to : “ Coupling image restoration and segmentation : a generalized linear model / Bregman perspective ”

This supplementary text provides details about optimization trajectories of the alternating minimization (AM) solver based on level-set (LS) and and on the alternating split Bregman (ASB) (section 4), collects the main results from Banerjee et al. (2005) about the duality between the natural and the mean parametrization in a one-dimensional regular exponential family (section 1), adapts the proofs from the generalized linear model literature of the photometric estimation results (section 2), and provides explicit formulae for the w1 sub-problem of the ASB for Gaussian, Poisson and, Bernoulli noise models (section 3). 1 Duality in the Regular Exponential Family We collect results about the duality relation in the Regular Exponential Family (REF) as stated by Banerjee et al. (2005) (cf. their Definition 4, Lemma 1 and, Theorem 2), but with our notation and for a one dimensional REF. We further gather required results about the duality relation induced by the Legendre-Fenchel transform of the cumulant-generating function b of a REF (Banerjee et al. 2005; Rockafellar 1997). We use the notation int(·) for the interior of a set and dom(ψ) for the effective domain of the function ψ, namely the set of points in the domain of definition of ψ for which the function is finite. Definition 1 Let b be a real-valued function on R. Its conjugate function b is defined as: b(μ) = sup θ∈dom(b) (θμ− b(θ)) . We can now state the relevant duality results induced by the cumulant-generating function b between the natural parameter θ and the mean parameter μ of a onedimensional REF. MOSAIC Group ETH Zurich Universitätstr. 6, CH–8092 Zürich, Switzerland E-mail: [email protected], [email protected], [email protected] 2 Grégory Paul, Janick Cardinale, and Ivo F. Sbalzarini Lemma 1 Let b be the cumulant-generating function of a REF with natural parameter space Θ = int(dom(b)). Then b is a proper, closed, and convex function with int(Θ) = Θ and (Θ, b) is a convex function of Legendre type (Rockafellar 1997). Its conjugate function (Θ, b) satisfies: 1. (Θ, b) is a convex function of Legendre type with Θ = int(dom(b)). 2. (Θ, b) and (Θ, b) are Legendre duals of each other. 3. The gradient mapping b′ : Θ → Θ is a one-to-one mapping from the open convex set Θ onto the open convex set Θ. 4. The gradient mappings b′ and (b?)′ are continuous and (b?)′ = (b′)−1. Therefore, two points (θ, μ) ∈ Θ×Θ in (Legendre) duality are uniquely related by the Legendre transformations induced by the diffeomorphism b′: b(θ) = μ(θ) and (b)(μ(θ)) = θ(μ) . The last result concerns the dual relationship between the Bregman divergences Bb and Bb? induced by b and its conjugate b . For any two pairs of points ((p, q), (p, q)) ∈ Θ × (Θ) in duality: Bb(p ‖ q) = Bb?(q ‖ p) . 2 Proofs We adapt classical proofs from the GLM literature (Nelder and Wedderburn 1972; McCullagh and Nelder 1989) to our image-processing problem. The results are not new and only provided here for the convenience of the reader. The only differences with the statistics literature are the continuous formulation of the results, requiring basics results about interchanging derivation and integration, and the sign convention. We recall that ` denotes the anti-log-likelihood function. Proof (Proof of Result 2) The proof is classical (McCullagh and Nelder 1989) and amounts to writing the appropriate chain rule in order to ease the substitutions of the mean function μ(x) and the variance function V . The only difference is the sign convention. s(β,x) = ∂` ∂θ dθ dμ dμ dη ∂η ∂β (1) = ∂` ∂θ ( dμ dθ )−1( dη dμ )−1 ∂η ∂β (2) = b′(θ)− u0(x) a(x, φ) 1 b′′(θ) ( g(μ(x,β)) )−1 X(x) (3) = μ(x,β)− u0(x) σ2(x,β)g′(μ(x,β))2 g(μ(x,β))X(x) . (4) 1 A convex function (Θ, f) of Legendre type has a non-empty domain, is strictly convex, differentiable, and the norm of its gradient is diverging to infinity for sequences in Θ converging to boundary points. Supplementary text 3 We emphasized here the chain rule (1) and the derivatives of interest (2). The following steps use the properties of the GLM: equations (3) and (4) use the definition of the link function and the properties of the mean and variance functions. Concerning the whole-image score function s(β), the regularity condition needed is that `(x,β) is regular enough such that differentiation and integration can be interchanged: