Mathematical Problems in Image Processing: Partial by Gilles Aubert

By Gilles Aubert

Partial differential equations and variational equipment have been brought into picture processing approximately 15 years in the past, and in depth examine has been conducted for the reason that then. the most objective of this paintings is to offer the diversity of photo research purposes and the best arithmetic concerned. it really is meant for 2 audiences. the 1st is the mathematical neighborhood, to teach the contribution of arithmetic to this area and to focus on a few unresolved theoretical questions. the second one is the pc imaginative and prescient group, to provide a transparent, self-contained, and international assessment of the maths curious about photo processing difficulties. This ebook might be beneficial to researchers and graduate scholars in arithmetic and machine imaginative and prescient.

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Extra info for Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations

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We answer the following questions: what are the good hypotheses ensuring the existence and the uniqueness of a minimizer? What can be said when some assumptions are missing? 3). 4 concerns the notion of Γ-convergence which is a notion of convergence for functionals. The Γ-convergence theory is particularly useful to approximate free discontinuity or ill-posed problems. This notion will be used several times in all the book. 2 presents the space BV (Ω) of functions of bounded variation. e. across edges).

The condition (iii) ensures that the problem has a meaning. c. while coercivity is related to the compactness. Before going further, let us illustrate on three examples the importance of coercivity, reflexivity and convexity. Examples Let Ω = ] 0, 1 ]. We propose below some classical examples where either coercivity, reflexivity or convexity are no longer true: (A) Weierstrass (N = m = 1) Let f defined by f (x, u, ξ) = xξ 2 and let us denote: m = inf 1 0 2 x (u (x)) dx with u(0) = 1 and u(1) = 0 .

E. x 48 2. Mathematical preliminaries u divϕ dx = − σ · ϕ dµ for all ϕ ∈ C01 (Ω)N Ω Ω which means that Du is a vector-valued Radon measure (Du = σ dµ). An important example is the case when u = χA , the characteristic function of a subset A of RN . Then: divϕ dx; ϕ ∈ C01 (Ω)N , |ϕ|L∞ (Ω) ≤ 1 . |Du| = sup Ω A If this supremum is finite, A is called a set of finite perimeter in Ω and we note: |Du| = PerΩ (A). Ω If ∂A is smooth, PerΩ (A) coincides with the classical length (N = 2) or surface area (N = 3).

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