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    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

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    离线niuhelen
     
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    只看楼主 倒序阅读 楼主  发表于: 2011-03-12
    小弟不是学光学的,所以想请各位大侠指点啊!谢谢啦 8Op^6rX4  
    就是我用ansys计算出了镜面的面型的数据,怎样可以得到zernike多项式系数,然后用zemax各阶得到像差!谢谢啦! 6NHP/bj<1V  
     
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    离线phility
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    只看该作者 1楼 发表于: 2011-03-12
    可以用matlab编程,用zernike多项式进行波面拟合,求出zernike多项式的系数,拟合的算法有很多种,最简单的是最小二乘法,你可以查下相关资料,挺简单的
    离线phility
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    只看该作者 2楼 发表于: 2011-03-12
    泽尼克多项式的前9项对应象差的
    离线niuhelen
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    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 MX7Ix{  
    function z = zernfun(n,m,r,theta,nflag) 3EY m@oZj  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. /!A"[Tyt  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N !.q 9:|oc  
    %   and angular frequency M, evaluated at positions (R,THETA) on the j(]O$""  
    %   unit circle.  N is a vector of positive integers (including 0), and "5O>egt  
    %   M is a vector with the same number of elements as N.  Each element /c 7z[|  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ;134$7!Y  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, %7w8M{I R3  
    %   and THETA is a vector of angles.  R and THETA must have the same ccPWfy_  
    %   length.  The output Z is a matrix with one column for every (N,M) #7}M\\$M  
    %   pair, and one row for every (R,THETA) pair. t u{~:Z(  
    % zUZET'Bm9  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike #62ThH~  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), MSeg7/MF  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral +PI}$c-|`  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, V45adDiZ  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized W*#5Sk  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. LL)t)  
    % DI2e%`$  
    %   The Zernike functions are an orthogonal basis on the unit circle. I"x|U[*B  
    %   They are used in disciplines such as astronomy, optics, and &GJVFr~z  
    %   optometry to describe functions on a circular domain. JMo r[*  
    % c$L1aZo  
    %   The following table lists the first 15 Zernike functions. 2nCc(F&+?  
    % u a_w5o7  
    %       n    m    Zernike function           Normalization y Rl   
    %       -------------------------------------------------- wy${EY^h  
    %       0    0    1                                 1 S-Vj$asv!  
    %       1    1    r * cos(theta)                    2 l&e$:=;8  
    %       1   -1    r * sin(theta)                    2 92A9gY  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) .Y?]r6CC/  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ,+6u6  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) SJMbYjn0J  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) uL1lB@G@  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) q>>1?hzA  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) qm:C1#<p   
    %       3    3    r^3 * sin(3*theta)             sqrt(8) X9]} UX  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Q1x&Zm1v  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9X;*GC;d  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) aGi`(|shW  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) lN,a+S/'  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ~wv$uL8y  
    %       -------------------------------------------------- q{f\_2[  
    % F`x_W;\  
    %   Example 1: n5.sx|bI?  
    % {cIk-nG -_  
    %       % Display the Zernike function Z(n=5,m=1) h4|}BGO  
    %       x = -1:0.01:1; QSa#}vCp*  
    %       [X,Y] = meshgrid(x,x); Rk#'^ }  
    %       [theta,r] = cart2pol(X,Y); Y:,C_^$w;  
    %       idx = r<=1; GWPBP-)0  
    %       z = nan(size(X)); c!7WRHJE_a  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 1 Ga3[ g  
    %       figure }8aqSD<:  
    %       pcolor(x,x,z), shading interp zb!1o0, J  
    %       axis square, colorbar _0'X!1"  
    %       title('Zernike function Z_5^1(r,\theta)') 6fo" k+S  
    % 'b}RFzEn  
    %   Example 2: _u$DcA8B  
    % LDHu10l  
    %       % Display the first 10 Zernike functions 8zj&e8&v  
    %       x = -1:0.01:1; 4=|Q2qgFV  
    %       [X,Y] = meshgrid(x,x); IjRUr\l  
    %       [theta,r] = cart2pol(X,Y); Z.Z;p/4F  
    %       idx = r<=1; $6wSqH?q  
    %       z = nan(size(X)); o^UOkxs.  
