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 O(x1Ja,& function z = zernfun(n,m,r,theta,nflag) Q*4{2oQ %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Y;2WY0eq % Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N )!z4LE % and angular frequency M, evaluated at positions (R,THETA) on the yp}J+/PX} % unit circle. N is a vector of positive integers (including 0), and 3v\69s % M is a vector with the same number of elements as N. Each element Qw>~]d,Z % k of M must be a positive integer, with possible values M(k) = -N(k) O0^m_ % to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, o%3i(H % and THETA is a vector of angles. R and THETA must have the same e}l F#$ % length. The output Z is a matrix with one column for every (N,M) Ckd
j| % pair, and one row for every (R,THETA) pair. ^UU@7cSi|G % kU:ge % Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike `1Ui % functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), zF:
:?L~ % with delta(m,0) the Kronecker delta, is chosen so that the integral g0({$2Q7R % of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, J9aqmQj(' % and theta=0 to theta=2*pi) is unity. For the non-normalized "x1?T+j4 % polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 1 S<E=7 % 3CL1Z\8To % The Zernike functions are an orthogonal basis on the unit circle. ~mBY_[_s= % They are used in disciplines such as astronomy, optics, and we:P_\6 % optometry to describe functions on a circular domain. wrP3:!= % arK(dg~S % The following table lists the first 15 Zernike functions. HxUJ 0Q % z)%Ke~)<\@ % n m Zernike function Normalization }H#C<:A % -------------------------------------------------- _oz1'}= % 0 0 1 1 /]U),LbN % 1 1 r * cos(theta) 2 %f)%FN.S % 1 -1 r * sin(theta) 2 GJs{t1
E % 2 -2 r^2 * cos(2*theta) sqrt(6) wjtFZGx& % 2 0 (2*r^2 - 1) sqrt(3) pyUzHF0 % 2 2 r^2 * sin(2*theta) sqrt(6) &/m0N\n?
% 3 -3 r^3 * cos(3*theta) sqrt(8) #W$6[#7=I % 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) d+qeZGg^A % 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Kz~E"? % 3 3 r^3 * sin(3*theta) sqrt(8) 8I8{xt4 % 4 -4 r^4 * cos(4*theta) sqrt(10) KWS\ iu % 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Six2{b)p % 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) |@W|nbAfX % 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U8S<wf& % 4 4 r^4 * sin(4*theta) sqrt(10) xiv8q/ % -------------------------------------------------- `y3*\l % cqYMzS
t % Example 1: :3N6Ej % _<Ip0?N % % Display the Zernike function Z(n=5,m=1) n +v(t % x = -1:0.01:1; n\GN}?4 % [X,Y] = meshgrid(x,x); ^*G
UcQ$ % [theta,r] = cart2pol(X,Y); t5CJG '!ql % idx = r<=1; c( _R
xLJ % z = nan(size(X)); t/lQSUip % z(idx) = zernfun(5,1,r(idx),theta(idx)); V=gu'~ % figure g& ou[_A % pcolor(x,x,z), shading interp 63`5A3rii % axis square, colorbar TO&^%d % title('Zernike function Z_5^1(r,\theta)') }wB!Bx2 % '2qbIYanh % Example 2: r}:Dg
fn % vs^)= % % Display the first 10 Zernike functions !k<k]^Z\ % x = -1:0.01:1; q*K[? % [X,Y] = meshgrid(x,x); zr ~4@JTS % [theta,r] = cart2pol(X,Y); :x,dYJm % idx = r<=1; ug_c}Nv=Y % z = nan(size(X)); *5u3d`bW % n = [0 1 1 2 2 2 3 3 3 3]; }#q0K % m = [0 -1 1 -2 0 2 -3 -1 1 3]; ')T*cLQ>< % Nplot = [4 10 12 16 18 20 22 24 26 28]; vL#I+_ 2 % y = zernfun(n,m,r(idx),theta(idx)); mGpBj9jr1 % figure('Units','normalized') mg< v9# % for k = 1:10 \WqC^Di % z(idx) = y(:,k); N(e>]ui % subplot(4,7,Nplot(k)) n5 <B* % pcolor(x,x,z), shading interp QYj*|p^x % set(gca,'XTick',[],'YTick',[]) e6>[Z C % axis square >E7s}bL" % title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 0h
kZ % end aA
