切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9070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, 3l3'bw2  
    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, DNj "SF(J  
    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? !V,{_(LT  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? YBP:q2H  
    Ce")[<:  
    kJ-*fe'S  
    v0ES;  
    %B)6$!x  
    function z = zernfun(n,m,r,theta,nflag) sSQs#+ &=[  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. E4W zU  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N SJ};TEA  
    %   and angular frequency M, evaluated at positions (R,THETA) on the mK [0L  
    %   unit circle.  N is a vector of positive integers (including 0), and *L'>U[Pl7  
    %   M is a vector with the same number of elements as N.  Each element /M*a,o  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) j~e;DO  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, \;mH(-  
    %   and THETA is a vector of angles.  R and THETA must have the same Iz{R}#8CZ  
    %   length.  The output Z is a matrix with one column for every (N,M) (<Th=Fns?  
    %   pair, and one row for every (R,THETA) pair. e4z1`YLsG  
    % Zt&6Ua[Y}  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike = pzn u+,  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), eU N"w,@y  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral l $jxLZ  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, FA }_(Hf.[  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ?x0pe4^If  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ZKOXI%~Mc  
    % 5n ^TRB  
    %   The Zernike functions are an orthogonal basis on the unit circle. RNhJ'&SYs  
    %   They are used in disciplines such as astronomy, optics, and EgDQ+( -  
    %   optometry to describe functions on a circular domain. ^+1#[E  
    % 9Y<#=C  
    %   The following table lists the first 15 Zernike functions. &%_& 8DkG  
    % 'D%w|Pe?Q  
    %       n    m    Zernike function           Normalization _C+b]r/E  
    %       -------------------------------------------------- kee|42E  
    %       0    0    1                                 1 -Z?Vd!H:  
    %       1    1    r * cos(theta)                    2 d)"?mD:m/M  
    %       1   -1    r * sin(theta)                    2 F|HJH"2*&q  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 4#'(" #R  
    %       2    0    (2*r^2 - 1)                    sqrt(3) i]#+1Hf  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) `WOYoec   
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 1<<kA:d  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 1 `7<2w  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) >R2SQA o  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 4 8{vE3JY  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 2]c {P\  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) N*@aDM07  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) MCP "GZK6W  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) zP;cTF(C  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 3J=Y9 }  
    %       -------------------------------------------------- ,= &B28Qe)  
    % ?9X&tK)E-  
    %   Example 1: _zu?.I0^  
    % 7'-j%!#w  
    %       % Display the Zernike function Z(n=5,m=1)  N<L`c/  
    %       x = -1:0.01:1; Jz!Z2c  
    %       [X,Y] = meshgrid(x,x); cf7v[ZZ}  
    %       [theta,r] = cart2pol(X,Y); DS -fjH\  
    %       idx = r<=1; 3F#+~^2  
    %       z = nan(size(X)); 4A3nO<o MF  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); )kJH5/  
    %       figure 1H,g=Y4f%  
    %       pcolor(x,x,z), shading interp q,2]5 '  
    %       axis square, colorbar oiH|uIsqR  
    %       title('Zernike function Z_5^1(r,\theta)') 8V-\e?&^  
    % 2nFy`|aA%  
    %   Example 2: f N "tA  
    % lI/0:|l  
    %       % Display the first 10 Zernike functions Z.wA@ ~e  
    %       x = -1:0.01:1; &|<xqt  
    %       [X,Y] = meshgrid(x,x); \Yoa:|%*y  
    %       [theta,r] = cart2pol(X,Y); ]}UgS+g>$  
    %       idx = r<=1; -[Qvg49jy  
    %       z = nan(size(X)); XIWm>IQ[)  
    %       n = [0  1  1  2  2  2  3  3  3  3]; <q,+ON\'  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 2 /y}a#s  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 8:.nEo'  
    %       y = zernfun(n,m,r(idx),theta(idx)); M- 0i7%  
    %       figure('Units','normalized') a? R[J==  
    %       for k = 1:10 m4[g6pNx~  
    %           z(idx) = y(:,k); %o}(sShS  
    %           subplot(4,7,Nplot(k)) t+nRw?Z  
    %           pcolor(x,x,z), shading interp vSW L$Y2  
    %           set(gca,'XTick',[],'YTick',[]) z\/53Sy<  
    %           axis square ba3-t;S  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) {$M;H+Foh  
    %       end C#w]4$/  
    % G]Jz"xH#  
    %   See also ZERNPOL, ZERNFUN2. kHJ96G  
    M3zDtN  
    #' hLb  
    %   Paul Fricker 11/13/2006 c {I"R8  
    8#` 6M5  
    * \HRw +cL  
    2>\\@ 1  
    =raA?Bp3;(  
    % Check and prepare the inputs: Yn9j-`  
    % ----------------------------- \nqo%5XL  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }xlKonk  
        error('zernfun:NMvectors','N and M must be vectors.') RH~3M0'0  
    end Z v0C@r  
    dZGbC9  
    =w<v3wWN4  
    if length(n)~=length(m) Zwe[_z!*D  
        error('zernfun:NMlength','N and M must be the same length.') k:#6^!b1  
    end s T3p>8n  
    (3*UPZv  
    ' 9J|=z9.  
