levenberg-marquardt algorithm

B'levenberg-Marquardt算法是一种非线性最小二乘优化算法，适用于求解复杂的非线性问题，特别是在数据噪声较大的情况下。该算法利用了Levenberg（1944年）和Marquardt（1963年）的思想，通过在牛顿法和梯度下降法之间进行平衡来快速优化相关参数。该算法具有收敛速度快、精度高、参数调整灵活等优点，在很多领域得到广泛应用，如信号处理、图像处理、机器学习、计算机视觉等。

相关问题

Levenberg-Marquardt pytorch

Levenberg-Marquardt (LM) is a widely used optimization algorithm for nonlinear least squares problems. In PyTorch, LM can be implemented using the `torch.autograd.functional` module. Here's an example of how to use LM to fit a curve to data:

import torch
from torch.autograd.functional import jacobian

# Define the model
def model(x, params):
    a, b, c = params
    return a * torch.sin(b * x) + c

# Define the objective function
def objective(params):
    y_pred = model(x, params)
    residuals = y - y_pred
    return torch.sum(residuals ** 2)

# Generate some data
x = torch.linspace(-1, 1, 100)
y = 2 * torch.sin(3 * x) + 0.5 * torch.randn(100)

# Initialize the parameters
params = torch.tensor([1.0, 1.0, 1.0], requires_grad=True)

# Set up the LM optimizer
optimizer = torch.optim.LBFGS([params])

# Train the model
for i in range(100):
    def closure():
        optimizer.zero_grad()
        loss = objective(params)
        loss.backward()
        return loss
    optimizer.step(closure)

# Print the final parameters
print(params)

In this example, we define a model that takes an input `x` and a set of parameters `params`, and returns a prediction `y_pred`. We then define an objective function that measures the difference between the predicted values and the actual values (`y` in this case). We generate some random data, initialize the parameters, and set up the LM optimizer. Finally, we train the model by minimizing the objective function using the LM optimizer. The final parameters are printed out.

Levenberg-Marquardt算法对相机标定结果的优化代码（matlab）

以下是在Matlab中使用LM算法进行相机标定的示例代码：

% 导入标定所需的数据
load('calibration_data.mat'); % calibration_data包含3D物体点和对应的2D图像点

% 定义初始内参矩阵和畸变系数
K = eye(3); % 内参矩阵
distCoeffs = zeros(5, 1); % 畸变系数

% 定义LM算法的参数
options = optimoptions('lsqnonlin', 'Algorithm', 'levenberg-marquardt', 'Display', 'iter');

% 定义优化函数
fun = @(params) calibrate(params, calibration_data);

% 使用LM算法进行内参优化
[params, ~, ~, ~, ~] = lsqnonlin(fun, [K(:); distCoeffs(:)], [], [], options);

% 将优化后的参数赋值给内参矩阵和畸变系数
K_optimized = reshape(params(1:9), 3, 3);
distCoeffs_optimized = params(10:end);

% 定义标定函数
function [residuals, J] = calibrate(params, calibration_data)
    % 将参数分解为内参矩阵和畸变系数
    K = reshape(params(1:9), 3, 3);
    distCoeffs = params(10:end);

    % 计算每个物体点的重投影误差
    nPoints = size(calibration_data.objectPoints, 1);
    residuals = zeros(2 * nPoints, 1);
    for i = 1:nPoints
        [projectedPoints, ~] = projectPoints(calibration_data.objectPoints(i, :), calibration_data.rvecs(i, :), calibration_data.tvecs(i, :), K, distCoeffs);
        residuals(2 * i - 1:2 * i) = calibration_data.imagePoints(i, :) - projectedPoints;
    end

    % 计算雅可比矩阵
    if nargout > 1
        J = zeros(2 * nPoints, 9 + size(distCoeffs, 1));
        for i = 1:nPoints
            [projectedPoints, J_i] = projectPoints(calibration_data.objectPoints(i, :), calibration_data.rvecs(i, :), calibration_data.tvecs(i, :), K, distCoeffs);
            J(2 * i - 1:2 * i, :) = [J_i(:, 1:3), J_i(:, 4:6) * skewSymmetric(projectedPoints), J_i(:, 7:9), J_i(:, 10:end)];
        end
    end
end

% 定义点的投影函数
function [projectedPoints, J] = projectPoints(objectPoints, rvec, tvec, K, distCoeffs)
    % 将旋转向量转换为旋转矩阵
    R = rodrigues(rvec);

    % 计算物体点在相机坐标系下的坐标
    cameraPoints = R * objectPoints' + tvec';

    % 计算投影点
    projectedPoints = (K * cameraPoints)';

    % 计算畸变系数
    k1 = distCoeffs(1);
    k2 = distCoeffs(2);
    p1 = distCoeffs(3);
    p2 = distCoeffs(4);
    k3 = distCoeffs(5);
    r2 = sum(projectedPoints .^ 2, 2);

    % 计算畸变后的坐标
    projectedPoints = projectedPoints .* (1 + k1 * r2 + k2 * r2 .^ 2 + k3 * r2 .^ 3) + [2 * p1 * projectedPoints(:, 1) .* projectedPoints(:, 2) + p2 * (r2 + 2 * projectedPoints(:, 1) .^ 2), p1 * (r2 + 2 * projectedPoints(:, 2) .^ 2) + 2 * p2 * projectedPoints(:, 1) .* projectedPoints(:, 2)];

    % 计算投影点的导数
    if nargout > 1
        x = projectedPoints(:, 1);
        y = projectedPoints(:, 2);
        xy = x .* y;
        r2 = sum(projectedPoints .^ 2, 2);
        r4 = r2 .^ 2;
        r6 = r2 .^ 3;

        J = zeros(size(projectedPoints, 1), 10);
        J(:, 1:2) = projectedPoints;
        J(:, 3) = 1;
        J(:, 4:5) = [2 * k1 * xy + k2 * (r2 + 2 * x .^ 2), k1 * (r2 + 2 * y .^ 2) + 2 * k2 * xy];
        J(:, 6:7) = [k2 * xy .* (7 * r2 + 4 * x .^ 2), k2 * xy .* (7 * r2 + 4 * y .^ 2)];
        J(:, 8) = k3 * xy .* r4;
        J(:, 9) = 2 * p1 * y + 6 * p2 * x;
        J(:, 10) = 2 * p2 * y + 6 * p1 * x;
    end
end

% 定义罗德里格斯旋转函数
function R = rodrigues(r)
    theta = norm(r);
    if theta < eps
        R = eye(3);
    else
        r = r / theta;
        R = cos(theta) * eye(3) + (1 - cos(theta)) * (r' * r) + sin(theta) * skewSymmetric(r);
    end
end

% 定义叉积矩阵函数
function M = skewSymmetric(v)
    M = [0, -v(3), v(2); v(3), 0, -v(1); -v(2), v(1), 0];
end

其中，`calibration_data`包含了标定所需的数据，包括3D物体点和对应的2D图像点，以及每幅图像对应的旋转向量和平移向量。`calibrate`函数为优化函数，它通过计算物体点的重投影误差，并返回误差向量和雅可比矩阵。`projectPoints`函数为点的投影函数，它将物体点从世界坐标系变换到相机坐标系，并计算出投影点和对应的导数。`rodrigues`函数为罗德里格斯旋转函数，它将旋转向量转换为旋转矩阵。`skewSymmetric`函数为叉积矩阵函数，它将向量转换为叉积矩阵。

最后，使用`lsqnonlin`函数进行LM算法的优化，得到优化后的内参矩阵和畸变系数。