How to compute the orthonormal basis of a non square (rectangular) matrix

Question

I need to find a way of computing the orthonormal basis for range of a matrix. In matlab this function does it.

I need to do this in c/c++ and I am actually working with OpenCV

However, I haven't found anything that provides this capability in OpenCV.

I've tried working with cvSVD, but my results aren't correct.

Any clues?

Answer 1

This is in openCV, and it works with rectangular matrix as long as m>n, according to this paper

- (CvMat *) buildOrthonormal:(CvMat *) matrix {

    NSInteger rows = matrix->rows;
    NSInteger cols = matrix->cols;

    CvMat *Q = cvCreateMat(rows, cols, kMatrixType);
    CvMat *R = cvCreateMat(cols, cols, kMatrixType);  

    for (NSInteger k = 0; k < cols; k++) {
        cvSetReal2D(R, k, k, 0.0f);

        for (NSInteger i = 0; i < rows; i++) {
            double value = cvGetReal2D(R, k, k) + cvGetReal2D(matrix, i, k) * cvGetReal2D(matrix, i, k);
            cvSetReal2D(R, k, k, value);
        }
        cvSetReal2D(R, k, k, sqrt(cvGetReal2D(R, k, k)));    

        for (NSInteger i = 0; i < rows; i++) {
            double value = cvGetReal2D(matrix, i, k) / cvGetReal2D(R, k, k);
            cvSetReal2D(Q, i, k, value);
        }

        for (NSInteger j = k + 1; j < cols; j++) {
            cvSetReal2D(R, k, j, 0.0f);
            for (NSInteger i = 0; i < rows; i++) {
                double value = cvGetReal2D(R, k, j) + cvGetReal2D(Q, i, k) * cvGetReal2D(matrix, i, j);
                cvSetReal2D(R, k, j, value);
            }

            for (NSInteger i = 0; i < rows; i++) {
                double value = cvGetReal2D(matrix, i, j) - cvGetReal2D(R, k, j) * cvGetReal2D(Q, i, k);
                cvSetReal2D(matrix, i, j, value);
            }
        }
    }
    cvReleaseMat(&R);

    return Q;
}