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， <y
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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， Wbq0K6X
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？ z#[PTqD-_
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ w)rd--9f
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function z = zernfun(n,m,r,theta,nflag) +]3kcm7 B
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. r|_@S[hZg
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N o=n F .y
% and angular frequency M, evaluated at positions (R,THETA) on the ;u8a %h!
% unit circle. N is a vector of positive integers (including 0), and ( < e q[(
% M is a vector with the same number of elements as N. Each element Sx0/Dm
% k of M must be a positive integer, with possible values M(k) = -N(k) =*5< w
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 0"F|)
% and THETA is a vector of angles. R and THETA must have the same Ke;eI+P[
% length. The output Z is a matrix with one column for every (N,M) gkM Q=;Nn
% pair, and one row for every (R,THETA) pair. 2il`'X
% K1+4W=|
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 6!`GUU
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), A_\`Gj!s%
% with delta(m,0) the Kronecker delta, is chosen so that the integral )n&6= Li
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, sTxgU !_
% and theta=0 to theta=2*pi) is unity. For the non-normalized g8SVuG<DI\
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. -U {CWn3G
% !X[P)/?b0+
% The Zernike functions are an orthogonal basis on the unit circle. S}b^_+UbP
% They are used in disciplines such as astronomy, optics, and '5m4kDs
% optometry to describe functions on a circular domain. &z]x\4#,
% kz*6%Cg*~
% The following table lists the first 15 Zernike functions. 5SMV3~*P
% 2<T/N
% n m Zernike function Normalization h"y~!NWn
% -------------------------------------------------- iG ,z3/~v
% 0 0 1 1 ]$,3vYBf
% 1 1 r * cos(theta) 2 #`ZBA>FLaQ
% 1 -1 r * sin(theta) 2 WM;5/;bB
% 2 -2 r^2 * cos(2*theta) sqrt(6) '~9w<dSB!r
% 2 0 (2*r^2 - 1) sqrt(3) 8RI'Fk{
% 2 2 r^2 * sin(2*theta) sqrt(6) <N:)Xf9`
% 3 -3 r^3 * cos(3*theta) sqrt(8) @ # p6C
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) I
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% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ?Dl; DE 1
% 3 3 r^3 * sin(3*theta) sqrt(8) M tVvi6T
% 4 -4 r^4 * cos(4*theta) sqrt(10) R.\]JvqO
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q Hr'r/0
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ;X
N Ahg7
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ou4 `#7FR
% 4 4 r^4 * sin(4*theta) sqrt(10) '
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% -------------------------------------------------- >"+ ho
% X`(fJ',
% Example 1: Q r_0
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% T/.U Mw
% % Display the Zernike function Z(n=5,m=1) lbX
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% x = -1:0.01:1; )}7X4g6X
% [X,Y] = meshgrid(x,x); WH"'Ju5}
% [theta,r] = cart2pol(X,Y); {;|pcx\L6~
% idx = r<=1; {b'
% z = nan(size(X)); =CW> ;h]
% z(idx) = zernfun(5,1,r(idx),theta(idx)); n2~WUK
% figure dC;&X
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% pcolor(x,x,z), shading interp / :^nG+
% axis square, colorbar +\*b?x
% title('Zernike function Z_5^1(r,\theta)') }Q*J!OH
% U)M &AYb
% Example 2: A.mFa 1lH
% &8pGq./lr=
% % Display the first 10 Zernike functions 6oq5C D oq
% x = -1:0.01:1; l =t/"M=
% [X,Y] = meshgrid(x,x); cs7^#/ 3<
% [theta,r] = cart2pol(X,Y); C=(Q0-+L|
% idx = r<=1; xkR S?Q g
% z = nan(size(X)); B9Mp3[
% n = [0 1 1 2 2 2 3 3 3 3]; +_k A&Q(t
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; +!W:gA
% Nplot = [4 10 12 16 18 20 22 24 26 28]; A|4om=MO
% y = zernfun(n,m,r(idx),theta(idx)); E=CA Wj\
% figure('Units','normalized') ?0M$p
% for k = 1:10 Lq$ig8V:O7
% z(idx) = y(:,k); Uf|uFGb
% subplot(4,7,Nplot(k)) }& W=
% pcolor(x,x,z), shading interp 7_P33l8y
% set(gca,'XTick',[],'YTick',[]) # S/n3
% axis square R?;mu^B
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) $)$ r
% end cWG%>.`5r
% d"IZt;s/,
% See also ZERNPOL, ZERNFUN2. Mtv{37k~
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% Paul Fricker 11/13/2006 4F9!3[}qF
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% Check and prepare the inputs: F\&{ >&
% ----------------------------- M)!"R [V
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~Kt1%&3{a?
error('zernfun:NMvectors','N and M must be vectors.') >e&
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end kW2DKr-[
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if length(n)~=length(m) @ MVul_ @6
error('zernfun:NMlength','N and M must be the same length.') kS &>g
end
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n = n(:); M49l2x=]9
m = m(:); K:jn^JN$
if any(mod(n-m,2)) ^\Z +Xq1~/
error('zernfun:NMmultiplesof2', ... ~-6_-Y|
'All N and M must differ by multiples of 2 (including 0).') SepwMB4@
end n[gE[kw
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if any(m>n) "^6Fh"]
error('zernfun:MlessthanN', ... KUYwc@si\
'Each M must be less than or equal to its corresponding N.') d4BzFGsW
end 5 ,-8oEUL
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if any( r>1 | r<0 ) !L9OJ1F
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ^Z#G_%\Y:
end .l_Nf9=
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) wpYk`L r
error('zernfun:RTHvector','R and THETA must be vectors.') +pme]V|<
end Ad>81=Z
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r = r(:); &l7E|.JE
theta = theta(:); KS 93v9|
length_r = length(r); yD7 }
if length_r~=length(theta) ap )B%9
error('zernfun:RTHlength', ... "lw|EpQk`
'The number of R- and THETA-values must be equal.') 5Y^"&h[/
end F/BR#J1
O# ZZ PJ"
X>=` l)ZR
% Check normalization: lTqlQ<`V
% -------------------- Y2QX <
if nargin==5 && ischar(nflag) 5[SwF& zZ
isnorm = strcmpi(nflag,'norm'); \alV #>J5
if ~isnorm $~.YB\3
error('zernfun:normalization','Unrecognized normalization flag.') NT*r7_e
end 9;U?_
else ;\2Z?Kq
isnorm = false; ap}p ?r
end 3r kcIVO
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9% T"W
% Compute the Zernike Polynomials ( ~5 M{Xh
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xt5/`C
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% Determine the required powers of r: )ciP6WzzbI
% ----------------------------------- 1]2]l*&3
m_abs = abs(m); 4kM/`g6?,q
rpowers = []; "g"a-{8
for j = 1:length(n) E@ U]k$M
rpowers = [rpowers m_abs(j):2:n(j)]; 0wv #AT
end Z*co\ pW
rpowers = unique(rpowers); [U zD3VPg
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.
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% Pre-compute the values of r raised to the required powers, l}z<q
% and compile them in a matrix: ( *+'k1Ea
% ----------------------------- ^ b +>r
if rpowers(1)==0 nL:&G'd
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ZiJF.(JS
rpowern = cat(2,rpowern{:}); Kt_oo[ey{
rpowern = [ones(length_r,1) rpowern]; mgjJNzclL
else `sYFQ+D#O
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); s h$-}1 ;
rpowern = cat(2,rpowern{:}); `3rwqcxA
end w'H' o!*/
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% Compute the values of the polynomials: 46 A sD
% -------------------------------------- R#d~a;j
y = zeros(length_r,length(n)); C:J;'[,S
for j = 1:length(n) +H2Jhgi
s = 0:(n(j)-m_abs(j))/2; ~ 1h#
pows = n(j):-2:m_abs(j); .c"nDCFVR
for k = length(s):-1:1 :]-oo*xP
p = (1-2*mod(s(k),2))* ... crM5&L9zF
prod(2:(n(j)-s(k)))/ ... 1(?4*v@B
prod(2:s(k))/ ... 2^WJ 1: A
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... k5S;G"i J
prod(2:((n(j)+m_abs(j))/2-s(k))); 8 c8`"i
idx = (pows(k)==rpowers); YO7U}6wBt
y(:,j) = y(:,j) + p*rpowern(:,idx); jfxNV2[
end &F&`y
p`Pa;=L
if isnorm 6$k#B ~~
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ebk>e*
end IK2da@V
end gpV4qDX V
% END: Compute the Zernike Polynomials [A-_?#cZ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r[L%ap\{
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D|LO!,=b
% Compute the Zernike functions: b' o] Y
% ------------------------------ %v0M~J}+
idx_pos = m>0; 2Xt4Rqk $
idx_neg = m<0; )O1 ]|r7v
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z = y; k)p` x "To
if any(idx_pos) ]" 'yf;g
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); *d 1Bp R%
end ;'"'|} xn
if any(idx_neg) }@r2 3g%
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ) O0Cz n
end tD K@?PfKz
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% EOF zernfun &`D$w?b eg
