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, 5rQu^6&
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, taO(\FOm
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? 'jye*
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? WWOjck#
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function z = zernfun(n,m,r,theta,nflag) m6gMVon
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 5as5{"l
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N um( xZ6&m
% and angular frequency M, evaluated at positions (R,THETA) on the <;1M!.)5
% unit circle. N is a vector of positive integers (including 0), and h1f 05
% M is a vector with the same number of elements as N. Each element ~JS@$ #
% k of M must be a positive integer, with possible values M(k) = -N(k) }-9 c1&m
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, VAqZ`y
% and THETA is a vector of angles. R and THETA must have the same 4#ikdjB;
% length. The output Z is a matrix with one column for every (N,M) PZ?kv 4
% pair, and one row for every (R,THETA) pair. TWfkr
% ,,ML^ey
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 9}a&:QTHR
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), _E/
% with delta(m,0) the Kronecker delta, is chosen so that the integral RfT)dS+rAh
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ,<s:*
k
% and theta=0 to theta=2*pi) is unity. For the non-normalized b+$wx~PLi
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. )FfS7 C\.
% T?tZ?!6
% The Zernike functions are an orthogonal basis on the unit circle. {)Shc;Qh
% They are used in disciplines such as astronomy, optics, and z 8#{=e
% optometry to describe functions on a circular domain. gplrJaH@
% ]xbMMax
% The following table lists the first 15 Zernike functions. j}fSz)`i
% &78lep
% n m Zernike function Normalization )Z\Zw~L
% -------------------------------------------------- m5,&;~
% 0 0 1 1 =hI;5KF
% 1 1 r * cos(theta) 2 >?ec"P%vS/
% 1 -1 r * sin(theta) 2 ]AN%#1++U
% 2 -2 r^2 * cos(2*theta) sqrt(6) 5Ux= 5a
% 2 0 (2*r^2 - 1) sqrt(3) ogJ';i/o
% 2 2 r^2 * sin(2*theta) sqrt(6) (''w$qq"D
% 3 -3 r^3 * cos(3*theta) sqrt(8) 152LdZevF
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) S/YHT)0x[
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) /R>YDout}
% 3 3 r^3 * sin(3*theta) sqrt(8) - "{hP
% 4 -4 r^4 * cos(4*theta) sqrt(10) aO
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% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 8~|v:qk
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ]x%sX|Rj
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Id8e%)
% 4 4 r^4 * sin(4*theta) sqrt(10) cu)B!#<!&
% -------------------------------------------------- K;>9K'n
% OXB 5W#$
% Example 1: iDltN]zS
% }
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% % Display the Zernike function Z(n=5,m=1) oqg +<m
% x = -1:0.01:1; 7=&+0@R#/d
% [X,Y] = meshgrid(x,x); 'Axe:8LA'
% [theta,r] = cart2pol(X,Y); G6xNR
% idx = r<=1; +Z]}ce
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% z = nan(size(X)); 6:?mz;oP
% z(idx) = zernfun(5,1,r(idx),theta(idx)); xP27j_*m>
% figure 2av=W
% pcolor(x,x,z), shading interp }U%T6~_wR
% axis square, colorbar r- Y7wM`TZ
% title('Zernike function Z_5^1(r,\theta)') @twi<U_
% u('`.dwkc
% Example 2: 31QDN0o!~
% #<#-B v
% % Display the first 10 Zernike functions Q9;VSF)
% x = -1:0.01:1; uh>"TeOi
% [X,Y] = meshgrid(x,x); t%@u)b p
% [theta,r] = cart2pol(X,Y); 6^2='y~e
% idx = r<=1; |Nadk(}
% z = nan(size(X)); F'JT7#eX
% n = [0 1 1 2 2 2 3 3 3 3]; ['3E'q,4&
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; $Yw~v36`t/
% Nplot = [4 10 12 16 18 20 22 24 26 28]; VA %lJ!$
% y = zernfun(n,m,r(idx),theta(idx)); ZoCk]hk
% figure('Units','normalized') aN!,\D
% for k = 1:10 NSq29#
% z(idx) = y(:,k); lwjA07i
% subplot(4,7,Nplot(k)) 9hJ
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% pcolor(x,x,z), shading interp =F5zU5`i
% set(gca,'XTick',[],'YTick',[]) /_yAd,^-+
% axis square ,| j\x
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) -<e_^
% end 8m#y>`
% 90ov[|MkM
% See also ZERNPOL, ZERNFUN2. }%^ 3
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% Paul Fricker 11/13/2006 Gh<#wa['}
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% Check and prepare the inputs: $bN_0s0:'
% ----------------------------- s{42_O?,c
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) by$mD_sr
error('zernfun:NMvectors','N and M must be vectors.') E?VOst&
end 9! yDZ<s
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if length(n)~=length(m) [+!+Yn6:
error('zernfun:NMlength','N and M must be the same length.') +
+Eu.W;
end Iv u'0vF
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n = n(:); ]Y$jc
m = m(:); S %wdXe
if any(mod(n-m,2)) E5Ls/ HK
error('zernfun:NMmultiplesof2', ... \FnR'ne
'All N and M must differ by multiples of 2 (including 0).')
