The purpose of this presentation is to describe a recent family of basis functions—the fractional B-splines—which appear to be intimately connected to fractional calculus. Among other properties, we show that they are the convolution kernels that link the discrete (finite differences) and continuous (derivatives) fractional differentiation operators. We also provide simple closed forms for the ...

Traditional wavelets have a number of vanishing moments that corresponds to their equivalent order of the derivation. They offer good energy compaction for piecewise smooth signals, but are less appropriate for more complex signals such as those originating in functional imaging ; e.g., the hemodynamic response after brain activation in functional magnetic resonance imaging (fMRI) and time acti...

We consider the problem of estimating a smooth function from noisy, sampled data. We use orthonormal bases of compactly supported wavelets to constriuct nonlinear function estimates which can significantly outperform evey linear method (kernel, smoothing spline, sieve, ...). Our estimates are simple nonlinear functions of the empirical wavelet coefficients and are asymptotically minimax over ce...

We propose a flexible and efficient wavelet construction for non-uniform B-spline curves and surfaces. The method allows to remove knots in arbitrary order minimizing the displacement of control points when a knot is re-inserted. Geometric detail subtracted from a shape by knot removal is represented by an associated wavelet coefficient replacing one of the control points at a coarser level of ...

The problem of change detection of time series with abrupt and smooth changes in the spectral characteristics is addressed. We first review the main characteristics of the discrete wavelet transform and the maximal overlap discrete wavelet transform. An algorithm for sequential change detection in time series is then reported based on the maximal overlap discrete wavelet transform and Bayesian ...

The projection method is a simple way of constructing functions and filters by integrating multidimensional functions and filters along parallel superplanes in the space domain. Equivalently expressed in the frequency domain, the projection method constructs a new function by simply taking a cross-section of the Fourier transform of a multidimensional function. The projection method is linked t...

Physical modelling of musical instruments is one possible approach to digital sound synthesis techniques. By the term physical modelling, we refer to the simulation of sound production mechanism of a musical instrument, which is modelled with reference to the physics using wave-guides. One of the fundamental parameters of such a physical model is the pitch, and so pitch period estimation is one...

We extend Schoenberg's B-splines to all fractional degrees α > − 2 . These splines are constructed using linear combinations of the integer shifts of the power functions x+ α (one-sided) or x * α (symmetric); in each case, they are αHölder continuous for α > 0. They satisfy most of the properties of the traditional B-splines; in particular, the Riesz basis condition and the two-scale relation, ...

We present multiresolution B-spline surfaces of arbitrary order defined over triangular domains. Unlike existing methods, the basic idea of our approach is to construct the triangular basis functions from their tensor product relatives in the spirit of box splines by projecting them onto the barycentric plane. The scheme works for splines of any order where the fundamental building blocks of th...

In a few-view or limited-angle computed tomography (CT), where the number of measurements is far fewer than image unknowns, the reconstruction task is an ill-posed problem. We present a spline-based sparse tomographic reconstruction algorithm where content-adaptive patch sparsity is integrated into the reconstruction process. The proposed method leverages closed-form Radon transforms of tensor-...