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jssylttc 2012-04-23 19:23

如何从zernike矩中提取出zernike系数啊

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, O1~7#nJ*4[  
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, Q~(Qh_Ff  
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? `xx.,;S  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? `^Ll@Cx"  
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function z = zernfun(n,m,r,theta,nflag) &Y^4>y%  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Ye]K 74M.  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N |Ogh-<|<  
%   and angular frequency M, evaluated at positions (R,THETA) on the ?#GTD?3d  
%   unit circle.  N is a vector of positive integers (including 0), and Pm6U:RL  
%   M is a vector with the same number of elements as N.  Each element :xHKbWz6j  
%   k of M must be a positive integer, with possible values M(k) = -N(k) gbYM1guiD  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 8?8V;   
%   and THETA is a vector of angles.  R and THETA must have the same tf6-DmMH  
%   length.  The output Z is a matrix with one column for every (N,M) \)5mO 8w  
%   pair, and one row for every (R,THETA) pair. sSfP.R  
% _`p-^ I  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike LpY{<:y  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 8Tg1 >q<  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral /fUdb=!Z  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, U\Y0v.11  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized }J6:D]Q  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. e|:\Ps`8  
% q]yw",muT  
%   The Zernike functions are an orthogonal basis on the unit circle. &[{sA;  
%   They are used in disciplines such as astronomy, optics, and E} ]=<8V  
%   optometry to describe functions on a circular domain. SrH::-{  
% 7k,BE2]"  
%   The following table lists the first 15 Zernike functions. w0;4O)H$O  
% Io*H}$Gf  
%       n    m    Zernike function           Normalization *lA+ -gkK*  
%       -------------------------------------------------- E`.hM}h  
%       0    0    1                                 1 r+m.! +  
%       1    1    r * cos(theta)                    2 OvQzMXU^I  
%       1   -1    r * sin(theta)                    2 Uhr2"Nuuy  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) [K,P)V>K  
%       2    0    (2*r^2 - 1)                    sqrt(3) S'^ q  
%       2    2    r^2 * sin(2*theta)             sqrt(6) kJl^,q  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) ML'y`S  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) @:RoYvk$  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) qE2VUEv5Y  
%       3    3    r^3 * sin(3*theta)             sqrt(8) \"$P :Uv  
%       4   -4    r^4 * cos(4*theta)             sqrt(10)  ?;v\wx  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) IagM#}m@  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) @. $- ^-  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) OU.}H $x"  
%       4    4    r^4 * sin(4*theta)             sqrt(10) #8M?y*<I  
%       -------------------------------------------------- CR23$<FC  
% c*7|>7C$i  
%   Example 1: ;G}  
% O >+=cg  
%       % Display the Zernike function Z(n=5,m=1) ,ja!OZ0$  
%       x = -1:0.01:1; B<L7`xL  
%       [X,Y] = meshgrid(x,x); k L6s49  
%       [theta,r] = cart2pol(X,Y); "~r)_Ko  
%       idx = r<=1; 'WhJ}Uo\  
%       z = nan(size(X)); %w[Z/  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); aL[6}U0(}  
%       figure ?Xvy0/s5  
%       pcolor(x,x,z), shading interp &*" *b\  
%       axis square, colorbar wdP(MkaV  
%       title('Zernike function Z_5^1(r,\theta)') Z@dVK`nD  
% s !?uLSEdb  
%   Example 2: ^?H|RAp  
% Dfzj/spFV  
%       % Display the first 10 Zernike functions U!-Nx9  
%       x = -1:0.01:1; +@^);b6  
%       [X,Y] = meshgrid(x,x); v[|W\y@H/3  
%       [theta,r] = cart2pol(X,Y); ^wWbW&<Tg  
%       idx = r<=1; vTx>z\7q,  
%       z = nan(size(X)); l+ >eb  
%       n = [0  1  1  2  2  2  3  3  3  3]; XfE9QA[  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; hT1JEu  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; P~{8L.w!>W  
%       y = zernfun(n,m,r(idx),theta(idx)); gZ^Qt.6Z  
%       figure('Units','normalized') DqQ p47kp  
%       for k = 1:10 S=-$:65  
%           z(idx) = y(:,k); 9o5D3 d K  
%           subplot(4,7,Nplot(k)) CR'%=N04^  
%           pcolor(x,x,z), shading interp w -o#=R_  
%           set(gca,'XTick',[],'YTick',[]) f%.Ngf9  
%           axis square xrvM}Il  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?110} [jw  
%       end I4jRz*Ufe?  
% s*la`(x  
%   See also ZERNPOL, ZERNFUN2. 0c`zg7|  
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R]X 0D.  
%   Paul Fricker 11/13/2006 AcuF0KWw/  
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HQ-+ +;Q  
% Check and prepare the inputs: =w+8q1!o  
