有没有一种简单的方法/算法来匹配2个2D点的云？

Question

我想知道是否有一种简单的方法来匹配（注册）2个点的2个云。

假设我有一个由点表示的对象和一个杂乱的第二个图像，其中包含对象点和噪声（噪声以一种无用的点的方式）。

基本上，对象可以是2d旋转以及翻译和缩放。

我知道有ICP算法，但我认为由于噪音很大，这​​不是一个好方法。

我希望你明白我的意思。 请询问（确定是）任何事情都不清楚。

干杯

Answer 1

这是查找平移和旋转的函数。 缩放，加权点和RANSAC的推广是直截了当的。 我使用openCV库进行可视化和SVD。 以下功能结合了数据生成，单元测试和实际解决方案。

 // rotation and translation in 2D from point correspondences
 void rigidTransform2D(const int N) {

// Algorithm: http://igl.ethz.ch/projects/ARAP/svd_rot.pdf

const bool debug = false;      // print more debug info
const bool add_noise = true; // add noise to imput and output
srand(time(NULL));           // randomize each time

/*********************************
 * Creat data with some noise
 **********************************/

// Simulated transformation
Point2f T(1.0f, -2.0f);
float a = 30.0; // [-180, 180], see atan2(y, x)
float noise_level = 0.1f;
cout<<"True parameters: rot = "<<a<<"deg., T = "<<T<<
        "; noise level = "<<noise_level<<endl;

// noise
vector<Point2f> noise_src(N), noise_dst(N);
for (int i=0; i<N; i++) {
    noise_src[i] = Point2f(randf(noise_level), randf(noise_level));
    noise_dst[i] = Point2f(randf(noise_level), randf(noise_level));
}

// create data with noise
vector<Point2f> src(N), dst(N);
float Rdata = 10.0f; // radius of data
float cosa = cos(a*DEG2RAD);
float sina = sin(a*DEG2RAD);
for (int i=0; i<N; i++) {

    // src
    float x1 = randf(Rdata);
    float y1 = randf(Rdata);
    src[i] = Point2f(x1,y1);
    if (add_noise)
        src[i] += noise_src[i];

    // dst
    float x2 = x1*cosa - y1*sina;
    float y2 = x1*sina + y1*cosa;
    dst[i] = Point2f(x2,y2) + T;
    if (add_noise)
        dst[i] += noise_dst[i];

    if (debug)
        cout<<i<<": "<<src[i]<<"---"<<dst[i]<<endl;
}

// Calculate data centroids
Scalar centroid_src = mean(src);
Scalar centroid_dst = mean(dst);
Point2f center_src(centroid_src[0], centroid_src[1]);
Point2f center_dst(centroid_dst[0], centroid_dst[1]);
if (debug)
    cout<<"Centers: "<<center_src<<", "<<center_dst<<endl;

/*********************************
 * Visualize data
 **********************************/

// Visualization
namedWindow("data", 1);
float w = 400, h = 400;
Mat Mdata(w, h, CV_8UC3); Mdata = Scalar(0);
Point2f center_img(w/2, h/2);

float scl = 0.4*min(w/Rdata, h/Rdata); // compensate for noise
scl/=sqrt(2); // compensate for rotation effect
Point2f dT = (center_src+center_dst)*0.5; // compensate for translation

for (int i=0; i<N; i++) {
    Point2f p1(scl*(src[i] - dT));
    Point2f p2(scl*(dst[i] - dT));
    // invert Y axis
    p1.y = -p1.y; p2.y = -p2.y;
    // add image center
    p1+=center_img; p2+=center_img;
    circle(Mdata, p1, 1, Scalar(0, 255, 0));
    circle(Mdata, p2, 1, Scalar(0, 0, 255));
    line(Mdata, p1, p2, Scalar(100, 100, 100));

}

/*********************************
 * Get 2D rotation and translation
 **********************************/

markTime();

// subtract centroids from data
for (int i=0; i<N; i++) {
    src[i] -= center_src;
    dst[i] -= center_dst;
}

// compute a covariance matrix
float Cxx = 0.0, Cxy = 0.0, Cyx = 0.0, Cyy = 0.0;
for (int i=0; i<N; i++) {
    Cxx += src[i].x*dst[i].x;
    Cxy += src[i].x*dst[i].y;
    Cyx += src[i].y*dst[i].x;
    Cyy += src[i].y*dst[i].y;
}
Mat Mcov = (Mat_<float>(2, 2)<<Cxx, Cxy, Cyx, Cyy);
if (debug)
    cout<<"Covariance Matrix "<<Mcov<<endl;

// SVD
cv::SVD svd;
svd = SVD(Mcov, SVD::FULL_UV);
if (debug) {
    cout<<"U = "<<svd.u<<endl;
    cout<<"W = "<<svd.w<<endl;
    cout<<"V transposed = "<<svd.vt<<endl;
}

// rotation = V*Ut
Mat V = svd.vt.t();
Mat Ut = svd.u.t();
float det_VUt = determinant(V*Ut);
Mat W = (Mat_<float>(2, 2)<<1.0, 0.0, 0.0, det_VUt);
float rot[4];
Mat R_est(2, 2, CV_32F, rot);
R_est = V*W*Ut;
if (debug)
    cout<<"Rotation matrix: "<<R_est<<endl;

float cos_est = rot[0];
float sin_est = rot[2];
float ang = atan2(sin_est, cos_est);

// translation = mean_dst - R*mean_src
Point2f center_srcRot = Point2f(
        cos_est*center_src.x - sin_est*center_src.y,
        sin_est*center_src.x + cos_est*center_src.y);
Point2f T_est = center_dst - center_srcRot;

// RMSE
double RMSE = 0.0;
for (int i=0; i<N; i++) {
    Point2f dst_est(
            cos_est*src[i].x - sin_est*src[i].y,
            sin_est*src[i].x + cos_est*src[i].y);
    RMSE += SQR(dst[i].x - dst_est.x) + SQR(dst[i].y - dst_est.y);
}
if (N>0)
    RMSE = sqrt(RMSE/N);

// Final estimate msg
cout<<"Estimate = "<<ang*RAD2DEG<<"deg., T = "<<T_est<<"; RMSE = "<<RMSE<<endl;

// show image
printTime(1);
imshow("data", Mdata);
waitKey(-1);

return;
} // rigidTransform2D()

// --------------------------- 3DOF

// calculates squared error from two point mapping; assumes rotation around Origin.
inline float sqErr_3Dof(Point2f p1, Point2f p2,
        float cos_alpha, float sin_alpha, Point2f T) {

    float x2_est = T.x + cos_alpha * p1.x - sin_alpha * p1.y;
    float y2_est = T.y + sin_alpha * p1.x + cos_alpha * p1.y;
    Point2f p2_est(x2_est, y2_est);
    Point2f dp = p2_est-p2;
    float sq_er = dp.dot(dp); // squared distance

    //cout<<dp<<endl;
    return sq_er;
}

// calculate RMSE for point-to-point metrics
float RMSE_3Dof(const vector<Point2f>& src, const vector<Point2f>& dst,
        const float* param, const bool* inliers, const Point2f center) {

    const bool all_inliers = (inliers==NULL); // handy when we run QUADRTATIC will all inliers
    unsigned int n = src.size();
    assert(n>0 && n==dst.size());

    float ang_rad = param[0];
    Point2f T(param[1], param[2]);
    float cos_alpha = cos(ang_rad);
    float sin_alpha = sin(ang_rad);

    double RMSE = 0.0;
    int ninliers = 0;
    for (unsigned int i=0; i<n; i++) {
        if (all_inliers || inliers[i]) {
            RMSE += sqErr_3Dof(src[i]-center, dst[i]-center, cos_alpha, sin_alpha, T);
            ninliers++;
        }
    }

    //cout<<"RMSE = "<<RMSE<<endl;
    if (ninliers>0)
        return sqrt(RMSE/ninliers);
    else
        return LARGE_NUMBER;
}

// Sets inliers and returns their count
inline int setInliers3Dof(const vector<Point2f>& src, const vector <Point2f>& dst,
        bool* inliers,
        const float* param,
        const float max_er,
        const Point2f center) {

    float ang_rad = param[0];
    Point2f T(param[1], param[2]);

