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， #~]zh HI
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， VD*6g%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？ "S[ 450%
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ u,ho7ht3(
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function z = zernfun(n,m,r,theta,nflag) (XTG8W sN
%ZERNFUN Zernike functions of order N and frequency M on the unit circle.
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% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N XSB"{H>&
% and angular frequency M, evaluated at positions (R,THETA) on the dlh)g p;
% unit circle. N is a vector of positive integers (including 0), and 5Pc;5
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% M is a vector with the same number of elements as N. Each element XT%nbh&y
% k of M must be a positive integer, with possible values M(k) = -N(k) Z?q] b SIT
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, :LQYo '@yB
% and THETA is a vector of angles. R and THETA must have the same QT 5TE: D
% length. The output Z is a matrix with one column for every (N,M) #lo6c;*m5
% pair, and one row for every (R,THETA) pair. dES"@?!^
% e(&v"}Ef`
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike QO:!p5 ^:
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), |*xA 8&/
% with delta(m,0) the Kronecker delta, is chosen so that the integral t.y2ff<[U
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, *8 A
% and theta=0 to theta=2*pi) is unity. For the non-normalized tKuwpT 1Qc
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. DCO\c9
% !?jrf ]
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% The Zernike functions are an orthogonal basis on the unit circle. Dj?> <@
% They are used in disciplines such as astronomy, optics, and }-{H Y
% optometry to describe functions on a circular domain. 3*XNV
% D /gw .XYL
% The following table lists the first 15 Zernike functions. m])y .T
% net@j#}j-
% n m Zernike function Normalization xIW3={b 3
% -------------------------------------------------- Z clQ
% 0 0 1 1 P`+{@@
% 1 1 r * cos(theta) 2 p`dU2gV
% 1 -1 r * sin(theta) 2 SHxNr(wJ<Q
% 2 -2 r^2 * cos(2*theta) sqrt(6) Mj3A5;#
% 2 0 (2*r^2 - 1) sqrt(3) 1-uxC^u?|#
% 2 2 r^2 * sin(2*theta) sqrt(6) %wg -=;d4
% 3 -3 r^3 * cos(3*theta) sqrt(8) K7B/s9/xs
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) :RTC!spy
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) \:'/'^=#|
% 3 3 r^3 * sin(3*theta) sqrt(8) Q8tL[>Xt
% 4 -4 r^4 * cos(4*theta) sqrt(10) U}[d_f
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) H2\;%K 2
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) |A~jsz6 pI
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) nHA S(
% 4 4 r^4 * sin(4*theta) sqrt(10) &{hL&BLr
% -------------------------------------------------- mDABH@ R
% 2]jn '4
% Example 1: /Iy]DU8
% 8 ^2oWC#U(
% % Display the Zernike function Z(n=5,m=1) n)-$e4u2
% x = -1:0.01:1; ek\ xx
% [X,Y] = meshgrid(x,x); 4[r0G +
% [theta,r] = cart2pol(X,Y); 'F3f+YD
% idx = r<=1; 2;`1h[, -^
% z = nan(size(X)); =:Fc;n>c<K
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 7IH@oMvE
% figure 6<SAa#@ey
% pcolor(x,x,z), shading interp xh,qNnGG i
% axis square, colorbar [PM 2\#K
% title('Zernike function Z_5^1(r,\theta)') }OR@~V{Gj
% )[6 U^j4
% Example 2: J?1 uKR
% ^ogt+6c
% % Display the first 10 Zernike functions 2 86;=rN]*
% x = -1:0.01:1; zT.7
% [X,Y] = meshgrid(x,x); Yui3+}Ms
% [theta,r] = cart2pol(X,Y); h bDXo :
% idx = r<=1; iL&fgF"'
% z = nan(size(X)); O,
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% n = [0 1 1 2 2 2 3 3 3 3]; -#[a7',Z;
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; TDKki(o=~
% Nplot = [4 10 12 16 18 20 22 24 26 28]; l`{\"#4
% y = zernfun(n,m,r(idx),theta(idx)); &j`} vg
% figure('Units','normalized') PI)+Jr%L
% for k = 1:10 d#Y^>"|$.
% z(idx) = y(:,k); OA1uY83"
% subplot(4,7,Nplot(k)) u;"TTN
% pcolor(x,x,z), shading interp Lc,Pom
% set(gca,'XTick',[],'YTick',[]) KnQ*vM*VM
% axis square 3 ?9I J5p
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) RD i]2
% end &MQmu,4
% ,/%=su x
% See also ZERNPOL, ZERNFUN2. Xm}/0g&7
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% Paul Fricker 11/13/2006 HxV=F66"
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% Check and prepare the inputs: gMmaK0uhS
% ----------------------------- ?