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end ?KE:KV[Y
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if any(m>n) X +/^s)
error('zernfun:MlessthanN', ... 7&(h_}Z
'Each M must be less than or equal to its corresponding N.') _T;Kn'Gz(&
end DU-dIqi
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if any( r>1 | r<0 ) (-@I'CFd
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ]H<}6}Gd
end } V"A;5j`
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Jn9{@??
error('zernfun:RTHvector','R and THETA must be vectors.') &gI*[5v
end 4.>y[_vu
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r = r(:); f9O_M1=|lo
theta = theta(:); ^,J>=>,1\
length_r = length(r); vOl3utu7
if length_r~=length(theta) a|k*A&5u2
error('zernfun:RTHlength', ... QoMa+QTuc
'The number of R- and THETA-values must be equal.') b27t-p8
end "
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% Check normalization: /74)c~.W
% -------------------- dki3(
if nargin==5 && ischar(nflag) OD?y
isnorm = strcmpi(nflag,'norm'); V5 Gy|X
if ~isnorm 4Vd[cRh2
error('zernfun:normalization','Unrecognized normalization flag.') TeyFq0j@'
end >A}ra ^gU
else KXBTJ&
isnorm = false; 2<d'!cm
end l(}l([rdQ
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M"$g*j
% Compute the Zernike Polynomials iaQFVROu
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2/x~w~3U
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% Determine the required powers of r: w*u.z(:a`
% ----------------------------------- {
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m_abs = abs(m); BU|m{YZ$
rpowers = []; i6O'UzD@T
for j = 1:length(n) hK3Twzte
rpowers = [rpowers m_abs(j):2:n(j)]; BLm}mb#/{
end oq(W|
rpowers = unique(rpowers); |{rhks~
%Kh}6
q}(f9
% Pre-compute the values of r raised to the required powers, Hdjp^O!
% and compile them in a matrix: .fK~IKA
% ----------------------------- 8rNf4]5@X(
if rpowers(1)==0 %PPkT]~\
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); r/QI-Cf&
rpowern = cat(2,rpowern{:}); )[=C@U
rpowern = [ones(length_r,1) rpowern]; eUD 5V
else qr~zTBT]
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rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); vJ;0%;eu[!
rpowern = cat(2,rpowern{:}); J@rBrKC
end Xod/GYG
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% Compute the values of the polynomials: Ctk1\quz
% -------------------------------------- Q{~;4+ZD
y = zeros(length_r,length(n)); xSq+>, b
for j = 1:length(n) MI`<U:-lP
s = 0:(n(j)-m_abs(j))/2; $4]4G=o
pows = n(j):-2:m_abs(j); i\*
b<V
for k = length(s):-1:1 FQ/z,it_i
p = (1-2*mod(s(k),2))* ... rgEN~e'
prod(2:(n(j)-s(k)))/ ... )?(_vrc<
prod(2:s(k))/ ... +}BKDEb
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... a24(9(yh
prod(2:((n(j)+m_abs(j))/2-s(k))); ^ JU#_
idx = (pows(k)==rpowers); z\K-KD{Ad
y(:,j) = y(:,j) + p*rpowern(:,idx); BNixp[Hc
end qI[AsM+
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if isnorm LVy (O9g
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 5K=>x<
end @2+'s;mUV
end (62Sc]
% END: Compute the Zernike Polynomials w(Q{;RNM;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O81'i2MJ9
<JW%h :\t
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% Compute the Zernike functions: u\f QaQV
% ------------------------------ $7p0<<Nck
idx_pos = m>0; 6s$h _$[X
idx_neg = m<0; `a@YbuLd
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z = y; GvL\%0Ibx
if any(idx_pos) LE g#W
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); %~N| RSec
end i,l$1g-i
if any(idx_neg) `L3{y/U'
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Z|d+1i
end Qn@[{%),4
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% EOF zernfun '.I0n