    // set inliers
    unsigned int ninliers = 0;
    unsigned int n = src.size();
    assert(n>0 && n==dst.size());

    float cos_ang = cos(ang_rad);
    float sin_ang = sin(ang_rad);
    float max_sqErr = max_er*max_er; // comparing squared values

    if (inliers==NULL) {

        // just get the number of inliers (e.g. after QUADRATIC fit only)
        for (unsigned int i=0; i<n; i++) {

            float sqErr = sqErr_3Dof(src[i]-center, dst[i]-center, cos_ang, sin_ang, T);
            if ( sqErr < max_sqErr)
                ninliers++;
        }
    } else  {

        // get the number of inliers and set them (e.g. for RANSAC)
        for (unsigned int i=0; i<n; i++) {

            float sqErr = sqErr_3Dof(src[i]-center, dst[i]-center, cos_ang, sin_ang, T);
            if ( sqErr < max_sqErr) {
                inliers[i] = 1;
                ninliers++;
            } else {
                inliers[i] = 0;
            }
        }
    }

    return ninliers;
}

// fits 3DOF (rotation and translation in 2D) with least squares.
float fit3DofQUADRATICold(const vector<Point2f>& src, const vector<Point2f>& dst,
        float* param, const bool* inliers, const Point2f center) {

    const bool all_inliers = (inliers==NULL); // handy when we run QUADRTATIC will all inliers
    unsigned int n = src.size();
    assert(dst.size() == n);

    // count inliers
    int ninliers;
    if (all_inliers) {
        ninliers = n;
    } else {
        ninliers = 0;
        for (unsigned int i=0; i<n; i++){
            if (inliers[i])
                ninliers++;
        }
    }

    // under-dermined system
    if (ninliers<2) {
        //      param[0] = 0.0f; // ?
        //      param[1] = 0.0f;
        //      param[2] = 0.0f;
        return LARGE_NUMBER;
    }

    /*
     * x1*cosx(a)-y1*sin(a) + Tx = X1
     * x1*sin(a)+y1*cos(a) + Ty = Y1
     *
     * approximation for small angle a (radians) sin(a)=a, cos(a)=1;
     *
     * x1*1 - y1*a + Tx = X1
     * x1*a + y1*1 + Ty = Y1
     *
     * in matrix form M1*h=M2
     *
     *  2n x 4       4 x 1   2n x 1
     *
     * -y1 1 0 x1  *   a   =  X1
     *  x1 0 1 y1      Tx     Y1
     *                 Ty
     *                 1=Z
     *  ----------------------------
     *  src1         res      src2
     */

    //  4 x 1
    float res_ar[4]; // alpha, Tx, Ty, 1
    Mat res(4, 1, CV_32F, res_ar); // 4 x 1

    // 2n x 4
    Mat src1(2*ninliers, 4, CV_32F); // 2n x 4

    // 2n x 1
    Mat src2(2*ninliers, 1, CV_32F); // 2n x 1: [X1, Y1, X2, Y2, X3, Y3]'

    for (unsigned int i=0, row_cnt = 0; i<n; i++) {

        // use inliers only
        if (all_inliers || inliers[i]) {

            float x = src[i].x - center.x;
            float y = src[i].y - center.y;

            // first row

            // src1
            float* rowPtr = src1.ptr<float>(row_cnt);
            rowPtr[0] = -y;
            rowPtr[1] = 1.0f;
            rowPtr[2] = 0.0f;
            rowPtr[3] = x;

            // src2
            src2.at<float> (0, row_cnt) = dst[i].x - center.x;

            // second row
            row_cnt++;

            // src1
            rowPtr = src1.ptr<float>(row_cnt);
            rowPtr[0] = x;
            rowPtr[1] = 0.0f;
            rowPtr[2] = 1.0f;
            rowPtr[3] = y;

            // src2
            src2.at<float> (0, row_cnt) = dst[i].y - center.y;
        }
    }

    cv::solve(src1, src2, res, DECOMP_SVD);

    // estimators
    float alpha_est;
    Point2f T_est;

