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Here is another way to achieve the same effect. Though it lacks the scale part which, I was trying to understand that by asking the question in the link in my comment. Anyway, here is the code:

static void meshgrid(InputArray _xgv, InputArray _ygv, OutputArray _X, OutputArray _Y)
{
    Mat xgv = _xgv.getMat();
    Mat ygv = _ygv.getMat();

  cv::repeat(xgv.reshape(1,1), ygv.total(), 1, _X);
  cv::repeat(ygv.reshape(1,1).t(), 1, xgv.total(), _Y);
}

// helper function (maybe that goes somehow easier)
static void meshgrid/*Test*/(const cv::Range &xgv, const cv::Range &ygv, OutputArray _X, OutputArray _Y)
{
  std::vector<float> t_x, t_y;
  for (int i = xgv.start; i <= xgv.end; i++) t_x.push_back(i);
  for (int i = ygv.start; i <= ygv.end; i++) t_y.push_back(i);

  Mat __X, __Y;
  meshgrid(cv::Mat(t_x), cv::Mat(t_y), __X, __Y);

  Mat(__X).copyTo(_X);
  Mat(__Y).copyTo(_Y);
}

void radialDistortion(InputArray _src, OutputArray _dst, double Cx, double Cy, double coef, bool scale = true)
{
    // die if distortion parameters are not correct
    CV_Assert(Cx >= 0 && Cy >= 0/* && k >= 0*/);

    Mat src = _src.getMat();

    int ncols = src.cols;
    int nrows = src.rows;

    Mat xi, yi;
    meshgrid(Range(1, ncols), Range(1, nrows), xi, yi);

    int imid_x = Cx;
    int imid_y = Cy;

    Mat xt = xi - imid_x;
    Mat yt = yi - imid_y;

    xt = xt.t();
    yt = yt.t();

    if (!xt.isContinuous())
        xt = xt.clone();

    if (!yt.isContinuous())
        yt = yt.clone();

    xt = xt.reshape(0, xt.rows * xt.cols);
    yt = yt.reshape(0, yt.rows * yt.cols);

    Mat r, theta;
    cartToPolar(xt, yt, r, theta);

    Mat s = r + coef * r.mul(r.mul(r));

    Mat ut, vt;
    polarToCart(s, theta, ut, vt);

    Mat u, v;

    if (!ut.isContinuous())
        ut = ut.clone();

    if (!vt.isContinuous())
        vt = vt.clone();

    u = ut.reshape(0, ncols) + imid_x;
    v = vt.reshape(0, ncols) + imid_y;
    u = u.t();
    v = v.t();

    remap(src, _dst, u, v, CV_INTER_LINEAR, BORDER_CONSTANT);
}

int main()
{
    Mat input = imread("lena.jpg"/*, CV_LOAD_IMAGE_GRAYSCALE*/); // it works with grayscale images as well ;-)

    if(!input.data || input.empty())
        cout << "Problem loading image!!!" << endl;

    Mat output;
    radialDistortion(input, output, round(static_cast<float>(input.cols) / 2), round(static_cast<float>(input.rows) / 2), 0.000003, false);

    Mat canvas = Mat::zeros(input.rows, input.cols*2+10, input.type());

    input.copyTo(canvas(Range::all(), Range(0, output.cols)));
    output.copyTo(canvas(Range::all(), Range(output.cols+10, output.cols*2+10)));

    // if it is too big to fit on the screen, then scale it down by 2, hopefully it'll fit :-)
    if(canvas.cols > 1920)
    {
        resize(canvas, canvas, Size(canvas.cols/2, canvas.rows/2));
    }

    imshow("canvas", canvas);

    waitKey(0);
    return 0;
}

if you have some time you can search more about it ;-).

Here is another way to achieve the same effect. Though it lacks the scale part which, part. I was trying to understand that part and implemented here as well by asking the question in the link in my comment. Anyway, here is the code:

static void meshgrid(InputArray _xgv, InputArray _ygv, OutputArray _X, OutputArray _Y)
{
    Mat xgv = _xgv.getMat();
    Mat ygv = _ygv.getMat();

  cv::repeat(xgv.reshape(1,1), ygv.total(), 1, _X);
  cv::repeat(ygv.reshape(1,1).t(), 1, xgv.total(), _Y);
}

// helper function (maybe that goes somehow easier)
static void meshgrid/*Test*/(const cv::Range &xgv, const cv::Range &ygv, OutputArray _X, OutputArray _Y)
{
  std::vector<float> t_x, t_y;
  for (int i = xgv.start; i <= xgv.end; i++) t_x.push_back(i);
  for (int i = ygv.start; i <= ygv.end; i++) t_y.push_back(i);

  Mat __X, __Y;
  meshgrid(cv::Mat(t_x), cv::Mat(t_y), __X, __Y);

  Mat(__X).copyTo(_X);
  Mat(__Y).copyTo(_Y);
}

void radialDistortion(InputArray _src, OutputArray _dst, double Cx, double Cy, double coef, bool scale = true)
{
    // die if distortion parameters are not correct
    CV_Assert(Cx >= 0 && Cy >= 0/* && k >= 0*/);

    Mat src = _src.getMat();

    int ncols = src.cols;
    int nrows = src.rows;

    Mat xi, yi;
    meshgrid(Range(1, ncols), Range(1, nrows), xi, yi);

    int imid_x = Cx;
    int imid_y = Cy;

    Mat xt = xi - imid_x;
    Mat yt = yi - imid_y;

    xt = xt.t();
    yt = yt.t();

    if (!xt.isContinuous())
        xt = xt.clone();

    if (!yt.isContinuous())
        yt = yt.clone();

    xt = xt.reshape(0, xt.rows * xt.cols);
    yt = yt.reshape(0, yt.rows * yt.cols);

    Mat r, theta;
    cartToPolar(xt, yt, r, theta);

    Mat s = r + coef * r.mul(r.mul(r));

    Mat ut, vt;
    polarToCart(s, theta, ut, vt);

    Mat u, v;

    if (!ut.isContinuous())
        ut = ut.clone();

    if (!vt.isContinuous())
        vt = vt.clone();

    u = ut.reshape(0, ncols) + imid_x;
    v = vt.reshape(0, ncols) + imid_y;
    u = u.t();
    v = v.t();

    remap(src, _dst, u, v, CV_INTER_LINEAR, BORDER_CONSTANT);
}

int main()
{
    Mat input = imread("lena.jpg"/*, CV_LOAD_IMAGE_GRAYSCALE*/); // it works with grayscale images as well ;-)

    if(!input.data || input.empty())
        cout << "Problem loading image!!!" << endl;

    Mat output;
    radialDistortion(input, output, round(static_cast<float>(input.cols) / 2), round(static_cast<float>(input.rows) / 2), 0.000003, false);

    Mat canvas = Mat::zeros(input.rows, input.cols*2+10, input.type());

    input.copyTo(canvas(Range::all(), Range(0, output.cols)));
    output.copyTo(canvas(Range::all(), Range(output.cols+10, output.cols*2+10)));

    // if it is too big to fit on the screen, then scale it down by 2, hopefully it'll fit :-)
    if(canvas.cols > 1920)
    {
        resize(canvas, canvas, Size(canvas.cols/2, canvas.rows/2));
    }

    imshow("canvas", canvas);

    waitKey(0);
    return 0;
}

if you have some time you can search more about it ;-).