Building multiresolution decompositions along multi ple directions permits characterization and compression of phenomena other than just horizontal and vertical ones, as in the standard separable wavelet transform.
a) b) c)
Fig. 1. A simple object and its standard and directional transform; (a) Original image, (b) standard, horizontalvertical transform, 1 step, (c) 3direction transform, 1 step.
Figure 1 conceptually shows this. One step of a three direction transform does isolate the different directions, as well as combinations thereof, and this in an intuitive way.
3. SEPARABLE DIRECTIONAL TRANSFORM
In this section, we concentrate on building some simple di rectional separable transforms. Among them, the simplest
has four directions, namely
,
,
,
.
A onedirection analysis leads to separation of an orig
inal image into two channels [8]. In the multidirectional case, onedirectional analyses are applied direction by di rection. Order is not important, since all combinations re sult in the same set of channels. For fourdirectional analy sis, application of horizontal, vertical and both and (or ) directional analysis gives 16 channels where each channel contains different combination of kept orientations.
Figure 2 shows an example of the multidirectional anal ysis. The original image contains four digital lines with dif
ferent angles ( form (along
,
,
,
,
and and
). The 4direction trans ) gives 16 subchannels
that keep different sets of angles each.
Iteration of such a transform is also possible. Some practical problems may appear in the downsampling of the transform subbands. The downsampling lattice may be come extremely complex for a combination of the angles other than horizontal or vertical.
An example of the iteration of the 4direction transform steps is shown in Figure 3a).
4. DIRECTIONAL ORTHOGONAL TRANSFORMS AND FRAMES
We show below that directional transforms, when built on top of onedimensional orthogonal transforms or tight frames, again lead to orthogonal transforms or tight frames [7].
First, we need the following result. Assume a set of matrices corresponding to orthogonal transforms or tight
a) b)
Fig. 2. 4direction transform of an image with 4 differently ori ented lines. (a) Original image, (b) 16 transform channels.
a)
b)
Fig. 3. Two decomposition trees: (a) several steps of horizon
tal and vertical analysis followed by iterated steps of

degree
analysis applied on highpass channels; (b) several steps of hori zontal and vertical analysis only.
frames, satisfying:
where fined as:
is some positive constant. Then, the frame
de
satisfies
, where
.
So for example, taking an orthonormal wavelet trans form on the rows of an image, and separately taking an or thonormal wavelet transform on the columns leads to a tight
frame with
.
For the purpose of proving the orthogonality of such directional transforms, consider an image as a set of real
numbers defined on a finite lattice
. We can divide
this set into a series of subsets by taking directional lines at some angle as in (1). The division can be represented as
and
, where is the number of parallel lines. For example, for a
transform along angle
,
.
If a onedimensional unitary orthogonal transform is ap plied on the subsets it results in the same division of the
space
. Namely, for a set of transforms
, the following holds: