1_5 optical

采用特征点法做VO存在耗时较大的问题，一般耗时情况：如下

(1) 在图像中提取特征点并计算特征描述，非常耗时 ~10+ms ORB，shift耗时更多；
(2) 在不同图像中寻找特征匹配，非常耗时 O(n^2) 暴力匹配
(3) 利用匹配点信息计算相机位姿，比较快速<1ms

因此，为了速度还有不使用特征匹配计算VO的方法，主要有两大类，分别是：

(1) 通过其他方式寻找配对点; 光流法
稀疏光流法：以Lucas-Kanade(LK)光流为代表
稠密光流法：以Horn-Schunk(HS)光流为代表
本质上是估计像素在不同时刻图像中的运动。
光流法推导过程依赖灰度不变假设： $I(x_{1},y_{1},t_{1})=I(x_{2},y_{2},t_{2})=I(x_{3},y_{3},t_{3})$
t时刻位于x,y处像素点的灰度值为 $I(x,y,t)$
$t+dt$ 时刻位于x+dx,y+dy处的像素点灰度值为 $I(x+dx,y+dy,t+dt)$
对t+dt时刻进行泰勒一级展开如下：
$I(x+dx,y+dy,t+dt)\approx I(x,y,t)+\frac{\partial I}{\partial x}dx+\frac{\partial I}{\partial y}dy+\frac{\partial I}{\partial t}dt$ ，由于灰度值两个时刻相等，所以可得：
$\frac{\partial I}{\partial x}\frac{dx}{dt}+\frac{\partial I}{\partial y}\frac{dy}{dt}=-\frac{\partial I}{\partial t}$ ，所以这里是希望求解dx/dt和dy/dt。

注意：灰度不变是一种理想假设，对于高光、阴影等情况很可能不成立。

由于上述的光流公式是一个二元一次方程，欠定的，所以需要引入额外的约束。假定一个窗口(w*w)内光度不变，则可以构建矩阵：

$\begin{bmatrix} I_{x} & I_{y} \end{bmatrix}_{k}\begin{bmatrix} u\\v \end{bmatrix}=-I_{tk}, k=1,2,...w^{2}$

通过超定最小二乘求解运动u，v

$A=\begin{bmatrix} \begin{bmatrix} I_{x} &I_{y} \end{bmatrix}_{1}\\... \\ \begin{bmatrix} I_{x} &I_{y} \end{bmatrix}_{k} \end{bmatrix}$ , $b=\begin{bmatrix} I_{t1}\\ ... \\ I_{tk} \end{bmatrix}$

$\begin{bmatrix} u\\ v \end{bmatrix}^*=-(A^{T}A)^{-1}A^{T}b$

具体代码如下：

//
// Created by Xiang on 2017/12/19.
//

#include <opencv2/opencv.hpp>
#include <string>
#include <chrono>
#include <Eigen/Core>
#include <Eigen/Dense>

using namespace std;
using namespace cv;

string file_1 = "./LK1.png";  // first image
string file_2 = "./LK2.png";  // second image

/// Optical flow tracker and interface
class OpticalFlowTracker {
public:
    OpticalFlowTracker(
        const Mat &img1_,
        const Mat &img2_,
        const vector<KeyPoint> &kp1_,
        vector<KeyPoint> &kp2_,
        vector<bool> &success_,
        bool inverse_ = true, bool has_initial_ = false) :
        img1(img1_), img2(img2_), kp1(kp1_), kp2(kp2_), success(success_), inverse(inverse_),
        has_initial(has_initial_) {}

    void calculateOpticalFlow(const Range &range);

private:
    const Mat &img1;
    const Mat &img2;
    const vector<KeyPoint> &kp1;
    vector<KeyPoint> &kp2;
    vector<bool> &success;
    bool inverse = true;
    bool has_initial = false;
};

