OpenCV templates in 2D point data set

I was wandering what the best approach would be for detecting 'figures' in an array of 2D points.

In this example I have two 'templates'. Figure 1 is a template and figure 2 is a template. Each of these templates exists only as a vector of points with an x,y coordinate.

Let's say we have a third vector with points with x,y coordinate

What would be the best way to find out and isolate points matching one of the first two arrays in the third one. (including scaling, rotation)?

I have been trying nearest neigbours(FlannBasedMatcehr) or even SVM implementation but it doesn't seem to get me any result, template matching doesn't seem to be the way to go either, I think. I am not working on images but only on 2D points in memory...

Especially because the input vector always has more points than the original data set to be compared with.

All it needs to do is find points in array that match a template.

I am not a 'specialist' in machine learning or opencv. I guess I am overlooking something from the beginning...

Thank you very much for your help/suggestions.

just for fun I tried this:

Choose two points of the point dataset and compute the transformation mapping the first two pattern points to those points.
Test whether all transformed pattern points can be found in the data set.

This approach is very naive and has a complexity of O(m*n²) with n data points and a single pattern of size m (points). This complexity might be increased for some nearest neighbor search methods. So you have to consider whether it's not efficient enough for your appplication.

Some improvements could include some heuristic to not choose all n² combinations of points but, but you need background information of maximal pattern scaling or something like that.

For evaluation I first created a pattern:

Then I create random points and add the pattern somewhere within (scaled, rotated and translated):

After some computation this method recognizes the pattern. The red line shows the chosen points for transformation computation.

Here's the code:

// draw a set of points on a given destination image
void drawPoints(cv::Mat & image, std::vector<cv::Point2f> points, cv::Scalar color = cv::Scalar(255,255,255), float size=10)
{
    for(unsigned int i=0; i<points.size(); ++i)
    {
        cv::circle(image, points[i], 0, color, size);
    }
}

// assumes a 2x3 (affine) transformation (CV_32FC1). does not change the input points
std::vector<cv::Point2f> applyTransformation(std::vector<cv::Point2f> points, cv::Mat transformation)
{
    for(unsigned int i=0; i<points.size(); ++i)
    {
        const cv::Point2f tmp = points[i];
        points[i].x = tmp.x * transformation.at<float>(0,0) + tmp.y * transformation.at<float>(0,1) + transformation.at<float>(0,2) ;
        points[i].y = tmp.x * transformation.at<float>(1,0) + tmp.y * transformation.at<float>(1,1) + transformation.at<float>(1,2) ;
    }

    return points;
}

const float PI = 3.14159265359;

// similarity transformation uses same scaling along both axes, rotation and a translation part
cv::Mat composeSimilarityTransformation(float s, float r, float tx, float ty)
{
    cv::Mat transformation = cv::Mat::zeros(2,3,CV_32FC1);

    // compute rotation matrix and scale entries
    float rRad = PI*r/180.0f;
    transformation.at<float>(0,0) = s*cosf(rRad);
    transformation.at<float>(0,1) = s*sinf(rRad);
    transformation.at<float>(1,0) = -s*sinf(rRad);
    transformation.at<float>(1,1) = s*cosf(rRad);

    // translation
    transformation.at<float>(0,2) = tx;
    transformation.at<float>(1,2) = ty;

    return transformation;

}

// create random points
std::vector<cv::Point2f> createPointSet(cv::Size2i imageSize, std::vector<cv::Point2f> pointPattern, unsigned int nRandomDots = 50)
{
    // subtract center of gravity to allow more intuitive rotation
    cv::Point2f centerOfGravity(0,0);
    for(unsigned int i=0; i<pointPattern.size(); ++i)
    {
        centerOfGravity.x += pointPattern[i].x;
        centerOfGravity.y += pointPattern[i].y;
    }
    centerOfGravity.x /= (float)pointPattern.size();
    centerOfGravity.y /= (float)pointPattern.size();
    pointPattern = applyTransformation(pointPattern, composeSimilarityTransformation(1,0,-centerOfGravity.x, -centerOfGravity.y));

