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We present complex rotation-covariant multiresolution families aimed for image analysis. Since they are complex-valued functions, they provide the important phase information, which is missing in the discrete wavelet transform with real wavelets. Our basis elements have nice properties in Hilbert space such as smoothness of fractional order α ∈

R ^{ + }

. The corresponding filters allow a FFT-based implementation and thus provide a fast algorithm for the wavelet transform.

Hex-splines are a novel family of bivariate splines, which are well suited to handle hexagonally sampled data. Similar to classical 1D B-splines, the spline coefficients need to be computed by a prefilter. Unfortunately, the elegant implementation of this prefilter by causal and anti-causal recursive filtering is not applicable for the (non-separable) hex-splines. Therefore, in this paper we introduce a novel approach from the viewpoint of approximation theory. We propose three different recursive filters a

We propose a complex generalization of Schoenberg's cardinal splines. To this end, we go back to the Fourier domain definition of the B-splines and extend it to complex-valued degrees. We show that the resulting complex B-splines are piecewise modulated po ...

Recently, we have proposed a novel family of bivariate, non-separable splines. These splines, called "hexsplines" have been designed to deal with hexagonally sampled data. Incorporating the shape of the Voronoi cell of a hexagonal lattice, they preserve the twelve-fold symmetry of the hexagon tiling cell. Similar to B-splines, we can use them to provide a link between the discrete and the continuous domain, which is required for many fundamental operations such as interpolation and resampling. The question

Consider the problem of sampling signals which are not bandlimited, but still have a finite number of degrees of freedom per unit of time, such as, for example, nonuniform splines or piecewise polynomials, and call the number of degrees of freedom per unit ...

Every now and then, a new design of an interpolation kernel shows up in the literature. While interesting results have emerged, the traditional design methodology proves laborious and is riddled with very large systems of linear equations that must be solved analytically. In this paper, we propose to ease this burden by providing an explicit formula that will generate every possible piecewise-polynomial kernel given its degree, its support, its regularity, and its order of approximation. This formula contai

We provide a new comparison between hexagonal and orthogonal lattices, based on approximation theory. For each of the lattices, we select the “natural” spline basis function as generator for a shift-invariant function space; i.e., the tensor-product B-spli ...

We present a new digital two-step reconstruction method for off-axis holograms recorded on a CCD camera. First, we retrieve the complex object wave in the acquisition plane from the hologram's samples. In a second step, if required, we propagate the wave front by using a digital Fresnel transform to achieve proper focus. This algorithm is sufficiently general to be applied to sophisticated optical setups that include a microscope objective. We characterize and evaluate the algorithm by using simulated data

Probably, the most important property of wavelets for signal processing is their multiscale derivative-like behavior when applied to functions. In order to extend the class of problems that can profit of wavelet-based techniques, we propose to build new families of wavelets that behave like an arbitrary scale-covariant operator. Our extension is general and includes many known wavelet bases. At the same time, the method takes advantage a fast filterbank decomposition-reconstruction algorithm. We give necess

In this paper, we present a new algorithm for the design of orthonormal two-band rational filter banks. Owing to the connection between iterated rational filter banks and rational wavelets, this is also a design algorithm for orthonormal rational wavelets. It is basically a simple iterative procedure, which explains its exponential convergence and adaptability under various linear constraints (e.g., regularity). Although the filters obtained from this algorithm are suboptimally designed, they show excellent

Recently, we have introduced an integrated framework that combines wavelet-based processing with statistical testing in the spatial domain. In this paper, we propose two important enhancements of the framework. First, we revisit the underlying paradigm; i. ...

Shannon's sampling theory and its variants provide effective solutions to the problem of reconstructing a signal from its samples in some “shift-invariant” space, which may or may not be bandlimited. In this paper, we present some further justification for ...

Frequency domain optical coherence tomography (FDOCT) is a new technique that is well-suited for fast imaging of biological specimens, as well as non-biological objects. The measurements are in the frequency domain, and the objective is to retrieve an arti ...

This paper presents an interpolation method based on shifted versions of two piecewise linear generators, which provides approximation order 2 like usual piecewise-linear interpolation; i.e., this method is able to represent the constant and the ramp exactly. Our interpolation is characterized by two real parameters: τ, the location of the generators, and α, related to their dissymmetry. By varying these parameters, we show that it is possible to optimize the quality of the approximation, independently of t

We define a new wavelet transform that is based on a recently defined family of scaling functions: the fractional B-splines. The interest of this family is that they interpolate between the integer degrees of polynomial B-splines and that they allow a fractional order of approximation. The orthogonal fractional spline wavelets essentially behave as a fractional differentiators. This property seems promising for the analysis of

1/f ^{ \alpha }

; noise that can be whitened by an appropriate choice of the

The central theme of this pair of papers (Parts I and II in this issue) is self-similarity, which is used as a bridge for connecting splines and fractals. The first part of the investigation is deterministic, and the context is that of L-splines; these are ...

In this paper, we use polyharmonic B-splines to build multidimensional wavelet bases. These functions are nonseparable, multidimensional basis functions that are localized versions of radial basis functions. We show that Rabut's elementary polyharmonic B-s ...

We consider the problem of locating point sources in the planar domain from overdetermined boundary measurements of solutions of Poisson's equation. In this paper, we propose a novel technique, termed "analytic sensing," which combines the application of G ...

We present a new class of wavelet bases—Fresnelets—which is obtained by applying the Fresnel transform operator to a wavelet basis of

L _{ 2 }

. The thus constructed wavelet family exhibits properties that are particularly useful for analyzing and processing optically generated holograms recorded on CCD-arrays. We first investigate the multiresolution properties (translation, dilation) of the Fresnel transform that are needed to construct our new wavelet. We derive a Heisenberg-like uncertainty relation

Due to the random nature of photon emission and the various internal noise sources of the detectors, real timelapse fluorescence microscopy images are usually modeled as the sum of a Poisson process plus some Gaussian white noise. In this paper, we propose ...

Thierry Blu

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A New Family of Complex Rotation-Covariant Multiresolution Bases in 2D

Recursive Filtering for Splines on Hexagonal Lattices

Complex B-Splines

On the Approximation Power of Splines: Orthogonal Versus Hexagonal Lattices

Sampling Moments and Reconstructing Signals of Finite Rate of Innovation: Shannon Meets Strang-Fix

Complete Parameterization of Piecewise-Polynomial Interpolation Kernels

Hexagonal Versus Orthogonal Lattices: A New Comparison Using Approximation Theory

Complex-Wave Retrieval from a Single Off-Axis Hologram

Construction of wavelet bases that mimic the behaviour of some given operator

A New Design Algorithm for Two-Band Orthonormal Rational Filter Banks and Orthonormal Rational Wavelets

WSPM: Wavelet-Based Statistical Parametric Mapping

Nonideal Sampling and Regularization Theory

A new technique for high-resolution frequency domain optical coherence tomography

Interpolation of Signals by Generalized Piecewise-Linear Multiple Generators

The Fractional Spline Wavelet Transform: Definition and Implementation

Self-Similarity: Part I—Splines and Operators

Isotropic Polyharmonic B-Splines: Scaling Functions and Wavelets

Analytic Sensing: Noniterative Retrieval Of Point Sources From Boundary Measurements

Fresnelets—A New Wavelet Basis for Digital Holography

Multiframe SURE-LET denoising of timelapse fluorescence microscopy images