Publications by Thierry Blu | EPFL Graph Search

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

We introduce a 3-D parametric active contour algorithm for the shape estimation of DNA molecules from stereo cryo-electron micrographs. We consider a 3-D filament (consisting of a B-spline skeleton and a specified radial profile) and match its projections with the micrographs using an optimization algorithm. To accelerate the evaluation of the projections, we approximate the global model locally by an elongated blob-like template that is designed to be projection-steerable. This means that the 2-D projectio

Complex wavelet transforms offer the opportunity to perform directional and coherent processing based on the local magnitude and phase of signals and images. Although denoising, segmentation, and image enhancement are significantly improved using complex wavelets, the redundancy of most current transforms hinders their application in compression and related problems. In this paper we introduce a new orthonormal complex wavelet transform with no redundancy for both real— and complex-valued signals. The trans

Statistical Parametric Mapping (SPM) is a widely deployed tool for detecting and analyzing brain activity from fMRI data. One of SPM's main features is smoothing the data by a Gaussian filter to increase the SNR. The subsequent statistical inference is based on the continuous Gaussian random field theory. Since the remaining spatial resolution has deteriorated due to smoothing, SPM introduces the concept of “resels” (resolution elements) or spatial information-containing cells. The number of resels turns ou

We propose a new orthonormal wavelet thresholding algorithm for denoising color images that are assumed to be corrupted by additive Gaussian white noise of known intercolor covariance matrix. The proposed wavelet denoiser consists of a linear expansion of thresholding (LET) functions, integrating both the interscale and intercolor dependencies. The linear parameters of the combination are then solved for by minimizing Stein's unbiased risk estimate (SURE), which is nothing but a robust unbiased estimate of

We address the problem of exact signal recovery in frequency domain optical coherence tomography (FDOCT) systems. Our technique relies on the fact that, in a spectral interferometry setup, the intensity of the total signal reflected from the object is smal ...

A filterbank decomposition can be seen as a series of projections onto several discrete wavelet subspaces. In this presentation, we analyze the projection onto one of them—the low-pass one, since many signals tend to be low-pass. We prove a general but simple formula that allows the computation of the

l _{ 2 }

-error made by approximating the signal by its projection. This result provides a norm for evaluating the accuracy of a complete decimation/interpolation branch for arbitrary analysis and synthesi

Our goal is to detect and localize areas of activation in the brain from sequences of fMRI images. The standard approach for reducing the noise contained in the fMRI images is to apply a spatial Gaussian filter which entails some loss of details. Here instead, we consider a wavelet solution to the problem, which has the advantage of retaining high-frequency information. We use fractional-spline orthogonal wavelets with a continuously-varying order parameter alpha; by adjusting alpha, we can balance spatial

This chapter presents a survey of interpolation and resampling techniques in the context of exact, separable interpolation of regularly sampled data. In this context, the traditional view of interpolation is to represent an arbitrary continuous function as a discrete sum of weighted and shifted synthesis functions—in other words, a mixed convolution equation. An important issue is the choice of adequate synthesis functions that satisfy interpolation properties. Examples of finite-support ones are the square

We present an exact expression for the

L _{ 2 }

error that occurs when one approximates a periodic signal in a basis of shifted and scaled versions of a generating function. This formulation is applicable to a wide variety of linear approximation schemes including wavelets, splines, and bandlimited signal expansions. The formula takes the simple form of a Parseval's-like relation, where the Fourier coefficients of the signal are weighted against a frequency kernel that characterizes the approximation op

The dilation matrix associated with the three-dimensional (3-D) face-centered cubic (FCC) sublattice is often considered to be the natural 3-D extension of the two-dimensional (2-D) quincunx dilation matrix. However, we demonstrate that both dilation matri ...

In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a B-spline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory—including some new extensions for fractional orders—in a self-contained, accessible fashion. In particular, we prove that the B-spline component is entirely responsible for f

In this paper, we present different solutions for improving spline-based snakes. First, we demonstrate their minimum curvature interpolation property, and use it as an argument to get rid of the explicit smoothness constraint. We also propose a new external energy obtained by integrating a non-linearly pre-processed image in the closed region bounded by the curve. We show that this energy, besides being efficiently computable, is sufficiently general to include the widely used gradient-based schemes, Bayesi

Our goal in this paper is to set a theoretical basis for the comparison of re-sampling and interpolation methods. We consider the general problem of the approximation of an arbitrary continuously-defined function f(x)—not necessarily bandlimited—when we vary the sampling step T. We present an accurate

