js实现高斯-约旦消元法求解Homography矩阵

最小二乘法来求解矩阵

// 计算Homography矩阵
function calculateHomography(srcPoints, dstPoints) {
    if (srcPoints.length !== dstPoints.length || srcPoints.length < 4) {
        throw new Error('需要至少四个点进行计算');
    }

    // 设置矩阵方程 Ax = b
    let A = [];
    let b = [];

    // 构建方程的A矩阵和b向量
    for (let i = 0; i < srcPoints.length; i++) {
        let x1 = srcPoints[i][0];
        let y1 = srcPoints[i][1];
        let x2 = dstPoints[i][0];
        let y2 = dstPoints[i][1];

        // 方程：x2 = H11 * x1 + H12 * y1 + H13
        // 方程：y2 = H21 * x1 + H22 * y1 + H23
        A.push([-x1, -y1, -1, 0, 0, 0, x1 * x2, y1 * x2]);
        A.push([0, 0, 0, -x1, -y1, -1, x1 * y2, y1 * y2]);

        b.push(x2);
        b.push(y2);
    }

    // 使用高斯-约旦消元法求解A * h = b
    let H = gaussJordan(A, b);

    // 构造Homography矩阵
    let homographyMatrix = [
        [H[0], H[1], H[2]],
        [H[3], H[4], H[5]],
        [H[6], H[7], 1]
    ];

    return homographyMatrix;
}

// 高斯-约旦消元法解线性方程组
function gaussJordan(A, b) {
    let n = A.length;
    let augmentedMatrix = A.map((row, i) => row.concat(b[i]));  // A 与 b 合并为增广矩阵

    for (let i = 0; i < n; i++) {
        // 找到最大值所在的行进行行交换
        let maxRow = i;
        for (let j = i + 1; j < n; j++) {
            if (Math.abs(augmentedMatrix[j][i]) > Math.abs(augmentedMatrix[maxRow][i])) {
                maxRow = j;
            }
        }
        [augmentedMatrix[i], augmentedMatrix[maxRow]] = [augmentedMatrix[maxRow], augmentedMatrix[i]];

        // 将主对角线上的元素变为1
        let scale = augmentedMatrix[i][i];
        for (let j = i; j < n + 1; j++) {
            augmentedMatrix[i][j] /= scale;
        }

        // 通过行操作消去其他列的元素
        for (let j = 0; j < n; j++) {
            if (j !== i) {
                scale = augmentedMatrix[j][i];
                for (let k = i; k < n + 1; k++) {
                    augmentedMatrix[j][k] -= augmentedMatrix[i][k] * scale;
                }
            }
        }
    }

    // 提取解向量（即矩阵的最后一列）
    let solution = augmentedMatrix.map(row => row[n]);
    return solution;
}

// 示例：输入源点和目标点
let srcPoints = [
    [7.606727752, 13.23349844],
    [5.727826596, 13.23044123],
    [4.788626754, 13.23106254],
    [2.906026488, 13.23060659]
];

let dstPoints = [
    [7.623758876, 13.24447489],
    [5.744538875, 13.24339615],
    [4.804678876, 13.24468678],
    [2.923728876, 13.24552804]
];

// 计算Homography矩阵
let homographyMatrix = calculateHomography(srcPoints, dstPoints);
console.log("Homography Matrix:");
console.log(homographyMatrix);