Cartoon+Texture Image Decomposition

The algorithm first proposed in [3] stems from a theory proposed by Yves Meyer in [1]. The cartoon+texture algorithm decomposes any image f into the sum of a cartoon part, u , where only the image contrasted shapes appear, and a textural v part with the oscillating patterns. Such a decomposition f=u+v is analogous to the classical signal processing low pass-high pass filter decomposition. However, the cartoon part of an image actually contains strong edges, and therefore all frequencies, up to the high ones, while a texture can also contain middle and high frequencies. Thus, linear decomposition algorithms cannot make a clear cut separation between cartoon and textures. They blur out edges and take their high frequencies into the texture part. Conversely, they leave behind some texture in the low pass filtered cartoon part. Yves Meyer proposed to solve the problem by a variational problem containing two norms: the right decomposition f=u+v is the one where the cartoon part u has minimal total variation while the oscillatory component has a minimal norm in a dual space of BV. This second norm does not penalize oscillation: the higher the frequency of the oscillation of v, the smaller its norm. In [3] one can find a detailed history and analysis of variational algorithms and variants for the original Meyer formulation, which seems to start with [2]. The many variants proposed in the literature consider various functional spaces for the textural part (dual of BV, Besov spaces, etc.).