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， L[9+xK^g
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， W SeRV?+T
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？ [}g5Z=l
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ @X / =.
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% L,,
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function z = zernfun(n,m,r,theta,nflag) B 0fo[Ev
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. : .o 0<
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N !"qEB2r
% and angular frequency M, evaluated at positions (R,THETA) on the j1C0LP8
% unit circle. N is a vector of positive integers (including 0), and i3\oy`GJ
% M is a vector with the same number of elements as N. Each element !c;p4 B)
% k of M must be a positive integer, with possible values M(k) = -N(k) (6_/n&mF
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 5Szo5
% and THETA is a vector of angles. R and THETA must have the same D2mAyU -
% length. The output Z is a matrix with one column for every (N,M) 53#5p;k
% pair, and one row for every (R,THETA) pair. X=7vUb,\gB
% Kof-;T
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike z:q'?{` I
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), d=Ihl30m
% with delta(m,0) the Kronecker delta, is chosen so that the integral
(Y?yG q/
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, x-P_}}K 79
% and theta=0 to theta=2*pi) is unity. For the non-normalized uqH! eN5
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. U%6lYna{M#
% = ~R3* GN
% The Zernike functions are an orthogonal basis on the unit circle. @o.i2iG
% They are used in disciplines such as astronomy, optics, and ?q8g< -?
% optometry to describe functions on a circular domain. qdnNapWnc
% 60gn`s,,
% The following table lists the first 15 Zernike functions. R }YryzV5
% zL=I-f Vq
% n m Zernike function Normalization JQv
ZTwSI
% -------------------------------------------------- Kd21:|!t^
% 0 0 1 1 #rL@
% 1 1 r * cos(theta) 2
0>J4O:k
% 1 -1 r * sin(theta) 2 &~ 6O;}\
% 2 -2 r^2 * cos(2*theta) sqrt(6) 'Z%aBCM
% 2 0 (2*r^2 - 1) sqrt(3) gM:oP.
% 2 2 r^2 * sin(2*theta) sqrt(6) y3$\ m
% 3 -3 r^3 * cos(3*theta) sqrt(8)
B#lj8I^|
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) =<tEc+!T3
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) O[J +dWyp
% 3 3 r^3 * sin(3*theta) sqrt(8) jWjK -q@Y
% 4 -4 r^4 * cos(4*theta) sqrt(10) ziip*<a !_
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) o=5uM
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 2{qG
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ] nGA1 S{
% 4 4 r^4 * sin(4*theta) sqrt(10) Q^;\!$:M
% -------------------------------------------------- 7 "U,N; y
% ij SYQ
% Example 1: "K=)J'/n
% `t"Kq+
% % Display the Zernike function Z(n=5,m=1) %&S]cEw
% x = -1:0.01:1; l"g% vS,;`
% [X,Y] = meshgrid(x,x); $G.|5sEk
% [theta,r] = cart2pol(X,Y); 9%veUvY
% idx = r<=1; eesLTy D2_
% z = nan(size(X)); yL,B\YCf8
% z(idx) = zernfun(5,1,r(idx),theta(idx)); p 5w g+K
% figure B(NL3WJ
% pcolor(x,x,z), shading interp ? =Qg
% axis square, colorbar FX% E7H
% title('Zernike function Z_5^1(r,\theta)') 3
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% WW z ns[$f
% Example 2: 2o}FB\4^i
% ;\0RXirk
% % Display the first 10 Zernike functions 8hV:bz"
% x = -1:0.01:1; 6!m#_z8qG3
% [X,Y] = meshgrid(x,x); Jk{2!uP
% [theta,r] = cart2pol(X,Y); .;Yei6H
% idx = r<=1; 09i[2n;O
% z = nan(size(X)); NX/)Z& Fx:
% n = [0 1 1 2 2 2 3 3 3 3]; @K> Pw arl
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; %^A++Z$`
% Nplot = [4 10 12 16 18 20 22 24 26 28]; x /v+7Pt_
% y = zernfun(n,m,r(idx),theta(idx)); $^GnY7$!>
% figure('Units','normalized') bsDUFXH]
% for k = 1:10 XA k l,Y
% z(idx) = y(:,k); TR7TF]i tb
% subplot(4,7,Nplot(k)) VUhu"h@w%
% pcolor(x,x,z), shading interp .w"O/6."
% set(gca,'XTick',[],'YTick',[]) J> |`
% axis square yx4c+(J^8
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) s_ $@N!
