Flavia Colonna

Shearlet-Based Total Variation Diffusion for Denoising

We propose a shearlet formulation of the total variation (TV) method for denoising images. Shearlets have been mathematically proven to represent distributed discontinuities such as edges better than traditional wavelets and are a suitable tool for edge characterization. Common approaches in combining wavelet-like representations such as curvelets with TV or diffusion methods aim at reducing Gibbs-type artifacts after obtaining a nearly optimal estimate. We show that it is possible to obtain much better estimates from a shearlet representation by constraining the residual coefficients using a projected adaptive total variation scheme in the shearlet domain. We also analyze the performance of a shearlet-based diffusion method. Numerical examples demonstrate that these schemes are highly effective at denoising complex images and outperform a related method based on the use of the curvelet transform. Furthermore, the shearlet-TV scheme requires far fewer iterations than similar competitors.

Characterisation of the isometric composition operators on the Bloch space

In this paper, we characterise the analytic functions ’ mapping the open unit disk into itself whose induced composition operator C’ : f 7! f ’ is an isometry on the Bloch space. We show that such functions are either rotations of the identity function or have a factorisation ’ = gB where g is a non-vanishing analytic function from into the closure of , and B is an infinite Blaschke product whose zeros form a sequence {zn} containing 0 and a subsequence {znj } satisfying the conditions g(znj ) ! 1, and lim j!1 Y k6nj znj zk 1 znj zk = 1.

Generalized Discrete Radon Transforms and Their Use in the Ridgelet Transform

We introduce and study a new class of Radon transforms in a discrete setting for the purpose of applying them to the ridgelet and curvelet transforms. We give a detailed analysis of the p-adic case and provide a closed-form formula for an inverse of the p-adic Radon transform. We give conditions for a scaled version of the generalized discrete Radon transform to yield a tight frame, and discuss a direct Radon matrix method for the implementation of a local ridgelet transform. We then study the effectiveness of some types of the generalized Radon transforms in reducing a type of noise known as speckle that is present in synthetic aperture radar (SAR) imagery.

Bloch and normal functions and their relation

Following a brief introduction to Bloch and normal functions, several conditions, including a convergence theorem, are shown for determining them. In addition, since an exponential of any constant multiple of a Bloch function is always normal, we investigate whether or not the converse holds, and construct an example of a non-Bloch function such that the exponential of any constant multiple of it is normal.

New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space

Let ψ and φ be analytic functions on the open unit disk

ISOMETRIES AND SPECTRA OF MULTIPLICATION OPERATORS ON THE BLOCH SPACE

\mathbb{D}

Embeddings of trees in the hyperbolic disk

with φ(

POLYHARMONIC FUNCTIONS ON TREES

\mathbb{D}

Multiplication Operators between Lipschitz-Type Spaces on a Tree

) ⊆

Multiplication Operators on the Bloch Space of Bounded Homogeneous Domains

\mathbb{D}