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    [讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

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    离线jssylttc
     
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    只看楼主 倒序阅读 楼主  发表于: 2012-04-23
    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, X u"R^  
    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, |CgnCUv+  
    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? }14 {2=!Q  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? eLwTaW !C  
    N-lGa@ j  
    ?6Cz[5\  
    ~/_9P Fk  
    -B#yy]8  
    function z = zernfun(n,m,r,theta,nflag) %zC[KE*~  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ogM%N  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ]!:oYAm  
    %   and angular frequency M, evaluated at positions (R,THETA) on the #5sD{:f`  
    %   unit circle.  N is a vector of positive integers (including 0), and E< 4l#Z<  
    %   M is a vector with the same number of elements as N.  Each element f0+2t.tj  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) >idBS  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ;vhyhP.oM  
    %   and THETA is a vector of angles.  R and THETA must have the same wI M{pK  
    %   length.  The output Z is a matrix with one column for every (N,M) [#" =yzR<3  
    %   pair, and one row for every (R,THETA) pair. O^LTD#}$a)  
    % DPe]daF  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike d "BW/%m|g  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^\ ?O4,L  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral g}&hl"j  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Y9SGRV(  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized PYB+FcR6?n  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. @J[6,$UVu  
    % `Yc _5&"  
    %   The Zernike functions are an orthogonal basis on the unit circle. %v5R#14[n  
    %   They are used in disciplines such as astronomy, optics, and #L crI  
    %   optometry to describe functions on a circular domain. JGiKBm;  
    % y<W8Q<9  
    %   The following table lists the first 15 Zernike functions. nGZX7Fx5  
    % F}Mhs17!|  
    %       n    m    Zernike function           Normalization hovGQHg  
    %       -------------------------------------------------- wYeB)1.  
    %       0    0    1                                 1 `|1MlRM9  
    %       1    1    r * cos(theta)                    2 KHKS$D  
    %       1   -1    r * sin(theta)                    2 t^=U*~  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 7>o .0  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 0X..e$ '  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ;N+$2w  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 0m[dP  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) C>^D*C(  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) fbrp#G71y  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ?{o/I\\  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) >QQ(m\a$  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (J$\-a7<f  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) /rB{[zk  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Mro4`GL  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) \`'KlF2  
    %       -------------------------------------------------- NQTnhiM7$  
    % |wxGpBau  
    %   Example 1: tury<*  
    % lYf+V8{  
    %       % Display the Zernike function Z(n=5,m=1) ~ <0Z>qr  
    %       x = -1:0.01:1; oR+-+-? ?$  
    %       [X,Y] = meshgrid(x,x); 1.@vS&Y7OE  
    %       [theta,r] = cart2pol(X,Y);  R)Q 4  
    %       idx = r<=1; P sjbR  
    %       z = nan(size(X)); Df07y<>7Q  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); S{F-ttS"  
    %       figure [um&X=1V8  
    %       pcolor(x,x,z), shading interp \jW)Xy  
    %       axis square, colorbar jX=lAs~6  
    %       title('Zernike function Z_5^1(r,\theta)') *ck}|RhR  
    % t *6loS0+  
    %   Example 2: S^RUw  
    % _68BP)nz>.  
    %       % Display the first 10 Zernike functions -=$2p0" R  
    %       x = -1:0.01:1; )yee2(S  
    %       [X,Y] = meshgrid(x,x); 'aJgLws*w  
    %       [theta,r] = cart2pol(X,Y); PY\PUMF>  
    %       idx = r<=1; -Q e~)7  
    %       z = nan(size(X)); tgFJZA  
    %       n = [0  1  1  2  2  2  3  3  3  3]; e&Y0}oY  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; jdRq6U^  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ,#u\l>&$  
    %       y = zernfun(n,m,r(idx),theta(idx)); O>r-]0DI[  
    %       figure('Units','normalized') a^nAZ  
    %       for k = 1:10 JXQPT  
    %           z(idx) = y(:,k); )-P!Ae_.v  
    %           subplot(4,7,Nplot(k)) Bl.u=I:Y4  
    %           pcolor(x,x,z), shading interp U)jUq_LX  
    %           set(gca,'XTick',[],'YTick',[]) vL_zvX A  
    %           axis square }F1s tDx  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) u4'z$>B  
    %       end kN9yO5 h7  
    % 1IH[g*f  
    %   See also ZERNPOL, ZERNFUN2. :{g7lTM  
    =WZ%H_oxi  
    I@7/jUO  
    %   Paul Fricker 11/13/2006 HQVh+(  
    Y)HbxFF`/  
    x/TGp?\g  
    w8M2N]&:  
    I=dGq;Jaz  
    % Check and prepare the inputs: ))#'4  
    % ----------------------------- QEJGnl676  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 9Ld9N;rWm#  
