根据论文“Analysis of the first variation and a numerical gradient flow for integral Menger curvature”中的计算 integral Menger curvature的公式编写计算integral Menger curvature的matlab代码

根据论文中的公式，计算integral Menger curvature的matlab代码如下：

function [IMC, grad] = integral_menger_curvature(X, T, lambda)
% X: 点云的坐标矩阵，每一行表示一个点的坐标
% T: 三角形面片的顶点索引矩阵，每一行表示一个三角形面片的三个顶点在X中的行索引
% lambda: 正则化参数，取值范围为[0,1]
% IMC: integral Menger curvature的值
% grad: IMC关于点云坐标的梯度

n = size(X, 1); % 点的个数
m = size(T, 1); % 面片的个数

% 计算每个面片的平面法向量
N = zeros(m, 3);
for i = 1:m
    V = X(T(i, :), :);
    N(i, :) = cross(V(2, :) - V(1, :), V(3, :) - V(1, :));
    N(i, :) = N(i, :) / norm(N(i, :));
end

% 计算每个点的邻域面片的集合
P = cell(n, 1);
for i = 1:m
    for j = 1:3
        P{T(i, j)} = [P{T(i, j)}, i];
    end
end

% 计算integral Menger curvature
IMC = 0;
grad = zeros(n, 3);
for i = 1:n
    Pi = P{i};
    Ni = length(Pi);
    if Ni >= 4
        IMC_i = 0;
        grad_i = zeros(1, 3);
        for j = 1:Ni-2
            for k = j+1:Ni-1
                for l = k+1:Ni
                    I_jkl = intersect(P{Pi(j)}, intersect(P{Pi(k)}, P{Pi(l)}));
                    if ~isempty(I_jkl)
                        X_jkl = X(T(I_jkl, :), :);
                        N_jkl = cross(X_jkl(2, :) - X_jkl(1, :), X_jkl(3, :) - X_jkl(1, :));
                        N_jkl = N_jkl / norm(N_jkl);
                        IMC_i = IMC_i + dot(N(Pi(j), :), cross(N(Pi(k), :), N(Pi(l), :))) / dot(N_jkl, cross(N(Pi(j), :), N(Pi(k), :)));
                        grad_jkl = -dot(cross(N(Pi(k), :), N(Pi(l), :)), N_jkl) / dot(N_jkl, cross(N(Pi(j), :), N(Pi(k), :)))^2;
                        grad_i = grad_i + grad_jkl * (cross(N(Pi(j), :), N(Pi(k), :)) + cross(N(Pi(j), :), N(Pi(l), :)) + cross(N(Pi(k), :), N(Pi(l), :)));
                    end
                end
            end
        end
        IMC_i = IMC_i / nchoosek(Ni, 3);
        grad_i = grad_i / nchoosek(Ni, 3);
        IMC = IMC + IMC_i;
        grad(i, :) = grad_i;
    end
end
IMC = IMC / m;

% 添加正则化项
IMC = IMC + lambda * sum(sum((X - mean(X)).^2));
grad = grad + 2 * lambda * (X - mean(X));
end

其中，IMC和grad分别为计算得到的integral Menger curvature的值和关于点云坐标的梯度。lambda为正则化参数，用于平衡IMC和点云的平均值之间的差异。