    %       n = [0  1  1  2  2  2  3  3  3  3]; J@_^]  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; vn$=be8l4  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; }s<;YC  
    %       y = zernfun(n,m,r(idx),theta(idx)); i.)n#@M2  
    %       figure('Units','normalized') s=jYQ5nv  
    %       for k = 1:10 `H$XO{w  
    %           z(idx) = y(:,k); #\Rxqh7  
    %           subplot(4,7,Nplot(k)) l:UKU!  
    %           pcolor(x,x,z), shading interp 1 @t.J>  
    %           set(gca,'XTick',[],'YTick',[]) ?yq=c  
    %           axis square HB5-B XBU  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 8uLS7\,$z  
    %       end IBJNs$  
    % !s1<)%Jt  
    %   See also ZERNPOL, ZERNFUN2. _&V,yp!|  
    nf"#F@dk  
    %   Paul Fricker 11/13/2006 tR'RB@kJ  
    cRrJZ9  
    0'pB7^y  
    % Check and prepare the inputs: a_5s'Dh  
    % ----------------------------- ?i#x13  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /Z^a, %1  
        error('zernfun:NMvectors','N and M must be vectors.') L@AFt)U  
    end o(Z~J}l({  
    7UW\|r  
    if length(n)~=length(m) {zm8`  
        error('zernfun:NMlength','N and M must be the same length.') Fovah4q%V  
    end zE$HHY2ovi  
    AJ*17w  
    n = n(:); h?SRX_  
    m = m(:); C@`#@1X  
    if any(mod(n-m,2)) T{+a48,;  
        error('zernfun:NMmultiplesof2', ... |LQ%sV  
              'All N and M must differ by multiples of 2 (including 0).') {*GBUv5  
    end |*g#7 YL  
    Lv%t*s2$/  
    if any(m>n) zytN leyc  
        error('zernfun:MlessthanN', ... ^"?a)KC  
              'Each M must be less than or equal to its corresponding N.') e3CFW_p  
    end  eu$VKLY*  
    ~$T>,^K y  
    if any( r>1 | r<0 ) ,(x` zpp _  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') $#D#ezvxe  
    end d>)=|  
    ZOV,yuD{8{  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) N)Q_z9b=  
        error('zernfun:RTHvector','R and THETA must be vectors.') jH<Sf: Y(  
    end i:jXh9+  
    +yfUB8Xw  
    r = r(:); }a5TY("d9H  
    theta = theta(:); v; #y^O  
    length_r = length(r); >KrI}>!9r  
    if length_r~=length(theta) ms}o[Z@n  
        error('zernfun:RTHlength', ... RNB&!NC  
              'The number of R- and THETA-values must be equal.') mq4Zy3H   
    end o}KVT%}  
    =h-E N_[  
    % Check normalization: =T2SJ)  
    % -------------------- v0)Y,hW  
    if nargin==5 && ischar(nflag) K(u pz n*a  
        isnorm = strcmpi(nflag,'norm'); S5>ztK.e  
        if ~isnorm PsNrCe%e  
            error('zernfun:normalization','Unrecognized normalization flag.') 7"'PfP4c  
        end -avxH?;?7  
    else Ss 5@n  
        isnorm = false; '1b8>L  
    end aIa<,  
    nD eVYK  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EL3X8H  
    % Compute the Zernike Polynomials 8493Sw  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ojl X<y.  