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 61>.vT8P
error('zernfun:NMvectors','N and M must be vectors.') @Z
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end Xll}x+'uZK
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if length(n)~=length(m) T{. pM4Hd
error('zernfun:NMlength','N and M must be the same length.') f!uw zHA`?
end 3g,` .I_
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n = n(:); 2d #1=+V
m = m(:); <I\/n<*
if any(mod(n-m,2)) kR-SE5`Jk
error('zernfun:NMmultiplesof2', ... 5|j<`()H
:
'All N and M must differ by multiples of 2 (including 0).') ^R7lom.
end EF[@$j
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if any(m>n) ThajH K|U
error('zernfun:MlessthanN', ... t7Iv?5]N
'Each M must be less than or equal to its corresponding N.') IqaT?+O\?r
end v!6
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if any( r>1 | r<0 ) \5:i;A E
error('zernfun:Rlessthan1','All R must be between 0 and 1.') xw,IJ/E$1
end $aDVG})
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 3tIVXtUCUk
error('zernfun:RTHvector','R and THETA must be vectors.') x;P_1J%Q
end \^J%sf${
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x:NY \._
r = r(:); f P
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theta = theta(:); A5I)^B<(
length_r = length(r); QC
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if length_r~=length(theta) X1x#6
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error('zernfun:RTHlength', ... 2> x F){`
'The number of R- and THETA-values must be equal.') ArI2 wM/v
end +s, =lL
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% Check normalization: w7.V6S$Ga
% -------------------- C\Wmq
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if nargin==5 && ischar(nflag) {0Yf]FQb-a
isnorm = strcmpi(nflag,'norm'); p J!
mw\:
if ~isnorm !21FR*
error('zernfun:normalization','Unrecognized normalization flag.') vA F
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end Q^9_' t}X
else n`B:;2X,
isnorm = false; sk<3`x+
end ^B.5G K)!
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% qdJ=lhHM}
% Compute the Zernike Polynomials .LnGL]/
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F3[T.sf
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% Determine the required powers of r: 5,lEx1{_
% ----------------------------------- X Swl Tg
m_abs = abs(m); 6EoMt@7g
rpowers = []; T9E+\D
for j = 1:length(n) z [ }v{
rpowers = [rpowers m_abs(j):2:n(j)]; x/I%2F
end ~OYiq}g
rpowers = unique(rpowers); m/@wh a
#>("CAB02T
b;B%q$sntC
% Pre-compute the values of r raised to the required powers, iJI }TVep#
% and compile them in a matrix: lV3x *4O=
% ----------------------------- \g&,@'uh
if rpowers(1)==0 \j}ZB<.>
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); d=$Mim
rpowern = cat(2,rpowern{:}); ^qvZXb
rpowern = [ones(length_r,1) rpowern]; $lfn(b,
else $D~0~gn~
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); >W=,j)MA
rpowern = cat(2,rpowern{:}); w _ "E*9
end 13$%,q )
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% Compute the values of the polynomials: t9GR69v:?
% -------------------------------------- xA2YG|RU=b
y = zeros(length_r,length(n)); K-^\"
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for j = 1:length(n) htO +z7
s = 0:(n(j)-m_abs(j))/2; .ljnDL/
pows = n(j):-2:m_abs(j); *2>&"B09`
for k = length(s):-1:1 8rAg \H3E
p = (1-2*mod(s(k),2))* ... zJKv'>?
prod(2:(n(j)-s(k)))/ ... 8? B! 2
prod(2:s(k))/ ... ihhDO mUto
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Hp|kQJ[L E
prod(2:((n(j)+m_abs(j))/2-s(k))); g>E LGG |Q
idx = (pows(k)==rpowers); xk9% F?)
y(:,j) = y(:,j) + p*rpowern(:,idx); 5 Aw"B
end <6%?OJhp
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if isnorm :;%2BSgFU
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); p}}R -D&K
end )W,aN)1)
end n K1Slg#U
% END: Compute the Zernike Polynomials D=A&+6B@-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F/,NDZN
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zH 72'"w
% Compute the Zernike functions: 7y'RFD9@{
% ------------------------------ l5Ui w2
idx_pos = m>0; &@X<zWg
idx_neg = m<0; Y Vt% 0
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z = y; \Gef \
if any(idx_pos) r8t}TU>C
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ]6k\)# %2
end E<rp7~#
if any(idx_neg) nUaJz Pl
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); .r=4pQ@#
end >>4qJ%bL
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% EOF zernfun ^ x ]r`b