-j % A 4*D3\>%u % See also ZERNPOL, ZERNFUN2. \=[38?QOY +W/{UddeKU % Paul Fricker 11/13/2006 zjTCq; G )xL_jSyh )8taMC:H^ % Check and prepare the inputs: 9e7):ZupO % ----------------------------- _&N:%;9uD if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }R~C<3u\2 error('zernfun:NMvectors','N and M must be vectors.') I* PxQ end T2A74>Nw 8PqlbLo1 if length(n)~=length(m) o4^#W;%w error('zernfun:NMlength','N and M must be the same length.') .zy2_3: end 7H4\AG\> VVEJE$ n = n(:); YkQ=rurE m = m(:); L*P*^I^1 if any(mod(n-m,2)) <'jygZ( error('zernfun:NMmultiplesof2', ... gk}.LE 'All N and M must differ by multiples of 2 (including 0).') mqBX1D`e2 end XM3~] Abpzf\F if any(m>n) K#N5S]2yb error('zernfun:MlessthanN', ... p`S~UBcL. 'Each M must be less than or equal to its corresponding N.') Gx|/
Jq end J!"m{ 8- x}f)P if any( r>1 | r<0 ) vos-[$ error('zernfun:Rlessthan1','All R must be between 0 and 1.') %C)|fDwN end {)4@rM rW)}$|-Z if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) F)50 6 error('zernfun:RTHvector','R and THETA must be vectors.') CHdYY7\{ end #UnGU,J "2}n(8 r = r(:); dCWq~[[ theta = theta(:); &!*p>Ns)e length_r = length(r); .4!wp& if length_r~=length(theta) orEb+ error('zernfun:RTHlength', ... }cIj1: 'The number of R- and THETA-values must be equal.') "VeNc,-nfQ end "^t;V+Io W,%qL6qV % Check normalization: 1y7$"N8Xo % -------------------- b:&=W>r if nargin==5 && ischar(nflag) '1lz`CAB+ isnorm = strcmpi(nflag,'norm'); <2\QY if ~isnorm I^O`#SA ( error('zernfun:normalization','Unrecognized normalization flag.') ?YM0VB,y end Iy2AJ|d. else 8WwLKZ} isnorm = false; 5?TjuGc end ?o(Y\YJf ,27=i>> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7D4I>N'T % Compute the Zernike Polynomials /j:-GJb*!u %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% UQ2;Dg G% #~6X9,x= % Determine the required powers of r: ^p~ 3H % ----------------------------------- C2?p>S/q m_abs = abs(m); peU1
t:k? rpowers = []; &^ =Y76 for j = 1:length(n) L_AQS9a^D rpowers = [rpowers m_abs(j):2:n(j)]; f q*V76F end (PnrY~9 rpowers = unique(rpowers); HTP~5J M5B?`mTl % Pre-compute the values of r raised to the required powers, T)cbpkH4 % and compile them in a matrix: 3]/Y=A % ----------------------------- YifTC-Q; if rpowers(1)==0 m6
a@Y< rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); [u8JqX rpowern = cat(2,rpowern{:}); GnW_^$Fs rpowern = [ones(length_r,1) rpowern]; Y.o-e)zX else E2|c;{c rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;<v9i#K5 rpowern = cat(2,rpowern{:}); bhT:MW! end !%YV0O0 H{*R(S<I % Compute the values of the polynomials: >c@1UEwkm % -------------------------------------- p:qj.ukw y = zeros(length_r,length(n)); 9/50+2F for j = 1:length(n) a~;`&Uj s = 0:(n(j)-m_abs(j))/2; a EqDxr6 pows = n(j):-2:m_abs(j); $g)X,iQu for k = length(s):-1:1 >l!DWi6 p = (1-2*mod(s(k),2))* ... +DP{ _x)t prod(2:(n(j)-s(k)))/ ... rxAb]~MMp prod(2:s(k))/ ... "ZFK-jn/ prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 24/ ^_Td prod(2:((n(j)+m_abs(j))/2-s(k))); .JL?RH2@8 idx = (pows(k)==rpowers); (yi{<$U* y(:,j) = y(:,j) + p*rpowern(:,idx); '|K408i end WUqfY?5 0Bhf(5 if isnorm TfqQh!Y y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 97(*-e= e end $F86Dwd end . xdSUe % END: Compute the Zernike Polynomials 8Dy;'BtT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~@bh[o~rF 2M+'9+k~ % Compute the Zernike functions: ~m.@{Do0p % ------------------------------ DU-&bm idx_pos = m>0; ]Syr{| idx_neg = m<0; v}\Nx[} xA2"i2k9 z = y; TwXqk>J if any(idx_pos) Q#rj>+? z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); S-k:+ 4 end .`K<Iug1 if any(idx_neg) S&YC" z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Do5)ilt end ]J7.d$7T (-U6woB6o % EOF zernfun
|
|