    // original
    alpha_est = res.at<float>(0, 0);
    T_est = Point2f(res.at<float>(1, 0), res.at<float>(2, 0));

    float Z = res.at<float>(3, 0);
    if (abs(Z-1.0) > 0.1) {
        //cout<<"Bad Z in fit3DOF(), Z should be close to 1.0 = "<<Z<<endl;
        //return LARGE_NUMBER;
    }
    param[0] = alpha_est; // rad
    param[1] = T_est.x;
    param[2] = T_est.y;

    // calculate RMSE
    float RMSE = RMSE_3Dof(src, dst, param, inliers, center);
    return RMSE;
} // fit3DofQUADRATICOLd()

// fits 3DOF (rotation and translation in 2D) with least squares.
float fit3DofQUADRATIC(const vector<Point2f>& src_, const vector<Point2f>& dst_,
        float* param, const bool* inliers, const Point2f center) {

    const bool debug = false;                   // print more debug info
    const bool all_inliers = (inliers==NULL);   // handy when we run QUADRTATIC will all inliers
    assert(dst_.size() == src_.size());
    int N = src_.size();

    // collect inliers
    vector<Point2f> src, dst;
    int ninliers;
    if (all_inliers) {
        ninliers = N;
        src = src_; // copy constructor
        dst = dst_;
    } else {
        ninliers = 0;
        for (int i=0; i<N; i++){
            if (inliers[i]) {
                ninliers++;
                src.push_back(src_[i]);
                dst.push_back(dst_[i]);
            }
        }
    }
    if (ninliers<2) {
        param[0] = 0.0f; // default return when there is not enough points
        param[1] = 0.0f;
        param[2] = 0.0f;
        return LARGE_NUMBER;
    }

    /* Algorithm: Least-Square Rigid Motion Using SVD by Olga Sorkine
     * http://igl.ethz.ch/projects/ARAP/svd_rot.pdf
     *
     * Subtract centroids, calculate SVD(cov),
     * R = V[1, det(VU')]'U', T = mean_q-R*mean_p
     */

    // Calculate data centroids
    Scalar centroid_src = mean(src);
    Scalar centroid_dst = mean(dst);
    Point2f center_src(centroid_src[0], centroid_src[1]);
    Point2f center_dst(centroid_dst[0], centroid_dst[1]);
    if (debug)
        cout<<"Centers: "<<center_src<<", "<<center_dst<<endl;

    // subtract centroids from data
    for (int i=0; i<ninliers; i++) {
        src[i] -= center_src;
        dst[i] -= center_dst;
    }

    // compute a covariance matrix
    float Cxx = 0.0, Cxy = 0.0, Cyx = 0.0, Cyy = 0.0;
    for (int i=0; i<ninliers; i++) {
        Cxx += src[i].x*dst[i].x;
        Cxy += src[i].x*dst[i].y;
        Cyx += src[i].y*dst[i].x;
        Cyy += src[i].y*dst[i].y;
    }
    Mat Mcov = (Mat_<float>(2, 2)<<Cxx, Cxy, Cyx, Cyy);
    Mcov /= (ninliers-1);
    if (debug)
        cout<<"Covariance-like Matrix "<<Mcov<<endl;

    // SVD of covariance
    cv::SVD svd;
    svd = SVD(Mcov, SVD::FULL_UV);
    if (debug) {
        cout<<"U = "<<svd.u<<endl;
        cout<<"W = "<<svd.w<<endl;
        cout<<"V transposed = "<<svd.vt<<endl;
    }

    // rotation (V*Ut)
    Mat V = svd.vt.t();
    Mat Ut = svd.u.t();
    float det_VUt = determinant(V*Ut);
    Mat W = (Mat_<float>(2, 2)<<1.0, 0.0, 0.0, det_VUt);
    float rot[4];
    Mat R_est(2, 2, CV_32F, rot);
    R_est = V*W*Ut;
    if (debug)
        cout<<"Rotation matrix: "<<R_est<<endl;

    float cos_est = rot[0];
    float sin_est = rot[2];
    float ang = atan2(sin_est, cos_est);

