# DIRECTIONAL WAVELET TRANSFORMS AND FRAMES

# Vladan Velisavljevic´ , Pier Luigi Dragotti , Martin Vetterli

´ Laboratoire de Communications Audiovisuelles (LCAV), Ecole Polytechnique Fe´de´ral de Lausanne

(EPFL), CH - 1015 Lausanne, Switzerland Electrical Engineering and Computer Science Departmant, University of California at Berkeley, Berkeley CA 94720

# ABSTRACT

The application of the wavelet transform in image process- ing is most frequently based on a separable transform. Lines and columns in an image are treated independently and the basis functions are simply products of corresponding one- dimensional functions. Such a method keeps simplicity in design and computation. In this paper, a new two-dimensio- nal approach is proposed, which retains the simplicity of separable processing, but allows more directionalities. The method can be applied in many areas like denoising, non- linear approximation and compression. The results on non- linear approximation and denoising show interesting gains compared to the standard two-dimensional analysis.

possible to apply orthogonal wavelet transforms or wavelet frame decompositions. Many useful properties are again in- herited from the one-dimensional case (e.g. orthogonality). Techniques like denoising can be applied easily again by using more directions.

The outline of the paper is as follows.

Section 2 de-

fines digital directions and associated partitions of

.

It

also shows how directional elements in an image are treated by a separable directional transforms. Section 3 develops simple directional transforms and shows how to build a hi- erarchy of directional decompositions. Section 4 shows how to obtain orthogonal transforms and tight frames from these elementary directional transforms. Finally, Section 5 shows applications in non-linear approximation and denoising. We

conclude in Section 6.

# 1. INTRODUCTION

# 2. DIGITAL DIRECTIONS

Most wavelet transforms applied on images are separable transforms, that is, they treat lines and columns indepen- dently. The resulting basis functions are simply products of their one-dimensional counter parts.

The advantages of such an approach are conceptual sim- plicity, low computational complexity and inheritance of most properties from the one-dimensional case (e.g. reg- ularity of the basis functions). The drawbacks are a par- tial treatment of the complexity inherent in two-dimensional images, which goes well beyond horizontal and vertical di- rections. Non-separable approaches [4], in particular us- ing directional filter banks [1, 5], have been investigated, showing the potential of truly non-separable methods. Such methods come at a price in terms of design and computa- tional complexity. Some separable approaches have been made in [9] but not on discrete space.

A line in

is a simple object, but its equivalent in

is

a bit more complicated, as well known in raster graphics for example. A solution has been proposed in discrete to- mography with the finite Radon transform [2] but involves wrap-around due to periodization. Here, we are interested in discrete approximations to real lines.

# Then, we define a digital line of angle

as a one-dimensi-

onal set of pixels approximately along a line of angle

. In

addition, the digital line and its shifts along orthogonal di- rection have to tile . While there are many solutions to this problem, a simple and tractable one is to use the analyt-

ical definition of a discrete line [3].

The line is determined

by its slope and shift by the following equation:

(1)

In the present paper, we wish to retain the simplicity of separable wavelet transforms while realizing some of the potential of non-separable schemes. We do this by intro- ducing a directional wavelet transform that acts much like a standard separable transform, but allows more direction- ality. This is done by introducing ”digital directions” that partition the discrete plane. Along these directions, it is then

where range

represents the slope and belongs to the

, and

represents the real-valued shift

parameter. The definition of a discrete approximation of a line insures that each pixel belongs exactly to one line for a chosen slope. Lines with slopes out of the range may be obtained by symmetry, rotating and flipping vertically the

space. This gives access to a wealth of directions in

.