Aaron Luttman

Total variation-penalized Poisson likelihood estimation for ill-posed problems

The noise contained in data measured by imaging instruments is often primarily of Poisson type. This motivates, in many cases, the use of the Poisson negative-log likelihood function in place of the ubiquitous least squares data fidelity when solving image deblurring problems. We assume that the underlying blurring operator is compact, so that, as in the least squares case, the resulting minimization problem is ill-posed and must be regularized. In this paper, we focus on total variation regularization and show that the problem of computing the minimizer of the resulting total variation-penalized Poisson likelihood functional is well-posed. We then prove that, as the errors in the data and in the blurring operator tend to zero, the resulting minimizers converge to the minimizer of the exact likelihood function. Finally, the practical effectiveness of the approach is demonstrated on synthetically generated data, and a nonnegatively constrained, projected quasi-Newton method is introduced.

Uniform algebra isomorphisms and peripheral multiplicativity

Let φ: A→ B be a surjective operator between two uniform alge-(1) =1 we Show that if φ satisfies the peripheral multiplicativity bras with φ(1) = 1. We show that if if satisfies the peripheral multiplicativity condition σ π (φ(f)φ(g)) = σ π (fg) for all f,g ∈ A, where σ π (f) is the peripheral spectrum of f, then φis an isometric algebra isomorphism from A onto B. One of the consequences of this result is that any surjective, unital, and multiplicative operator that preserves the peripheral ranges of algebra elements is an isometric algebra isomorphism. We describe also the structure of general, not necessarily unital, surjective and peripherally multiplicative operators between uniform algebras.

Norm conditions for uniform algebra isomorphisms

AbstractIn recent years much work has been done analyzing maps, not assumed to be linear, between uniform algebras that preserve the norm, spectrum, or subsets of the spectra of algebra elements, and it is shown that such maps must be linear and/or multiplicative. Letting A and B be uniform algebras on compact Hausdorff spaces X and Y, respectively, it is shown here that if λ ∈ ℂ / {0} and T: A → B is a surjective map, not assumed to be linear, satisfying

Generalized weak peripheral multiplicativity in algebras of Lipschitz functions

\left\| {T(f)T(g) + \lambda } \right\| = \left\| {fg + \lambda } \right\|\forall f,g \in A,

Bayesian Abel Inversion in Quantitative X-Ray Radiography

then T is an ℝ-linear isometry and there exist an idempotent e ∈ B, a function κ ∈ B with κ2 = 1, and an isometric algebra isomorphism

A Variational Approach to Video Segmentation for Botanical Data

\tilde T:{\rm A} \to Be \oplus \bar B(1 - e)