在 OpenCV C++ 中估计基本矩阵之前标准化对应点的正确方法是什么？

Question

我正在尝试手动实现一个基本矩阵估计函数用于对应点（基于两个图像之间的相似性）。 ORB特征检测、提取、匹配、比值检验后得到对应点。

有很多关于此主题的良好来源的文献。然而，其中 none 似乎提供了执行此过程的良好伪代码。我浏览了多视图几何书的各个章节；以及许多在线资源。

这个 source 似乎给出了一个进行标准化的公式，我遵循了该来源第 13 页上提到的公式。

基于这个公式，我创建了以下算法。不过，我不确定我的做法是否正确！

Normalization.hpp

class Normalization {
    typedef std::vector <cv::Point2f> intercepts;
    typedef std::vector<cv::Mat> matVec;
 public:
    Normalization () {}
    ~Normalization () {}

    void makeAverage(intercepts pointsVec);

    std::tuple <cv::Mat, cv::Mat> normalize(intercepts pointsVec);

    matVec getNormalizedPoints(intercepts pointsVec);

 private:
    double xAvg = 0;
    double yAvg = 0;
    double count = 0;
    matVec normalizedPts;
    double distance = 0;
    matVec matVecData;
    cv::Mat forwardTransform;
    cv::Mat reverseTransform;
};

Normalization.cpp

#include "Normalization.hpp"

typedef std::vector <cv::Point2f> intercepts;
typedef std::vector<cv::Mat> matVec;

/*******
*@brief  : The makeAverage function receives the input 2D coordinates (x, y)
*          and creates the average of x and y
*@params : The input parameter is a set of all matches (x, y pairs) in image A
************/
void Normalization::makeAverage(intercepts pointsVec) {
    count = pointsVec.size();
    for (auto& member : pointsVec) {
        xAvg = xAvg + member.x;
        yAvg = yAvg + member.y;
    }
    xAvg = xAvg / count;
    yAvg = yAvg / count;
}

/*******
*@brief  : The normalize function accesses the average distance calculated
*          in the previous step and calculates the forward and inverse transformation
*          matrices
*@params : The input to this function is a vector of corresponding points in given image
*@return : The returned data is a tuple of forward and inverse transformation matrices
*************/
std::tuple <cv::Mat, cv::Mat> Normalization::normalize(intercepts pointsVec) {
    for (auto& member : pointsVec) {
        //  Accumulate the distance for every point

        distance += ((1 / (count * std::sqrt(2))) *\
                     (std::sqrt(std::pow((member.x - xAvg), 2)\
                                + std::pow((member.y - yAvg), 2))));
    }
    forwardTransform = (cv::Mat_<double>(3, 3) << (1 / distance), \
                        0, -(xAvg / distance), 0, (1 / distance), \
                        -(yAvg / distance), 0, 0, 1);

    reverseTransform = (cv::Mat_<double>(3, 3) << distance, 0, xAvg, \
                        0, distance, yAvg, 0, 0, 1);

    return std::make_tuple(forwardTransform, reverseTransform);
}

/*******
*@brief  : The getNormalizedPoints function trannsforms the raw image coordinates into
*          transformed coordinates using the forwardTransform matrix estimated in previous step
*@params : The input to this function is a vector of corresponding points in given image
*@return : The returned data is vector of transformed coordinates
*************/
matVec Normalization::getNormalizedPoints(intercepts pointsVec) {
    cv::Mat triplet;
    for (auto& member : pointsVec) {
        triplet = (cv::Mat_<double>(3, 1) << member.x, member.y, 1);
        matVecData.emplace_back(forwardTransform * triplet);
    }
    return matVecData;
}

这是正确的方法吗？还有其他规范化方法吗？

Answer 1

我认为您可以按照自己的方式进行，但在 "Multiple View Geometry in Computer Vision" Hartley 和 Zisserman 中建议各向同性缩放（第 107 页）：

Isotropic scaling. As a first step of normalization, the coordinates in each image are translated (by a different translation for each image) so as to bring the centroid of the set of all points to the origin. The coordinates are also scaled so that on the average a point x is of the form x = (x, y,w)T, with each of x, y and w having the same average magnitude. Rather than choose different scale factors for each coordinate direction, an isotropic scaling factor is chosen so that the x and y-coordinates of a point are scaled equally. To this end, we choose to scale the coordinates so that the average distance of a point x from the origin is equal to √ 2. This means that the “average” point is equal to (1, 1, 1)T. In summary the transformation is as follows:
(i) The points are translated so that their centroid is at the origin.
(ii) The points are then scaled so that the average distance from the origin is equal to √2.
(iii) This transformation is applied to each of the two images independently.

他们说这对于直接线性变换 (DLT) 很重要，但对于像您想要的那样计算基本矩阵更重要。您选择的算法将点坐标归一化为 (1, 1, 1) 但未应用缩放，以便与原点的平均距离等于 √2。

下面是这种规范化的一些代码。平均步骤保持不变：

std::vector<cv::Mat> normalize(std::vector<cv::Point2d> pointsVec) {
    // Averaging
    double count = (double) pointsVec.size();
    double xAvg = 0;
    double yAvg = 0;
    for (auto& member : pointsVec) {
        xAvg = xAvg + member.x;
        yAvg = yAvg + member.y;
    }
    xAvg = xAvg / count;
    yAvg = yAvg / count;

    // Normalization
    std::vector<cv::Mat> points3d;
    std::vector<double> distances;
    for (auto& member : pointsVec) {

        double distance = (std::sqrt(std::pow((member.x - xAvg), 2) + std::pow((member.y - yAvg), 2)));
        distances.push_back(distance);
    }
    double xy_norm = std::accumulate(distances.begin(), distances.end(), 0.0) / distances.size();

    // Create a matrix transforming the points into having mean (0,0) and mean distance to the center equal to sqrt(2)
    cv::Mat_<double> Normalization_matrix(3, 3); 
    double diagonal_element = sqrt(2) / xy_norm;
    double element_13 = -sqrt(2) * xAvg / xy_norm;
    double element_23 = -sqrt(2)* yAvg/ xy_norm;

    Normalization_matrix << diagonal_element, 0, element_13,
        0, diagonal_element, element_23,
        0, 0, 1;

    // Multiply the original points with the normalization matrix
    for (auto& member : pointsVec) {
        cv::Mat triplet = (cv::Mat_<double>(3, 1) << member.x, member.y, 1);
        points3d.emplace_back(Normalization_matrix * triplet);
    }
    return points3d;
}

在 OpenCV C++ 中估计基本矩阵之前标准化对应点的正确方法是什么？

What is the correct way to Normalize corresponding points before estimation of Fundamental matrix in OpenCV C++?

c++

c++11

opencv3.0

normalize