L ^{ 2 }

computation of the induced approximation error as a function of T for a general class of linear approximation operators including interpolation and other kinds of projectors. This new quantitative

We extend Schoenberg's B-splines to all fractional degrees α > -1/2. These splines are constructed using linear combinations of the integer shifts of the power functions

x _{ + } ^{ \alpha }

(one-sided) or

|x| _{ * } ^{ \alpha }

(symmetric); in each case, they are α-Hölder 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, which makes them suitable for the construction of new families of wavelet

Wavelet-based statistical analysis methods for fMRI are able to detect brain activity without smoothing the data. Typically, the statistical inference is performed in the wavelet domain by testing the t-values of each wavelet coefficient; subsequently, an activity map is reconstructed from the significant coefficients. The limitation of this approach is that there is no direct statistical interpretation of the reconstructed map. In this paper, we propose a new methodology that takes advantage of wavelet pro

We propose to design the reduction operator of an image pyramid so as to minimize the approximation error in the

l _{ p }

sense (not restricted to the usual p = 2), where p can take non-integer values. The underlying image model is specified using arbitrary shift-invariant basis functions such as splines. The solution is determined by an iterative optimization algorithm, based on digital filtering. Its convergence is accelerated by the use of first and second derivatives. For p = 1, our modified pyramid

The ARSIS concept is designed to increase the spatial resolution of an image without modification of its spectral contents, by merging structures extracted from a higher resolution image of the same scene, but in a different spectral band. It makes use of wavelet transforms and multiresolution analysis. It is currently applied in an operational way with dyadic wavelet transforms that limit the merging of images whose ratio of their resolution is a power of 2. Rational discrete wavelet transforms can be appr

We propose to design the reduction operator of an image pyramid so as to minimize the approximation error in the

l _{ p }

-sense (not restricted to the usual p=2), where p can take non-integer values. The underlying image model is specified using shift-invariant basis functions, such as B-splines. The solution is well-defined and determined by an iterative optimization algorithm based on digital filtering. Its convergence is accelerated by the use of first and second order derivatives. For p close to 1,

This paper proposes a new family of bivariate, non-separable splines, called hex-splines, especially designed for hexagonal lattices. The starting point of the construction is the indicator function of the Voronoi cell, which is used to define in a natural way the first-order hex-spline. Higher order hex-splines are obtained by successive convolutions. A mathematical analysis of this new bivariate spline family is presented. In particular, we derive a closed form for a hex-spline of arbitrary order. We also

We consider the problem of interpolating a signal using a linear combination of shifted versions of a compactly-supported basis function φ(x). We first give the expression of the φ's that have minimal support for a given accuracy (also known as "approximation order"). This class of functions, which we call maximal-order-minimal-support functions (MOMS), is made of linear combinations of the B-spline of same order and of its derivatives. We provide the explicit form of the MOMS that maximize the approximatio

Thierry Blu

Chat with Graph Search

Shape Estimation of 3-D DNA Molecules from Stereo Cryo-Electron Micro-Graphs

Orthogonal Hilbert transform filter banks and wavelets

Wavelets Versus Resels in the Context of fMRI: Establishing the Link with SPM

SURE-LET interscale-intercolor wavelet thresholding for color image denoising

Non-iterative exact signal recovery in frequency domain optical coherence tomography

A Theoretical Analysis of the Projection Error onto Discrete Wavelet Subspaces

Analysis of fMRI Data Using Spline Wavelets

Image Interpolation and Resampling

Sampling of Periodic Signals: A Quantitative Error Analysis

On the Multidimensional Extension of the Quincunx Subsampling Matrix

Wavelet Theory Demystified

A Unifying Approach and Interface for Spline-Based Snakes

Quantitative $ L ^{ 2 } $ Error Analysis for Interpolation Methods and Wavelet Expansions

Construction of Fractional Spline Wavelet Bases

Wavelet-Based fMRI Statistical Analysis and Spatial Interpretation: A Unifying Approach

Non-Euclidean Pyramids

Using Iterated Rational Filter Banks Within the ARSIS Concept for Producing 10 m Landsat Multispectral Images

$ l _{ p } $ -Multiresolution Analysis: How to Reduce Ringing and Sparsify the Error

Hex-Splines: A Novel Spline Family for Hexagonal Lattices

MOMS: Maximal-Order Interpolation of Minimal Support