    @uRJl$3  
    % Determine the required powers of r: M 1 m]1<  
    % ----------------------------------- G5U?]& I8  
    m_abs = abs(m); Sq,>^|v4&e  
    rpowers = []; s1cu5eCt  
    for j = 1:length(n) t6+W  
        rpowers = [rpowers m_abs(j):2:n(j)]; xP_%d,  
    end y'^U4# (  
    rpowers = unique(rpowers); rMIX{K)'f  
    l@GJcCufE  
    % Pre-compute the values of r raised to the required powers, W3UxFs]$  
    % and compile them in a matrix: #p*uk  
    % ----------------------------- FvVC 2Z  
    if rpowers(1)==0 C=&n1/  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); hL/u5h%$  
        rpowern = cat(2,rpowern{:}); =6ru%.8U,  
        rpowern = [ones(length_r,1) rpowern]; Ip7#${f5M  
    else IowXVdm@6  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); d*Mqs}8  
        rpowern = cat(2,rpowern{:}); 8~Zw"  
    end oCkG  
    F!hjtIkPj  
    % Compute the values of the polynomials: }Em{?Hqy  
    % -------------------------------------- diu"Nt  
    y = zeros(length_r,length(n)); 4s:M}=]N  
    for j = 1:length(n) -V4{tIQY  
        s = 0:(n(j)-m_abs(j))/2; xP>cQELot  
        pows = n(j):-2:m_abs(j); ]3,9 ."^  
        for k = length(s):-1:1 L$O\fhO?  
            p = (1-2*mod(s(k),2))* ... ;Z0&sFm  
                       prod(2:(n(j)-s(k)))/              ... g9 ^\Q Yh!  
                       prod(2:s(k))/                     ... 3]kM&lK5\  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 5%9Uh'y#  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); :t`W&z41  
            idx = (pows(k)==rpowers); U'F}k0h?\'  
            y(:,j) = y(:,j) + p*rpowern(:,idx); V]J"v#!{  
        end 7)<Ib j<M  
         -7w}+iS  
        if isnorm K:< Viz  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); `qmwAT  
        end qgl-,3GY%N  
    end iP9]b&  
    % END: Compute the Zernike Polynomials :^`j:B  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {:"<E?+  
    j~\FDcG*ed  
    % Compute the Zernike functions: &uE )Vr4R  
    % ------------------------------ Dx /w&v  
    idx_pos = m>0; ?/MkH0[G=  
    idx_neg = m<0; _I;hM  
    V2?{ebx`  
    z = y; )?radg  
    if any(idx_pos) p2l@6\m\  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); (Q||5  
    end g,WTXRy  
    if any(idx_neg) -eK0 +beQ  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); a"xRc  
    end *jc >?)k  
    Y1r'\@L w  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) "dv\ 9O  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. d!<>Fh^6,  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated c %Y *XJ'  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive [V?HK_~  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, rC|nE=i  
    %   and THETA is a vector of angles.  R and THETA must have the same k~<Ozx^AyY  
    %   length.  The output Z is a matrix with one column for every P-value, z"7?I$N Q  
    %   and one row for every (R,THETA) pair. AX{<d@z`j  
    % LC=M{\  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike tq}sedYhee  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) * wqR.n?  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) fl4@5AVY  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 &,p6lbP  
    %   for all p. 3C=QWw?  
    % pK{G2]OK{U  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 0hkYexX73  
    %   Zernike functions (order N<=7).  In some disciplines it is ?\4kV*/Cqz  
    %   traditional to label the first 36 functions using a single mode ]S?G]/k}  
    %   number P instead of separate numbers for the order N and azimuthal 3VMaD@nYa  
    %   frequency M. @/As|)  
    % dmkGIg}  
    %   Example: S]fkA6v  
    % 7!`1K_v6  
    %       % Display the first 16 Zernike functions g1F9IB42@<  
    %       x = -1:0.01:1; wb#[&2i  
    %       [X,Y] = meshgrid(x,x); Z[z" v  
    %       [theta,r] = cart2pol(X,Y); G DBV  
    %       idx = r<=1; :SN?t  
    %       p = 0:15; ?en-_'}~a  
    %       z = nan(size(X)); ?^-fivzS>  
    %       y = zernfun2(p,r(idx),theta(idx)); 2XBHo (  
    %       figure('Units','normalized') dwvc;f-  
    %       for k = 1:length(p) [=. iJ5,{2  
    %           z(idx) = y(:,k); j1Sjw6}GCH  
    %           subplot(4,4,k) B "4A1!  
