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， ^Kb9@lz/
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， 5c*p2:]
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？ kbD*=d}3{
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ 2&suo!ig
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function z = zernfun(n,m,r,theta,nflag) X+/{%P!w
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. no mu$|I
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N yp( ?1
% and angular frequency M, evaluated at positions (R,THETA) on the sj #{TTW
% unit circle. N is a vector of positive integers (including 0), and c1 gz #,
% M is a vector with the same number of elements as N. Each element
h4 J{j h.
% k of M must be a positive integer, with possible values M(k) = -N(k) p)K9 ZI
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, {yGZc3e1j
% and THETA is a vector of angles. R and THETA must have the same ^A:!ni@3
% length. The output Z is a matrix with one column for every (N,M) Nck!z8
% pair, and one row for every (R,THETA) pair. ,?P8m"
% L3-<K op
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike e5]&1^+
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), _%AJmt}
% with delta(m,0) the Kronecker delta, is chosen so that the integral hWl""66+5
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 6GvhEulYR
% and theta=0 to theta=2*pi) is unity. For the non-normalized ;5,`Jpca
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2&zn^\%"
% ?6_"nT*}
% The Zernike functions are an orthogonal basis on the unit circle. 6R3"L]J
% They are used in disciplines such as astronomy, optics, and S7@ZtFf
% optometry to describe functions on a circular domain. t;Fbt("]:
% O('i*o4!}
% The following table lists the first 15 Zernike functions. IM l9\U
% 'vqj5 YTj
% n m Zernike function Normalization zav*
% -------------------------------------------------- f\U? :8 3
% 0 0 1 1 )Tyky%P+iI
% 1 1 r * cos(theta) 2 G2U5[\
% 1 -1 r * sin(theta) 2 8=ukS_?Vy
% 2 -2 r^2 * cos(2*theta) sqrt(6) +?4*,8Tmmz
% 2 0 (2*r^2 - 1) sqrt(3) * K0j5dx
% 2 2 r^2 * sin(2*theta) sqrt(6) F^/~@^{P
% 3 -3 r^3 * cos(3*theta) sqrt(8) E.5*Jr=J
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) R#[Qoy J
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) (ffOu#RQ3
% 3 3 r^3 * sin(3*theta) sqrt(8) uFA|r X
% 4 -4 r^4 * cos(4*theta) sqrt(10) N3S,33
8s
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) , }xpYq_/
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) A>&>6O4
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) m!FM+kge
% 4 4 r^4 * sin(4*theta) sqrt(10) [[.&, 6
% -------------------------------------------------- ~T;a jvJ
% #*ZnA,
% Example 1: b.w(x*a
% pw(U< )
% % Display the Zernike function Z(n=5,m=1) Vsm%h^]d
% x = -1:0.01:1; 5 b#"
G"
% [X,Y] = meshgrid(x,x); sqMNo n`5
% [theta,r] = cart2pol(X,Y); Gdc ~Lh
% idx = r<=1; 8CN7+V
% z = nan(size(X)); 7DC0 W|Fe
% z(idx) = zernfun(5,1,r(idx),theta(idx)); K~fDv i
% figure p;c_<>ws-Y
% pcolor(x,x,z), shading interp + !E{L
% axis square, colorbar Uy_}@50"l
% title('Zernike function Z_5^1(r,\theta)') 0k] ju
% )ZQ9 a4%
% Example 2: 5~ kW-x
% / ut~jf`
% % Display the first 10 Zernike functions % BKR}
% x = -1:0.01:1; >? A `C!i
% [X,Y] = meshgrid(x,x); f)ucC$1=
% [theta,r] = cart2pol(X,Y); !4b; >y=m
% idx = r<=1; I/ e2,
% z = nan(size(X)); x1 &b@u
% n = [0 1 1 2 2 2 3 3 3 3]; { C,1w
% m = [0 -1 1 -2 0 2 -3 -1 1 3];
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% Nplot = [4 10 12 16 18 20 22 24 26 28]; .SKNIct
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% y = zernfun(n,m,r(idx),theta(idx)); ]y)R C-N
% figure('Units','normalized') YiQeI|{oN
% for k = 1:10 # ZYid t
% z(idx) = y(:,k); @88z{
% subplot(4,7,Nplot(k)) 4E>/*F!
% pcolor(x,x,z), shading interp fjK]m.w
% set(gca,'XTick',[],'YTick',[]) 9 FFfRIVY
% axis square k1 LtqV
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) J}Z_.:JO(w
% end 6{Cu~G{]N
% 71n uTE%!
% See also ZERNPOL, ZERNFUN2. >1)@n3. <O
u;'<- _
w'zO(6 `
% Paul Fricker 11/13/2006 Dry ;$C}P
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% Check and prepare the inputs: ja,L)b :
% ----------------------------- mSfkyw.
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #&`WMLl+ 8
error('zernfun:NMvectors','N and M must be vectors.') AN:RY/ %Wo
end Q\/":ISq1
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if length(n)~=length(m) >!v,` O1
error('zernfun:NMlength','N and M must be the same length.') @)juP- o%
end HTtGpTsF
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n = n(:); -z~ V
m = m(:); =R24 h
if any(mod(n-m,2)) m 'H
error('zernfun:NMmultiplesof2', ... id[>! fQ=Y
'All N and M must differ by multiples of 2 (including 0).') @ vYN 7
end p7=^m>Z6
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if any(m>n) ka9v2tE \
error('zernfun:MlessthanN', ... ht74h
'Each M must be less than or equal to its corresponding N.') l<MCmKuYp
end U %PMV?L{
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if any( r>1 | r<0 ) o)' =D(
error('zernfun:Rlessthan1','All R must be between 0 and 1.') o?
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end <~8f0+"
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) =smY/q^3
error('zernfun:RTHvector','R and THETA must be vectors.') uY%3X/^j
end ]O(HZ D%
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r = r(:); cZK?kz_Y
theta = theta(:); S 0QU@e
length_r = length(r); T+{ 'W
if length_r~=length(theta) XxU}|jTO#
error('zernfun:RTHlength', ... P}u<NPy3Q
'The number of R- and THETA-values must be equal.') Ex&RR< 5
end 0c;"bA0>Sx
n\)f.}YD8d
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% Check normalization: ,`2xfVa-
% -------------------- 3eDx@8N
}
if nargin==5 && ischar(nflag) -a^sX%|Bl
isnorm = strcmpi(nflag,'norm'); OZ]3OL,
if ~isnorm e5 \1k#@
error('zernfun:normalization','Unrecognized normalization flag.') eDZ3SIZ
end #7:9XID /
else l:C0:m%
isnorm = false; g wjv&.T6^
end G ,*
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%_LHD|<
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% J3J RWy@?P
% Compute the Zernike Polynomials ]vyF &`phb
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Oua/NF)
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% Determine the required powers of r: w [x+2
% -----------------------------------
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m_abs = abs(m); IHMyP~{
rpowers = []; BTQC1;;N
for j = 1:length(n) WC&Ltw8
rpowers = [rpowers m_abs(j):2:n(j)]; c oz}VMp
end BP s
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rpowers = unique(rpowers); s-DL=MD
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% Pre-compute the values of r raised to the required powers, _C#( )#
% and compile them in a matrix: KT?s\w
% ----------------------------- QlXF:Gx"=
if rpowers(1)==0 m1Z8SM+
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); i 58C A?
rpowern = cat(2,rpowern{:}); $1
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rpowern = [ones(length_r,1) rpowern]; ! \Kh\
else j_ <n~ri-
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3&2q\]Y,
rpowern = cat(2,rpowern{:}); \ku{-^ 7
end Q9V4 -MC9
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% Compute the values of the polynomials: ^>28>!"1
% --------------------------------------
';V+~pi
y = zeros(length_r,length(n)); 6Ky"4\e
for j = 1:length(n) daNIP1Qn
s = 0:(n(j)-m_abs(j))/2; 2DQC)Pe+z
pows = n(j):-2:m_abs(j); a'~y '6
for k = length(s):-1:1 Jxp'.oo[
p = (1-2*mod(s(k),2))* ... ]bSt[
prod(2:(n(j)-s(k)))/ ... A84HaRlkF5
prod(2:s(k))/ ... _kLoDju%
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... aE$p;I
prod(2:((n(j)+m_abs(j))/2-s(k))); l.)!jWY
idx = (pows(k)==rpowers); # nhAW
y(:,j) = y(:,j) + p*rpowern(:,idx); 9R3=h5Y
end Agf!6kh
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if isnorm b)@x@3"O
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); /_(Dq8^g@
end Zt=X
%M|aw
end a{,t@G
% END: Compute the Zernike Polynomials &6EfybAt^_
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u'> CU
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% Compute the Zernike functions: giq` L1<
% ------------------------------ ?[">%^
idx_pos = m>0; 1vb0G ;a;|
idx_neg = m<0; D1 k]
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z = y; jd*H$ BU^
if any(idx_pos) \O~P
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z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); aQ.
\!&U
end WI3!?>d
if any(idx_neg) 2S/ 7f:
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); H[C n@XE
end w6 .HvH-@?
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% EOF zernfun V
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