How to determine rotation of a shape?

Question

I have following shape.

It may be rotated by unknown angle. I want to determine its rotation in reference to horizontal axis (so shape abov

Answer 1

I adapted my answer from here: https://stackoverflow.com/a/23993030/2393191 It gives quite good results:

inline void getCircle(cv::Point2f& p1, cv::Point2f& p2, cv::Point2f& p3, cv::Point2f& center, float& radius)
{
    float x1 = p1.x;
    float x2 = p2.x;
    float x3 = p3.x;

    float y1 = p1.y;
    float y2 = p2.y;
    float y3 = p3.y;

    // PLEASE CHECK FOR TYPOS IN THE FORMULA :)
    center.x = (x1*x1 + y1*y1)*(y2 - y3) + (x2*x2 + y2*y2)*(y3 - y1) + (x3*x3 + y3*y3)*(y1 - y2);
    center.x /= (2 * (x1*(y2 - y3) - y1*(x2 - x3) + x2*y3 - x3*y2));

    center.y = (x1*x1 + y1*y1)*(x3 - x2) + (x2*x2 + y2*y2)*(x1 - x3) + (x3*x3 + y3*y3)*(x2 - x1);
    center.y /= (2 * (x1*(y2 - y3) - y1*(x2 - x3) + x2*y3 - x3*y2));

    radius = sqrt((center.x - x1)*(center.x - x1) + (center.y - y1)*(center.y - y1));
}



std::vector<cv::Point2f> getPointPositions(cv::Mat binaryImage)
{
    std::vector<cv::Point2f> pointPositions;

    for (unsigned int y = 0; y<binaryImage.rows; ++y)
    {
        //unsigned char* rowPtr = binaryImage.ptr<unsigned char>(y);
        for (unsigned int x = 0; x<binaryImage.cols; ++x)
        {
            //if(rowPtr[x] > 0) pointPositions.push_back(cv::Point2i(x,y));
            if (binaryImage.at<unsigned char>(y, x) > 0) pointPositions.push_back(cv::Point2f(x, y));
        }
    }

    return pointPositions;
}


float verifyCircle(cv::Mat dt, cv::Point2f center, float radius, std::vector<cv::Point2f> & inlierSet)
{
    unsigned int counter = 0;
    unsigned int inlier = 0;
    float minInlierDist = 2.0f;
    float maxInlierDistMax = 100.0f;
    float maxInlierDist = radius / 25.0f;
    if (maxInlierDist<minInlierDist) maxInlierDist = minInlierDist;
    if (maxInlierDist>maxInlierDistMax) maxInlierDist = maxInlierDistMax;

    // choose samples along the circle and count inlier percentage
    for (float t = 0; t<2 * 3.14159265359f; t += 0.05f)
    {
        counter++;
        float cX = radius*cos(t) + center.x;
        float cY = radius*sin(t) + center.y;

        if (cX < dt.cols)
            if (cX >= 0)
                if (cY < dt.rows)
                    if (cY >= 0)
                        if (dt.at<float>(cY, cX) < maxInlierDist)
                        {
                            inlier++;
                            inlierSet.push_back(cv::Point2f(cX, cY));
                        }
    }

    return (float)inlier / float(counter);
}

float evaluateCircle(cv::Mat dt, cv::Point2f center, float radius)
{

    float completeDistance = 0.0f;
    int counter = 0;

    float maxDist = 1.0f;   //TODO: this might depend on the size of the circle!

    float minStep = 0.001f;
    // choose samples along the circle and count inlier percentage

    //HERE IS THE TRICK that no minimum/maximum circle is used, the number of generated points along the circle depends on the radius.
    // if this is too slow for you (e.g. too many points created for each circle), increase the step parameter, but only by factor so that it still depends on the radius

    // the parameter step depends on the circle size, otherwise small circles will create more inlier on the circle
    float step = 2 * 3.14159265359f / (6.0f * radius);
    if (step < minStep) step = minStep; // TODO: find a good value here.