        error('zernfun:NMvectors','N and M must be vectors.') M=!i>(yG  
    end Z[#IfbYt  
    O&?.&h  
    z_SagU,\  
    if length(n)~=length(m) XF,<i1ZlM  
        error('zernfun:NMlength','N and M must be the same length.') P;91~``b-  
    end (i`(>I.(/  
    @ RR\lZ  
    b](o]O{v  
    n = n(:); hY;_/!_  
    m = m(:); jz:gr=* z  
    if any(mod(n-m,2)) Y(i?M~3\t  
        error('zernfun:NMmultiplesof2', ... |qUrEGjiSS  
              'All N and M must differ by multiples of 2 (including 0).') B4W\ t{  
    end (Pi-uL<[a  
    av'*u  
    2_pz3<,\  
    if any(m>n) L7q |^`  
        error('zernfun:MlessthanN', ... #s"B-sWE  
              'Each M must be less than or equal to its corresponding N.') V/y=6wUiSl  
    end D1"7s,Hmu  
    4,}GyVJFb`  
    "EPD2,%S  
    if any( r>1 | r<0 ) 0-xCp ~vE  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') d'zT:g  
    end m6n hC  
    U</+.$b  
    960qvz!  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) !wh=dQgMe  
        error('zernfun:RTHvector','R and THETA must be vectors.')  (K #A  
    end ~mH+DV3  
    J+2R&3;_O  
    `SOhG?Zo  
    r = r(:); ^ }#f()  
    theta = theta(:); hx!`F  
    length_r = length(r); vjTwv+B"  
    if length_r~=length(theta) 6E+=Xi  
        error('zernfun:RTHlength', ... .hN3`>*V  
              'The number of R- and THETA-values must be equal.') 1%eLs=u?  
    end JSjYC0e  
    lgT?{,>RkW  
    =lrN'$z?%  
    % Check normalization: OV|Z=EwJ  
    % -------------------- 79tJV  
    if nargin==5 && ischar(nflag) E~He~wHWe  
        isnorm = strcmpi(nflag,'norm'); M {xie  
        if ~isnorm t<lyg0f  
            error('zernfun:normalization','Unrecognized normalization flag.') ,OB&nN t>  
        end G%OpO.Wf  
    else /=M.-MU2  
        isnorm = false; 4A~)b"j5  
    end 6y@<?08Q  
    Y'_ D<Mp  
    cEi<}9r  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% OK\]*r  
    % Compute the Zernike Polynomials |Ow$n  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lIl9ypikg  
    r5)f82pQ  
    ,4Y sZ  
    % Determine the required powers of r: ayH>XwY6  
    % ----------------------------------- 4~WlP,,M  
    m_abs = abs(m); M9g1d7%  
    rpowers = []; IMR$x(g= F  
    for j = 1:length(n) '%O\E{h  
        rpowers = [rpowers m_abs(j):2:n(j)]; X,53c$  
    end s}!"a8hU`  
    rpowers = unique(rpowers); M=Is9)y  
    \[E-:  
    o}R|tOe  
    % Pre-compute the values of r raised to the required powers, K z^hQd  
    % and compile them in a matrix: ^z?=?%{  
    % ----------------------------- -9Xw]I#QR  
    if rpowers(1)==0 Bcm=G""  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); hGKdGu`0  
        rpowern = cat(2,rpowern{:}); 9oD#t~+F4  
        rpowern = [ones(length_r,1) rpowern]; bgXc_>T6_y  
    else _Fvsi3d/  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Sl~C0eO  
        rpowern = cat(2,rpowern{:}); [r~~=b7*[  
    end / GZV_H%v  
    c$.T<r)Z  
    &@Yoj%%  
    % Compute the values of the polynomials: [M2Dy{dh  
    % -------------------------------------- +{bh  
    y = zeros(length_r,length(n)); 6KBHRt  
    for j = 1:length(n) "lb\c  
        s = 0:(n(j)-m_abs(j))/2; #|D:f~"d3  
        pows = n(j):-2:m_abs(j); g&8.A(  
        for k = length(s):-1:1 {B v`i8e  
            p = (1-2*mod(s(k),2))* ... o}W7.7^2  
                       prod(2:(n(j)-s(k)))/              ... m&{rBz0  
                       prod(2:s(k))/                     ... 33S`aJ  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 4t(QvIydA  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); )%1&/uN)  
            idx = (pows(k)==rpowers); B)(w%\M4^  
            y(:,j) = y(:,j) + p*rpowern(:,idx); akY6D]M  
        end j[BgP\&,  
         D`5: JR-{  
        if isnorm C(ZcR_+r$,  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); yl|R:/2V  
        end ,7/\&X<`B  
    end 0c{Gr 0[>  
    % END: Compute the Zernike Polynomials T&e%/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |Yg}WHm  
    KN|'|2/|  
    U@MOvW)  
    % Compute the Zernike functions: 7YSuB9{M  
    % ------------------------------ M|aQ)ivh3  
    idx_pos = m>0; lp 3(&p<:  
    idx_neg = m<0; eq7C]i rH  
    *GB$sXF  
    ook' u }h  
    z = y; qRWJ-T:!F  
    if any(idx_pos) ],WwqD=  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); il<gjlyR]L  
    end d u _O}x  
    if any(idx_neg) agGgJ@  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); </~1p~=hAt  
    end %,h!: Ec^c  
    an #jZ[  
    +X{cN5Y K  
    % EOF zernfun oTZo[T@zRx  
     
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    离线phoenixzqy
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    只看该作者 1楼 发表于: 2012-04-23
    慢慢研究,这个专业性很强的。用的人又少。
    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)
    离线sansummer
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    只看该作者 2楼 发表于: 2012-04-27
    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
    离线jssylttc
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    只看该作者 3楼 发表于: 2012-05-14
    回 sansummer 的帖子
    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)  $'mB8 S  
    Xxsnpb>  
    DDE还是手动输入的呢? E[htB><  
    "8iyMP%8  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    离线18257342135
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    只看该作者 5楼 发表于: 2016-12-13
    支持一下,慢慢研究