Gabor frames and directional time-frequency analysis

We introduce a directionally sensitive time-frequency decomposition and representation of functions. The coefficients of this representation allow us to measure the "amount" of frequency a function (signal, image) contains in a certain time interval, and also in a certain direction. This has been previously achieved using a version of wavelets called ridgelets [E.J. Candes, Harmonic analysis of neural networks, Appl. Comput. Harmon. Anal. 6 (1999) 197-218. M; E.J. Candes, D.L. Donoho, New tight frames of curvelets and optimal representations of objects with piesewise-C-2 singularities, Comm. Pure Appl. Math. 57 (2004) 219-266. [3]] but in this work we discuss an approach based on time-frequency or Gabor elements. For such elements, a Parseval formula and a continuous frame-type representation together with boundedness properties of a semi-discrete frame operator are obtained. Spaces of functions tailored to measure quantitative properties of the time-frequency-direction analysis coefficients are introduced and some of their basic properties are discussed. Applications to image processing and medical imaging are presented. (c) 2007 Elsevier Inc. All rights reserved.