Features and Matching - Computer Vision for Visual Effects - page 120

Gaussian, σ = 1 Gaussian, σ = 1.2

0.2 0.2

0.15 0.15

0.1 0.1

0.05 0.05

0 0

5 5

5 5

0 0

0 0

y −5 −5 y −5 −5

x x

(a) (b)

Laplacian of Gaussian Difference of Gaussians

0.01 0.01

−0.01 −0.01

−0.03 −0.03

−0.05 −0.05

5 5

5 5

0 0

0 0

y y

−5 −5 −5 −5

x x

(c) (d)

Figure 4.8. (a) A Gaussian with σ = 1. (b) A Gaussian with σ = 1.2. (c) The Laplacian of Gaussian
with σ = 1. (d) The difference of the Gaussians in (b) and (a), normalized to have the same
maximum value as (c). We can see that the DoG is a good approximation to the LoG.

As with LoG detection, we seek maxima in both the spatial and scale dimensions to
detect features.
Lowe proposed a rearrangement of scale space to make the detection of local
maxima more computationally efficient. We define an octave of a Gaussian scale
space as a pair of images whose smoothing factors

differ by a factor of two. 8 Lowe's
idea was to define the constant factor k characterizing the scale space as k

σ

2 s ,so =

that each set of s 1 images results in an octave. The image beginning each octave

is downsampled by a factor of 2 in both directions, but the sizes of the sequence
of Gaussian filters applied at each octave remains the same. In this way, we avoid
applying very large filters to original-sized images, and instead apply same-sized
filters to increasingly smaller images. If we just took the first image in each octave, we
would have a Gaussian pyramid, as discussed in Section 3.1.2 . Figure 4.9 illustrates
the idea. Lowe suggested using s

+

= 3 intervals (that is, four images per octave), with a

base scale of σ = 1.6 applied to the first image in each octave (after downsampling).

0

8 While the word “octave” might imply a factor of eight is involved, the term originally comes from
music theory (i.e., there are eight notes in a major scale, which corresponds to a doubling of
frequency).