% =========================================================================
% 3D Zernike 函数计算
% =========================================================================
% By Gu Jinjin 2013/09/01 - 08 程序为原创，转载请声明
%------ 参数解释 ------
%{
Input args:
n 表示3D Zernike矩的具体阶
l 表示重复度（球函数的阶）, n-l 为偶数，且 0 =< l <= n
m （球函数的重复度）, -l =< m <= l
X 单坐标列向量，表示单位球（US）中x的值
Y，Z 同上
Output args:
Z_nlm n阶3D Zernike 多项式的值
%}
%======================================
% 主函数
%======================================
%
function [Z_nlm] = zernikeFunction(n,l,m,X,Y,Z)
if nargin == 6
% 思想来源于文章 Precipitate shape fitting and reconstruction by means of 3D Zernike functions
v_up = (n-l)/2;
u_up = fix((l-m)/2);
v_sum = 0;
for v=0:v_up
q_nlv = calculateQ_klv(v_up,l,v);
u_sum = 0;
for u = 0:u_up
u_p1 = (-1)^(m+u)/2^(m+2*u)*combin(l,u)*combin(l-u,m+u);
a_sum = 0;
for a = 0:m
k_sum = 0;
for k=0:v
b_sum = 0;
for b = 0:k+u
b_sum = b_sum + combin(k+u,b).*X.^(2*b+a).*Y.^(2*(k+u-b)+m-a).*Z.^(2*(v-k-u)+l-m);
end
k_sum = k_sum + combin(v,k).*b_sum;
end
a_sum = a_sum + i^(m-a)*combin(m,a).*k_sum;
end
u_sum = u_sum+u_p1.*a_sum;
end
v_sum = v_sum + q_nlv.*u_sum;
end
Z_nlm = calculateC_lm(l,m).*v_sum;
% 思想来源于文章 3DZernike Descriptor for Content Based Shape Retrieval
% 6重求和计算，但重构有问题，因为论文公式表示 存在印刷错误
%{
k = (n-l)/2;
mu_up = fix((l-m)/2);
V_sum = 0;
for V=0:k
q_klV = calculateQ_klv(k,l,V);
a_sum = 0;
for a=0:V
b_sum = 0;
for b=0:V-a
u_sum = 0;
for u=0:m
mu_sum = 0;
for mu=0:mu_up
v_sum = 0;
for v=0:mu
v_sum = v_sum+combin(mu,v).*X.^(2*(v+a)+u).*Y.^(2*(mu-v+b)+m-u).*Z.^(2*(V-a-b-mu)+l-m);
end
mu_sum = mu_sum+(-1)^mu*2^(-2*mu)*combin(l,mu)*combin(l-mu,m+mu).*v_sum;
end
u_sum = u_sum+(-1)^(m-u)*combin(m,u)*i^u.*mu_sum;
end
b_sum = b_sum+combin(V-a,b).*u_sum;
end
a_sum = a_sum+combin(V,a).*b_sum;
end
V_sum = V_sum+q_klV.*a_sum;
end
Z_nlm = 2^(-m)*calculateC_lm(l,m).*V_sum;
%}
else
disp 'Wrong Input Args: No Img Info!'
end
%======================================
% 计算 C_lm 的值
%======================================
function [c_lm] = calculateC_lm(l,m)
%den = 2*sqrt(pi)*fac(l); % 分母
den = fac(l);
t = (2*l+1).*fac(l+m).*fac(l-m);
num = sqrt(t); % 分子
c_lm = num./den;
return;
%======================================
% 计算 Q_klv 的值
%======================================
function [q_klv] = calculateQ_klv(k,l,v)
p1 = (-1)^k/(2^(2*k));
p2 = sqrt((2*l+4*k+3)/3);
p3 = combin(2*k,k)*(-1)^v;
p4 = combin(k,v)*combin(2*(k+l+v)+1,2*k)/combin(k+l+v,k);
q_klv = p1*p2*p3*p4;
return;
%======================================
% 计算组合数 - Combinatorial
%======================================
% fac(n)/((fac(n-m)*fac(m))
function [combin] = combin(n,m)
num = fac(n);
den = fac(n-m).*fac(m);
combin = num./den;
return;
%======================================
% 求解n! - factorial
%======================================
% n可以为数，向量或者矩阵
function [factorial] = fac(n)
max_n = max(max(n));
zeros_n = find(n<=0); % 找到n中小于或者等于0的元素位置
n(zeros_n) = ones(size(zeros_n));
factorial = n;
n_1 = n;
% 保证n-1次乘法，1乘以任何数皆为1，所以不需要计算
for k= max_n:-1:2
cand = find(n_1>2); % 这里 1,2的阶乘皆为其本身，所以不需要计算
n_1(cand) = n_1(cand) - 1;
factorial(cand) = factorial(cand).*n_1(cand);
end
return;