    //for(float t =0; t<2*3.14159265359f; t+= 0.05f) // this one which doesnt depend on the radius, is much worse!
    for (float t = 0; t<2 * 3.14159265359f; t += step)
    {
        float cX = radius*cos(t) + center.x;
        float cY = radius*sin(t) + center.y;

        if (cX < dt.cols)
            if (cX >= 0)
                if (cY < dt.rows)
                    if (cY >= 0)
                        if (dt.at<float>(cY, cX) <= maxDist)
                        {
                            completeDistance += dt.at<float>(cY, cX);
                            counter++;
                        }

    }

    return counter;
}




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

    cv::Mat input = cv::imread("C:/StackOverflow/Input/rotatedShape1.png", cv::IMREAD_GRAYSCALE);
    std::string outString = "C:/StackOverflow/Output/rotatedShape1.png";

    cv::Mat output;
    cv::cvtColor(input, output, cv::COLOR_GRAY2BGR);

    std::vector<std::vector<cv::Point> > contours;
    cv::findContours(input, contours, cv::RETR_EXTERNAL, cv::CHAIN_APPROX_NONE);

    std::vector<cv::Point> biggestContour;
    double biggestArea = 0;
    for (int i = 0; i < contours.size(); ++i)
    {
        double cArea = cv::contourArea(contours[i]);
        if (cArea > biggestArea)
        {
            biggestArea = cArea;
            biggestContour = contours[i];
        }
    }

    if (biggestContour.size() == 0)
    {
        std::cout << "error: no contour found. Press enter to quit." << std::endl;
        std::cin.get();
        return 0;
    }



    cv::Mat mask = cv::Mat::zeros(input.size(), input.type());
    std::vector < std::vector<cv::Point> > tmp;
    tmp.push_back(biggestContour);
    cv::drawContours(mask, tmp, 0, cv::Scalar::all(255), 1); // contour points in the image

    std::vector<cv::Point2f> circlesList;

    unsigned int numberOfCirclesToDetect = 2;   // TODO: if unknown, you'll have to find some nice criteria to stop finding more (semi-) circles

    for (unsigned int j = 0; j<numberOfCirclesToDetect; ++j)
    {
        std::vector<cv::Point2f> edgePositions;
        //for (int i = 0; i < biggestContour.size(); ++i) edgePositions.push_back(biggestContour[i]);
        edgePositions = getPointPositions(mask);



        std::cout << "number of edge positions: " << edgePositions.size() << std::endl;

        // create distance transform to efficiently evaluate distance to nearest edge
        cv::Mat dt;
        cv::distanceTransform(255 - mask, dt, CV_DIST_L1, 3);



        unsigned int nIterations = 0;

        cv::Point2f bestCircleCenter;
        float bestCircleRadius;
        //float bestCVal = FLT_MAX;
        float bestCVal = -1;

        //float minCircleRadius = 20.0f; // TODO: if you have some knowledge about your image you might be able to adjust the minimum circle radius parameter.
        float minCircleRadius = 0.0f;

        //TODO: implement some more intelligent ransac without fixed number of iterations
        for (unsigned int i = 0; i<2000; ++i)
        {
            //RANSAC: randomly choose 3 point and create a circle:
            //TODO: choose randomly but more intelligent,
            //so that it is more likely to choose three points of a circle.
            //For example if there are many small circles, it is unlikely to randomly choose 3 points of the same circle.
            unsigned int idx1 = rand() % edgePositions.size();
            unsigned int idx2 = rand() % edgePositions.size();
            unsigned int idx3 = rand() % edgePositions.size();

            // we need 3 different samples:
            if (idx1 == idx2) continue;
            if (idx1 == idx3) continue;
            if (idx3 == idx2) continue;

            // create circle from 3 points:
            cv::Point2f center; float radius;
            getCircle(edgePositions[idx1], edgePositions[idx2], edgePositions[idx3], center, radius);

            if (radius < minCircleRadius)continue;


            //verify or falsify the circle by inlier counting:
            //float cPerc = verifyCircle(dt,center,radius, inlierSet);
            float cVal = evaluateCircle(dt, center, radius);

            if (cVal > bestCVal)
            {
                bestCVal = cVal;
                bestCircleRadius = radius;
                bestCircleCenter = center;
            }

            ++nIterations;
        }
        std::cout << "current best circle: " << bestCircleCenter << " with radius: " << bestCircleRadius << " and nInlier " << bestCVal << std::endl;
        cv::circle(output, bestCircleCenter, bestCircleRadius, cv::Scalar(0, 0, 255));

        //TODO: hold and save the detected circle.

        //TODO: instead of overwriting the mask with a drawn circle it might be better to hold and ignore detected circles and dont count new circles which are too close to the old one.
        // in this current version the chosen radius to overwrite the mask is fixed and might remove parts of other circles too!

