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, |a /cw"
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, !Q%r4Nr
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? qPUACuF'
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? <&B]p
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function z = zernfun(n,m,r,theta,nflag) x
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. {- MhhRa5
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N )[&j&AI
% and angular frequency M, evaluated at positions (R,THETA) on the prIJjy-F
% unit circle. N is a vector of positive integers (including 0), and %wu,ce]*
% M is a vector with the same number of elements as N. Each element Aq(,
% k of M must be a positive integer, with possible values M(k) = -N(k) (U.VCSn
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, =KnHa.%
% and THETA is a vector of angles. R and THETA must have the same \MmB+'f&R
% length. The output Z is a matrix with one column for every (N,M) VzcW9'"#
% pair, and one row for every (R,THETA) pair. eISHV.QV
% j
*N^.2
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike M3GFKWQI,`
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), $SniQ
% with delta(m,0) the Kronecker delta, is chosen so that the integral i
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% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ^;\6ju2
% and theta=0 to theta=2*pi) is unity. For the non-normalized rXe+#`m2
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. d)$seZB
% 5$$]ZMof
% The Zernike functions are an orthogonal basis on the unit circle. Ur""&@
% They are used in disciplines such as astronomy, optics, and F:0 E-
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% optometry to describe functions on a circular domain. b+CvA(*
% Na.e1A&?j
% The following table lists the first 15 Zernike functions. )^E6VD&6
% f|yq~3x)
% n m Zernike function Normalization REk^pZ3B
% -------------------------------------------------- XFww|SG$
% 0 0 1 1 Fy_~~nI0
% 1 1 r * cos(theta) 2 x^pHP|<3`
% 1 -1 r * sin(theta) 2 5(Xq58nhxI
% 2 -2 r^2 * cos(2*theta) sqrt(6) g^\>hjNX
% 2 0 (2*r^2 - 1) sqrt(3) f-}_
% 2 2 r^2 * sin(2*theta) sqrt(6)
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% 3 -3 r^3 * cos(3*theta) sqrt(8) gvP.\,U
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) n{c-3w.uD
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Mt121Q&"
% 3 3 r^3 * sin(3*theta) sqrt(8) R_:-Z.
% 4 -4 r^4 * cos(4*theta) sqrt(10) GMob&0l8_
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) T=pKen/
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) u)P)r,
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) oYeFOw`
% 4 4 r^4 * sin(4*theta) sqrt(10) z}7U>y6`
% -------------------------------------------------- <<1_rRL]
% "fd'~e$S#
% Example 1: m W4tW
% GIUyW
% % Display the Zernike function Z(n=5,m=1) tZD^<Q7}\
% x = -1:0.01:1; Z2k5qs7g
% [X,Y] = meshgrid(x,x); B :1r;8{j
% [theta,r] = cart2pol(X,Y); `{S4_'
% idx = r<=1; @#5?tk0
% z = nan(size(X)); U }}E
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% z(idx) = zernfun(5,1,r(idx),theta(idx)); ? ~_h3bHH
% figure S'AS,'EnY
% pcolor(x,x,z), shading interp I{u+=0^Y
% axis square, colorbar hB]<li)"C
% title('Zernike function Z_5^1(r,\theta)') ery{>|k
% X,+N/nku
% Example 2: ,aSK L1
% 0av2w5>af
% % Display the first 10 Zernike functions !f8]gT zN
% x = -1:0.01:1; k=5v
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% [X,Y] = meshgrid(x,x); mDIN%/S'
% [theta,r] = cart2pol(X,Y); G\S_e7$/
% idx = r<=1; Dt+uf5o(
% z = nan(size(X)); 1f5;^T
I
% n = [0 1 1 2 2 2 3 3 3 3]; 8d\/
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ZL- ` 3x
% Nplot = [4 10 12 16 18 20 22 24 26 28]; GG`;c?d@
% y = zernfun(n,m,r(idx),theta(idx)); L>2gx$f
% figure('Units','normalized') &vS @-K
% for k = 1:10 k.#[h@Pm
% z(idx) = y(:,k); G%fNGQwT
% subplot(4,7,Nplot(k)) (0bXsfe
% pcolor(x,x,z), shading interp ]4-t*Em
% set(gca,'XTick',[],'YTick',[]) _VAX~Y]
% axis square 1VO>Bh.Wm
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) -gLU>I7wV
% end zB)wYKwZ
% I~U;M+n*y
% See also ZERNPOL, ZERNFUN2. 'xc=N
=:v5`
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% Paul Fricker 11/13/2006 $PKUcT0N9
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%KmhR2v
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% Check and prepare the inputs: BYsQu.N
% ----------------------------- WzO[-csy
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) -VRKQNT
error('zernfun:NMvectors','N and M must be vectors.') WEB enGQ
end "Bbd[ZI8
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if length(n)~=length(m) G[[<-[C]5
error('zernfun:NMlength','N and M must be the same length.') ++M%PF [
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end )u(Dq u\t
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n = n(:); q#!c6lG
m = m(:); _'DZoOH|VE
if any(mod(n-m,2)) @
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error('zernfun:NMmultiplesof2', ... Av^<_`L:
'All N and M must differ by multiples of 2 (including 0).') p3z%Y$!Tm
end 5 iP{)
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if any(m>n) \KPwh]0
error('zernfun:MlessthanN', ... 9jTm g%
'Each M must be less than or equal to its corresponding N.') dW>$C_`?