/**
 * single level optical flow
 * @param [in] img1 the first image
 * @param [in] img2 the second image
 * @param [in] kp1 keypoints in img1
 * @param [in|out] kp2 keypoints in img2, if empty, use initial guess in kp1
 * @param [out] success true if a keypoint is tracked successfully
 * @param [in] inverse use inverse formulation?
 */
void OpticalFlowSingleLevel(
    const Mat &img1,
    const Mat &img2,
    const vector<KeyPoint> &kp1,
    vector<KeyPoint> &kp2,
    vector<bool> &success,
    bool inverse = false,
    bool has_initial_guess = false
);

/**
 * multi level optical flow, scale of pyramid is set to 2 by default
 * the image pyramid will be create inside the function
 * @param [in] img1 the first pyramid
 * @param [in] img2 the second pyramid
 * @param [in] kp1 keypoints in img1
 * @param [out] kp2 keypoints in img2
 * @param [out] success true if a keypoint is tracked successfully
 * @param [in] inverse set true to enable inverse formulation
 */
void OpticalFlowMultiLevel(
    const Mat &img1,
    const Mat &img2,
    const vector<KeyPoint> &kp1,
    vector<KeyPoint> &kp2,
    vector<bool> &success,
    bool inverse = false
);

/**
 * get a gray scale value from reference image (bi-linear interpolated)
 * @param img
 * @param x
 * @param y
 * @return the interpolated value of this pixel
 */

inline float GetPixelValue(const cv::Mat &img, float x, float y) {
    // boundary check
    if (x < 0) x = 0;
    if (y < 0) y = 0;
    if (x >= img.cols - 1) x = img.cols - 2;
    if (y >= img.rows - 1) y = img.rows - 2;
    
    float xx = x - floor(x);
    float yy = y - floor(y);
    int x_a1 = std::min(img.cols - 1, int(x) + 1);
    int y_a1 = std::min(img.rows - 1, int(y) + 1);
    
    return (1 - xx) * (1 - yy) * img.at<uchar>(y, x)
    + xx * (1 - yy) * img.at<uchar>(y, x_a1)
    + (1 - xx) * yy * img.at<uchar>(y_a1, x)
    + xx * yy * img.at<uchar>(y_a1, x_a1);
}

int main(int argc, char **argv) {

    // images, note they are CV_8UC1, not CV_8UC3
    Mat img1 = imread(file_1, 0);
    Mat img2 = imread(file_2, 0);

    // key points, using GFTT here.
    vector<KeyPoint> kp1;
    Ptr<GFTTDetector> detector = GFTTDetector::create(500, 0.01, 20); // maximum 500 keypoints
    detector->detect(img1, kp1);

    // now lets track these key points in the second image
    // first use single level LK in the validation picture
    vector<KeyPoint> kp2_single;
    vector<bool> success_single;
    OpticalFlowSingleLevel(img1, img2, kp1, kp2_single, success_single);

    // then test multi-level LK
    vector<KeyPoint> kp2_multi;
    vector<bool> success_multi;
    chrono::steady_clock::time_point t1 = chrono::steady_clock::now();
    OpticalFlowMultiLevel(img1, img2, kp1, kp2_multi, success_multi, true);
    chrono::steady_clock::time_point t2 = chrono::steady_clock::now();
    auto time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
    cout << "optical flow by gauss-newton: " << time_used.count() << endl;

    // use opencv's flow for validation
    vector<Point2f> pt1, pt2;
    for (auto &kp: kp1) pt1.push_back(kp.pt);
    vector<uchar> status;
    vector<float> error;
    t1 = chrono::steady_clock::now();
    cv::calcOpticalFlowPyrLK(img1, img2, pt1, pt2, status, error);
    t2 = chrono::steady_clock::now();
    time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
    cout << "optical flow by opencv: " << time_used.count() << endl;

    // plot the differences of those functions
    Mat img2_single;
    cv::cvtColor(img2, img2_single, CV_GRAY2BGR);
    for (int i = 0; i < kp2_single.size(); i++) {
        if (success_single[i]) {
            cv::circle(img2_single, kp2_single[i].pt, 2, cv::Scalar(0, 250, 0), 2);
            cv::line(img2_single, kp1[i].pt, kp2_single[i].pt, cv::Scalar(0, 250, 0));
        }
    }

