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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

.

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