Computing the Scale Space using the Laplace-Beltrami Operator (LPO)

Using the LBO we compute the scale space of omnidirectional images. We use the initial scale scale_0 = 0.8 and the definition of Arican's t_i = k^(2i)*scale_0^2. In opposition to other approaches that use the standard deviation. The full set of scales used are:

scale 1 = 0.8

scale 2 = 1.0159

scale 3 = 1.61.27

scale 4 = 2.56

We use the last image in each scale downsampled as the first one in the next scale. Here we show the first and last octaves.

FIRST OCTAVE

LAST OCTAVE

When the scale space is built, we perform the steps of the normal SIFT. We compute the differences of smoothed images to approximate the Laplacian of Gaussians (LoG). We observe the differences correspondig to the first and last octaves.

FIRST OCTAVE

LAST OCTAVE

Then the extrema are computed. Here we observe some results for different rotations and scales.

ORIGINAL AND DOUBLE SCALED:

ORIGINAL AND ROTATED