如下程序，求关于matlab中的zernike函数的解释？

Question

function [A_nm,zmlist,cidx,V_nm] = zernike(img,n,m)
if nargin>0
if nargin==1
n = 0;
end
img=imread('A.png');
d = size(img);
img = double(img);
xstep = 2/(d(1)-1); %每一行的步长
ystep = 2/(d(2)-1);

[x,y] = meshgrid(-1:xstep:1,-1:ystep:1);% 设置X为从-1到1且变换步长为xstep
circle1 = x.^2 + y.^2;
inside = find(circle1<=1);% 存贮满足该条件的所有X,Y的值
mask = zeros(d);
mask(inside) = ones(size(inside));

[cimg,cidx] = clipimg(img,mask);
z = clipimg(x+i*y,mask);
p = 0.9*abs(z);
theta = angle(z);
c = 1;
for order=1:length(n)
n1 = n(order);
if nargin<3
m = zpossible(n1);
end
for r=1:length(m)
V_nmt = zpoly(n1,m(r),p,theta);
zprod = cimg.*conj(V_nmt);
A_nm(c) = (n1+1)*sum(sum(zprod))/pi;
zmlist(c,1:2) = [n1 m(r)];
if nargout==4
V_nm(:,c) = V_nmt;
end
c = c+1;
end
end
else
end

function [cimg,cindex,dim] = clipimg(img,mask)

dim = size(img);
cindex = find(mask~=0);
cimg = img(cindex);
return;

function [m] = zpossible(n)
if iseven(n)
m = 0:2:n;
else
m = 1:2:n;
end
return;

function [V_nm,mag,phase] = zpoly(n,m,p,theta)
R_nm = zeros(size(p));
a = (n+abs(m))/2;
b = (n-abs(m))/2;
total = b;
for s=0:total
num = ((-1)^s)*fac(n-s)*(p.^(n-2*s));
den = fac(s)*fac(a-s)*fac(b-s);
R_nm = R_nm + num/den;
end
mag = R_nm;
phase = m*theta;
V_nm = mag.*exp(i*phase);
return;

function [factorial] = fac(n)
maxno = max(max(n));
zerosi = find(n<=0);
n(zerosi) = ones(size(zerosi));
factorial = n;
findex = n;
for i=maxno:-1:2
cand = find(findex>2);
candidates = findex(cand);
findex(cand) = candidates-1;
factorial(cand) = factorial(cand).*findex(cand);
end
return;

function [verdict] = iseven(candy)
verdict = zeros(size(candy));
isint = find(isint(candy)==1);
divided2 = candy(isint)/2;
evens = (divided2==floor(divided2));
verdict(isint) = evens;
return;

function [verdict] = isint(candy)
verdict = double(round(candy))==candy;
return;

Answer 1

% [A_nm,zmlist,cidx,V_nm] = zernike(img,n,m) 计算Zernike矩% V_nm为n阶的Zernike多项式，定义为在极坐标系中p，theta的函数% cidx 表示虚部值% A_nm为zernike矩function [A_nm,zmlist,cidx,V_nm] =zernike(img,n,m)if nargin>0if nargin==1n=0;endd=size(img);img=double(img);xstep=2/(d(1)-1); %取步长ystep=2/(d(2)-1);[x,y]=meshgrid(-1:xstep:1,-1:ystep:1); %画方格circle1= x.^2 + y.^2;inside=find(circle1<=1); %提取符合circle1<=1的数mask=zeros(d); %构造size（d）*size(d)的矩阵。mask(inside)=ones(size(inside)); %构造size(inside)*size(inside)的全为1的矩阵赋值给mask（inside）[cimg,cidx]=clipimg(img,mask);% 计算图像的复数表示int i;z=clipimg(x+i*y,mask); % 计算Z的实部和虚部p=0.9*abs(z); %计算复数的模，sqrt(x,y),z=x+iy;theta=angle(z); % 计算复数z的辐角值（tanz）c=1;fororder=1:length(n)n1=n(order);% if input arguments less than3if nargin<3m=zpossible(n1);endfor r=1:length(m)V_nmt=zpoly(n1,m(r),p,theta); % V_nm为n阶的Zernike多项式，定义为在极坐标系中p，theta的函数zprod=cimg.*conj(V_nmt); % conj是求复数的共轭A_nm(c)=(n1+1)*sum(sum(zprod))/pi; % (n1+1)/π*∑∑(zprod); 对于图像而言求和代替了求积分zmlist(c,1:2)=[n1 m(r)];if nargout==4V_nm(:,c)=V_nmt;endc=c+1;endendelseend% 计算复数的实部和虚部function[cimg,cindex,dim]=clipimg(img,mask)dim=size(img);cindex=find(mask~=0);cimg=img(cindex);return;%判断n是偶数还是奇数，是偶数时，m取0,2,4,6等,否则取奇数赋值mfunction [m]=zpossible(n)ifiseven(n)m=0:2:n;elsem=1:2:n;endreturn;%计算Zernike矩多项式function[V_nm,mag,phase]=zpoly(n,m,p,theta)R_nm=zeros(size(p)); %产生size(p)*size(p)的零矩阵赋给R_nma=(n+abs(m))/2;b=(n-abs(m))/2;total=b;fors=0:totalnum=((-1)^s)*fac(n-s)*(p.^(n-2*s)); %(-1).-1*(n-s)!r.^(n-2*s)den=fac(s)*fac(a-s)*fac(b-s); %s!*(a-s)!*(b-s)!R_nm=R_nm + num/den; %R_nm是一个实数值的径向多项式endmag=R_nm; %赋值phase=m*theta;V_nm=mag.*exp(i*phase); %V_nm为n阶的Zernike多项式，定义为在极坐标系中p，theta的函数return;% 求n的阶乘function[factorial]=fac(n)maxno=max(max(n));zerosi=find(n<=0);%取n小于等于0的数n(zerosi)=ones(size(zerosi));factorial=n;findex=n;fori=maxno:-1:2cand=find(findex>2);candidates=findex(cand);findex(cand)=candidates-1;factorial(cand)=factorial(cand).*findex(cand);endreturn;function[verdict]=iseven(candy)verdict=zeros(size(candy));isint=find(isint(candy)==1);divided2=candy(isint)/2;evens=(divided2==floor(divided2));verdict(isint)=evens;return;function[verdict]=isint(candy)verdict =double(round(candy))==candy;return;