% end KLB?GN?Pb
% G(e?]{(
% See also ZERNPOL, ZERNFUN2. yIP
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Cl=ExpX/O
SesO$=y
% Paul Fricker 11/13/2006 {%dQV#'c
s)=7tHoqB)
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b9jm= U
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% Check and prepare the inputs: / GNYv*
% ----------------------------- zc 5_;!t
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) =0 |evC
error('zernfun:NMvectors','N and M must be vectors.') l1-FL-1
end ggW fk
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/vKDlCH*
if length(n)~=length(m) *a4eL [
error('zernfun:NMlength','N and M must be the same length.') Z]CH8GS~<
end L x&ZWF$
Vy VC#AK,
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n = n(:); 6cvm\ opH
m = m(:); (w fZ !
if any(mod(n-m,2)) 64cmv}d _
error('zernfun:NMmultiplesof2', ... KYaf7qy]
'All N and M must differ by multiples of 2 (including 0).') =lnz5H
end f
#14%?/
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if any(m>n) sa($3`d
error('zernfun:MlessthanN', ... dE~ns
,+
'Each M must be less than or equal to its corresponding N.') u ""= 9>0
end 0v ?,:]A0E
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8j k*N
if any( r>1 | r<0 ) H?m2|.
error('zernfun:Rlessthan1','All R must be between 0 and 1.') -1:asM7
end %K4-V 5f
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q Z.\GHS
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) L
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error('zernfun:RTHvector','R and THETA must be vectors.') l/5/|UE9
end [ 0Sd +{Q
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N C& 1l]
r = r(:); jn'8F$GU
theta = theta(:); <|@9]>z
length_r = length(r); bhRpYP%x
if length_r~=length(theta) SzDi= lY
error('zernfun:RTHlength', ... >JhQ =j
'The number of R- and THETA-values must be equal.') ,>Q,0bVhH0
end *4bV8T>0Z
l`k3!EZDS
R!=XMV3$PH
% Check normalization: TBzM~y
% -------------------- ,yoT3_%P
if nargin==5 && ischar(nflag) /[p4. F L
isnorm = strcmpi(nflag,'norm'); 8I'?9rt2M
if ~isnorm GU xhCoxb
error('zernfun:normalization','Unrecognized normalization flag.') K(?7E6\vO
end NNT9\JRv_
else z{ 8!3>:E
isnorm = false; Kt-@a%O0
end `'/8ifKz
R *U>T$
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% eP)RP6ON{
% Compute the Zernike Polynomials |7 argk+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vc<8ApK3V
9 }=Fdt
*\/ U T
% Determine the required powers of r: @RjLDj+)S
% ----------------------------------- Y<B| e91C
m_abs = abs(m); yC
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rpowers = []; ?^hC|IR$
for j = 1:length(n) !@Ox %v K
rpowers = [rpowers m_abs(j):2:n(j)]; D`ZYF)[}J
end z) ydQw>
rpowers = unique(rpowers); /N $T[
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&h;
q nOAIP:0
% Pre-compute the values of r raised to the required powers, 7-^d4P+|g
% and compile them in a matrix: a^22H
% ----------------------------- =hA/;
if rpowers(1)==0 8WAg{lVs
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); h:|aQJG5
rpowern = cat(2,rpowern{:}); $V[ ob
rpowern = [ones(length_r,1) rpowern]; A9"ho}<
else "Kqe4$
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); {AZW."?
rpowern = cat(2,rpowern{:}); wm} i+ApK
end xd* kNY
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% Compute the values of the polynomials: rIXAn4,dTv
% -------------------------------------- WPPmh~:
y = zeros(length_r,length(n)); Eq|_> f@@8
for j = 1:length(n) Z@ 1rs#
s = 0:(n(j)-m_abs(j))/2; 9N9;EY -U
pows = n(j):-2:m_abs(j); t ({:TQ
for k = length(s):-1:1 :5ji.g* 0
p = (1-2*mod(s(k),2))* ... N(D_*% 96
prod(2:(n(j)-s(k)))/ ... ~($h9* \
prod(2:s(k))/ ... n04Zji(F@
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... /vBp R m
prod(2:((n(j)+m_abs(j))/2-s(k))); k }/0 B
idx = (pows(k)==rpowers); "Li"NxObCA
y(:,j) = y(:,j) + p*rpowern(:,idx); 1:8ZS
end C\1Dy5
. uh P (
if isnorm <@oK ^ja
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); xC|7"N^/
end <h( t W
end s{gdTG6v`
% END: Compute the Zernike Polynomials Up8#Nz
T
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +YP,LDJ!v
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% Compute the Zernike functions: <CeDIX t
% ------------------------------ ZMbv1*Vt
idx_pos = m>0; (}'0K?
idx_neg = m<0; pZXva9bE
cd \0
oM J5;
z = y; /']Gnt G.
if any(idx_pos) I"r*p?
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); lE /"
end !}U&%2<69
if any(idx_neg) h " j{B
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); t lc&Wx
end &Jq ?tnNd
f.Jz]WXw,
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% EOF zernfun \T>f+0=4
