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， ywk RH
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， {~j/sto-:
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？ )n 1[#x^I
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ 1c429 &-
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function z = zernfun(n,m,r,theta,nflag) }@ ktAt
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Y)$%-'=b+
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N _uL [
Z
% and angular frequency M, evaluated at positions (R,THETA) on the >Yk|(!v
% unit circle. N is a vector of positive integers (including 0), and "frioi`a2
% M is a vector with the same number of elements as N. Each element HRjbGc|[
% k of M must be a positive integer, with possible values M(k) = -N(k) I{WP:]"Yf
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, c0rU&+:Ry
% and THETA is a vector of angles. R and THETA must have the same }
XhL`%
% length. The output Z is a matrix with one column for every (N,M) * h=>*t?I2
% pair, and one row for every (R,THETA) pair. : ir3u
% }&v-<qC^
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike HC 1<zW[
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ( xWsyo(4
% with delta(m,0) the Kronecker delta, is chosen so that the integral 5 r_Z3 /%
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 9wGsHf8]
% and theta=0 to theta=2*pi) is unity. For the non-normalized /D yeMC Y-
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. FE^/us7r
% zS|4@t\__
% The Zernike functions are an orthogonal basis on the unit circle.
N- &ZaK
% They are used in disciplines such as astronomy, optics, and D)DD 6
% optometry to describe functions on a circular domain. )"hd"
% T U2oQ1
% The following table lists the first 15 Zernike functions. /Z!$b D
% CDXN%~0h
% n m Zernike function Normalization XksI .]tfj
% -------------------------------------------------- jF
j'6LT9/
% 0 0 1 1 DO~[VK%|
% 1 1 r * cos(theta) 2 @ <2y+_e
% 1 -1 r * sin(theta) 2 s3nt2$=:t
% 2 -2 r^2 * cos(2*theta) sqrt(6) }!> \Ja<\
% 2 0 (2*r^2 - 1) sqrt(3) TQNdBq5I6
% 2 2 r^2 * sin(2*theta) sqrt(6) M*D_p n&
% 3 -3 r^3 * cos(3*theta) sqrt(8) |2 n*Ds'
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) MN5}}@
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) K@xMPB8in
% 3 3 r^3 * sin(3*theta) sqrt(8) *i#N50k*j'
% 4 -4 r^4 * cos(4*theta) sqrt(10) zTfjuI|R
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) '[%Pdd]!
E
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) do.>Y}d
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +HRtuRv0T
% 4 4 r^4 * sin(4*theta) sqrt(10) }cGILH%
% -------------------------------------------------- 77s G;8HE
% Vn:v{-i
% Example 1: 7-n HPDp'
% dTCLE t.
% % Display the Zernike function Z(n=5,m=1) =uNc\a (
% x = -1:0.01:1; 8v8- 5N
% [X,Y] = meshgrid(x,x); n 3&h1-
% [theta,r] = cart2pol(X,Y); hCF_pt+
% idx = r<=1; T ,!CDm$=
% z = nan(size(X)); *{k{
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ss }-YnG
% figure .|g@#XIwe#
% pcolor(x,x,z), shading interp NB'G{),)Z
% axis square, colorbar D]aQt%TL
% title('Zernike function Z_5^1(r,\theta)') Gf9sexn]l
% d}Guj/cx,
% Example 2: @&&} J
% *y7 Yf7
% % Display the first 10 Zernike functions bV2a2#kj
% x = -1:0.01:1; K0C"s 'q
% [X,Y] = meshgrid(x,x); 3fpaTue|x
% [theta,r] = cart2pol(X,Y); xg^%8Ls^
% idx = r<=1; MZf?48"f
% z = nan(size(X)); .E+O,@?<
% n = [0 1 1 2 2 2 3 3 3 3]; pM+9K:^B
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; }a, ycFt
% Nplot = [4 10 12 16 18 20 22 24 26 28]; cr ]b #z
% y = zernfun(n,m,r(idx),theta(idx)); A-3^~aEgx
% figure('Units','normalized') :=+YZ|& j
% for k = 1:10 .57F h)Y
% z(idx) = y(:,k); ) :Z #!O<
% subplot(4,7,Nplot(k)) p^q/ u
% pcolor(x,x,z), shading interp lg2I|Z6DH
% set(gca,'XTick',[],'YTick',[]) Yy]TU} PY
% axis square 1,$"'lK wt
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) [_3&
% end lfCr `[!E
% WjR2:kT
% See also ZERNPOL, ZERNFUN2. *,:2O&P
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% Paul Fricker 11/13/2006 )p;t
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% Check and prepare the inputs: L >"O[ @
% ----------------------------- ??P\v0E