    %           pcolor(x,x,z), shading interp \N?lG q  
    %           set(gca,'XTick',[],'YTick',[]) #>CWee;  
    %           axis square qS}{O0  
    %           title(['Z_{' num2str(p(k)) '}']) j";L{  
    %       end s;Zi   
    % 0V"(}!=2a  
    %   See also ZERNPOL, ZERNFUN. {Ho_U&<  
    tQxAZ0B^  
    %   Paul Fricker 11/13/2006 g(jn /Cx  
    ]B&jMj~y&  
    k+@ :+ RL  
    % Check and prepare the inputs: I )% bOK]  
    % ----------------------------- g rQ,J  
    if min(size(p))~=1 4yMi9Ri4H  
        error('zernfun2:Pvector','Input P must be vector.') I L&PN`#  
    end {}Afah  
    W1M Bk[:Q  
    if any(p)>35 _iqaKYT$  
        error('zernfun2:P36', ... |l|$ Q;  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... j~Ci*'*L  
               '(P = 0 to 35).']) /8dRql-Ne  
    end c2gZ<[~  
    5P);t9O6  
    % Get the order and frequency corresonding to the function number: ] :](xW%  
    % ---------------------------------------------------------------- 0yUn~'+(Sp  
    p = p(:); ^'sy hI\  
    n = ceil((-3+sqrt(9+8*p))/2); 4 ;6,h6a  
    m = 2*p - n.*(n+2); 6: R1jF*eG  
    FhEfW7]0,  
    % Pass the inputs to the function ZERNFUN: SrMfd7H8f  
    % ---------------------------------------- yi-)4#YN  
    switch nargin =ZV+*cCC=q  
        case 3 .nA9irc  
            z = zernfun(n,m,r,theta); sA"B/C|(g  
        case 4 se_Oi$VZ{  
            z = zernfun(n,m,r,theta,nflag); j->5%y  
        otherwise a|dn3R>vX  
            error('zernfun2:nargin','Incorrect number of inputs.') _>t6]?*  
    end T!&VT;   
    [].euDrX  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) ~MK%^5y?  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. `q y@Qo  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 9$R}GK  
    %   order N and frequency M, evaluated at R.  N is a vector of v?q)E%5j  
    %   positive integers (including 0), and M is a vector with the ) @f6  
    %   same number of elements as N.  Each element k of M must be a s%zdP  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) lxLEYDGFS  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is {%Q+Pzl.  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix O 0P4uq  
    %   with one column for every (N,M) pair, and one row for every thh0~g0/  
    %   element in R. 0*/[z~Z-1  
    % 5q}7#{A  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- Ch&2{ ng  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is q+ 9c81b  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to $r(9'm}W  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 0{47TX*YX  
    %   for all [n,m]. X_wPuU%  
    % 5mI}IS|@  
    %   The radial Zernike polynomials are the radial portion of the E^Z?X2Z  
    %   Zernike functions, which are an orthogonal basis on the unit F*,RDM'M  
    %   circle.  The series representation of the radial Zernike ]RI+:f  
    %   polynomials is KN\tRE  
    % p}a0z?  
    %          (n-m)/2 zW; sr.  
