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， } $'_%,
我输入10阶的n、m，r,theta为38025*1向量，最后得到的z是29525*10阶的矩阵， o5NmNOXm
这个，跟我们用zygo干涉仪直接拟合出的36项zernike系数，有何关系呢？ Rqp#-04 *W
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ )H{1 Xjh -
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function z = zernfun(n,m,r,theta,nflag) H>+])~#
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. , 6X;YY
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N #X?[")R
% and angular frequency M, evaluated at positions (R,THETA) on the h72/03 !
% unit circle. N is a vector of positive integers (including 0), and 1BU 97 !
% M is a vector with the same number of elements as N. Each element xd ^Pkf
% k of M must be a positive integer, with possible values M(k) = -N(k) e&d$kUJrq
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, to < /
% and THETA is a vector of angles. R and THETA must have the same dX@ic,?
% length. The output Z is a matrix with one column for every (N,M) #?>)5C\Hqy
% pair, and one row for every (R,THETA) pair. dB0#EJaE
% %\HPYnIe
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ^Z?m)qxvB
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), d$3md<lIB
% with delta(m,0) the Kronecker delta, is chosen so that the integral abR<( H12
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, m1Y a
% and theta=0 to theta=2*pi) is unity. For the non-normalized w|s2f`!
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. : #CWiq("%
% Pg(Y}Tu
% The Zernike functions are an orthogonal basis on the unit circle. $jE<n/8
% They are used in disciplines such as astronomy, optics, and @ztT1?!e
% optometry to describe functions on a circular domain. hQm=9gS
% vjx'yh|
% The following table lists the first 15 Zernike functions. $Z#~wsw
% 8:V,>PH
% n m Zernike function Normalization VPYLDg.'
% -------------------------------------------------- w
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% 0 0 1 1 `D~oY =
% 1 1 r * cos(theta) 2 x-CjxU3
% 1 -1 r * sin(theta) 2 >,]a>V
% 2 -2 r^2 * cos(2*theta) sqrt(6) uhfK \.3
% 2 0 (2*r^2 - 1) sqrt(3) D5P-$1KPt
% 2 2 r^2 * sin(2*theta) sqrt(6) h$!YKfhq}
% 3 -3 r^3 * cos(3*theta) sqrt(8) mnK<5KLg1
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) )LFbz#;Y
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 3Z9Yzv)A
% 3 3 r^3 * sin(3*theta) sqrt(8) C?gqX0[ q
% 4 -4 r^4 * cos(4*theta) sqrt(10) 9S@x
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) M1- tRF
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) DPxx9lN_rx
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5.{= O p!
% 4 4 r^4 * sin(4*theta) sqrt(10) XKky-LeJ
% -------------------------------------------------- }'eef"DJ9
% e&VC }%m
% Example 1: $`3yImv+w
% O|8@cO
% % Display the Zernike function Z(n=5,m=1) M> WWP3
% x = -1:0.01:1; 5S!#^>_
% [X,Y] = meshgrid(x,x); vkTu:3Qe
% [theta,r] = cart2pol(X,Y); 94#,dA,M
% idx = r<=1; >
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% z = nan(size(X)); -@(LN%7!C
% z(idx) = zernfun(5,1,r(idx),theta(idx)); F,~BhKkbV
% figure { . 9BG&
% pcolor(x,x,z), shading interp lOVcXAe}
% axis square, colorbar qSr]d`7@
% title('Zernike function Z_5^1(r,\theta)') @rbd`7$%
% NgyEy n
\
% Example 2: ;O`f+rG~
% ';FJs&=I
% % Display the first 10 Zernike functions '=E;^'Rl
% x = -1:0.01:1; j;`Q82V\
% [X,Y] = meshgrid(x,x); q"2APvsvp
% [theta,r] = cart2pol(X,Y); TS6xF?
% idx = r<=1; m)p|NdTZc8
% z = nan(size(X)); 2.%)OC!q&5
% n = [0 1 1 2 2 2 3 3 3 3]; _{k*JT2
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; MQwxQ{
% Nplot = [4 10 12 16 18 20 22 24 26 28]; zb9G&'7
% y = zernfun(n,m,r(idx),theta(idx)); RQ8d1US
% figure('Units','normalized') vlkw Wm
% for k = 1:10 xcW\U^1d
% z(idx) = y(:,k); K{ DC{yLu
% subplot(4,7,Nplot(k)) {UP[iw$~
% pcolor(x,x,z), shading interp d9S/_iCI
% set(gca,'XTick',[],'YTick',[]) (7G4 v
% axis square A| f6H6UUx
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) )]C]K B
% end "ZGP,=?y2
% Li5&^RAo|J
% See also ZERNPOL, ZERNFUN2. WBWW7 HK
n o<$=(11i
n5d8^c! 2
% Paul Fricker 11/13/2006 *xNc^ &.
