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, kmT5g gy
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, 8(ej]9RObU
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? ce:p*
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? @jY=b<
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function z = zernfun(n,m,r,theta,nflag) 7]H<ou
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 'M!M$<j
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N T7~H|%
% and angular frequency M, evaluated at positions (R,THETA) on the Mqv[7.|
% unit circle. N is a vector of positive integers (including 0), and B-UsMO
% M is a vector with the same number of elements as N. Each element }\0ei(%H
% k of M must be a positive integer, with possible values M(k) = -N(k) *WaqNMD[%
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, qs Wy
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% and THETA is a vector of angles. R and THETA must have the same YpI|=mv
% length. The output Z is a matrix with one column for every (N,M) zd|n!3;
% pair, and one row for every (R,THETA) pair. 0TWd.+
% `br$kB
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike yQ0:M/r;0
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), sOVU>tb\'
% with delta(m,0) the Kronecker delta, is chosen so that the integral TyhO+;
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Kv9Z.DY
% and theta=0 to theta=2*pi) is unity. For the non-normalized 0p]v#z}
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. z3I
|jy1
% %X|u({(zb
% The Zernike functions are an orthogonal basis on the unit circle. !|Wf
mU
% They are used in disciplines such as astronomy, optics, and rld8hFj
% optometry to describe functions on a circular domain. )M><09
% 8PR\a!"
% The following table lists the first 15 Zernike functions. nvQTJ4,,
% #/B g5:
% n m Zernike function Normalization EKus0"|
% -------------------------------------------------- :g ~_
% 0 0 1 1 Vh{(*p
% 1 1 r * cos(theta) 2 LU/;`In
% 1 -1 r * sin(theta) 2 BU#3fPl
% 2 -2 r^2 * cos(2*theta) sqrt(6) !_P&SmK3
% 2 0 (2*r^2 - 1) sqrt(3) N "}N>xe2
% 2 2 r^2 * sin(2*theta) sqrt(6) A `{hKS
% 3 -3 r^3 * cos(3*theta) sqrt(8) -Xx4:S
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 0X3yfrim
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) RqX^$C8M
% 3 3 r^3 * sin(3*theta) sqrt(8) T+e*' <!O
% 4 -4 r^4 * cos(4*theta) sqrt(10) "hi03k
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) z]7 /Gc,j
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) [ ou$*
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?lML+
% 4 4 r^4 * sin(4*theta) sqrt(10) 6?'7`p
% -------------------------------------------------- <RKT
|
% Ec2;?pvd%J
% Example 1: DD2K>1A1
% pH3<QNq5
% % Display the Zernike function Z(n=5,m=1) o7t{?|
% x = -1:0.01:1; T*nP-b
% [X,Y] = meshgrid(x,x); K)U[xS;<
% [theta,r] = cart2pol(X,Y); &r\8VEZq"
% idx = r<=1; 4jt(tZS
% z = nan(size(X)); AmC?qoEWQ7
% z(idx) = zernfun(5,1,r(idx),theta(idx)); G[1\5dK*uR
% figure c]zFZJ6M
% pcolor(x,x,z), shading interp 3~VV2O
% axis square, colorbar C~R
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% title('Zernike function Z_5^1(r,\theta)') J#t-."f6^
% w@<II-9L)<
% Example 2: +IO>%
% L$BV`JWPw
% % Display the first 10 Zernike functions K_@?Q@#YhR
% x = -1:0.01:1; }Ba_epM
% [X,Y] = meshgrid(x,x); Qe{w)e0}`
% [theta,r] = cart2pol(X,Y); Q;J(
5;
% idx = r<=1; M~N/er
% z = nan(size(X)); 5'c#pm\Q
% n = [0 1 1 2 2 2 3 3 3 3]; 2;u
i'B
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; $dF3@(p
% Nplot = [4 10 12 16 18 20 22 24 26 28]; :eSsqt9]9
% y = zernfun(n,m,r(idx),theta(idx)); 2j}DI"|h
% figure('Units','normalized') R3;%eyu
% for k = 1:10 3H`{
A/r
% z(idx) = y(:,k); mf)+ 5On
% subplot(4,7,Nplot(k)) 1I{8 |
% pcolor(x,x,z), shading interp a eeor
% set(gca,'XTick',[],'YTick',[]) !1fZ7a
% axis square 9 @xl{S-
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) !nCq8~#
% end N@L{9ak1
% (
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% See also ZERNPOL, ZERNFUN2. E^zfI9R
naW!b&:
y?3.W
% Paul Fricker 11/13/2006 //_H_ue$
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% Check and prepare the inputs: |PlNVd2
% ----------------------------- kJp~'\b
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) O|~C qb
error('zernfun:NMvectors','N and M must be vectors.') c%J6!\
end qS2Nk.e]o
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if length(n)~=length(m) Vh;zV Y
error('zernfun:NMlength','N and M must be the same length.') weSq|f
end {VL@U$'oI
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Pps-,*m
n = n(:); R2gV(L(!!
