Computation of the Tree of Bilinear Level Lines

Abstract

Chapter 6 proposed an algorithm computing the tree of shapes based on an interpretation of the discrete image as a piecewise constant bivariate function. The corresponding interpolation operator, commonly known as nearest neighbor or order 0 interpolation, does not removes the pixelization effect. Bilinear or order 1 interpolation is notoriously preferable in this respect. This interpretation of the discrete image as a continuous function by bilinear interpolation is also amenable to a Fast Level Set Transform, which is presented in this chapter. The behavior of level lines of the interpolated function is proven to be easily classified, leading to an algorithm that borrows the same ideas as the FLST with appropriate modifications to handle the singularities. An implementation in the open source software suite MegaWave ( http://megawave.cmla.ens-cachan.fr ) is in the module flst bilinear. At this point, the reader may well wonder why more regular interpolations, like higher order spline interpolations (for example bicubic) or even zero padding of the Fourier transform, are not considered. The reason is that the behavior of the level lines becomes much more complex, as they are implicit solutions of higher degree polynomial equations. We are not aware of any work attempting the classification of these behaviors.