% ----------------------------- ?nW>' z  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) JXJ+lZmsz  
    error('zernfun:NMvectors','N and M must be vectors.') :CE4< {V  
end a)ry}E =f  
z4SJxL  
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if length(n)~=length(m) x ']'ODs  
    error('zernfun:NMlength','N and M must be the same length.') `5@F'tKQ  
end 5_'lu  
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To>,8E+GAb  
n = n(:); `Fn"QL-  
m = m(:); }?9&xVh?\  
if any(mod(n-m,2)) DO80HS3ZD  
    error('zernfun:NMmultiplesof2', ... zw+aZDcV(  
          'All N and M must differ by multiples of 2 (including 0).') l{Df{1b.  
end b&F9<XLqq  
_aPAn|.  
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if any(m>n) swF{}S"  
    error('zernfun:MlessthanN', ... 0h@FHw2d  
          'Each M must be less than or equal to its corresponding N.') nU_O|l9  
end Io.RT+slB  
AChz}N$C  
;_(f(8BO   
if any( r>1 | r<0 ) iwJ_~   
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') d>hv-n D  
end geR+v+B,  
.}!.4J%q2  
Gc|)4c  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) *z0d~j*W;  
    error('zernfun:RTHvector','R and THETA must be vectors.') ~+dps i  
end YGyv)\  
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Ax=HDW}  
r = r(:); hN'])[+V  
theta = theta(:); pIlEoG=[_  
length_r = length(r); (P)G|2=  
if length_r~=length(theta) LQR2T5S/Q,  
    error('zernfun:RTHlength', ... i 6G40!G=)  
          'The number of R- and THETA-values must be equal.') Tzex\]fw  
end B`}um;T#~,  
f,HUr% @  
2 kDsIEA  
% Check normalization: J3_aHI  
% -------------------- r9 @=d  
if nargin==5 && ischar(nflag) 6CBk=)qH  
    isnorm = strcmpi(nflag,'norm'); gN=.}$Kfu  
    if ~isnorm -G,}f\Cg  
        error('zernfun:normalization','Unrecognized normalization flag.') WBE>0L  
    end T^}UE<  
else @z@%vr=vX  
    isnorm = false; KG'i#(u[  
end !K>iSF<  
=j,WQ66r3  
)Z/"P\qo  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &P?2H66s  
% Compute the Zernike Polynomials iQ/~?'PB  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F_ F"3'[  
Tz]R}DKB&  
2zTi/&K&  
% Determine the required powers of r: ,S-h~x  
% ----------------------------------- @RoZd?  
m_abs = abs(m); KrE 'M  
rpowers = []; <gp?}Lk  
for j = 1:length(n) TLdlPBnr8  
    rpowers = [rpowers m_abs(j):2:n(j)]; 3"y 6|e/5  
end xl\Kj2^  
rpowers = unique(rpowers); /v R>.'  
75^6?#GS  
":Dm/g  
% Pre-compute the values of r raised to the required powers, &3Zq1o  
% and compile them in a matrix: Zzlf1#26\  
% ----------------------------- >d/H4;8  
if rpowers(1)==0 =hPXLCeC  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); "%-Vrb=:Y  
    rpowern = cat(2,rpowern{:}); o<`hj&s  
    rpowern = [ones(length_r,1) rpowern]; "D(Lp*3hj&  
else g$nS6w|5H  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); kWzN {]v  
    rpowern = cat(2,rpowern{:}); 8%[pno |0I  
end @]@|H?  
,EB}IG ]  
j_Szw w-  
% Compute the values of the polynomials: %**f`L%jN  
% -------------------------------------- HK@ij,px  
y = zeros(length_r,length(n)); cl4E6\?z  
for j = 1:length(n) L|=5jn9 :  
    s = 0:(n(j)-m_abs(j))/2; f`9Mcli !  
    pows = n(j):-2:m_abs(j); wcGK *sWG-  
    for k = length(s):-1:1 jj5S+ >4  
        p = (1-2*mod(s(k),2))* ... !J`lA  
                   prod(2:(n(j)-s(k)))/              ... )[Y B&  
                   prod(2:s(k))/                     ... :L[>!~YG_n  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... {K,In)4  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); 4DA34m(  
        idx = (pows(k)==rpowers); )kD/ 8  
        y(:,j) = y(:,j) + p*rpowern(:,idx); Wl j&_~  
    end t'qYM5  
     ,lyW'<~gA  
    if isnorm }#XFa#  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Jup)m/  
    end uDF;_bli)H  
end FcDS*ZEk!  
% END: Compute the Zernike Polynomials \(o"/*  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G=zWhqieh  
Z~5) )5Ye;  
tdy2ZPVtTV  
% Compute the Zernike functions: zsFzg.$3&  
% ------------------------------ 0I&k_7_   
idx_pos = m>0; Gz[yD ~6a  
idx_neg = m<0; F@w; .e!  
"W|A^@r}  
}Mc b\+[  
z = y; "LMj,qZ1!  
if any(idx_pos) x]~TGzS  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); YG p+[|'  
end )9QtnM  
if any(idx_neg) yIMqQSt79z  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); #/)t]&n  
end rqdwQ  
]MbPivM  
|c_qq Bd  
% EOF zernfun {T){!UVp!  
phoenixzqy 2012-04-23 20:38
慢慢研究,这个专业性很强的。用的人又少。
sansummer 2012-04-27 10:22
这个太牛了,我目前只能把zygo中的zernike的36项参数带入到zemax中,但是我目前对其结果的可信性表示质疑,以后多交流啊
jssylttc 2012-05-14 11:28
sansummer:这个太牛了,我目前只能把zygo中的zernike的36项参数带入到zemax中,但是我目前对其结果的可信性表示质疑,以后多交流啊 (2012-04-27 10:22)  +2tQ FV;  
+^)v"@,VP  
DDE还是手动输入的呢? b}J%4Lx%m  
%|Ps|iV  
zygo和zemax的zernike系数,类型对应好就没问题了吧
jssylttc 2012-05-14 11:37
顶顶·········
18257342135 2016-12-13 10:03
支持一下,慢慢研究
查看本帖完整版本: [-- 如何从zernike矩中提取出zernike系数啊 --] [-- top --]

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