    %            __ K|wB0TiXP  
    %    m      \       s                                          n-2s rhwjsC6  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r kfs[*ku  
    %    n      s=0 1n>(CwLG"  
    % 'iEu1! t\0  
    %   The following table shows the first 12 polynomials. yRldPk_  
    % 3ZL<6`YF  
    %       n    m    Zernike polynomial    Normalization /E5>cqX4A  
    %       --------------------------------------------- `R_;n#3F0  
    %       0    0    1                        sqrt(2) 9.l*#A^  
    %       1    1    r                           2 zHQSx7Ow 5  
    %       2    0    2*r^2 - 1                sqrt(6) vqJq=\ .m  
    %       2    2    r^2                      sqrt(6) FWQNO(  
    %       3    1    3*r^3 - 2*r              sqrt(8) /G!M\teeF  
    %       3    3    r^3                      sqrt(8) "l-R|>6~  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) p']oy;t  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 43BqNQ0  
    %       4    4    r^4                      sqrt(10) +(8Z8]Jf  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) zXv2plw(  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) SH1)@K-  
    %       5    5    r^5                      sqrt(12) @`:z$52  
    %       --------------------------------------------- JR4fJG  
    % o#=O5@>ai  
    %   Example: 6{?B`gm7g  
    % mW 'sdb  
    %       % Display three example Zernike radial polynomials LZ1)zoJ  
    %       r = 0:0.01:1; t'bhA20Z\  
    %       n = [3 2 5]; *f3? 0w  
    %       m = [1 2 1]; mBg$eiGTB  
    %       z = zernpol(n,m,r); OCbwV7q:  
    %       figure ")eY{C  
    %       plot(r,z) 8QFg6#"O  
    %       grid on )cBO_  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') Fz@9 @  
    % }\aJ%9X02  
    %   See also ZERNFUN, ZERNFUN2. "<yJ<lS&>  
    |sPUb;&~  
    % A note on the algorithm. Isg\ fSK<j  
    % ------------------------ Zd8`95  
    % The radial Zernike polynomials are computed using the series `z<I<  
    % representation shown in the Help section above. For many special trMwFpfu  
    % functions, direct evaluation using the series representation can fsUZG6  
    % produce poor numerical results (floating point errors), because V5bB$tL}3  
    % the summation often involves computing small differences between NWII?X#T}  
    % large successive terms in the series. (In such cases, the functions }5lC8{wZ  
    % are often evaluated using alternative methods such as recurrence M.fA5rJ^  
    % relations: see the Legendre functions, for example). For the Zernike z?'z{+HY  
    % polynomials, however, this problem does not arise, because the b VcA#7 uA  
    % polynomials are evaluated over the finite domain r = (0,1), and ugS  
    % because the coefficients for a given polynomial are generally all k83S.*9Mx  
    % of similar magnitude. CK(`]-q>,  
    % }sW%i#CV  
    % ZERNPOL has been written using a vectorized implementation: multiple &ZI-#(P  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] M^jEp  
    % values can be passed as inputs) for a vector of points R.  To achieve dnCurWjdk  
    % this vectorization most efficiently, the algorithm in ZERNPOL ,d$D0w  
    % involves pre-determining all the powers p of R that are required to Nyl)B7/w  
    % compute the outputs, and then compiling the {R^p} into a single y=SVS3D  
    % matrix.  This avoids any redundant computation of the R^p, and g,s^qW0vds  
    % minimizes the sizes of certain intermediate variables. `{9bf)vP6  
    % yVgHu#?PM  
    %   Paul Fricker 11/13/2006 q0VR&b`?>D  
    sI6coe5n  
    C!W0L`r  
    % Check and prepare the inputs:  N}KL'  
    % ----------------------------- I9*BT T]  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /-Z}=  
        error('zernpol:NMvectors','N and M must be vectors.') @IV,sz e  
    end J7+[+Y  
    w[OUGn'  
    if length(n)~=length(m) KRb'kW  
        error('zernpol:NMlength','N and M must be the same length.') a6\`r^@  
    end L"}tJM.d  
    Q2<v: *L  
    n = n(:); heQyz|o  
    m = m(:); [I`:%y  
    length_n = length(n); jGe%'A N\  
    z[' 2  
    if any(mod(n-m,2)) }/M muPp  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') (h'Bz6K  
    end pKaU [1x?%  
    3? F~ H  
    if any(m<0) XvVi)`8!u  
        error('zernpol:Mpositive','All M must be positive.') ~]SCf@pRk  
    end  Lr0:y o  
    st)qw]Dn;Y  
    if any(m>n) {%c&T S@s  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') b*1yvkX5  
    end 2WC$r8E  
    ]EdZ,`B4  
    if any( r>1 | r<0 ) B[9y<FB+  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') fNz(z\  
    end CV)K=Br5&_  
    0X5b32  
    if ~any(size(r)==1) UjS+Ddp  
        error('zernpol:Rvector','R must be a vector.') 3:T~$M`]  
    end nWA>u J5  
    Zxh<pd25Y  
    r = r(:); "r1 !hfIYf  
    length_r = length(r); _{$<s[S  
    L8(2or  
    if nargin==4 pfHfw,[  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); #_WkV  
        if ~isnorm TjHt:%7.  
