切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9357阅读
    • 5回复

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

    上一主题 下一主题
    离线jssylttc
     
    发帖
    25
    光币
    13
    光券
    0
    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, ?5%|YsJP_  
    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, ~ "] 6  
    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? .Jt&6N  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? SOyE$GoOsx  
    O1.a=O  
    *CA7 {2CX  
    );^] is~  
    dnby&-+T  
    function z = zernfun(n,m,r,theta,nflag) FuZ7xM,  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. M~/%V NX  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N HqW|  
    %   and angular frequency M, evaluated at positions (R,THETA) on the {-sy,EYcw  
    %   unit circle.  N is a vector of positive integers (including 0), and w%no6 ;  
    %   M is a vector with the same number of elements as N.  Each element N{]|!#  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) w,\#)<boyb  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, yTDlDOmV!  
    %   and THETA is a vector of angles.  R and THETA must have the same <uugT9By  
    %   length.  The output Z is a matrix with one column for every (N,M) |]5g+sd  
    %   pair, and one row for every (R,THETA) pair. ,3k"J4|d  
    %  *q8L$D  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike x,\PV>   
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), hCX}*  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral y[*Bw)F\N  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, -ISI!EU$  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized %bnDxCj"  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. nj*B-M\p  
    % eCY gi7?  
    %   The Zernike functions are an orthogonal basis on the unit circle. #'Q_eBX  
    %   They are used in disciplines such as astronomy, optics, and +"!,rZ7,A  
    %   optometry to describe functions on a circular domain. t@Qs&DZ7k  
    % _MZqH8  
    %   The following table lists the first 15 Zernike functions. PrIS L[@  
    % N#')Qz:P  
    %       n    m    Zernike function           Normalization Hnwir!=7  
    %       -------------------------------------------------- yfS`g-j{~  
    %       0    0    1                                 1 C:n55BE9  
    %       1    1    r * cos(theta)                    2 y ?FKou'  
    %       1   -1    r * sin(theta)                    2 3A_7R-sQ  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) R qS2Qo]  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 0kI.d X)  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) TxYxB1C)  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) $cri"G  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ~y+QL{P4~  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) o$4n D#P3  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Vcg$H8m  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ,TTt<&c  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) NTk"W!<Cl2  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) n&=3Knbd@d  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) L$7 NT}L  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) [-cYFdt"V  
    %       -------------------------------------------------- L  &F0^  
    % dA[Z\  
    %   Example 1: v\#69J5.>)  
    % d18%zY>  
    %       % Display the Zernike function Z(n=5,m=1) Nhv~f0  
    %       x = -1:0.01:1; U}7 a;4?  
    %       [X,Y] = meshgrid(x,x); NZ/>nNs  
    %       [theta,r] = cart2pol(X,Y); ~A+D H  
    %       idx = r<=1; x68$?CD  
    %       z = nan(size(X)); tY<D\T   
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); !tGXh9g  
    %       figure C6=7zYhR  
    %       pcolor(x,x,z), shading interp A-vK0l+  
    %       axis square, colorbar 95;q ] =U  
    %       title('Zernike function Z_5^1(r,\theta)') ~xqRCf{8  
    % 5V\\w~&/  
    %   Example 2: Z |uII#lq  
    % '{j.5~4y  
    %       % Display the first 10 Zernike functions w{3 B  
    %       x = -1:0.01:1; %ci/(wL  
    %       [X,Y] = meshgrid(x,x); PuAcsYQhN  
    %       [theta,r] = cart2pol(X,Y); Dh0`t@  
    %       idx = r<=1; ;"=a-$vm  
    %       z = nan(size(X)); DG&14c>g  
    %       n = [0  1  1  2  2  2  3  3  3  3]; P ?dE\Po7  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; $VYMAk&\  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; t%<nS=u  
    %       y = zernfun(n,m,r(idx),theta(idx)); a_/\.  
    %       figure('Units','normalized') X62h7?'Pd  
    %       for k = 1:10 {w.rcObIw+  
    %           z(idx) = y(:,k); bNR}Mk]?  
    %           subplot(4,7,Nplot(k)) |a#4  
    %           pcolor(x,x,z), shading interp CRvUD.D  
    %           set(gca,'XTick',[],'YTick',[]) _>B0q|]j4'  
    %           axis square EoqUFa,  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 8 M3Q8&  
    %       end O:3pp8  
    % q bb:)>  
    %   See also ZERNPOL, ZERNFUN2. jQOY\1SR  
    @a) x^d  
    %zQME6WELz  
    %   Paul Fricker 11/13/2006 '/kSUvd  
    ~M%r.WFpA  
    >bWsUG9  
    306C_ M\$  
    CZv.$H"lW  
    % Check and prepare the inputs: Me[T=Tt`@w  
    % ----------------------------- -J4?Km  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #Yi,EwD  
