function z = zernpol(n,m,r,nflag) -O-qEQd
%ZERNPOL Radial Zernike polynomials of order N and frequency M. S<V__Sv
% Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of |4s`;4c&
% order N and frequency M, evaluated at R. N is a vector of `+/xA\X]
% positive integers (including 0), and M is a vector with the uM9[
% same number of elements as N. Each element k of M must be a vQpR0IEf]e
% positive integer, with possible values M(k) = 0,2,4,...,N(k) >-{)wk;1&
% for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd. R is ki^c)Tqn
% a vector of numbers between 0 and 1. The output Z is a matrix Ll
!J!{
% with one column for every (N,M) pair, and one row for every RjS&^uaP
% element in R. $Z;?d@6yI
% //}[(9b'\
% Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 3+2&@:$t
% nomials. The normalization factor Nnm = sqrt(2*(n+1)) is oG )JH)!
% chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ~"+Fp&[9f
% r=1 is unity. For the non-normalized polynomials, Znm(r=1)=1 RrDNEwAr
% for all [n,m]. <([1(SY2e
% AZCbUkq
% The radial Zernike polynomials are the radial portion of the ^"h`U'YC
% Zernike functions, which are an orthogonal basis on the unit FV&&
% circle. The series representation of the radial Zernike t$z
FsFTQ
% polynomials is jtk2>Ol
% r>hkm53
% (n-m)/2 #Pz},!7
% __ ,afh]#
% m \ s n-2s 3P!Jw7e
% Z(r) = /__ (-1) [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r FSqS]6b3
% n s=0 z6K"}C%
% 1YA_`_@w
% The following table shows the first 12 polynomials. 8S>T1st
% Gv!*
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% n m Zernike polynomial Normalization %jK-}0Tu
% --------------------------------------------- P{eL;^I
% 0 0 1 sqrt(2) ,?+rM ;
% 1 1 r 2 XQu~/{A=
% 2 0 2*r^2 - 1 sqrt(6) mACj>0Z'
% 2 2 r^2 sqrt(6) CqUK[#kW(
% 3 1 3*r^3 - 2*r sqrt(8) l("Dw8H
% 3 3 r^3 sqrt(8) h,q%MZ==^s
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(10)
?6!7fs,
% 4 2 4*r^4 - 3*r^2 sqrt(10) JBCcR,\kM*
% 4 4 r^4 sqrt(10) f!~gfnn
% 5 1 10*r^5 - 12*r^3 + 3*r sqrt(12) ?#^_yd|<
% 5 3 5*r^5 - 4*r^3 sqrt(12) Dq!Vo ;s2
% 5 5 r^5 sqrt(12) &WIiw$@
% --------------------------------------------- Z~tOR{q
% 8Hf!@p6R+
% Example: Nw}y_Qf{
% dlC)&Ai
% % Display three example Zernike radial polynomials TE4{W4I
% r = 0:0.01:1; nBGFa
% n = [3 2 5]; kmM1)- v
% m = [1 2 1];
&j,rq?eh$
% z = zernpol(n,m,r); *]fBd<(8
% figure Bl-nS{9"
% plot(r,z) adh=Kp e!w
% grid on VpJ/M(UD-
% legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') A
e&t#,)
% E8WOXoP(
% See also ZERNFUN, ZERNFUN2. yVm~5Y&Z
rS>JzbWa
% A note on the algorithm. q28i9$Yqj\
% ------------------------ 0A@'w*=
% The radial Zernike polynomials are computed using the series 3~\mP\/4v
% representation shown in the Help section above. For many special oQ= Q}
% functions, direct evaluation using the series representation can ewqfs/
% produce poor numerical results (floating point errors), because ]5lp.#EB
% the summation often involves computing small differences between Y&aFAjj
% large successive terms in the series. (In such cases, the functions lvIKL!;H
% are often evaluated using alternative methods such as recurrence V?v,q'? $
% relations: see the Legendre functions, for example). For the Zernike R74kt36M
% polynomials, however, this problem does not arise, because the @kUCc1LT
% polynomials are evaluated over the finite domain r = (0,1), and &dZ-}.
af
% because the coefficients for a given polynomial are generally all :04sB]H
% of similar magnitude. +qe!KPk2
% ja}_u}:
% ZERNPOL has been written using a vectorized implementation: multiple q_5k2'4K
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] R:98'`X=
% values can be passed as inputs) for a vector of points R. To achieve T9\wkb.