            error('zernpol:normalization','Unrecognized normalization flag.') 8Oz9 UcG  
        end oRZe?h^r#  
    else m|3 Q'  
        isnorm = false; g'cVsO)S  
    end aJ}hlM>  
    _|T{2LvwT  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |K9*><P?)2  
    % Compute the Zernike Polynomials 2_}oOt?qiM  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FC WF$'cO  
    A]ZQ?- L/  
    % Determine the required powers of r: Gn<0Fy2  
    % ----------------------------------- %xr'96d  
    rpowers = []; E1SWZ&';  
    for j = 1:length(n) o`G6!  
        rpowers = [rpowers m(j):2:n(j)]; -[}Aka,f!  
    end m^ Epw4eg  
    rpowers = unique(rpowers); "Mz#1Laby`  
    X,bhX/h  
    % Pre-compute the values of r raised to the required powers, ;hF}"shJN  
    % and compile them in a matrix: hV:++g  
    % ----------------------------- e4|a^lS;  
    if rpowers(1)==0 [5ncBY*A7  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); O p1TsRm5L  
        rpowern = cat(2,rpowern{:}); %a FZbLK  
        rpowern = [ones(length_r,1) rpowern]; {{?MO{Mh*  
    else (V1;`sI8  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Jg)( F|>o  
        rpowern = cat(2,rpowern{:}); $0vWC#.A]  
    end %!eRR  
    -}PE(c1%?q  
    % Compute the values of the polynomials: Bh'!aipk  
    % -------------------------------------- HB`'S7Q  
    z = zeros(length_r,length_n); :!hO9ho  
    for j = 1:length_n TQb@szp:|  
        s = 0:(n(j)-m(j))/2; l fF RqZ  
        pows = n(j):-2:m(j); {~w(pAx  
        for k = length(s):-1:1 V^4v`}Wgx  
            p = (1-2*mod(s(k),2))* ... bDudETl  
                       prod(2:(n(j)-s(k)))/          ... 3Gr"YG{,  
                       prod(2:s(k))/                 ... dm]g:KWg  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... yYSmmgrX0  
                       prod(2:((n(j)+m(j))/2-s(k))); &iZt(XD  
            idx = (pows(k)==rpowers); (>E/C^Tc%  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ("F$r$9S  
        end E)`0(Z:E  
         5gV,^[E-z  
        if isnorm $gN\%X/n"1  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); kvuRT`/  
        end :y`LF <  
    end I?E+  
    ]uF7HX7F  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  Bqa_l|  
    i>G:*?a  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 2++$ Ql/  
    &.z-itiV  
    07年就写过这方面的计算程序了。
    2024年6月28-30日于上海组织线下成像光学设计培训,欢迎报名参加。请关注子在川上光学公众号。详细内容请咨询13661915143(同微信号)