theta,r] = cart2pol(X,Y); \b$pH
% idx = r<=1; IAGY-+8e
% z = nan(size(X)); 2]9
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% z(idx) = zernfun(5,1,r(idx),theta(idx)); ~+0IFJ `}
% figure G1e_pszD{o
% pcolor(x,x,z), shading interp 8@LWg d
% axis square, colorbar 9O-~Ws ;
% title('Zernike function Z_5^1(r,\theta)') C7vBa<a
% =^rp=
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% Example 2: #k)z5vZ$h
% R_g(6l"3R^
% % Display the first 10 Zernike functions )sdHJ
% x = -1:0.01:1; Z}0xK6
% [X,Y] = meshgrid(x,x); ezL1,GT
% [theta,r] = cart2pol(X,Y); '"\n,3h
% idx = r<=1; ;A@DE@^5w
% z = nan(size(X)); XC~"T6F
% n = [0 1 1 2 2 2 3 3 3 3]; -N^Ah_9ek
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; *A8*FX>\F
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Spx%`O<
% y = zernfun(n,m,r(idx),theta(idx)); ]g ;+7
% figure('Units','normalized') \/j,
% for k = 1:10 c CDT27@
% z(idx) = y(:,k); !',%kvJI
% subplot(4,7,Nplot(k)) "u4x#7n|
% pcolor(x,x,z), shading interp #[x*0K-h
% set(gca,'XTick',[],'YTick',[]) /D;ugc*3
% axis square CC"a2Hu/
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) DMsqTB`
% end Rrry;Hr
% gt.F[q3
% See also ZERNPOL, ZERNFUN2. ?t6wozib2
T }msF
X\H P{$fY_
% Paul Fricker 11/13/2006 8]vut{
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xSM1b5=Pu
% Check and prepare the inputs: ge?or]T1S
% ----------------------------- w0j'>4
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) h
x5M)8#+
error('zernfun:NMvectors','N and M must be vectors.') nt()UC`5
end V[*>}XQER
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if length(n)~=length(m) c u";rnj
error('zernfun:NMlength','N and M must be the same length.') Da8gOZ
end .xT{Rz
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n = n(:); J!~kqNI
m = m(:); 1QD49)
if any(mod(n-m,2)) =X5w=(&
error('zernfun:NMmultiplesof2', ... LVdR,'lS
'All N and M must differ by multiples of 2 (including 0).') 2p;I<C:Eo
end Uvc$&j^k
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if any(m>n) `<v$+mG
error('zernfun:MlessthanN', ... g)$KN,gGuO
'Each M must be less than or equal to its corresponding N.') k\SqDmv
end rA?<\*
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,
if any( r>1 | r<0 ) =$y J66e
error('zernfun:Rlessthan1','All R must be between 0 and 1.') O"o|8
l}M/
end #*y.C[^5{
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) M1NdlAAf
error('zernfun:RTHvector','R and THETA must be vectors.') m6K7D([f
end EHhc2^e
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r = r(:); .IY@Q
theta = theta(:); ,66(*\xT
length_r = length(r); p&<n_b
if length_r~=length(theta) d(RMD
error('zernfun:RTHlength', ... C:^
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'The number of R- and THETA-values must be equal.') C|IHRw`[
end u]O}Ub`
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% Check normalization:
n=~!x
% -------------------- .L%pWRxA[
if nargin==5 && ischar(nflag) VrfEa d
isnorm = strcmpi(nflag,'norm'); &3"ODAp'
if ~isnorm ZWS:-]P.
error('zernfun:normalization','Unrecognized normalization flag.') +IbV
end b5]<!~Fv:`
else <Dgf'GrJ
isnorm = false; }dMX1e1h8
end jP}Ry=V/
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HRbv%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% toD!RE
% Compute the Zernike Polynomials ~}ifwm'7 a
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PcZ<JJ16F$
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% Determine the required powers of r: Yuf+d-%
% ----------------------------------- 6+ptL-Zt<
m_abs = abs(m); 1~E4]Ef:W
rpowers = []; %1#|>^
for j = 1:length(n) vyWx{@
rpowers = [rpowers m_abs(j):2:n(j)]; bxL'k/Y$
end <v_Wh@m
rpowers = unique(rpowers); .L1[Rv3
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d
{ P$}b
% Pre-compute the values of r raised to the required powers, WnOYU9;%
% and compile them in a matrix: d^tY?*n
% ----------------------------- W]bytsl
if rpowers(1)==0 7 u Q +]d
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); GJE+sqMX1
rpowern = cat(2,rpowern{:}); FGc#_4SiL
rpowern = [ones(length_r,1) rpowern]; m*)jndXY
else 3 @O/#CP+
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 1lA? 5:
rpowern = cat(2,rpowern{:}); L_:~{jV
end /GJL&RMx
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n|Y}M]u,
% Compute the values of the polynomials: dikX_ Q>D
% -------------------------------------- KX!/n`2u
y = zeros(length_r,length(n)); n[i:$! ,
for j = 1:length(n) 7iv g3*
s = 0:(n(j)-m_abs(j))/2; w&es N$2
pows = n(j):-2:m_abs(j); x+%> 2qgj"
for k = length(s):-1:1 KC9VQeSc
p = (1-2*mod(s(k),2))* ... o,J8n;"l
prod(2:(n(j)-s(k)))/ ... 5oB#{h
prod(2:s(k))/ ... fo>_*6i74
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... IvQuxs&a
prod(2:((n(j)+m_abs(j))/2-s(k))); :~s*yznf
idx = (pows(k)==rpowers); As^eL/m2L
y(:,j) = y(:,j) + p*rpowern(:,idx); #ifjQ7(:
end ih75C"
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if isnorm n^qwE
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); e iH&<AH
end Abmi=]\bx
end ^aJ]|*m
% END: Compute the Zernike Polynomials vGJw/ij'X
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +m~3InWq
e_rEu'[av
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% Compute the Zernike functions: 4 s9^%K\8{
% ------------------------------ e&[~}f?
idx_pos = m>0; |L}tAS`8
idx_neg = m<0; |VyN>&r~6
CSWA/#&8>
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z = y; A+\rGVNH'S
if any(idx_pos) ,ag*
/
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); M[iWWCX
end T| (w-)mv
if any(idx_neg) D=5%lL
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Y|/,*,u+
end j#p3<V S4
s{Y-Vdx
pA@R,O>zr
% EOF zernfun ,CqGO %DY