    Mat img2_multi;
    cv::cvtColor(img2, img2_multi, CV_GRAY2BGR);
    for (int i = 0; i < kp2_multi.size(); i++) {
        if (success_multi[i]) {
            cv::circle(img2_multi, kp2_multi[i].pt, 2, cv::Scalar(0, 250, 0), 2);
            cv::line(img2_multi, kp1[i].pt, kp2_multi[i].pt, cv::Scalar(0, 250, 0));
        }
    }

    Mat img2_CV;
    cv::cvtColor(img2, img2_CV, CV_GRAY2BGR);
    for (int i = 0; i < pt2.size(); i++) {
        if (status[i]) {
            cv::circle(img2_CV, pt2[i], 2, cv::Scalar(0, 250, 0), 2);
            cv::line(img2_CV, pt1[i], pt2[i], cv::Scalar(0, 250, 0));
        }
    }

    cv::imshow("tracked single level", img2_single);
    cv::imshow("tracked multi level", img2_multi);
    cv::imshow("tracked by opencv", img2_CV);
    cv::waitKey(0);

    return 0;
}

void OpticalFlowSingleLevel(
    const Mat &img1,
    const Mat &img2,
    const vector<KeyPoint> &kp1,
    vector<KeyPoint> &kp2,
    vector<bool> &success,
    bool inverse, bool has_initial) {
    kp2.resize(kp1.size());
    success.resize(kp1.size());
    OpticalFlowTracker tracker(img1, img2, kp1, kp2, success, inverse, has_initial);
    parallel_for_(Range(0, kp1.size()),
                  std::bind(&OpticalFlowTracker::calculateOpticalFlow, &tracker, placeholders::_1));
}

void OpticalFlowTracker::calculateOpticalFlow(const Range &range) {
    // parameters
    int half_patch_size = 4;
    int iterations = 10;
    for (size_t i = range.start; i < range.end; i++) {
        auto kp = kp1[i];
        double dx = 0, dy = 0; // dx,dy need to be estimated
        if (has_initial) {
            dx = kp2[i].pt.x - kp.pt.x;
            dy = kp2[i].pt.y - kp.pt.y;
        }

        double cost = 0, lastCost = 0;
        bool succ = true; // indicate if this point succeeded

        // Gauss-Newton iterations
        Eigen::Matrix2d H = Eigen::Matrix2d::Zero();    // hessian
        Eigen::Vector2d b = Eigen::Vector2d::Zero();    // bias
        Eigen::Vector2d J;  // jacobian
        for (int iter = 0; iter < iterations; iter++) {
            if (inverse == false) {
                H = Eigen::Matrix2d::Zero();
                b = Eigen::Vector2d::Zero();
            } else {
                // only reset b
                b = Eigen::Vector2d::Zero();
            }

            cost = 0;

            // compute cost and jacobian
            for (int x = -half_patch_size; x < half_patch_size; x++)
                for (int y = -half_patch_size; y < half_patch_size; y++) {
                    double error = GetPixelValue(img1, kp.pt.x + x, kp.pt.y + y) -
                                   GetPixelValue(img2, kp.pt.x + x + dx, kp.pt.y + y + dy);;  // Jacobian
                    if (inverse == false) {
                        J = -1.0 * Eigen::Vector2d(
                            0.5 * (GetPixelValue(img2, kp.pt.x + dx + x + 1, kp.pt.y + dy + y) -
                                   GetPixelValue(img2, kp.pt.x + dx + x - 1, kp.pt.y + dy + y)),
                            0.5 * (GetPixelValue(img2, kp.pt.x + dx + x, kp.pt.y + dy + y + 1) -
                                   GetPixelValue(img2, kp.pt.x + dx + x, kp.pt.y + dy + y - 1))
                        );
                    } else if (iter == 0) {
                        // in inverse mode, J keeps same for all iterations
                        // NOTE this J does not change when dx, dy is updated, so we can store it and only compute error
                        J = -1.0 * Eigen::Vector2d(
                            0.5 * (GetPixelValue(img1, kp.pt.x + x + 1, kp.pt.y + y) -
                                   GetPixelValue(img1, kp.pt.x + x - 1, kp.pt.y + y)),
                            0.5 * (GetPixelValue(img1, kp.pt.x + x, kp.pt.y + y + 1) -
                                   GetPixelValue(img1, kp.pt.x + x, kp.pt.y + y - 1))
                        );
                    }
                    // compute H, b and set cost;
                    b += -error * J;
                    cost += error * error;
                    if (inverse == false || iter == 0) {
                        // also update H
                        H += J * J.transpose();
                    }
                }

