Serena Morigi

Cascadic Multiresolution Methods for Image Deblurring

This paper investigates the use of cascadic multiresolution methods for image deblurring. Iterations with a conjugate gradient-type method are carried out on each level, and terminated by a stopping rule based on the discrepancy principle. Prolongation is carried out by nonlinear edge-preserving operators, which are defined via PDEs associated with Perona-Malik or total variation-type models. Computed examples demonstrate the effectiveness of the methods proposed.

A fast interactive reverse-engineering system

A new method of reverse engineering for fast, simple and interactive acquisition and reconstruction of a virtual three-dimensional (3D) model is presented. We propose an active stereo acquisition system, which makes use of two infrared cameras and a wireless active-pen device, supported by a reconstruction method based on subdivision surfaces. In the 3D interactive hand sketching process the user draws and refines the 3D style-curves, which characterize the shape to be constructed, by simply dragging the active-pen device; then the system automatically produces a low-resolution mesh that is naturally refined through subdivision surfaces. Several examples demonstrate the ability of the proposed advanced design methodology to produce complex 3D geometric models by the interactive and iterative process that provides the user with a real-time visual feedback on the ongoing work.

Segmentation of 3D Tubular Structures by a PDE-Based Anisotropic Diffusion Model

Many different approaches have been proposed for segmenting vessels, or more generally tubular-like structures from 2D/3D images. In this work we propose to reconstruct the boundaries of 2D/3D tubular structures by continuously deforming an initial distance function following the Partial Differential Equation (PDE)-based diffusion model derived from a minimal volume-like variational formulation. The gradient flow for this functional leads to a non-linear curvature motion model. An anisotropic variant is provided which includes a diffusion tensor aimed to follow the tube geometry. Space discretization of the PDE model is obtained by finite volume schemes and semi-implicit approach is used in time/scale. The use of an efficient strategy to apply the linear system iterative solver allows us to reduce significantly the numerical effort by limiting the computation near the structure boundaries. We illustrate how the proposed method works to segment 2D/3D images of synthetic and medical real data representing branching tubular structures.

An adaptive norm algorithm for image restoration

We propose an adaptive norm strategy designed for the re-storation of images contaminated by blur and noise. Standard Tikhonov regularization can give good results with Gaussian noise and smooth images, but can over-smooth the output. On the other hand, L1 -TV (Total Variation) regularization has superior performance with some non-Gaussian noise and controls both the size of jumps and the geometry of the object boundaries in the image but smooth parts of the recovered images can be blocky. According to a coherence map of the image which is obtained by a threshold structure tensor, and can detect smooth regions and edges in the image, we apply L2 -norm or L1 -norm regularization to different parts of the image. The solution of the resulting minimization problem is obtained by a fast algorithm based on the half-quadratic technique recently proposed in [2] for L1 -TV regularization. Some numerical results show the effectiveness of our adaptive norm image restoration strategy.

Nonlocal surface fairing

We propose a new variational model for surface fairing. We extend nonlocal smoothing techniques for image regularization to surface smoothing or fairing, with surfaces represented by triangular meshes. Our method is able to smooth the surfaces and preserve features due to geometric similarities using a mean curvature based local geometric descriptor. We present an efficient two step approach that first smoothes the mean curvature normal map, and then corrects the surface to fit the smoothed normal field. This leads to a fast implementation of a feature preserving fourth order geometric flow. We demonstrate the efficacy of the model with several surface fairing examples.

Edge-driven Image Interpolation using Adaptive Anisotropic Radial Basis Functions

This paper investigates the image interpolation problem, where the objective is to improve the resolution of an image by dilating it according to a given enlargement factor. We present a novel interpolation method based on Radial Basis Functions (RBF) which recovers a continuous intensity function from discrete image data samples. The proposed anisotropic RBF interpolant is designed to easily deal with the local anisotropy in the data, such as edge-structures in the image. Considering the underlying geometry of the image, this algorithm allows us to remove the artifacts that may arise when performing interpolation, such as blocking and blurring. Computed examples demonstrate the effectiveness of the method proposed by visual comparisons and quantitative measures.

Framelet-Based algorithm for segmentation of tubular structures

Framelets have been used successfully in various problems in image processing, including inpainting, impulse noise removal, super-resolution image restoration, etc. Segmentation is the process of identifying object outlines within images. There are quite a few efficient algorithms for segmentation that depend on the partial differential equation modeling. In this paper, we apply the framelet-based approach to identify tube-like structures such as blood vessels in medical images. Our method iteratively refines a region that encloses the possible boundary or surface of the vessels. In each iteration, we apply the framelet-based algorithm to denoise and smooth the possible boundary and sharpen the region. Numerical experiments of real 2D/3D images demonstrate that the proposed method is very efficient and outperforms other existing methods.

Geometric surface evolution with tangential contribution

Surface processing tools based on Partial Differential Equations (PDEs) are useful in a variety of applications in computer graphics, digital animation, computer aided modelling, and computer vision. In this work, we deal with computational issues arising from the discretization of geometric PDE models for the evolution of surfaces, considering both normal and tangential velocities. The evolution of the surface is formulated in a Lagrangian framework. We propose several strategies for tangential velocities, yielding uniform redistribution of mesh points along the evolving family of surfaces, preventing computational instabilities and increasing the mesh regularity. Numerical schemes based on finite co-volume approximation in space will be considered. Finally, we describe how this framework may be employed in applications such as mesh regularization, morphing, and features preserving surface smoothing.

Reconstructing surfaces from sketched 3D irregular curve networks

This paper presents a system for designing free-form surfaces starting from a sketched 3D irregular curve network. By simply dragging a smart-pen device in space, the user draws and refines arbitrary 3D style-curves that define an outline of the desired shape. Unlike previous touch-based sketching systems, the user-drawn strokes can both stay on the model surface to reconstruct parts of an existing object, or freely sketch 3D style-lines of non-existing parts to design new geometry. The wireless smart-pen device is supported by an active stereo acquisition system which makes use of two infrared cameras. For a given set of 3D curves, the system automatically constructs a low-resolution mesh that is naturally refined to produce a smooth surface which preserves curvature features defined by the user on the curve network. The interpolating surface is obtained by applying a high-order diffusion flow. We present an efficient two step approach that first diffuses curvature values preserving the curvature constraints, and then corrects the surface to fit the resulting curvature vector field and interpolating the 3D curve network. This leads to fast implementation of a feature preserving fourth order geometric flow. We show several examples to demonstrate the ability of the proposed advanced design methodology to create sophisticated models possibly having sharp creases and corners.

Noise-reducing cascadic multilevel methods for linear discrete ill-posed problems

Cascadic multilevel methods for the solution of linear discrete ill-posed problems with noise-reducing restriction and prolongation operators recently have been developed for the restoration of blur- and noise-contaminated images. This is a particular ill-posed problem. The multilevel methods were found to determine accurate restorations with fairly little computational work. This paper describes noise-reducing multilevel methods for the solution of general linear discrete ill-posed problems.