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niuhelen	2011-03-12 18:40
ansys分析后面型数据如何进行zernike多项式拟合？小弟不是学光学的，所以想请各位大侠指点啊！谢谢啦 ~>\|o3&G{ 就是我用ansys计算出了镜面的面型的数据，怎样可以得到zernike多项式的系数，然后用zemax各阶得到像差！谢谢啦！ Ce_ES.

phility	2011-03-12 22:31
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

phility	2011-03-12 22:41
泽尼克多项式的前9项对应象差的

niuhelen

2011-03-12 23:00

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
function z = zernfun(n,m,r,theta,nflag)
%ZERNFUN Zernike functions of order N and frequency M on the unit circle.
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N
%   and angular frequency M, evaluated at positions (R,THETA) on the
%   unit circle.  N is a vector of positive integers (including 0), and
%   M is a vector with the same number of elements as N.  Each element
%   k of M must be a positive integer, with possible values M(k) = -N(k)
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1,
%   and THETA is a vector of angles.  R and THETA must have the same
%   length.  The output Z is a matrix with one column for every (N,M)
%   pair, and one row for every (R,THETA) pair.
%
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi),
%   with delta(m,0) the Kronecker delta, is chosen so that the integral
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1,
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m].
%
%   The Zernike functions are an orthogonal basis on the unit circle.
%   They are used in disciplines such as astronomy, optics, and
%   optometry to describe functions on a circular domain.
%
%   The following table lists the first 15 Zernike functions.
%
%       n    m    Zernike function           Normalization
%       --------------------------------------------------
%       0    0    1                                 1
%       1    1    r * cos(theta)                    2
%       1   -1    r * sin(theta)                    2
%       2   -2    r^2 * cos(2*theta)             sqrt(6)
%       2    0    (2*r^2 - 1)                    sqrt(3)
%       2    2    r^2 * sin(2*theta)             sqrt(6)
%       3   -3    r^3 * cos(3*theta)             sqrt(8)
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8)
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8)
%       3    3    r^3 * sin(3*theta)             sqrt(8)
%       4   -4    r^4 * cos(4*theta)             sqrt(10)
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5)
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)
%       4    4    r^4 * sin(4*theta)             sqrt(10)
%       --------------------------------------------------
%
%   Example 1:
%
%       % Display the Zernike function Z(n=5,m=1)
%       x = -1:0.01:1;
%       [X,Y] = meshgrid(x,x);
%       [theta,r] = cart2pol(X,Y);
%       idx = r<=1;
%       z = nan(size(X));
%       z(idx) = zernfun(5,1,r(idx),theta(idx));
%       figure
%       pcolor(x,x,z), shading interp
%       axis square, colorbar
%       title('Zernike function Z_5^1(r,\theta)')
%
%   Example 2:
%
%       % Display the first 10 Zernike functions
%       x = -1:0.01:1;
%       [X,Y] = meshgrid(x,x);
%       [theta,r] = cart2pol(X,Y);
%       idx = r<=1;
%       z = nan(size(X));
%       n = [0  1  1  2  2  2  3  3  3  3];
%       m = [0 -1  1 -2  0  2 -3 -1  1  3];
%       Nplot = [4 10 12 16 18 20 22 24 26 28];
%       y = zernfun(n,m,r(idx),theta(idx));
%       figure('Units','normalized')
%       for k = 1:10
%           z(idx) = y(:,k);
%           subplot(4,7,Nplot(k))
%           pcolor(x,x,z), shading interp
%           set(gca,'XTick',[],'YTick',[])
%           axis square
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])
%       end
%
%   See also ZERNPOL, ZERNFUN2.

%   Paul Fricker 11/13/2006

% Check and prepare the inputs:
% -----------------------------
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) )
    error('zernfun:NMvectors','N and M must be vectors.')
end

if length(n)~=length(m)
    error('zernfun:NMlength','N and M must be the same length.')
end

n = n(:);
m = m(:);
if any(mod(n-m,2))
    error('zernfun:NMmultiplesof2', ...
          'All N and M must differ by multiples of 2 (including 0).')
end

if any(m>n)
    error('zernfun:MlessthanN', ...
          'Each M must be less than or equal to its corresponding N.')
end

if any( r>1 | r<0 )
    error('zernfun:Rlessthan1','All R must be between 0 and 1.')
end

if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) )
    error('zernfun:RTHvector','R and THETA must be vectors.')
end

r = r(:);
theta = theta(:);
length_r = length(r);
if length_r~=length(theta)
    error('zernfun:RTHlength', ...
          'The number of R- and THETA-values must be equal.')
end