        error('zernfun:NMvectors','N and M must be vectors.') 7f_4qb8  
    end #q40  >)]  
    S P)$K=  
    wxEFM)zr  
    if length(n)~=length(m) &*RJh'o|N(  
        error('zernfun:NMlength','N and M must be the same length.') ma>{((N  
    end  Ok[y3S  
    r Ip84}  
     @*'|8%  
    n = n(:); *xXa4HB  
    m = m(:); 7%L%dyN  
    if any(mod(n-m,2)) ,T?8??bZ  
        error('zernfun:NMmultiplesof2', ... .Y[sQO~%  
              'All N and M must differ by multiples of 2 (including 0).') ZurQr}  
    end ]kx)/n-K  
    "TA r\; [  
    7(lR$,bE;=  
    if any(m>n) ;LNFPo   
        error('zernfun:MlessthanN', ... -8; ,#  
              'Each M must be less than or equal to its corresponding N.') s2L|J[Y"s  
    end iD#HB o  
    Urur/_]-%  
    " & 'Jw  
    if any( r>1 | r<0 ) 48Y5ppcS  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') X*VHi  
    end Q[`J=  
    \^vf`-uG  
    _@jBz"aq\  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) y-O# +{7  
        error('zernfun:RTHvector','R and THETA must be vectors.') *IUw$|Z6z)  
    end Px5ArSS  
    +ia  F$  
    ZvEcExA-  
    r = r(:); l j*ELy  
    theta = theta(:); dHc38zp  
    length_r = length(r); I^ sWf3'db  
    if length_r~=length(theta) |\"vHt?@G  
        error('zernfun:RTHlength', ... Ffk$8"   
              'The number of R- and THETA-values must be equal.') h[72iVn  
    end ork/:y9*y  
    R4GmUCKB=  
    <T{2a\i 4f  
    % Check normalization: z.n`0`^  
    % -------------------- x nWCio>M  
    if nargin==5 && ischar(nflag) SHS:>V  
        isnorm = strcmpi(nflag,'norm'); =( b;Cow  
        if ~isnorm |&+g,A _w  
            error('zernfun:normalization','Unrecognized normalization flag.') XbdoTriE  
        end e|u|b  
    else ).@8+}`  
        isnorm = false; J"'2zg1&  
    end .f 4a+w  
    jca7Cx`sm  
    {ve86 POY  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Va,M9)F  
    % Compute the Zernike Polynomials  uZ][#[u  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~Fv&z'R  
    sL|lfc'bB  
    2P`QS@v0a=  
    % Determine the required powers of r:  c'?4*O  
    % ----------------------------------- 4Z>hP]7  
    m_abs = abs(m); &WAO.*:y  
    rpowers = []; E;\XZ<E  
    for j = 1:length(n) B MU@J  
        rpowers = [rpowers m_abs(j):2:n(j)]; 0mo^I==J1  
    end k.? aq  
    rpowers = unique(rpowers); B~oSKM%8R  
    V0+D{|thh6  
    hWpn~q  
    % Pre-compute the values of r raised to the required powers, ^/\OS@CT\  
    % and compile them in a matrix: V_jVVy30Ji  
    % ----------------------------- _l,?Y;OF  
    if rpowers(1)==0 -G&>b D  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); T677d.zaT  
        rpowern = cat(2,rpowern{:}); ^p(t*%LM  
        rpowern = [ones(length_r,1) rpowern]; rks+\e}^Z  
    else 7qSlqA<Hs  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); bHE'R!*  
        rpowern = cat(2,rpowern{:}); 3?I^D /K^  
    end GgkljF@{}  
    <cG .V |B  
    9frP`4<)  
    % Compute the values of the polynomials:  s#om  
    % -------------------------------------- % INRds  
    y = zeros(length_r,length(n)); H6?ZE  
    for j = 1:length(n) :Z(?Ct&8  
        s = 0:(n(j)-m_abs(j))/2; d!/@+i  
        pows = n(j):-2:m_abs(j); ?Z%Ja_}8ma  
        for k = length(s):-1:1 s mub> V  
            p = (1-2*mod(s(k),2))* ... [o8a(oC  
                       prod(2:(n(j)-s(k)))/              ... jq(3y|6,  
                       prod(2:s(k))/                     ... OD<0,r0f,  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ^c{}G<U^  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 2%\Nq:; T  
            idx = (pows(k)==rpowers); ZxkX\gl91  
            y(:,j) = y(:,j) + p*rpowern(:,idx); @!6eRp>Z  
        end {H s" "/sb  
         k7P~*ll$  
        if isnorm 6W$ #`N>  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); <$Q\vCR  
        end Ib.`2@ o&  
    end kb1{ ;c:  
    % END: Compute the Zernike Polynomials |8}f  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Frn#?n)S9  
    /G`&k{SiK  
    p.i$[6M  
    % Compute the Zernike functions: )l*H$8  
    % ------------------------------  SzkF-yRd  
    idx_pos = m>0; Yf Udpa0  
    idx_neg = m<0; _`Ey),c_  
    eU_|.2  
    Yu=4j9e_mG  
    z = y; L^rtypkJ  
    if any(idx_pos) ~J!a?]  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); x-+[gNc 6  
    end pWH8ex+  
    if any(idx_neg) hABC rd Em  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); E (tdL,m'  
    end !OM9aITv[  
    "T5?<c  
    kH*l83  
    % EOF zernfun wqBGJ   
     
    分享到
    离线phoenixzqy
    发帖
    4352
    光币
    5478
    光券
    1
    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
    发帖
    959
    光币
    1087
    光券
    1
    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
    发帖
    25
    光币
    13
    光券
    0
    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)  qOZe\<.V<  
    7R<<}dA]  
    DDE还是手动输入的呢? 4xT(Uj  
    A[XEbfDO  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    离线jssylttc
    发帖
    25
    光币
    13
    光券
    0
    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    离线18257342135
    发帖
    51
    光币
    1518
    光券
    0
    只看该作者 5楼 发表于: 2016-12-13
    支持一下,慢慢研究