Graphics Reference
In-Depth Information
(a)
(b)
(c)
(d)
Figure 4.6.
(a) An original image. (b) Harris-Laplace features. (c) Laplacian-of-Gaussian features
(i.e., obtained with the trace of the Hessian). Note that the detector responds to both blobs and
edges. (d) Hessian-Laplace features (i.e., obtained with the determinant of the Hessian). The
detector does not respond to edges.
We can substitute the Hessian matrix and either its trace or determinant in Step 2
on p.
115
to obtain a feature detector that responds strongly to
blobs
, as illustrated in
the bottom row of Figure
4.6
. Using the trace, that is, the Laplacian, has a pleas-
ing symmetry in that we can view the detector as selecting local maxima of the
same function in both the spatial and scale dimensions. These features are called
Laplacian-of-Gaussian
or
LoG
features, since we're computing the Laplacian of a
Gaussian-smoothed image at a given scale. That is, to detect LoG features, we com-
pute the quantity in Equation (
4.24
) at every
(
x
,
y
,
σ
)
, and find points where this
D
function of three parameters is locally maximal.
Figures
4.6
c-d illustrate that the determinant of the Hessian does a better job than
the trace for rejecting long, thin structures and finding well-proportioned blobs. This
approach (using the determinant of the Hessian for detection and its trace for scale
selection) produces what are called
Hessian-Laplace
features. One can also require
that the trace and determinant of the Hessian are simultaneously maximized [
326
].
All of these features are scale-covariant.
Bay et al. [
33
] noted that the discreteGaussianfilters used in the computationof the
scale-normalized Hessian could be approximated by extremely simple box filters, as
illustrated in Figure
4.7
. Since box filters only involve simple sums and differences of
pixels, they can be applied very quickly compared to filters with floating-point coeffi-
cients. If integral images [
516
] are used for the computation, the speed of applying the
box filter is independent of the filter size. Bay et al. proposed to use the box filters in
an approximation of the Hessian's determinant, resulting inwhat they called the
Fast
Hessian
detector. We will discuss additional fast feature detectors in Section
4.1.6
.
