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3.2 Linear filtering 105

(a) (b) (c)
Figure 3.15 Second-order steerable filter (Freeman 1992) c 1992 IEEE: (a) original image of Einstein; (b)
orientation map computed from the second-order oriented energy; (c) original image with oriented structures
enhanced.

directional derivative ∇ = ∂ , which is obtained by taking the dot product between the
ˆ u ∂ˆu
gradient field ∇ and a unit direction ˆu = (cos θ, sin θ),
ˆ u ·` (G ∗ f)= ∇ (G ∗ f)=(∇ G) ∗ f. (3.27)
ˆ u
ˆ u

The smoothed directional derivative filter,
∂G ∂G
G = uG x + vG y = u + v , (3.28)
ˆ u ∂x ∂y
where ˆu =(u, v), is an example of a steerable filter, since the value of an image convolved
with G can be computed by first convolving with the pair of filters (G x ,G y ) and then
ˆ u
steering the filter (potentially locally) by multiplying this gradient field with a unit vector ˆu
(Freeman and Adelson 1991). The advantage of this approach is that a whole family of filters
can be evaluated with very little cost.
G , which is the result
How about steering a directional second derivative filter ∇ ·` ˆ u ˆu
ˆ u
of taking a (smoothed) directional derivative and then taking the directional derivative again?
For example, G xx is the second directional derivative in the x direction.
At first glance, it would appear that the steering trick will not work, since for every di-
rection ˆu, we need to compute a different first directional derivative. Somewhat surprisingly,
Freeman and Adelson (1991) showed that, for directional Gaussian derivatives, it is possible
to steer any order of derivative with a relatively small number of basis functions. For example,
only three basis functions are required for the second-order directional derivative,

2
2
G ˆ uˆu = u G xx +2uvG xy + v G yy . (3.29)
Furthermore, each of the basis filters, while not itself necessarily separable, can be computed
using a linear combination of a small number of separable filters (Freeman and Adelson
1991).
This remarkable result makes it possible to construct directional derivative filters of in-
creasingly greater directional selectivity, i.e., filters that only respond to edges that have
strong local consistency in orientation (Figure 3.15). Furthermore, higher order steerable