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, >M2~p&Si
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, ;"Gy5
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? @l j|
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 06Wqfzceb
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function z = zernfun(n,m,r,theta,nflag) 9DA|;|
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Nksm&{=6S
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N %htI!b+"@
% and angular frequency M, evaluated at positions (R,THETA) on the 7/~=[#]*
% unit circle. N is a vector of positive integers (including 0), and bfA>kn0C
% M is a vector with the same number of elements as N. Each element Ps@']]4>W
% k of M must be a positive integer, with possible values M(k) = -N(k) DehjV6t
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, B%\&Q@X
% and THETA is a vector of angles. R and THETA must have the same bI
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% length. The output Z is a matrix with one column for every (N,M) Cik1~5iF
% pair, and one row for every (R,THETA) pair. i24k
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% q3#[6!
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Cqnuf5e>L
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), .@)vJtH)
% with delta(m,0) the Kronecker delta, is chosen so that the integral ?:$
q~[LY
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, o~XK*f=(
% and theta=0 to theta=2*pi) is unity. For the non-normalized 5{b;wLi$X2
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2ul8]=
% 4q] 6[/
% The Zernike functions are an orthogonal basis on the unit circle. 1@OpvO5
% They are used in disciplines such as astronomy, optics, and `$> Y
% optometry to describe functions on a circular domain. kV1L.Xg
% BmV`<Q,
% The following table lists the first 15 Zernike functions. .4v?/t1
% q~> +x?30
% n m Zernike function Normalization fhN\AjB6Td
% -------------------------------------------------- B{Vc-qJ
% 0 0 1 1 a9e0lW:=c
% 1 1 r * cos(theta) 2 %}TJr]'F
% 1 -1 r * sin(theta) 2 a^l)vh{+
% 2 -2 r^2 * cos(2*theta) sqrt(6) H-pf8
% 2 0 (2*r^2 - 1) sqrt(3) "yQBHYP
% 2 2 r^2 * sin(2*theta) sqrt(6) {*+J`H_G2a
% 3 -3 r^3 * cos(3*theta) sqrt(8) ;av!fK
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) F3(SbM-
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) &fB=&jc*j
% 3 3 r^3 * sin(3*theta) sqrt(8) `C: 7N=9
% 4 -4 r^4 * cos(4*theta) sqrt(10) YtvDayR>
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7<WUjK|
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ~RVlc;W
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) m
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% 4 4 r^4 * sin(4*theta) sqrt(10) V$hL\`e
% -------------------------------------------------- Kfb(wW
% "T=j\/Q
% Example 1: 15jQ87)
% v K{2
% % Display the Zernike function Z(n=5,m=1) .9x*YS
% x = -1:0.01:1; K*5gb^Ul
% [X,Y] = meshgrid(x,x); zlEI_th:~
% [theta,r] = cart2pol(X,Y); yQ/O[(
% idx = r<=1; VLm\P S
% z = nan(size(X)); ~4+Y BN
% z(idx) = zernfun(5,1,r(idx),theta(idx)); me2vR#
% figure ?rOj?J9
% pcolor(x,x,z), shading interp G AY?F
% axis square, colorbar UY9*)pEE
% title('Zernike function Z_5^1(r,\theta)') ;MGm,F,o
% -}<Ru)
% Example 2: a%c <3'
% % WDTnEm
% % Display the first 10 Zernike functions ?n{m2.H
% x = -1:0.01:1; k-jFT3b$
% [X,Y] = meshgrid(x,x); Y$v d@Q
% [theta,r] = cart2pol(X,Y); ;O)*!yA(GG
% idx = r<=1; yL
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% z = nan(size(X)); >8{w0hh;
% n = [0 1 1 2 2 2 3 3 3 3]; xKE=$SV(
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; BC!) g+8
% Nplot = [4 10 12 16 18 20 22 24 26 28]; \h'7[vkr
% y = zernfun(n,m,r(idx),theta(idx)); hkl0N%[
% figure('Units','normalized') kO}%Y?9d
% for k = 1:10 <xeB9
% z(idx) = y(:,k); \LJ!X3TZ
% subplot(4,7,Nplot(k)) 3q`f|r
% pcolor(x,x,z), shading interp >QYx9`x&
% set(gca,'XTick',[],'YTick',[]) F-ZTy"z
% axis square ffk>IOH
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) j_,/U^Ws|f
% end I*%3E.Z@g
% OP+*%$wR
% See also ZERNPOL, ZERNFUN2. axmq/8X
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% Paul Fricker 11/13/2006 ;5;>f)diS
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% Check and prepare the inputs: afHRy:<+%
% ----------------------------- G?v<-=I