    n = n(:); Pj7gGf6v  
    m = m(:); 0p fnV%  
    if any(mod(n-m,2)) eFTX6XB:i  
        error('zernfun:NMmultiplesof2', ... V)D-pV V  
              'All N and M must differ by multiples of 2 (including 0).') K%}}fw2RMN  
    end oJ78jGTnb  
    <E':[.zC  
    uv4 _:   
    if any(m>n) |)@N-f:E  
        error('zernfun:MlessthanN', ... i=v]:TOu  
              'Each M must be less than or equal to its corresponding N.') M+sj}  
    end 1h"_[`L'  
    uC~g#[I QM  
    v9}[$HWx  
    if any( r>1 | r<0 ) #B\=Aa`*  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') i ilyw_$H  
    end =:h3w#_c  
    s0{ NsK>  
    DM3B]Yl  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) U |F>W~%  
        error('zernfun:RTHvector','R and THETA must be vectors.') .#^0pv!  
    end LD+f'^>>Z  
    MB:n~>ga  
    Nm8w/Q5D`  
    r = r(:); NMjnL&P`  
    theta = theta(:); N"DY?6  
    length_r = length(r); 'o\;x"YJ  
    if length_r~=length(theta) $<e +r$1  
        error('zernfun:RTHlength', ... {e]NU<G ,  
              'The number of R- and THETA-values must be equal.') Sb^ b)q"  
    end F(CRq`  
    GYgWf1$8_D  
    K="I<bK  
    % Check normalization: $ u2Cd4  
    % -------------------- Sa] mm/ G  
    if nargin==5 && ischar(nflag) PO ko]@~!i  
        isnorm = strcmpi(nflag,'norm'); 5_G'68;OV  
        if ~isnorm  a@|.;#FF  
            error('zernfun:normalization','Unrecognized normalization flag.') bNvAyKc-  
        end <q7s`,rG  
    else 0~xaUM`  
        isnorm = false; @ t@|q  
    end ;]h.m)~|  
    MOV =n75  
    + x_ wYv  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3c 28!3p  
    % Compute the Zernike Polynomials R^9"N?Q7;`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6H;kJHn  
    T|f_~#?eV  
    tL8't]M,  
    % Determine the required powers of r: onzA7Gre  
    % ----------------------------------- dy_.(r5[L]  
    m_abs = abs(m); z\[(g  
    rpowers = []; hCLk#_  
    for j = 1:length(n) 5c~'!:7  
        rpowers = [rpowers m_abs(j):2:n(j)]; n.;3X  
    end zXgkcq)  
    rpowers = unique(rpowers); |p'i,.(c_W  
    yGV{^?yoP  
    ,#%SK;1<  
    % Pre-compute the values of r raised to the required powers, tG 7+7Z =  
    % and compile them in a matrix: t/Z!O z6ZE  
    % ----------------------------- <t6 d)mJ%  
    if rpowers(1)==0 3ExVZu$  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); }9Q f#&o  
        rpowern = cat(2,rpowern{:}); t/LgHb:)  
        rpowern = [ones(length_r,1) rpowern]; RQ5P}A 3H  
    else >'0lw+a  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8g5.7{ky  
        rpowern = cat(2,rpowern{:}); IuWX*b`v  
    end X0x_+b? _  
    Qy%xL9  
    3qcpf:  
    % Compute the values of the polynomials: 9R:(^8P8  
    % -------------------------------------- tD^a5qPh  
    y = zeros(length_r,length(n)); 2e\Kw+(>{  
    for j = 1:length(n) 6+#,=!hF{  
        s = 0:(n(j)-m_abs(j))/2; % 9YA^ri  
        pows = n(j):-2:m_abs(j); &O{t^D)F  
        for k = length(s):-1:1 &`sR){R  
            p = (1-2*mod(s(k),2))* ... DD6'M U4  
                       prod(2:(n(j)-s(k)))/              ... 7?]!Ecr"  
                       prod(2:s(k))/                     ... -~]^5aa5n  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... PE7t_iSV  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); `L">"V`$Bj  
            idx = (pows(k)==rpowers); }Y$VB%&Hy  
            y(:,j) = y(:,j) + p*rpowern(:,idx); HqDa2q4  
        end +ks$UvtY  
         +9}' s{  
        if isnorm o7QK8#  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); PJ6$);9}6  
        end R''Sfz>8  
    end :`j"Sj !t3  
    % END: Compute the Zernike Polynomials *U2Ck<"]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X{|k<^:  
    1[# =,  
    fX$6;Ae  
    % Compute the Zernike functions: kz|[*%10  
    % ------------------------------ 3V!W@[ }:  
    idx_pos = m>0; =/f74s t  
    idx_neg = m<0; OT"lP(,  
    n_ OUWvs  
    @>Biyb  
    z = y; G)K9la<p  
    if any(idx_pos) 1(D1}fcul  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Q)9369<A  
    end E'fX&[  
    if any(idx_neg) r6<ArX$Yl  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 2@4MC`&  
    end r ufRaar  
    uJ7,rq  
    u'{sB5_H  
    % EOF zernfun ~mW>_[RT;  
     
    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)
    在线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)  Z-;uzx  
    Ozh^Q$>u  
    DDE还是手动输入的呢? Wy0a2Ve  
    MX< ($M  
    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