Wavelets and fractals location:Esneux (BE) date:26-28 April 2010
We shall consider the compression of piecewise smooth images. This means images that consist of (relatively few) smooth regions that have a smooth boundary. These boundaries are the only singularities in the image-function. It is well known that classical wavelets have a poor approximation behaviour on this kind of images. We present an approach that has its origin in computer graphics. An image can be considered as a geometric object when the pixel values are z-values perpendicular to the xy-plane. A triangulation of the xy-plane gives a mesh whose vertices interpolate the object in 3D space. That mesh can be decorated for example by polynomial wavelets. Successive resolution levels are added as differences to that mesh when the triangulation is refined. In particular we shall measure the difference for each edge of the mesh along the normal bisector of the edge in a plane (though that edge) normal to the xy-plane. That is what is called a normal offset. The piercing point is where that bisector pierces the surface. When the piercing points are projected onto the xy-plane, then that will define what the next finer triangulation will be, etc. That triangulation will be anisotropic and will naturally adapt itself by stretching along the line(s) of singulartites in the image. This method is still in its infancy. We hall give a survey of this idea, some problems and some ideas about the proper coding which approaches the performance of JPEG2000.