% Check normalization:
% --------------------
if nargin==5 && ischar(nflag)
    isnorm = strcmpi(nflag,'norm');
    if ~isnorm
        error('zernfun:normalization','Unrecognized normalization flag.')
    end
else
    isnorm = false;
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Compute the Zernike Polynomials
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Determine the required powers of r:
% -----------------------------------
m_abs = abs(m);
rpowers = [];
for j = 1:length(n)
    rpowers = [rpowers m_abs(j):2:n(j)];
end
rpowers = unique(rpowers);

% Pre-compute the values of r raised to the required powers,
% and compile them in a matrix:
% -----------------------------
if rpowers(1)==0
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);
    rpowern = cat(2,rpowern{:});
    rpowern = [ones(length_r,1) rpowern];
else
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);
    rpowern = cat(2,rpowern{:});
end

% Compute the values of the polynomials:
% --------------------------------------
y = zeros(length_r,length(n));
for j = 1:length(n)
    s = 0:(n(j)-m_abs(j))/2;
    pows = n(j):-2:m_abs(j);
    for k = length(s):-1:1
        p = (1-2*mod(s(k),2))* ...
                   prod(2:(n(j)-s(k)))/              ...
                   prod(2:s(k))/                     ...
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ...
                   prod(2:((n(j)+m_abs(j))/2-s(k)));
        idx = (pows(k)==rpowers);
        y(:,j) = y(:,j) + p*rpowern(:,idx);
    end

    if isnorm
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi);
    end
end
% END: Compute the Zernike Polynomials
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Compute the Zernike functions:
% ------------------------------
idx_pos = m>0;
idx_neg = m<0;

z = y;
if any(idx_pos)
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)');
end
if any(idx_neg)
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)');
end

% EOF zernfun

niuhelen

2011-03-12 23:01

function z = zernfun2(p,r,theta,nflag)
%ZERNFUN2 Single-index Zernike functions on the unit circle.
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated
%   at positions (R,THETA) on the unit circle.  P is a vector of positive
%   integers between 0 and 35, R is a vector of numbers between 0 and 1,
%   and THETA is a vector of angles.  R and THETA must have the same
%   length.  The output Z is a matrix with one column for every P-value,
%   and one row for every (R,THETA) pair.
%
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2)
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi)
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1
%   for all p.
%
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36
%   Zernike functions (order N<=7).  In some disciplines it is
%   traditional to label the first 36 functions using a single mode
%   number P instead of separate numbers for the order N and azimuthal
%   frequency M.
%
%   Example:
%
%       % Display the first 16 Zernike functions
%       x = -1:0.01:1;
%       [X,Y] = meshgrid(x,x);
%       [theta,r] = cart2pol(X,Y);
%       idx = r<=1;
%       p = 0:15;
%       z = nan(size(X));
%       y = zernfun2(p,r(idx),theta(idx));
%       figure('Units','normalized')
%       for k = 1:length(p)
%           z(idx) = y(:,k);
%           subplot(4,4,k)
%           pcolor(x,x,z), shading interp
%           set(gca,'XTick',[],'YTick',[])
%           axis square
%           title(['Z_{' num2str(p(k)) '}'])
%       end
%
%   See also ZERNPOL, ZERNFUN.

%   Paul Fricker 11/13/2006

% Check and prepare the inputs:
% -----------------------------
if min(size(p))~=1
    error('zernfun2:Pvector','Input P must be vector.')
end

if any(p)>35
    error('zernfun2:P36', ...
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ...
           '(P = 0 to 35).'])
end

% Get the order and frequency corresonding to the function number:
% ----------------------------------------------------------------
p = p(:);
n = ceil((-3+sqrt(9+8*p))/2);
m = 2*p - n.*(n+2);

% Pass the inputs to the function ZERNFUN:
% ----------------------------------------
switch nargin
    case 3
        z = zernfun(n,m,r,theta);
    case 4
        z = zernfun(n,m,r,theta,nflag);
    otherwise
        error('zernfun2:nargin','Incorrect number of inputs.')
end

