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Building multi-resolution 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, horizontal-vertical transform, 1 step, (c) 3-direction 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 one-direction analysis leads to separation of an orig-

inal image into two channels [8]. In the multi-directional case, one-directional analyses are applied direction by di- rection. Order is not important, since all combinations re- sult in the same set of channels. For four-directional 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 multi-directional anal- ysis. The original image contains four digital lines with dif-

ferent angles ( form (along

,

,

,

,

and and

). The 4-direction trans- ) gives 16 sub-channels

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 sub-bands. 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 4-direction transform steps is shown in Figure 3a).

4. DIRECTIONAL ORTHOGONAL TRANSFORMS AND FRAMES

We show below that directional transforms, when built on top of one-dimensional 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. 4-direction 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 high-pass 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 one-dimensional 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:

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