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, 8'[7
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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, f}ji?p
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? #G|RnV%t$~
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? /Iy]DU8
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function z = zernfun(n,m,r,theta,nflag) R?|.pq/Ln
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. TER=*"!
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ?
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% and angular frequency M, evaluated at positions (R,THETA) on the 7>0o&
% unit circle. N is a vector of positive integers (including 0), and %lhEM}Sm
% M is a vector with the same number of elements as N. Each element ^zmG0EH,
% k of M must be a positive integer, with possible values M(k) = -N(k) Qj.#)R
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, @V sG'
% and THETA is a vector of angles. R and THETA must have the same J?1 uKR
% length. The output Z is a matrix with one column for every (N,M) ZY55|eE
% pair, and one row for every (R,THETA) pair. 33x{CY15
% jXx<`I+]
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike rQs)O<jl
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), dr}`H,X"3
% with delta(m,0) the Kronecker delta, is chosen so that the integral mHTXni<!
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ZohCP
% and theta=0 to theta=2*pi) is unity. For the non-normalized TDKki(o=~
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. l`{\"#4
% }5[qo`M
% The Zernike functions are an orthogonal basis on the unit circle. BwGfTua
% They are used in disciplines such as astronomy, optics, and qvsd5P eCO
% optometry to describe functions on a circular domain. sN*N&XG
% X1|njJGO1
% The following table lists the first 15 Zernike functions. drP=A~?&:
% &K.d'$q
% n m Zernike function Normalization ,j{,h_Op
% -------------------------------------------------- hGe/;@%
% 0 0 1 1 J.b9F:&}
% 1 1 r * cos(theta) 2 AaOuL,l
% 1 -1 r * sin(theta) 2 )gIKH{JYL
% 2 -2 r^2 * cos(2*theta) sqrt(6) Q7\w+ANf0
% 2 0 (2*r^2 - 1) sqrt(3) *8Xh(`
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% 2 2 r^2 * sin(2*theta) sqrt(6) _\G"9,)u'
% 3 -3 r^3 * cos(3*theta) sqrt(8)
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% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) *[Tz![|
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Y@vTaE^w3
% 3 3 r^3 * sin(3*theta) sqrt(8) Y|f[bw
% 4 -4 r^4 * cos(4*theta) sqrt(10) 0/MtYIYk
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1\~ "VF*{
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) VcO0sa f`
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -q1??u
% 4 4 r^4 * sin(4*theta) sqrt(10) |(E
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% -------------------------------------------------- oXh#a8
% \BTODZ:h
% Example 1: uAJx.>$b
% D 6Ui!
% % Display the Zernike function Z(n=5,m=1) 9igiZmM
% x = -1:0.01:1; m)t;9J5
% [X,Y] = meshgrid(x,x); Y-_`23x`
% [theta,r] = cart2pol(X,Y); jh%Eq+#S
% idx = r<=1; z6=Z\P+
% z = nan(size(X)); RuA*YV
% z(idx) = zernfun(5,1,r(idx),theta(idx)); @ $ ;q;
% figure { ]{/t-=
% pcolor(x,x,z), shading interp #ym'AN
% axis square, colorbar /wEhVR`=
% title('Zernike function Z_5^1(r,\theta)') v5#jZ$<F
% %COX7gV
% Example 2: JN-y)L/>
% qZtzO2Mt
% % Display the first 10 Zernike functions v6bGjVK[
% x = -1:0.01:1; C=L>zOZ
% [X,Y] = meshgrid(x,x); DS(}<HK{
% [theta,r] = cart2pol(X,Y); {j?FNOJn
% idx = r<=1; P|tO<t6/9*
% z = nan(size(X)); %~H-)_d20
% n = [0 1 1 2 2 2 3 3 3 3]; yy^q2P
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; qpP=K $
% Nplot = [4 10 12 16 18 20 22 24 26 28]; p
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% y = zernfun(n,m,r(idx),theta(idx)); M#4pE_G
% figure('Units','normalized') i(%W_d!
% for k = 1:10 #uG%j
% z(idx) = y(:,k); XH 4
% subplot(4,7,Nplot(k)) J s@hLP`
% pcolor(x,x,z), shading interp UT~4x|b:O
% set(gca,'XTick',[],'YTick',[]) ; ; OAQ`
% axis square s 8jV(P(O
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) A Ru2W1g
% end TCwFPlF|
% GX!G>
% See also ZERNPOL, ZERNFUN2. a
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% Paul Fricker 11/13/2006 +K:Dx!9
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% Check and prepare the inputs: ,F8 Yn5h
% ----------------------------- )1J R#
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 8sWJcmVo
error('zernfun:NMvectors','N and M must be vectors.') r"3=44St
end *MhRW,=
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if length(n)~=length(m) qdJ=lhHM}
error('zernfun:NMlength','N and M must be the same length.') .LnGL]/
end F3[T.sf
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n = n(:); KEo,m
m = m(:); ` xEx^P^7
if any(mod(n-m,2)) O_muD\
error('zernfun:NMmultiplesof2', ... 1Kw+,.@d
'All N and M must differ by multiples of 2 (including 0).') E!)xj.aS$
end c(f
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if any(m>n) t:x\kp
error('zernfun:MlessthanN', ... Hh3X
\
'Each M must be less than or equal to its corresponding N.') YlJ@XpKM
end \$~|ZwV{
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if any( r>1 | r<0 ) }<0BX \@I
error('zernfun:Rlessthan1','All R must be between 0 and 1.') j;+b0(53
end 7FP*oN?
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 1Z/(G1
error('zernfun:RTHvector','R and THETA must be vectors.') e9Wa<i8
end e }?db
gS!:+G%
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r = r(:); oz\!V*CtK
theta = theta(:); HYD'.uj
length_r = length(r); fZGX}T<)p-
if length_r~=length(theta) xjUT{iwS
error('zernfun:RTHlength', ... g{]0sn#
'The number of R- and THETA-values must be equal.') Y#ap*
end 3V+] 9;
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% Check normalization: ('4_
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% -------------------- ;0]aq0_#(
if nargin==5 && ischar(nflag) T8?Ghbn
isnorm = strcmpi(nflag,'norm'); imhwY#D
if ~isnorm j1Y~_
error('zernfun:normalization','Unrecognized normalization flag.') P8OaoPj
end wQ:)KjhHH
else {Y(zd[
isnorm = false; "=HA Y
end <VMGTBVQ
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% W]$w@.oW[
% Compute the Zernike Polynomials k>Is:P
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]\-A;}\e
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% Determine the required powers of r: ]#iigPZ7
% ----------------------------------- nmee 'oEw
m_abs = abs(m); \Gef \
rpowers = []; Ko| d+
for j = 1:length(n) np|Sy;:
rpowers = [rpowers m_abs(j):2:n(j)]; yt+L0wzzB
end r5S[-`s;
rpowers = unique(rpowers); WMDl=6
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% Pre-compute the values of r raised to the required powers, d(ZO6Nr Q
% and compile them in a matrix: ~gJwW+
% ----------------------------- R+hU8 pu
if rpowers(1)==0 udK%>
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); #H&|*lr
rpowern = cat(2,rpowern{:}); 4Co6(
rpowern = [ones(length_r,1) rpowern]; pHGYQ;:L
else RT4x\&q
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Uk[b|<U-`d
rpowern = cat(2,rpowern{:}); SBu"3ym
end Ve$o}h-
#"6Qj'/h
(!u~CZ;
% Compute the values of the polynomials: l ~"^7H?4e
% -------------------------------------- 5;Czu(iH$
y = zeros(length_r,length(n)); .|KyNBn
for j = 1:length(n) U7,e/?a
s = 0:(n(j)-m_abs(j))/2; Df-DRi
pows = n(j):-2:m_abs(j); b}$+H/V
for k = length(s):-1:1 vQG5*pR*w
p = (1-2*mod(s(k),2))* ... 4d4ZT?V[
prod(2:(n(j)-s(k)))/ ... d UE,U=
prod(2:s(k))/ ... [C 7^r3w
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 94`7a<&ZNL
prod(2:((n(j)+m_abs(j))/2-s(k))); ^]Y>[[
idx = (pows(k)==rpowers); R{`(c/%8
y(:,j) = y(:,j) + p*rpowern(:,idx); h%na>G
end W\$`w
FW;?s+Uyx
if isnorm caR<Kb:;*
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ];$L &5^
end Wx%H%FeK
end ,Q$q=E;X
% END: Compute the Zernike Polynomials ;vR4XHl|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `6(S^P
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% Compute the Zernike functions: |)DGkOtd
% ------------------------------ RZ?jJm$
idx_pos = m>0; Xh"n]TK
idx_neg = m<0; 7vKK%H_P
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z = y; ;W
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if any(idx_pos) <]t%8GB2V
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); e;q!6%
end 2eS~/Pq5=i
if any(idx_neg) `:fZ)$sY
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); LzKj=5'Y
end ./Zk`-OBT
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% EOF zernfun (UD@q>c