% this vectorization most efficiently, the algorithm in ZERNPOL OS6 l*S('
% involves pre-determining all the powers p of R that are required to V<AT"vU[
% compute the outputs, and then compiling the {R^p} into a single ua*k{0[
% matrix. This avoids any redundant computation of the R^p, and PD12gUU?
% minimizes the sizes of certain intermediate variables. 0&Q-y&$7
% s)#FqB8
% Paul Fricker 11/13/2006 ^SB?NRk
Fd-PjW/E8
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% Check and prepare the inputs: Mxn>WCPo
% ----------------------------- ;wIpch e
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) jpZ, $
error('zernpol:NMvectors','N and M must be vectors.') kt.z,<w5O
end ps"DL4*
^ElUU ?rX
if length(n)~=length(m) D(D:/L8T,
error('zernpol:NMlength','N and M must be the same length.') yazC2Enes8
end EAU6z(X$
4[|^78
n = n(:); EA9`-xs|
m = m(:); QWv+Ja
length_n = length(n); bB'iK4
@FKNB.>
if any(mod(n-m,2)) *z;4.
OX
error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 9l9nT
end 4bqi&h3
0&2(1
if any(m<0) I.TdYSB
error('zernpol:Mpositive','All M must be positive.') EV|
6._Z(D
end $Zp\^cIE+
1GKd*z
if any(m>n) :zZK%}G<
error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ~~k_A|&
end 6Y0k}+j|>E
{^2``NYM_
if any( r>1 | r<0 ) .ml24SeC
error('zernpol:Rlessthan1','All R must be between 0 and 1.') }z1aKa9
end -hw^3Af
MW8GM }Ho[
if ~any(size(r)==1) 9 o6ig>C
error('zernpol:Rvector','R must be a vector.') nS)U+q-x&o
end JsI`#
6/Y3#d
r = r(:); HtB>#`'
length_r = length(r); Hj't.lg+j
p9Zi}!
if nargin==4 )WavG1
isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ;rYL\`6L
if ~isnorm `"zXf -qeE
error('zernpol:normalization','Unrecognized normalization flag.') +<7~yZ[Z8
end yEIM58l
else ?U.+SQ
isnorm = false; hAtf)
end 9HrT>{@
FIhq>L.q4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =B@+[b0Z
% Compute the Zernike Polynomials @S\!wjl]C
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :UM>`Y
|l)SX\Qf`@
% Determine the required powers of r: LgXc}3
% ----------------------------------- $(B|$e^:(
rpowers = []; =V~pQbZ
for j = 1:length(n) "1|n]0BF
rpowers = [rpowers m(j):2:n(j)]; {Qbg'|HO=l
end V:HxRMF2X
rpowers = unique(rpowers); =i[ _C>U
p&dpDJ?d:=
% Pre-compute the values of r raised to the required powers, wm~35cF(
% and compile them in a matrix: jWk1FQte
% ----------------------------- 5e=9~].7
if rpowers(1)==0 *Z'*^Y1le
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ,]RMa\Q4Wg
rpowern = cat(2,rpowern{:}); Q`-JRY-
rpowern = [ones(length_r,1) rpowern]; }-Q FMPXhG
else =p~k5k4
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 6D3hX>K4
rpowern = cat(2,rpowern{:}); LG3D3{H(.
end o;5 J=
j<|I@0
% Compute the values of the polynomials: {2"8^;
% -------------------------------------- &iR3]FNI
z = zeros(length_r,length_n); >dO1)
for j = 1:length_n T40&a(hXQ
s = 0:(n(j)-m(j))/2; U4;r.#qw,
pows = n(j):-2:m(j); :"QR;O@
for k = length(s):-1:1 M ,!Dhuas
p = (1-2*mod(s(k),2))* ... MiHa'90{K
prod(2:(n(j)-s(k)))/ ... D>tex/Of3
prod(2:s(k))/ ... }#%3y&7M7
prod(2:((n(j)-m(j))/2-s(k)))/ ... *-Y`7=^$
prod(2:((n(j)+m(j))/2-s(k))); q,S[[{("
idx = (pows(k)==rpowers); b7-M'-Km0_
z(:,j) = z(:,j) + p*rpowern(:,idx); 2OT
RP4U
end ?RW7TWf
v'3.`aZ!
if isnorm i/UDda"E
z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); Z*uv~0a>9Q
end ) 0NKL:u
end })#VO-J
8(d Hn
% EOF zernpol