% EOF zernfun2

niuhelen

2011-03-12 23:01

function z = zernpol(n,m,r,nflag)
%ZERNPOL Radial Zernike polynomials of order N and frequency M.
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of
%   order N and frequency M, evaluated at R.  N is a vector of
%   positive integers (including 0), and M is a vector with the
%   same number of elements as N.  Each element k of M must be a
%   positive integer, with possible values M(k) = 0,2,4,...,N(k)
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is
%   a vector of numbers between 0 and 1.  The output Z is a matrix
%   with one column for every (N,M) pair, and one row for every
%   element in R.
%
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly-
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1
%   for all [n,m].
%
%   The radial Zernike polynomials are the radial portion of the
%   Zernike functions, which are an orthogonal basis on the unit
%   circle.  The series representation of the radial Zernike
%   polynomials is
%
%          (n-m)/2
%            __
%    m      \       s                                          n-2s
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r
%    n      s=0
%
%   The following table shows the first 12 polynomials.
%
%       n    m    Zernike polynomial    Normalization
%       ---------------------------------------------
%       0    0    1                        sqrt(2)
%       1    1    r                           2
%       2    0    2*r^2 - 1                sqrt(6)
%       2    2    r^2                      sqrt(6)
%       3    1    3*r^3 - 2*r              sqrt(8)
%       3    3    r^3                      sqrt(8)
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10)
%       4    2    4*r^4 - 3*r^2            sqrt(10)
%       4    4    r^4                      sqrt(10)
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12)
%       5    3    5*r^5 - 4*r^3            sqrt(12)
%       5    5    r^5                      sqrt(12)
%       ---------------------------------------------
%
%   Example:
%
%       % Display three example Zernike radial polynomials
%       r = 0:0.01:1;
%       n = [3 2 5];
%       m = [1 2 1];
%       z = zernpol(n,m,r);
%       figure
%       plot(r,z)
%       grid on
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest')
%
%   See also ZERNFUN, ZERNFUN2.

% A note on the algorithm.
% ------------------------
% The radial Zernike polynomials are computed using the series
% representation shown in the Help section above. For many special
% functions, direct evaluation using the series representation can
% produce poor numerical results (floating point errors), because
% the summation often involves computing small differences between
% large successive terms in the series. (In such cases, the functions
% are often evaluated using alternative methods such as recurrence
% relations: see the Legendre functions, for example). For the Zernike
% polynomials, however, this problem does not arise, because the
% polynomials are evaluated over the finite domain r = (0,1), and
% because the coefficients for a given polynomial are generally all
% of similar magnitude.
%
% ZERNPOL has been written using a vectorized implementation: multiple
% Zernike polynomials can be computed (i.e., multiple sets of [N,M]
% values can be passed as inputs) for a vector of points R.  To achieve
% this vectorization most efficiently, the algorithm in ZERNPOL
% involves pre-determining all the powers p of R that are required to
% compute the outputs, and then compiling the {R^p} into a single
% matrix.  This avoids any redundant computation of the R^p, and
% minimizes the sizes of certain intermediate variables.
%
%   Paul Fricker 11/13/2006

% Check and prepare the inputs:
% -----------------------------
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) )
    error('zernpol:NMvectors','N and M must be vectors.')
end

if length(n)~=length(m)
    error('zernpol:NMlength','N and M must be the same length.')
end

n = n(:);
m = m(:);
length_n = length(n);

if any(mod(n-m,2))
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).')
end

if any(m<0)
    error('zernpol:Mpositive','All M must be positive.')
end

if any(m>n)
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.')
end

if any( r>1 | r<0 )
    error('zernpol:Rlessthan1','All R must be between 0 and 1.')
end

if ~any(size(r)==1)
    error('zernpol:Rvector','R must be a vector.')
end

r = r(:);
length_r = length(r);

if nargin==4
    isnorm = ischar(nflag) & strcmpi(nflag,'norm');
    if ~isnorm
        error('zernpol:normalization','Unrecognized normalization flag.')
    end
else
    isnorm = false;
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Compute the Zernike Polynomials
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Determine the required powers of r:
% -----------------------------------
rpowers = [];
for j = 1:length(n)
    rpowers = [rpowers m(j):2:n(j)];
end
rpowers = unique(rpowers);

% Pre-compute the values of r raised to the required powers,
% and compile them in a matrix:
% -----------------------------
if rpowers(1)==0
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);
    rpowern = cat(2,rpowern{:});
    rpowern = [ones(length_r,1) rpowern];
else
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);
    rpowern = cat(2,rpowern{:});
end

% Compute the values of the polynomials:
% --------------------------------------
z = zeros(length_r,length_n);
for j = 1:length_n
    s = 0:(n(j)-m(j))/2;
    pows = n(j):-2:m(j);
    for k = length(s):-1:1
        p = (1-2*mod(s(k),2))* ...
                   prod(2:(n(j)-s(k)))/          ...
                   prod(2:s(k))/                 ...
                   prod(2:((n(j)-m(j))/2-s(k)))/ ...
                   prod(2:((n(j)+m(j))/2-s(k)));
        idx = (pows(k)==rpowers);
        z(:,j) = z(:,j) + p*rpowern(:,idx);
    end

    if isnorm
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1));
    end
end

% EOF zernpol

niuhelen	2011-03-12 23:03
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！

li_xin_feng	2012-09-28 10:52
我也正在找啊

guapiqlh	2014-03-04 11:35
我也一直想了解这个多项式的应用，还没用过呢

phoenixzqy	2014-04-22 23:39
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) ^[}0&_L w 8V\|jL?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。 B9IXa; ]Wd`GI 07年就写过这方面的计算程序了。

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