m = m(:); 1XMR7liE
if any(mod(n-m,2)) m&Mupl
error('zernfun:NMmultiplesof2', ... dy&UF,l6
'All N and M must differ by multiples of 2 (including 0).') $KO2+^%y
end w_xca(
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:Ko6.|
if any(m>n) q.VYPkEib
error('zernfun:MlessthanN', ... u]};QR
'Each M must be less than or equal to its corresponding N.') AO$AT_s
end a+E&{pV
&~
y)b`r
kkF)Tro\
if any( r>1 | r<0 ) >sfg`4
error('zernfun:Rlessthan1','All R must be between 0 and 1.') {P]C>
end 6:]N%
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&)%+DUV|
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) S{rltT-
error('zernfun:RTHvector','R and THETA must be vectors.') `za,sRFR
end CwA_jOp
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[|DKBJ
r = r(:); $(aq;DR
theta = theta(:); //U1mDFT
length_r = length(r); aa`(2%(:
if length_r~=length(theta) U]iI8c
error('zernfun:RTHlength', ... hm`=wceK
'The number of R- and THETA-values must be equal.') kI^*
'=:
end 5^u$zfR
uZS :
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% Check normalization: gBm'9|?
% -------------------- PgWWa*Ew
if nargin==5 && ischar(nflag) NXU:b"G
S
isnorm = strcmpi(nflag,'norm'); :8A+2ra&
if ~isnorm <W80A J
error('zernfun:normalization','Unrecognized normalization flag.') QF#w$%7
end Nr~$i% [
else <(L@@.87R
isnorm = false; {LO Pm1K8Y
end ?Z7`TnG$uf
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3!"N;Q"
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m+kP"]v
% Compute the Zernike Polynomials *qd:f!Q3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Gk,Bx1y
%ou,|Dww
XA>W>|
% Determine the required powers of r: K4c:k;
V
% ----------------------------------- 'o>)E>
m_abs = abs(m); >cu%C s=m
rpowers = []; #z*,CU#S9d
for j = 1:length(n) _ E;T"SC
rpowers = [rpowers m_abs(j):2:n(j)]; +$dJA
end J D\tt-
rpowers = unique(rpowers); RP4/:sO
yn4T!r "
wVs?E
% Pre-compute the values of r raised to the required powers, eUyF<j
% and compile them in a matrix: Ott6y
% ----------------------------- -8TJ:#|N
if rpowers(1)==0 :!`"GaTy
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); d7OygDb <
rpowern = cat(2,rpowern{:}); hi7_jl6
rpowern = [ones(length_r,1) rpowern]; `ONjEl
else m&.LJ*uM\K
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); -aoYoJ '
rpowern = cat(2,rpowern{:}); rf.pT+g.P
end N9e'jM>Oos
w?Te%/s.
h)KHc/S
% Compute the values of the polynomials: diq}\'f
% -------------------------------------- f98,2I(>`+
y = zeros(length_r,length(n)); TlqHj
for j = 1:length(n) SK<Rk
s = 0:(n(j)-m_abs(j))/2; b$ G{^
pows = n(j):-2:m_abs(j); }u Y2-l
for k = length(s):-1:1 /k#-OXP~
p = (1-2*mod(s(k),2))* ... $^Fl*:6
prod(2:(n(j)-s(k)))/ ... {keZ_2
prod(2:s(k))/ ... .Ro/ioq
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... :cT)M(o
prod(2:((n(j)+m_abs(j))/2-s(k))); $@g]?*L:
idx = (pows(k)==rpowers); D -}>28
y(:,j) = y(:,j) + p*rpowern(:,idx); S$6|KY u
end D!<F^mtl
Kl1v^3\{
if isnorm 3<0b_b
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); JzyCeM =
end kB7vc>@1
end [GwAm>k
% END: Compute the Zernike Polynomials
TBj 2(Z
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vB:\ZX4
FXQWT9Kk~_
IkrB}
% Compute the Zernike functions: eV;r /4
% ------------------------------ \Z-th,t
idx_pos = m>0; KkvcZs'4m
idx_neg = m<0; BZq#OAp
-^_m(@A<~
om3
%\
z = y; 3]Z1kB
if any(idx_pos) YagfCi ?
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); a)_3r]sv^
end 'o AmA=
if any(idx_neg) ^&zCPUH
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); BI'>\hX/V
end b?H"/Mu.
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% EOF zernfun #mxOwvJ