    // translation (mean_dst - R*mean_src)
    Point2f center_srcRot = Point2f(
            cos_est*center_src.x - sin_est*center_src.y,
            sin_est*center_src.x + cos_est*center_src.y);
    Point2f T_est = center_dst - center_srcRot;

    // Final estimate msg
    if (debug)
        cout<<"Estimate = "<<ang*RAD2DEG<<"deg., T = "<<T_est<<endl;

    param[0] = ang; // rad
    param[1] = T_est.x;
    param[2] = T_est.y;

    // calculate RMSE
    float RMSE = RMSE_3Dof(src_, dst_, param, inliers, center);
    return RMSE;
} // fit3DofQUADRATIC()

// RANSAC fit in 3DOF: 1D rot and 2D translation (maximizes the number of inliers)
// NOTE: no data normalization is currently performed
float fit3DofRANSAC(const vector<Point2f>& src, const vector<Point2f>& dst,
        float* best_param,  bool* inliers,
        const Point2f center ,
        const float inlierMaxEr,
        const int niter) {

    const int ITERATION_TO_SETTLE = 2;    // iterations to settle inliers and param
    const float INLIERS_RATIO_OK = 0.95f;  // stopping criterion

    // size of data vector
    unsigned int N = src.size();
    assert(N==dst.size());

    // unrealistic case
    if(N<2) {
        best_param[0] = 0.0f; // ?
        best_param[1] = 0.0f;
        best_param[2] = 0.0f;
        return LARGE_NUMBER;
    }

    unsigned int ninliers;                 // current number of inliers
    unsigned int best_ninliers = 0;     // number of inliers
    float best_rmse = LARGE_NUMBER;         // error
    float cur_rmse;                         // current distance error
    float param[3];                         // rad, Tx, Ty
    vector <Point2f> src_2pt(2), dst_2pt(2);// min set of 2 points (1 correspondence generates 2 equations)
    srand (time(NULL));

    // iterations
    for (int iter = 0; iter<niter; iter++) {

#ifdef DEBUG_RANSAC
        cout<<"iteration "<<iter<<": ";
#endif

        // 1. Select a random set of 2 points (not obligatory inliers but valid)
        int i1, i2;
        i1 = rand() % N; // [0, N[
        i2 = i1;
        while (i2==i1) {
            i2 = rand() % N;
        }
        src_2pt[0] = src[i1]; // corresponding points
        src_2pt[1] = src[i2];
        dst_2pt[0] = dst[i1];
        dst_2pt[1] = dst[i2];
        bool two_inliers[] = {true, true};

        // 2. Quadratic fit for 2 points
        cur_rmse = fit3DofQUADRATIC(src_2pt, dst_2pt, param, two_inliers, center);

        // 3. Recalculate to settle params and inliers using a larger set
        for (int iter2=0; iter2<ITERATION_TO_SETTLE; iter2++) {
            ninliers = setInliers3Dof(src, dst, inliers, param, inlierMaxEr, center);   // changes inliers
            cur_rmse = fit3DofQUADRATIC(src, dst, param, inliers, center);              // changes cur_param
        }

        // potential ill-condition or large error
        if (ninliers<2) {
#ifdef DEBUG_RANSAC
            cout<<" !!! less than 2 inliers "<<endl;
#endif
            continue;
        } else {
#ifdef DEBUG_RANSAC
            cout<<" "<<ninliers<<" inliers; ";
#endif
        }


#ifdef DEBUG_RANSAC
        cout<<"; recalculate: RMSE = "<<cur_rmse<<", "<<ninliers <<" inliers";
#endif


        // 4. found a better solution?
        if (ninliers > best_ninliers) {
            best_ninliers = ninliers;
            best_param[0] = param[0];
            best_param[1] = param[1];
            best_param[2] = param[2];
            best_rmse = cur_rmse;


#ifdef DEBUG_RANSAC
            cout<<" --- Solution improved: "<<
                    best_param[0]<<", "<<best_param[1]<<", "<<param[2]<<endl;
#endif
            // exit condition
            float inlier_ratio = (float)best_ninliers/N;
            if (inlier_ratio > INLIERS_RATIO_OK) {
#ifdef DEBUG_RANSAC
                cout<<"Breaking early after "<< iter+1<<
                        " iterations; inlier ratio = "<<inlier_ratio<<endl;
#endif
                break;
            }
        } else {
#ifdef DEBUG_RANSAC
            cout<<endl;
#endif
        }