        // update mask: remove the detected circle!
        cv::circle(mask, bestCircleCenter, bestCircleRadius, 0, 10); // here the thickness is fixed which isnt so nice.

        circlesList.push_back(bestCircleCenter);
    }



    if (circlesList.size() < 2)
    {
        std::cout << "error: not enough circles found. Press enter." << std::endl;
        std::cin.get();
        return 0;
    }

    cv::Point2f centerOfMass = circlesList[0];
    cv::Point2f cogFP = circlesList[1];
    std::cout << cogFP - centerOfMass << std::endl;
    float angle = acos((cogFP - centerOfMass).x / cv::norm(cogFP - centerOfMass)); // scalar product of [1,0] and point
    std::cout << angle * 180 / CV_PI << std::endl;

    cv::line(output, centerOfMass, cogFP, cv::Scalar(0, 255, 0), 1);
    cv::circle(output, centerOfMass, 5, cv::Scalar(0, 0, 255), 1);
    cv::circle(output, cogFP, 3, cv::Scalar(255, 0, 0), 1);


    cv::imwrite(outString, output);

    cv::imshow("input", input);
    cv::imshow("output", output);
    cv::waitKey(0);
    return 0;
}

results:

Answer 2

here's the simple logic of finding the center of gravity and the furthest contour point from it. It has an offset of 6 degrees for that contour, either because of the actual contour shape, or because of a slightly wrong center of gravity.

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

    //cv::Mat input = cv::imread("C:/StackOverflow/Input/rotatedShape1.png", cv::IMREAD_GRAYSCALE);
    cv::Mat input = cv::imread("C:/StackOverflow/Input/rotatedShape5.png", cv::IMREAD_GRAYSCALE);
    std::string outString = "C:/StackOverflow/Output/rotatedShape5.png";

    cv::Mat output;
    cv::cvtColor(input, output, cv::COLOR_GRAY2BGR);

    std::vector<std::vector<cv::Point> > contours;
    cv::findContours(input, contours, cv::RETR_EXTERNAL, cv::CHAIN_APPROX_NONE);

    std::vector<cv::Point> biggestContour;
    double biggestArea = 0;
    for (int i = 0; i < contours.size(); ++i)
    {
        double cArea = cv::contourArea(contours[i]);
        if (cArea > biggestArea)
        {
            biggestArea = cArea;
            biggestContour = contours[i];
        }
    }

    if (biggestContour.size() == 0)
    {
        std::cout << "error: no contour found. Press enter to quit." << std::endl;
        std::cin.get();
        return 0;
    }

    cv::Point2f centerOfMass(0,0);
    float invContourSize = 1.0f / biggestContour.size();
    for (int i = 0; i < biggestContour.size(); ++i)
    {
        centerOfMass = centerOfMass + (invContourSize * cv::Point2f(biggestContour[i]));
    }

    float furthestDist = 0;
    cv::Point2f furthestPoint = centerOfMass;
    for (int i = 0; i < biggestContour.size(); ++i)
    {
        float cDist = cv::norm(cv::Point2f(biggestContour[i]) - centerOfMass);
        if (cDist > furthestDist)
        {
            furthestDist = cDist;
            furthestPoint = biggestContour[i];
        }
    }

    // find points with very similar distance
    float maxDifference = 20; // magic number
    std::vector<cv::Point2f> listOfFurthestPoints;
    for (int i = 0; i < biggestContour.size(); ++i)
    {
        float cDist = cv::norm(cv::Point2f(biggestContour[i]) - furthestPoint);
        if (cDist < maxDifference)
        {
            listOfFurthestPoints.push_back( biggestContour[i] );
            // render:
            cv::circle(output, biggestContour[i], 0, cv::Scalar(255, 0, 255), 0);
        }
    }

    cv::Point2f cogFP(0, 0);
    float invListSize = 1.0f / listOfFurthestPoints.size();
    for (int i = 0; i < listOfFurthestPoints.size(); ++i)
    {
        cogFP = cogFP + (invListSize * cv::Point2f(listOfFurthestPoints[i]));
    }

    std::cout << cogFP - centerOfMass << std::endl;
    float angle = acos((cogFP - centerOfMass).x / cv::norm(cogFP - centerOfMass)); // scalar product of [1,0] and point
    std::cout << angle * 180 / CV_PI << std::endl;

    cv::line(output, centerOfMass, cogFP, cv::Scalar(0, 255, 0), 1);
    cv::circle(output, centerOfMass, 5, cv::Scalar(0, 0, 255), 1);
    cv::circle(output, cogFP, 3, cv::Scalar(255, 0, 0), 1);


    cv::imwrite(outString, output);
    cv::imshow("input", input);
    cv::imshow("output", output);
    cv::waitKey(0);
    return 0;
}

this is the ouput for several rotations:

I would love to try the circle method, using RANSAC to find the best 2 circles, but maybe won't have the time...

Another way could be to find the turning points of the smoothed contour.