Piecewise H +H +H Images and the Mumford-Shah-Sobolev Model for Segmented Image Decomposition

Image modeling stays at the very core of image processing and low-level vision analysis. In the Bayesian framework, image modeling amounts to the introduction of suitable prior models, while in the Tikhonov framework [47], to the specification of regularity structures in order to better condition ill-posed inverse tasks such as denoising or deblurring. In the variational approach [3, 10, 40, 41, 42], images are assumed to belong to certain function spaces on the associated continuum domain Ω (or hypothesis classes as in the learning theory [13, 43]). Common classes include (1) Sobolev spaces W(Ω) with p ≥ 1, (2) the BV space BV(Ω) first introduced for images by Rudin, Osher, and Fatemi [38, 39] and its subspace—special BV functions SBV(Ω) [1], (3) Besov spaces Bq(L (Ω)) as in wavelet analysis [7, 15, 18, 29], and (4) Mumford and Shah’s free-boundary model H(Γ) ⊕ H(Ω | Γ) [31, 35], where H(Γ) denotes the class of closed “edges” (denoted by Γ ’s) with finite 1D Hausdorff measures, and H(Ω | Γ) Sobolev functions on Ω \ Γ for any given “edge” Γ (or using the terminology in topology and geometry, H(Ω | Γ) is a fiber at a given Γ). Mumford remarked in 1999 (also see [34]) that it is insufficient to model images merely using ordinary (or equivalently, measurable) functions. As an alternative, generalized functions or distributions may better serve the purpose, for example, to better