end 5X"WgR;
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if any( r>1 | r<0 ) >4eZ%</D5
error('zernfun:Rlessthan1','All R must be between 0 and 1.') nfzKUJY
end :\8&Th}Se
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 5Dlx]_
error('zernfun:RTHvector','R and THETA must be vectors.') Qp]-4%^Vz
end '2.11cM3
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r = r(:); gdfG3d$4
theta = theta(:); y153ax
length_r = length(r); VyL|d^'f_
if length_r~=length(theta) n^Sc*7
error('zernfun:RTHlength', ... ;Q} H'Wg,
'The number of R- and THETA-values must be equal.')
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end j>t*k!db
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% Check normalization: K~JXP5`(
% -------------------- N`%f+eT(
if nargin==5 && ischar(nflag) 0al8%z9e@
isnorm = strcmpi(nflag,'norm'); [v$NxmRu
if ~isnorm +4%:q~C
error('zernfun:normalization','Unrecognized normalization flag.') M,b^W:('4
end %!e;sL~&
else Co#_Cyxg=9
isnorm = false; *X4$'LSx1
end z7P]g
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h8lI#Gs
% Compute the Zernike Polynomials edy6WzxBcm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CAD:ifV
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T{%'"mm;
% Determine the required powers of r: /4Lmu+G4
% ----------------------------------- E[RLBO[*n
m_abs = abs(m); %d\|a~p:
rpowers = []; gwepaW
for j = 1:length(n) d4#Ra%
rpowers = [rpowers m_abs(j):2:n(j)]; z.7'yJIP#
end _ooSMp|
rpowers = unique(rpowers); (\6R"2
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% Pre-compute the values of r raised to the required powers, YQ+Kl[ec
% and compile them in a matrix: SLze) ?.
% ----------------------------- Ag!#epi{0
if rpowers(1)==0 8/y~3~A{D
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); bu2'JIDR
rpowern = cat(2,rpowern{:}); E |A,NPf%I
rpowern = [ones(length_r,1) rpowern]; .{|AHW&0<
else hoQ?8}r:
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); MXxE)"G*a
rpowern = cat(2,rpowern{:}); Ay Obaa5
end F^]?'`7md
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% Compute the values of the polynomials: [,|Z<
% -------------------------------------- plY`lqm
y = zeros(length_r,length(n)); 2F[;Z*&
for j = 1:length(n) |UO1v A@
s = 0:(n(j)-m_abs(j))/2; FDAREE\j
pows = n(j):-2:m_abs(j); _z!0ab
for k = length(s):-1:1 q$Ol"K@
p = (1-2*mod(s(k),2))* ... QJG]z'c+
prod(2:(n(j)-s(k)))/ ... j{nkus2
prod(2:s(k))/ ... @Yq!
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... _5nQe
!
prod(2:((n(j)+m_abs(j))/2-s(k))); d\ &jl`8*
idx = (pows(k)==rpowers); +"]'h~W
y(:,j) = y(:,j) + p*rpowern(:,idx); 3o'SY@'W
end ?ExfxR!~
n]B)\D+V^
if isnorm uxto:6),P<
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); '7F`qL\/#(
end Z6b3gV
end C%P"Ds=w0N
% END: Compute the Zernike Polynomials o4kNDXP#S
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -BV&u(
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% Compute the Zernike functions: z^4\?R50yO
% ------------------------------ nDvny0^a
idx_pos = m>0; b)u9#%Q
idx_neg = m<0; oh;F]*k6
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z = y; @T1>%oi
if any(idx_pos) ?.A6HrAPB
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); IBVP4&}x$
end 0nAeeVz|
if any(idx_neg) tS2lex%
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); lb1(1|#
end 4(JxZ49
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% EOF zernfun UhCd,