FMRI time course processing is traditionally performed using linear regression followed by statistical hypothesis
testing. While this analysis method is robust against noise, it relies strongly on the signal model. In this paper, we
propose a non-parametric framework that is based on two main ideas. First, we introduce a problem-specific type
of wavelet basis, for which we coin the term "activelets". The design of these wavelets is inspired by the form of
the canonical hemodynamic response function. Second, we take advantage of sparsity-pursuing search techniques
to find the most compact representation for the BOLD signal under investigation. The non-linear optimization
allows to overcome the sensitivity-specificity trade-off that limits most standard techniques. Remarkably, the
activelet framework does not require the knowledge of stimulus onset times; this property can be exploited to
answer to new questions in neuroscience.
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 necessary conditions for the scale-covariant operator to admit
our wavelet construction, and we provide examples of new wavelets that can be obtained with our method.
Magnetic resonance spectroscopy imaging (MRSI) is a promising and developing tool in medical imaging. Because of various difficulties imposed by the imperfections of the scanner and the reconstruction algorithms, its applicability in clinical practice is rather limited. In this paper, we suggest an extension of the constrained
reconstruction technique (SLIM). Our algorithm, named B-SLIM, takes into account the the measured field inhomogeneity map, which contains both the scanner's main field inhomogeneity and the object-dependent magnetic susceptibility effects. The method is implemented and tested both with synthetic and physical two-compartment phantom data. The results demonstrate significant performance improvement over the SLIM technique. At the same time, the algorithm has the same computational complexity as SLIM.
We build wavelet-like functions based on a parametrized family of pseudo-differential operators L<sub>v</sub><sup>→</sup> that satisfy some admissibility and scalability conditions. The shifts of the generalized B-splines, which are localized versions of the Green function of L<sub>v</sub><sup>→</sup>, generate a family of L-spline spaces. These spaces have the approximation order equal to the order of the underlying operator. A sequence of embedded spaces is obtained by choosing a dyadic scale progression <i>a</i>=2<sup>i</sup>. The consecutive inclusion of the spaces yields the refinement equation, where the scaling filter depends on scale. The generalized L-wavelets are then constructed as basis functions for the orthogonal complements of spline spaces. The vanishing moment property of conventional wavelets is generalized to the vanishing null space element property. In spite of the scale dependence of the filters, the wavelet decomposition can be performed using an adapted version of Mallat's filterbank algorithm.