Recent advances in high-resolution fluorescence microscopy have enabled the systematicstudy of morphological changes in large populations of cells induced by chemical andgenetic perturbations, facilitating the discovery of signaling pathways underlyingdiseases and the development of new pharmacological treatments. In these studies, though,due to the complexity of the data, quantification and analysis of morphological featuresare for the vast majority handled manually, slowing significantly data processing andlimiting often the information gained to a descriptive level. Thus, there is an urgentneed for developing highly efficient automated analysis and processing tools forfluorescent images. In this paper, we present the application of a method based on theshearlet representation for confocal image analysis of neurons. The shearletrepresentation is a newly emerged method designed to combine multiscale data analysis withsuperior directional sensitivity, making this approach particularly effective for therepresentation of objects defined over a wide range of scales and with highly anisotropicfeatures. Here, we apply the shearlet representation to problems of soma detection ofneurons in culture and extraction of geometrical features of neuronal processes in braintissue, and propose it as a new framework for large-scale fluorescent image analysis ofbiomedical data.

The shearlet representation has gained increasing recognition in recent years as aframework for the efficient representation of multidimensional data. This representationconsists of a countable collection of functions defined at various locations, scales andorientations, where the orientations are obtained through the use of shear matrices. Whileshear matrices offer the advantage of preserving the integer lattice and being moreappropriate than rotations for digital implementations, the drawback is that the action ofthe shear matrices is restricted to cone-shaped regions in the frequency domain. Hence, inthe standard construction, a Parseval frame of shearlets is obtained by combiningdifferent systems of cone-based shearlets which are projected onto certain subspaces ofL2(ℝD) with the consequence thatthe elements of the shearlet system corresponding to the boundary of the cone regions losetheir good spatial localization property. In this paper, we present a new constructionyielding smooth Parseval frame of shearlets forL2(ℝD). Specifically, allelements of the shearlet systems obtained from this construction are compactly supportedand C∞ in the frequency domain, hence ensuring that the systemhas also excellent spatial localization.

Directional multiscale representations such as shearlets and curvelets have gainedincreasing recognition in recent years as superior methods for the sparse representationof data. Thanks to their ability to sparsely encode images and other multidimensionaldata, transform-domain denoising algorithms based on these representations are among thebest performing methods currently available. As already observed in the literature, theperformance of many sparsity-based data processing methods can be further improved byusing appropriate combinations of dictionaries. In this paper, we consider the problem of3D data denoising and introduce a denoising algorithm which uses combined sparsedictionaries. Our numerical demonstrations show that the realization of the algorithmwhich combines 3D shearlets and local Fourier bases provides highly competitive results ascompared to other 3D sparsity-based denosing algorithms based on both single and combineddictionaries.