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) )
: *[mvF
error('zernfun:NMvectors','N and M must be vectors.') 5Uy *^C7M^
end .{? ; #Cdn
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Q x|HvT2P
if length(n)~=length(m) * HHL a
error('zernfun:NMlength','N and M must be the same length.') p p1Kor
end BQ[R)o
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W+=j@JY}q9
n = n(:); *>zOWocxD
m = m(:); K8-1?-W
if any(mod(n-m,2)) eNi#% ?=WB
error('zernfun:NMmultiplesof2', ... Eul3 {+]
'All N and M must differ by multiples of 2 (including 0).') Y?0x /2<
end xW9R -J \W
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if any(m>n) \r{ wN qyv
error('zernfun:MlessthanN', ... :(/1,] bF
'Each M must be less than or equal to its corresponding N.') c2 :,
end _dAn/rj
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z
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if any( r>1 | r<0 ) 4Wa$>vz
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0LzS #J+
end lFIaC}
i,Z-UA|f=T
#hs&)6S f
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) G)b:UJa"
error('zernfun:RTHvector','R and THETA must be vectors.') hv>Xr=RE
end QqW N7y_9
5&L*'kV@
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r = r(:); &dj/Dq@
theta = theta(:); "d~<{(:N^
length_r = length(r); ^!k_"C)B
if length_r~=length(theta) ']c;$wP
error('zernfun:RTHlength', ... AA ~7 "2 e
'The number of R- and THETA-values must be equal.') sRcS-Yw[S
end [J eq ?X9
jw\4`NZ]
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% Check normalization: Q;=4']hYU
% -------------------- A~k:
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if nargin==5 && ischar(nflag) #wvGS%
isnorm = strcmpi(nflag,'norm'); rP"Y.;s
if ~isnorm q%f90
error('zernfun:normalization','Unrecognized normalization flag.') rAW7Zp~KK
end R\5fl[
else <~v4BiQ3l^
isnorm = false; qQo*:3/];
end YbWz!.WPe
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4 8 J{Y3F
% Compute the Zernike Polynomials .ty2! .
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (8o;Cm
J?Q@f
sH1 ucZ>9Y
% Determine the required powers of r: 3&c'3y:b
% ----------------------------------- eD NY|}$}v
m_abs = abs(m); 3]'h (C
rpowers = []; 6wq%4RI0
for j = 1:length(n) vKdS1Dn1
rpowers = [rpowers m_abs(j):2:n(j)]; i^IL o,Q
end oHS Di
rpowers = unique(rpowers); P&Xy6@%[Z
!rqs!-cCQ
=Bh,>Kg
% Pre-compute the values of r raised to the required powers, v!<FeL W
% and compile them in a matrix: \fUVWXv
% ----------------------------- - \ew,y
if rpowers(1)==0 ;r]!
qv :
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); +[S< "}ls7
rpowern = cat(2,rpowern{:}); l#+@!2z
rpowern = [ones(length_r,1) rpowern]; vt(n: Xk
else "8(8]GgYx
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); YCzH@94QeV
rpowern = cat(2,rpowern{:}); ~ \u>jel
end ^$oEM0h
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% Compute the values of the polynomials: 56N DU>j$
% -------------------------------------- * "?,.
y = zeros(length_r,length(n)); duCXCX^n
T
for j = 1:length(n) { M[iYF g=
s = 0:(n(j)-m_abs(j))/2; ?&U~X)Q
pows = n(j):-2:m_abs(j); %JA^b5''
for k = length(s):-1:1 cauKG@:2F
p = (1-2*mod(s(k),2))* ... %/s+-j@s:
prod(2:(n(j)-s(k)))/ ... pg<c vok
prod(2:s(k))/ ... EF 8rh
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ...
^ fS_h `B
prod(2:((n(j)+m_abs(j))/2-s(k))); ~_-+Q=3
idx = (pows(k)==rpowers); <
r b5'
y(:,j) = y(:,j) + p*rpowern(:,idx); >7W 8_6sC<
end /B{c L`<
Ac
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if isnorm ~"R;p}5 "
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); O#vIn }
end /" &Jf}r
end `j.-hy>s
% END: Compute the Zernike Polynomials -b
)~
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fj <a;oV
v:9Vp{ )
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% Compute the Zernike functions: bD 1IY1
% ------------------------------ zj1_# =]
idx_pos = m>0; c+1<3)Q<
idx_neg = m<0; :pP l|"
= o1 &.v2j
*zX^Sg-[
z = y; dFnu&u"
if any(idx_pos) ;,B $lgF
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); vFgnb WxG
end x$b Cbg
if any(idx_neg) !T]bz+
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); M>jk"*hA|
end 7 /DDQ
xw1n;IO4
0INlo
% EOF zernfun :&O6Y-/B