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% Check and prepare the inputs: ~_EDJp1J
% ----------------------------- }X{rE|@
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Q ")Xg:
error('zernfun:NMvectors','N and M must be vectors.') :%sBY0 yF
end AA=Ob$2$
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* `+zf7-f
if length(n)~=length(m) G"F O%3&|
error('zernfun:NMlength','N and M must be the same length.') %9>w|%+;U+
end ,A` | jF
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n = n(:); 8HKv _vl
m = m(:); e&
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if any(mod(n-m,2)) j ^j"w( a
error('zernfun:NMmultiplesof2', ... N0S^{ j,i
'All N and M must differ by multiples of 2 (including 0).') 4O-LLH
end 6{.U7="
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if any(m>n) a 4ViVy
error('zernfun:MlessthanN', ... b Sw^a{~)
'Each M must be less than or equal to its corresponding N.') " YI,
end _ VuWo
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if any( r>1 | r<0 ) .@)vJtH)
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ?:$
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end o~XK*f=(
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) )
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error('zernfun:RTHvector','R and THETA must be vectors.') 4E.9CjN1>
end U 2YY
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r = r(:); 4\-11!'08
theta = theta(:); `]W9Fj<1j
length_r = length(r); 'zm5wqrkAd
if length_r~=length(theta) 6,YoP|@0
error('zernfun:RTHlength', ... >G|RVB
'The number of R- and THETA-values must be equal.')
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end KE,.Evyu=
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% Check normalization: %@C8EFl% 3
% -------------------- I^A>YJW
if nargin==5 && ischar(nflag) .Qrpz ^wdt
isnorm = strcmpi(nflag,'norm'); ]|!| 3lQ
if ~isnorm TXi|
error('zernfun:normalization','Unrecognized normalization flag.') -&
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end %/I:r7UR{
else i
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isnorm = false; H Y5R
end iHNQxLkk{:
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ({rcH.:
% Compute the Zernike Polynomials j.] ]VA
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sP Qj B[
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w ? _8OJ
% Determine the required powers of r: L~PiDQr?r
% ----------------------------------- z` 6$ p1U
m_abs = abs(m); IoOOS5a
rpowers = []; Brxnl, %\
for j = 1:length(n) @@*x/"GJG
rpowers = [rpowers m_abs(j):2:n(j)]; K@=u F 1?
end 82,^ Pu
rpowers = unique(rpowers); >g !Z|j u
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a%c <3'
% Pre-compute the values of r raised to the required powers, % WDTnEm
% and compile them in a matrix: ?n{m2.H
% ----------------------------- k -jFT3b$
if rpowers(1)==0 wA $? e}
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @cIYS%iZ
rpowern = cat(2,rpowern{:}); kAp#6->(q
rpowern = [ones(length_r,1) rpowern]; .b_ppieNY
else Ry}4M Eq]
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); :2xGfy??
rpowern = cat(2,rpowern{:}); =SmU ;t>t/
end S8AbLl9G@>
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% Compute the values of the polynomials: =KQIrS:
% -------------------------------------- %' WC7s
y = zeros(length_r,length(n)); mRAt5a#is
for j = 1:length(n) ? <.a>"!
s = 0:(n(j)-m_abs(j))/2; ^@/wXj:
pows = n(j):-2:m_abs(j); +)yoQRekX
for k = length(s):-1:1 EXeV @kg
p = (1-2*mod(s(k),2))* ... >dK0&+A
prod(2:(n(j)-s(k)))/ ... xkFa
prod(2:s(k))/ ...
yHE\Q
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 07>m*1G
prod(2:((n(j)+m_abs(j))/2-s(k))); +mBS&FK
idx = (pows(k)==rpowers); &i3SB[|
y(:,j) = y(:,j) + p*rpowern(:,idx); "gNi}dB<]
end OMk3\FV2Z
zf)*W#+
if isnorm q 1xSylE
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); }=f\WWJf 0
end y(<{e~
end <;#gcF[7>
% END: Compute the Zernike Polynomials \3ydNgl
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KsIHJr7-
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0 ;LF>+fJ
% Compute the Zernike functions: 8aHE=x/TL
% ------------------------------ >!Y#2]@}o
idx_pos = m>0; W2- l_{
idx_neg = m<0; *>?N>f"
PdVY tK%
pvl];w
z = y; !L;_f'\)6
if any(idx_pos) VTR 4uT-
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 'wFhfZB1!B
end mI<s f?.
if any(idx_neg) "4xo,JUf
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); XBX`L"0
end 4/{ pz$
lE%KzX?&
Ch l^LEN:
% EOF zernfun ! W,LG$=/
