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>0if nargin==1n = 0;endimg=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xstepcircle1 = 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<3m = zpossible(n1);endfor 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==4V_nm(:,c) = V_nmt;endc = c+1;endendelseendfunction [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;elsem = 1:2:n;endreturn;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:totalnum = ((-1)^s)*fac(n-s)*(p.^(n-2*s));den = fac(s)*fac(a-s)*fac(b-s);R_nm = R_nm + num/den;endmag = 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:-1:2cand = 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;

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>0
if nargin==1

n=0;
end
d=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;
for
order=1:length(n)
n1=n(order);
% if input arguments less than
3
if nargin<3
m=zpossible(n1);

end
for 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==4

V_nm(:,c)=V_nmt;
end

c=c+1;
end
end
else
end

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