    } // iterations

    // 5. recreate inliers for the best parameters
    ninliers = setInliers3Dof(src, dst, inliers, best_param, inlierMaxEr, center);

    return best_rmse;
} // fit3DofRANSAC()

Answer 2

让我首先确保我正确地解释你的问题。 您有两组2D点，其中一组包含对应于某些感兴趣对象的所有“好”点，其中一组包含添加了噪声点的仿射变换下的那些点。 对？

如果这是正确的，那么有一种相当可靠和有效的方法来拒绝噪声点并确定您感兴趣的点之间的转换。 通常用于拒绝噪声点（“异常值”）的算法称为RANSAC ，用于确定变换的算法可以采用多种形式，但是现有技术的最新状态称为五点算法，可以在这里找到 - 可以在这里找到MATLAB实现。

不幸的是，我不知道两者结合的成熟实施; 您可能需要自己做一些工作来实现RANSAC并将其与五点算法集成。

编辑：

实际上，OpenCV的实现对你的任务来说是过度的（意味着它可以工作，但需要的时间比必要的多），但是它已经准备好开箱即用。 感兴趣的函数称为cv :: findFundamentalMat 。

Answer 3

我相信你正在寻找像David Lowe的SIFT （尺度不变特征变换）这样的东西。 其他选择是SURF （SIFT受专利保护）。 OpenCV计算机库提供了SURF实现

Answer 4

我会尝试使用距离几何（http://en.wikipedia.org/wiki/Distance_geometry）

通过将距离与某个半径内的所有邻居相加，为每个点生成一个标量。 虽然不完美，但这对于每一点都是很好的鉴别器。

然后将所有标量放在一个映射中，该映射允许通过标量（+）加上/减去一些delta来检索点（p）
M（s + delta）= p（例如KD树）（http://en.wikipedia.org/wiki/Kd-tree）

将所有2D点的参考集放在地图中

在另一个（测试）2D点集上：
foreach测试缩放（特别是如果你很清楚典型的缩放值是什么）
...用S表示每个点
...重新计算测试点集的标量
......对于测试集中的每个点P（或者可能是更快的方法的样本）
.........某些delta内参考标量图中的查找点
.........如果没有找到映射，则丢弃P.
.........其他事先发现P'点
............检查P的邻居并查看他们是否在某个增量内的参考地图中有相应的标量（即参考点具有大约相同值的邻居）
.........如果测试的所有点都在参考集中有映射，您已经找到了测试点P到参考点P'的映射 - >测试点到参考点的记录映射..... .discard缩放，如果没有记录映射

请注意，这在几个不同的地方是平行的

根据我多年前做过的研究，这是我的头脑。 它缺乏精细的细节，但总体思路很明确：在噪声（测试）图中找到与其最近邻居的距离与参考集大致相同的点。 噪声图必须测量距离，允许误差较大，噪声图较少。

该算法适用于没有噪声的图形。

编辑 ：算法的细化不需要查看不同的缩放。 计算每个点的标量时，请改用相对距离度量。 这将是变换的不变量

Answer 5

从C ++开始，您可以使用ITK进行图像注册。 它包括许多注册功能，可在存在噪音的情况下工作。

Answer 6

KLT（Kanade Lucas Tomasi）功能跟踪器对跟踪功能进行仿射一致性检查 。 仿射一致性检查考虑了平移，旋转和缩放。 我不知道它对你有帮助，因为你不能直接使用函数（它计算矩形区域的仿射变换）。 但也许你可以从文档和源代码中学习 ，如何计算仿射变换并使其适应你的问题（点云而不是矩形区域）。

Answer 7

你想要Denton-Beveridge点匹配算法。 下面链接的页面底部的源代码，还有一篇文章解释了算法以及为什么Ransac是这个问题的错误选择。

http://jasondenton.me/pntmatch.html