    // create random points
    //unsigned int nRandomDots = 0;
    std::vector<cv::Point2f> pointset;
    srand (time(NULL));
    for(unsigned int i =0; i<nRandomDots; ++i)
    {
        pointset.push_back( cv::Point2f(rand()%imageSize.width, rand()%imageSize.height) );
    }

    cv::Mat image = cv::Mat::ones(imageSize,CV_8UC3);
    image = cv::Scalar(255,255,255);

    drawPoints(image, pointset, cv::Scalar(0,0,0));
    cv::namedWindow("pointset"); cv::imshow("pointset", image);

    // add point pattern to a random location

    float scaleFactor = rand()%30 + 10.0f;
    float translationX = rand()%(imageSize.width/2)+ imageSize.width/4;
    float translationY = rand()%(imageSize.height/2)+ imageSize.height/4;
    float rotationAngle = rand()%360;

    std::cout << "s: " << scaleFactor << " r: " << rotationAngle << " t: " << translationX << "/" << translationY << std::endl;


    std::vector<cv::Point2f> transformedPattern = applyTransformation(pointPattern,composeSimilarityTransformation(scaleFactor,rotationAngle,translationX,translationY));
    //std::vector<cv::Point2f> transformedPattern = applyTransformation(pointPattern,trans);

    drawPoints(image, transformedPattern, cv::Scalar(0,0,0));
    drawPoints(image, transformedPattern, cv::Scalar(0,255,0),3);
    cv::imwrite("dataPoints.png", image);
    cv::namedWindow("pointset + pattern"); cv::imshow("pointset + pattern", image);



    for(unsigned int i=0; i<transformedPattern.size(); ++i)
        pointset.push_back(transformedPattern[i]);

    return pointset;

}

void programDetectPointPattern()
{
    cv::Size2i imageSize(640,480);

    // create a point pattern, this can be in any scale and any relative location
    std::vector<cv::Point2f> pointPattern;
    pointPattern.push_back(cv::Point2f(0,0));
    pointPattern.push_back(cv::Point2f(2,0));
    pointPattern.push_back(cv::Point2f(4,0));
    pointPattern.push_back(cv::Point2f(1,2));
    pointPattern.push_back(cv::Point2f(3,2));
    pointPattern.push_back(cv::Point2f(2,4));

    // transform the pattern so it can be drawn
    cv::Mat trans = cv::Mat::ones(2,3,CV_32FC1);
    trans.at<float>(0,0) = 20.0f;   // scale x
    trans.at<float>(1,1) = 20.0f;   // scale y
    trans.at<float>(0,2) = 20.0f;   // translation x
    trans.at<float>(1,2) = 20.0f;   // translation y

    // draw the pattern
    cv::Mat drawnPattern = cv::Mat::ones(cv::Size2i(128,128),CV_8U);
    drawnPattern *= 255;
    drawPoints(drawnPattern,applyTransformation(pointPattern, trans), cv::Scalar(0),5);

    // display and save pattern
    cv::imwrite("patternToDetect.png", drawnPattern);
    cv::namedWindow("pattern"); cv::imshow("pattern", drawnPattern);

    // draw the points and the included pattern
    std::vector<cv::Point2f> pointset = createPointSet(imageSize, pointPattern);
    cv::Mat image = cv::Mat(imageSize, CV_8UC3);
    image = cv::Scalar(255,255,255);
    drawPoints(image,pointset, cv::Scalar(0,0,0));


    // normally we would have to use some nearest neighbor distance computation, but to make it easier here,
    // we create a small area around every point, which allows to test for point existence in a small neighborhood very efficiently (for small images)
    // in the real application this "inlier" check should be performed by k-nearest neighbor search and threshold the distance,
    // efficiently evaluated by a kd-tree
    cv::Mat pointImage = cv::Mat::zeros(imageSize,CV_8U);
    float maxDist = 3.0f; // how exact must the pattern be recognized, can there be some "noise" in the position of the data points?
    drawPoints(pointImage, pointset, cv::Scalar(255),maxDist);
    cv::namedWindow("pointImage"); cv::imshow("pointImage", pointImage);

