  We consider the task of image decomposition and we introduce a new model
coined directional global three-part decomposition (DG3PD) for solving it. As
key ingredients of the DG3PD model, we introduce a discrete multi-directional
total variation norm and a discrete multi-directional G-norm. Using these novel
norms, the proposed discrete DG3PD model can decompose an image into two parts
or into three parts. Existing models for image decomposition by Vese and Osher,
by Aujol and Chambolle, by Starck et al., and by Thai and Gottschlich are
included as special cases in the new model. Decomposition of an image by DG3PD
results in a cartoon image, a texture image and a residual image. Advantages of
the DG3PD model over existing ones lie in the properties enforced on the
cartoon and texture images. The geometric objects in the cartoon image have a
very smooth surface and sharp edges. The texture image yields oscillating
patterns on a defined scale which is both smooth and sparse. Moreover, the
DG3PD method achieves the goal of perfect reconstruction by summation of all
components better than the other considered methods. Relevant applications of
DG3PD are a novel way of image compression as well as feature extraction for
applications such as latent fingerprint processing and optical character
recognition.
