Multiscale texture separation

TL;DR

A new theorem is shown which shows that, by combining the decomposition model and a well-chosen Littlewood--Paley filter, it is possible to extract almost perfectly a certain class of textures.

Abstract

In this paper, we investigate theoretically the behavior of Meyer's image cartoon + texture decomposition model. Our main results is a new theorem which shows that, by combining the decomposition model and a well chosen Littlewood-Paley filter, it is possible to extract almost perfectly a certain class of textures. This theorem leads us to the construction of a parameterless multiscale texture separation algorithm. Finally, we propose to extend this algorithm into a directional multiscale texture separation algorithm by designing a directional Littlewood-Paley filter bank. Several experiments show the efficiency of the proposed method both on synthetic and real images.

Figures (14)

• Figure 4.1: Illustration of the Fourier transform magnitude of a $\Delta_j$ operator in one dimension.
• Figure 4.2: Reference synthetic components.
• Figure 4.3: The whole synthetic test image $f$.
• Figure 4.4: Component $w$ provided by the decomposition and its Littlewood-Paley filtered version $\Delta_j[w](x)$.
• Figure 4.5: Log-amplitude of Fourier transforms of $\Delta_j[w](x)$ (left) and $\Delta_j[f](x)$ (right).

Theorems & Definitions (14)

• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• proof
• Corollary 3.4
• Definition 4.1
• Lemma 4.2
• proof
• Lemma 4.3
• Theorem 4.4