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， f}t8V% ^E
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， AGG T]
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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？ Miz?t*|{[
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ n'@*RvI:
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function z = zernfun(n,m,r,theta,nflag) mKBO<l{S
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ij,Rq`}l
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ka_( 8
% and angular frequency M, evaluated at positions (R,THETA) on the ubv>* i O
% unit circle. N is a vector of positive integers (including 0), and bq2f?uD-}
% M is a vector with the same number of elements as N. Each element E/zclD5S
% k of M must be a positive integer, with possible values M(k) = -N(k) 3rY\y+m
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, fC".K
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% and THETA is a vector of angles. R and THETA must have the same DNr*|A2<
% length. The output Z is a matrix with one column for every (N,M) n?778Wo}
% pair, and one row for every (R,THETA) pair. k<|}&<h
% >xXC=z+g]
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike R GL2S]UFs
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), zI 0d
% with delta(m,0) the Kronecker delta, is chosen so that the integral |R2p^!m
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, l,* 5*1lM
% and theta=0 to theta=2*pi) is unity. For the non-normalized @ ^R l{p
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. _X|prIOb=
% J 5(^VKj
% The Zernike functions are an orthogonal basis on the unit circle. .DI?-=p|_#
% They are used in disciplines such as astronomy, optics, and ?N(<w?Gat
% optometry to describe functions on a circular domain. wB bC GU
% d'"|Qg_'
% The following table lists the first 15 Zernike functions. d_Jj&:"l
% Qvty;2$o@
% n m Zernike function Normalization W4,'?o
% -------------------------------------------------- !TivQB
% 0 0 1 1 W*Si"s2
% 1 1 r * cos(theta) 2 Ze[,0Y!u&
% 1 -1 r * sin(theta) 2 `{|w*) mD
% 2 -2 r^2 * cos(2*theta) sqrt(6) 0'HQ=pP
% 2 0 (2*r^2 - 1) sqrt(3) *7E#=xb
% 2 2 r^2 * sin(2*theta) sqrt(6) T(q Tipq0
% 3 -3 r^3 * cos(3*theta) sqrt(8) P2@Z7DhQ
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Wb>;L@jB7
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) @mJ~?d95v
% 3 3 r^3 * sin(3*theta) sqrt(8) y M `u]p1
% 4 -4 r^4 * cos(4*theta) sqrt(10) d@ >i=l [
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) L+*:VP6WD
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 8ok=&Gq4
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) OIJT~Z}
% 4 4 r^4 * sin(4*theta) sqrt(10) @H<*|3J
% -------------------------------------------------- #N "u 0
% 2n$Wey[
% Example 1: |Iw glb!k
% 4&=</ok6`0
% % Display the Zernike function Z(n=5,m=1) uIbAl E
% x = -1:0.01:1; <=V{tl
% [X,Y] = meshgrid(x,x); E%D T;1
% [theta,r] = cart2pol(X,Y); 9|lLce$
% idx = r<=1; 4=o vm[
% z = nan(size(X)); -pIz-*
% z(idx) = zernfun(5,1,r(idx),theta(idx)); W7Y@]QMX
% figure S2 e3d
% pcolor(x,x,z), shading interp =kfa1kD&{
% axis square, colorbar 6UqAs<c9
% title('Zernike function Z_5^1(r,\theta)') 71y{Dwya
% <zL_6Y2
% Example 2: Ix6\5}.c 9
% ^;'8yE/
% % Display the first 10 Zernike functions 8>t,n,k
% x = -1:0.01:1; /OWwC%tM/
% [X,Y] = meshgrid(x,x); Q #G x o
% [theta,r] = cart2pol(X,Y); 8}m J )9<7
% idx = r<=1; ol*,&C:{
% z = nan(size(X)); + C8O"
% n = [0 1 1 2 2 2 3 3 3 3]; Eamt_/LKf
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; :09NZ
!!
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 9OV@z6
% y = zernfun(n,m,r(idx),theta(idx)); |$b8(g$s)
% figure('Units','normalized') F_ (~b
% for k = 1:10 0U@#&pUc
% z(idx) = y(:,k); ~1%*w*
% subplot(4,7,Nplot(k)) ]c~yMA+]FZ
% pcolor(x,x,z), shading interp L FkDb}
% set(gca,'XTick',[],'YTick',[]) K^U ="
% axis square B=r DU$z
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) O7Jux-E 1C
% end 2t9UJu4
% OemY'M? ZQ
% See also ZERNPOL, ZERNFUN2. p X{wEc6 }
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% Paul Fricker 11/13/2006 yAryw{(
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% Check and prepare the inputs: } Xo #/9
% ----------------------------- 7%i'F=LzT
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) B` Z3e %g#
error('zernfun:NMvectors','N and M must be vectors.')