            // compute update
            Eigen::Vector2d update = H.ldlt().solve(b);

            if (std::isnan(update[0])) {
                // sometimes occurred when we have a black or white patch and H is irreversible
                cout << "update is nan" << endl;
                succ = false;
                break;
            }

            if (iter > 0 && cost > lastCost) {
                break;
            }

            // update dx, dy
            dx += update[0];
            dy += update[1];
            lastCost = cost;
            succ = true;

            if (update.norm() < 1e-2) {
                // converge
                break;
            }
        }

        success[i] = succ;

        // set kp2
        kp2[i].pt = kp.pt + Point2f(dx, dy);
    }
}

void OpticalFlowMultiLevel(
    const Mat &img1,
    const Mat &img2,
    const vector<KeyPoint> &kp1,
    vector<KeyPoint> &kp2,
    vector<bool> &success,
    bool inverse) {

    // parameters
    int pyramids = 4;
    double pyramid_scale = 0.5;
    double scales[] = {1.0, 0.5, 0.25, 0.125};

    // create pyramids
    chrono::steady_clock::time_point t1 = chrono::steady_clock::now();
    vector<Mat> pyr1, pyr2; // image pyramids
    for (int i = 0; i < pyramids; i++) {
        if (i == 0) {
            pyr1.push_back(img1);
            pyr2.push_back(img2);
        } else {
            Mat img1_pyr, img2_pyr;
            cv::resize(pyr1[i - 1], img1_pyr,
                       cv::Size(pyr1[i - 1].cols * pyramid_scale, pyr1[i - 1].rows * pyramid_scale));
            cv::resize(pyr2[i - 1], img2_pyr,
                       cv::Size(pyr2[i - 1].cols * pyramid_scale, pyr2[i - 1].rows * pyramid_scale));
            pyr1.push_back(img1_pyr);
            pyr2.push_back(img2_pyr);
        }
    }
    chrono::steady_clock::time_point t2 = chrono::steady_clock::now();
    auto time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
    cout << "build pyramid time: " << time_used.count() << endl;

    // coarse-to-fine LK tracking in pyramids
    vector<KeyPoint> kp1_pyr, kp2_pyr;
    for (auto &kp:kp1) {
        auto kp_top = kp;
        kp_top.pt *= scales[pyramids - 1];
        kp1_pyr.push_back(kp_top);
        kp2_pyr.push_back(kp_top);
    }

    for (int level = pyramids - 1; level >= 0; level--) {
        // from coarse to fine
        success.clear();
        t1 = chrono::steady_clock::now();
        OpticalFlowSingleLevel(pyr1[level], pyr2[level], kp1_pyr, kp2_pyr, success, inverse, true);
        t2 = chrono::steady_clock::now();
        auto time_used = chrono::duration_cast<chrono::duration<double>>(t2 - t1);
        cout << "track pyr " << level << " cost time: " << time_used.count() << endl;

        if (level > 0) {
            for (auto &kp: kp1_pyr)
                kp.pt /= pyramid_scale;
            for (auto &kp: kp2_pyr)
                kp.pt /= pyramid_scale;
        }
    }

    for (auto &kp: kp2_pyr)
        kp2.push_back(kp);
}

(2) 无配对点方法；直接法