Sparse 3D-Data Representations from Compactly Supported Atoms for Rigid Motion Invariant Classification with Applications to Neuroscience Imaging

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The quantitative characterization of neuronal morphology is currently among the most fundamental objectives in neuroscience, as it is essential to precisely correlate structure, activity and neuronal communication at the cellular level. It has been long established that neurons respond to external stimuli through significant structural changes. Hence, the ability to quantitatively capture and track these changes is fundamental for understanding the cell-level biology of brain functions. Dendritic spines, in particular, are are sub-cellular structures, which play a key-role in how neurons talk to each other; these structural changes are referred to as synaptic plasticity. Although, this type of microanatomical plasticity has been associated with drug addiction or even with autism, the change in the morphology and density of spines in addiction, autism and in neurodegenerative diseases is still unclear. Recent developments of high-resolution confocal single photon and multi-photon microscopy, together with the ability to mark subcellular structures, provide a new window for the observation of live single or small groups of neurons which never existed before. However, extracting useful information using this novel imaging is a challenging task. In particular, observing the variations of spine populations, which are in the order of several thousands for even a single neuron, categorizing them in different types and maintaining the timeline of changes are largely laborious manual tasks that are subject to the inevitable inaccuracies native to repetitive and tedious manual work.Our project aims to develop the algorithmic foundations for a new generation of software tools for extracting global spine morphometric characteristics and population dynamics from high-resolution confocal single photon and multi-photon microscopy images with minimal human intervention. Such tools will have a transformative effect in the logistics of spine studies, because they slash the required labor cost. To achieve our goal, we will make contributions to both mathematical analysis and computer vision. We aim to develop robust methods for the detection, identification of type and estimation of volume regardless of the position and spatial orientation of spines in a 3D image of a neuron acquired with the said microscopes. Finally, our project offers educational opportunities to graduate students in a blend of abstract and computational mathematics and computer vision. We also plan to continue our outreach activities to local high schools located in disadvantaged areas of Houston, offering to top students sneak peeks of the life of the mathematician, the computer scientist and the biologist researcher.