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
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
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.
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 , 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  but not on discrete space.
A line in
is a simple object, but its equivalent in
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  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
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 .
The line is determined
by its slope and shift by the following equation:
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
represents the slope and belongs to the
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