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end b^Z2Vf:k]
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if length(n)~=length(m) L@{'J
error('zernfun:NMlength','N and M must be the same length.') IQ @9S
end Tv DSs])
h(HpeN%`#
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n = n(:); dGY R
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m = m(:); M5ZH6X@5
if any(mod(n-m,2)) 5[jcw`
error('zernfun:NMmultiplesof2', ... 7K\v=
'All N and M must differ by multiples of 2 (including 0).') /=S@3?cQAB
end ~j'D%:[+VH
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k_,7#:+
if any(m>n) Xx{| [2`
error('zernfun:MlessthanN', ... ICN>kJ\;M
'Each M must be less than or equal to its corresponding N.') ;[}OZt
end &T,|?0>~=J
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if any( r>1 | r<0 ) EwzR4,r\M
error('zernfun:Rlessthan1','All R must be between 0 and 1.') k9}8xpH
end k~ 8-E u1
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) k sv ]
error('zernfun:RTHvector','R and THETA must be vectors.') Iw`tb N
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end o1ZVEvp
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r = r(:); V1]GOmXz
theta = theta(:); [f_^B U&
length_r = length(r); z< L2W",
if length_r~=length(theta) U3{<+vSR`
error('zernfun:RTHlength', ... KEOk%'c,
'The number of R- and THETA-values must be equal.') JD$g%hcVZa
end 1%+-}yo<
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% Check normalization: r%.k,FzGZY
% -------------------- eTa_R O,x
if nargin==5 && ischar(nflag) i<"lXu
isnorm = strcmpi(nflag,'norm'); +-j-)WU?,
if ~isnorm G ?$ @6
error('zernfun:normalization','Unrecognized normalization flag.') -Q n-w3~&
end sG`x |%t
else D.a>i?W
isnorm = false; dipfsH]p
end OT
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G!Zb27u+
% Compute the Zernike Polynomials y!=,u
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% bTum|GWf
wb$uq/|
mXRkR.zu+
% Determine the required powers of r: q 6> eb
% ----------------------------------- . $&^yp
m_abs = abs(m); :0r ,.)
rpowers = []; #d@wjQ0DW
for j = 1:length(n) Ol>q(-ea
rpowers = [rpowers m_abs(j):2:n(j)]; U!(.i1^n
end 5s[nE\oaG
rpowers = unique(rpowers); pp@
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f7x2"&?vg
% Pre-compute the values of r raised to the required powers, 7_I83$p'
% and compile them in a matrix: Ek L2nI
% ----------------------------- %+~\I\)1
if rpowers(1)==0 D~C'1C&W
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ab6I*DbF
rpowern = cat(2,rpowern{:}); $%~ JG (
rpowern = [ones(length_r,1) rpowern]; zgw ez$
else v6*0@/L
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rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); RCWmdR#}V
rpowern = cat(2,rpowern{:}); q^aDZzx,z
end : "85w#r
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% Compute the values of the polynomials: JIyS e:p3
% -------------------------------------- w)EY j+L
y = zeros(length_r,length(n)); AQ'%}(#0
for j = 1:length(n) fp [gKRSF
s = 0:(n(j)-m_abs(j))/2; ]}v]j`9m%
pows = n(j):-2:m_abs(j); < A,V/']
for k = length(s):-1:1 pkn^K+<n,
p = (1-2*mod(s(k),2))* ... {:1j>4m 2
prod(2:(n(j)-s(k)))/ ... ` g]
prod(2:s(k))/ ... tAv3 +
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... QHNyH
prod(2:((n(j)+m_abs(j))/2-s(k))); Tc^
0W=h
idx = (pows(k)==rpowers); n\ "6ol}>E
y(:,j) = y(:,j) + p*rpowern(:,idx); h1_Z&VJ
end i;O_B5
d
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if isnorm F rc kA
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); (tg.]q_=u
end tpJA~!mG3
end Jq/itsg
% END: Compute the Zernike Polynomials &e^;;<*w
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %,@pV%2
UA$IVK&{
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% Compute the Zernike functions: .a@12J(I
% ------------------------------ @lN\.O
idx_pos = m>0; RcYUO*
idx_neg = m<0; \ Bo$
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gC1LQ!:;Oi
z = y; u9"=t
if any(idx_pos) ZO< ,V
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); OFtaOjsyUa
end &ksuk9M
if any(idx_neg) >PA*L(Dh%
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ,U\ s89
end a}y b~:TC
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% EOF zernfun 5MnP6(3$
