How to apply taylor series expansion to an image in opencv?

Question

For computer vision project, I am trying to implement D.Lowe paper(SIFT). I am stuck in translating this(check the ppt image) into code. Can you please tell me how to apply taylor series expansion into code(the formula is below). Do you know any functions in numpy I can use to calculate this D(x)? can you give an exmple code for D(x) in python.

Video Credits

Does this help? http://firsttimeprogrammer.blogspot.co.uk/2015/03/taylor-series-with-python-and-sympy.html — pookie, Nov 05 '17 at 16:45
Or this: https://pythonhosted.org/algopy/examples/series_expansion.html — pookie, Nov 05 '17 at 16:46
@pookie - the first link you mentioned are using for sin(x). I am still not sure how I can apply that here, since its `x = (x, y, sigma)`. Thanks for those links but its not helping much. — ajayramesh, Nov 05 '17 at 17:29
Have you had a look at the "Related" list? This one https://stackoverflow.com/questions/9532415/sift-taylor-expansion-working-out-subpixel-locations?rq=1 looks promising, doesn't it? — gboffi, Nov 05 '17 at 23:08

score -1 · Accepted Answer · answered Nov 09 '17 at 16:03

To understand better go through SIFT Taylor Expansion working out subpixel locations, later find below c++ code which is real implementation of SIFT.

static void interp_step( IplImage*** dog_pyr, int octv, int intvl, int r, int c,
             double* xi, double* xr, double* xc )
{
  CvMat* dD, * H, * H_inv, X;
  double x[3] = { 0 };

  dD = deriv_3D( dog_pyr, octv, intvl, r, c );
  H = hessian_3D( dog_pyr, octv, intvl, r, c );
  H_inv = cvCreateMat( 3, 3, CV_64FC1 );
  cvInvert( H, H_inv, CV_SVD );
  cvInitMatHeader( &X, 3, 1, CV_64FC1, x, CV_AUTOSTEP );
  cvGEMM( H_inv, dD, -1, NULL, 0, &X, 0 );

  cvReleaseMat( &dD );
  cvReleaseMat( &H );
  cvReleaseMat( &H_inv );

  *xi = x[2];
  *xr = x[1];
  *xc = x[0];
}


static CvMat* hessian_3D( IplImage*** dog_pyr, int octv, int intvl, int r,
              int c )
{
  CvMat* H;
  double v, dxx, dyy, dss, dxy, dxs, dys;

  v = pixval32f( dog_pyr[octv][intvl], r, c );
  dxx = ( pixval32f( dog_pyr[octv][intvl], r, c+1 ) + 
      pixval32f( dog_pyr[octv][intvl], r, c-1 ) - 2 * v );
  dyy = ( pixval32f( dog_pyr[octv][intvl], r+1, c ) +
      pixval32f( dog_pyr[octv][intvl], r-1, c ) - 2 * v );
  dss = ( pixval32f( dog_pyr[octv][intvl+1], r, c ) +
      pixval32f( dog_pyr[octv][intvl-1], r, c ) - 2 * v );
  dxy = ( pixval32f( dog_pyr[octv][intvl], r+1, c+1 ) -
      pixval32f( dog_pyr[octv][intvl], r+1, c-1 ) -
      pixval32f( dog_pyr[octv][intvl], r-1, c+1 ) +
      pixval32f( dog_pyr[octv][intvl], r-1, c-1 ) ) / 4.0;
  dxs = ( pixval32f( dog_pyr[octv][intvl+1], r, c+1 ) -
      pixval32f( dog_pyr[octv][intvl+1], r, c-1 ) -
      pixval32f( dog_pyr[octv][intvl-1], r, c+1 ) +
      pixval32f( dog_pyr[octv][intvl-1], r, c-1 ) ) / 4.0;
  dys = ( pixval32f( dog_pyr[octv][intvl+1], r+1, c ) -
      pixval32f( dog_pyr[octv][intvl+1], r-1, c ) -
      pixval32f( dog_pyr[octv][intvl-1], r+1, c ) +
      pixval32f( dog_pyr[octv][intvl-1], r-1, c ) ) / 4.0;

  H = cvCreateMat( 3, 3, CV_64FC1 );
  cvmSet( H, 0, 0, dxx );
  cvmSet( H, 0, 1, dxy );
  cvmSet( H, 0, 2, dxs );
  cvmSet( H, 1, 0, dxy );
  cvmSet( H, 1, 1, dyy );
  cvmSet( H, 1, 2, dys );
  cvmSet( H, 2, 0, dxs );
  cvmSet( H, 2, 1, dys );
  cvmSet( H, 2, 2, dss );

  return H;
}

reference: https://github.com/robwhess/opensift/blob/master/src/sift.c

How to apply taylor series expansion to an image in opencv?

1 Answers1