    // choose two points from the pattern (can be arbitrary so just take the first two)
    cv::Point2f referencePoint1 = pointPattern[0];
    cv::Point2f referencePoint2 = pointPattern[1];
    cv::Point2f diff1;  // difference vector
    diff1.x = referencePoint2.x - referencePoint1.x;
    diff1.y = referencePoint2.y - referencePoint1.y;
    float referenceLength = sqrt(diff1.x*diff1.x + diff1.y*diff1.y);
    diff1.x = diff1.x/referenceLength; diff1.y = diff1.y/referenceLength;

    std::cout << "reference: " << std::endl;
    std::cout << referencePoint1 << std::endl;

    // now try to find the pattern
    for(unsigned int j=0; j<pointset.size(); ++j)
    {
        cv::Point2f targetPoint1 =  pointset[j];

        for(unsigned int i=0; i<pointset.size(); ++i)
        {
            cv::Point2f targetPoint2 =  pointset[i];

            cv::Point2f diff2;
            diff2.x = targetPoint2.x - targetPoint1.x;
            diff2.y = targetPoint2.y - targetPoint1.y;
            float targetLength = sqrt(diff2.x*diff2.x + diff2.y*diff2.y);
            diff2.x = diff2.x/targetLength; diff2.y = diff2.y/targetLength;

            // with nearest-neighborhood search this line will be similar or the maximal neighbor distance must be relative to targetLength!
            if(targetLength < maxDist) continue;

            // scale:
            float s = targetLength/referenceLength;

            // rotation:
            float r = -180.0f/PI*(atan2(diff2.y,diff2.x) + atan2(diff1.y,diff1.x));

            // scale and rotate the reference point to compute the translation needed
            std::vector<cv::Point2f> origin;
            origin.push_back(referencePoint1);
            origin = applyTransformation(origin, composeSimilarityTransformation(s,r,0,0));
            // compute the translation which maps the two reference points on the two target points
            float tx = targetPoint1.x - origin[0].x;
            float ty = targetPoint1.y - origin[0].y;

            std::vector<cv::Point2f> transformedPattern = applyTransformation(pointPattern,composeSimilarityTransformation(s,r,tx,ty));


            // now test if all transformed pattern points can be found in the dataset
            bool found = true;
            for(unsigned int i=0; i<transformedPattern.size(); ++i)
            {
                cv::Point2f curr = transformedPattern[i];
                // here we check whether there is a point drawn in the image. If you have no image you will have to perform a nearest neighbor search.
                // this can be done with a balanced kd-tree in O(log n) time
                // building such a balanced kd-tree has to be done once for the whole dataset and needs O(n*(log n)) afair
                if((curr.x >= 0)&&(curr.x <= pointImage.cols-1)&&(curr.y>=0)&&(curr.y <= pointImage.rows-1))
                {
                    if(pointImage.at<unsigned char>(curr.y, curr.x) == 0) found = false;
                    // if working with kd-tree: if nearest neighbor distance > maxDist => found = false;
                }
                else found = false;

            }



            if(found)
            {
                std::cout << composeSimilarityTransformation(s,r,tx,ty) << std::endl;
                cv::Mat currentIteration;
                image.copyTo(currentIteration);
                cv::circle(currentIteration,targetPoint1,5, cv::Scalar(255,0,0),1);
                cv::circle(currentIteration,targetPoint2,5, cv::Scalar(255,0,255),1);
                cv::line(currentIteration,targetPoint1,targetPoint2,cv::Scalar(0,0,255));
                drawPoints(currentIteration, transformedPattern, cv::Scalar(0,0,255),4);

                cv::imwrite("detectedPattern.png", currentIteration);
                cv::namedWindow("iteration"); cv::imshow("iteration", currentIteration); cv::waitKey(-1);
            }

        }
    }


}