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) nW]CA~
error('zernfun:NMvectors','N and M must be vectors.') 6, j60`f)
end #Ev}Gf+5Q
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if length(n)~=length(m) r{?qvl!q
error('zernfun:NMlength','N and M must be the same length.') BYdGK@ouk
end KW'nW
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n = n(:); !otseI!!/
m = m(:); 5-0&`,
if any(mod(n-m,2)) Q'jGNWep
error('zernfun:NMmultiplesof2', ... ylos6]zS8
'All N and M must differ by multiples of 2 (including 0).') v$@1q9 5J
end fk15O_#3
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if any(m>n) I/upiq y
error('zernfun:MlessthanN', ... %h0BA.r
'Each M must be less than or equal to its corresponding N.') 0J[B3JO@M
end kK4+K74B
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if any( r>1 | r<0 ) |g \_xl
error('zernfun:Rlessthan1','All R must be between 0 and 1.') A#']e 8
end unFm~rcf
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) bZUw^{~)D
error('zernfun:RTHvector','R and THETA must be vectors.') d]K8*a%[-
end H(Wiy@cJn
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r = r(:); 8[,R4@
theta = theta(:); |Wck-+}U
length_r = length(r); 5`&@3
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if length_r~=length(theta) I+W,%)vb
error('zernfun:RTHlength', ... ?z|Bf@TJ[+
'The number of R- and THETA-values must be equal.') `K@N\VM
end ]qZj@0#7n
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% Check normalization: -#;ZZ\fdj
% -------------------- _IEbRVpb
if nargin==5 && ischar(nflag) y+$vHnS/jC
isnorm = strcmpi(nflag,'norm'); @\gE{;a8
if ~isnorm pUmT?N!
error('zernfun:normalization','Unrecognized normalization flag.') /g%RIzgW
end vMX\q
else +B8oW3v# )
isnorm = false; U7/
=|Z
end _qOynW
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RWu<
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% Compute the Zernike Polynomials )#4(4
@R h
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jp}.W
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hxJKYU^%m
% Determine the required powers of r: #~m^RoE
% ----------------------------------- N&G(`]
m_abs = abs(m); Q A~F
rpowers = []; Z
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for j = 1:length(n) F#Pn]
rpowers = [rpowers m_abs(j):2:n(j)]; 4/\Ynb.L
end o[JZ>nm
rpowers = unique(rpowers); N|"q6M!ZL
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% Pre-compute the values of r raised to the required powers, 2|1CGHj\
% and compile them in a matrix: 45Zh8 k
% ----------------------------- 9T$%^H9
if rpowers(1)==0 >D##94PZ
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); afaQb
rpowern = cat(2,rpowern{:}); {#@[ttw$U
rpowern = [ones(length_r,1) rpowern]; dci,[TEGu
else XmVst*2=
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); S+eu3nMq
rpowern = cat(2,rpowern{:}); dF! B5(
end p}I\H
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% Compute the values of the polynomials: 6_rgRo&
% -------------------------------------- e8_EB/)_Z
y = zeros(length_r,length(n)); I3Z\]BI
for j = 1:length(n) i-WP#\s
s = 0:(n(j)-m_abs(j))/2; C[ KMaB
pows = n(j):-2:m_abs(j); .DnG}884
for k = length(s):-1:1 9&kPcFX B
p = (1-2*mod(s(k),2))* ... XdlA)0S)
prod(2:(n(j)-s(k)))/ ... })PU`?f
prod(2:s(k))/ ... hCX/k<}I
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 8OS^3JS3"
prod(2:((n(j)+m_abs(j))/2-s(k))); 2}.~
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idx = (pows(k)==rpowers); =kOo(
y(:,j) = y(:,j) + p*rpowern(:,idx); !w!k0z]
end wJgH15oB
!-SI &qy
if isnorm \|]+sQ WQ
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 7;6'=0(
end D#Mz#\4o
end gCL}Ba
% END: Compute the Zernike Polynomials U:
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .UN?Ak*R
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% Compute the Zernike functions: X*Q<REDB
% ------------------------------ BOdlz#&s
idx_pos = m>0; Hy'EbQ
idx_neg = m<0; j39"iAn
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z = y; @o<B>$tbu4
if any(idx_pos) x],XiSyp
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); CqX*.j{
end ;kG"m7-/
if any(idx_neg) HYJEz2RF
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); p~(STHDe#
end iK5[P
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% EOF zernfun X
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