Fine-Scale Singularity Detection in Multi-Dimensional Imaging with Regular, Orientable, Symmetric, Frame Atoms with Small Support

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One of life's essential characteristic is movement. Whether it is the spectacular, delicate dance of the unblemished, white swan in Tchaikovsky's Swan Lake, or the early attempts of a toddler to use his or her hands, the neurology of movement is uniquely common for all animals and humans: learning a motor skill, and the necessary muscle coordination. With practice the skill is perfected. Finally, the retainment of this experience-based learning process is the conclusion of this learning process. The ultimate goal of this project is to provide new tools to neuroscientists who study the biological basis of learning at the cell level using live animals. This function is facilitated by a number of anatomical changes in the structure of the cytoplasm of neural cells, such as the formation of lengthy branches known as axons and dendrites, and at a fine scale of dendritic spines and axonal buttons. The latter are less anatomically permanent structures arising on the surface of these cytoplasmic extensions. Dendritic spines and axonal buttons form synapses, which are the communication gateways between neurons. The research team will develop mathematical and computational tools for automatizing the study of spine populations in live neurons, and of their time-evolution during learning. The anticipated outcomes will provide neuroscientists with a number of software tools which will automatize the analysis of synaptic strength and its evolution with learning. These findings will contributed to the better understanding the biological mechanisms of autism and drug addictions.The investigators on this project will develop algorithms for the 3D digital segmentations of dendritic surfaces including spines from 3D images acquired with a certain type of microscope, which uses laser light and works as a scanner by exploiting the natural ability of neurons to fluoresce. They aim to generate accurate binary reconstructions of a dendritic arbor including its spines. The primary challenge in this project is that image acquisition of live neurons has a resolution which provides limited detail of the spines. Often, images contain noise which further complicates the extraction of accurate, binary 3D reconstructions of dendritic surfaces showing spine details. Overcoming this problem is a core goal of the project because spine volume estimation quantifies synaptic strength. These unique challenges lead the investigators to the development of novel mathematical tools for fine scale analysis. They will build ensembles of short in size 3D imaging, 3D-orientation selective, frame-based filters, suitable for sensing curves and surfaces in noisy images. These filters will be designed to respond to local changes of image smoothness. Information obtained from these filters at various scales, will be utilized as input for multilayer, deep-learning inspired neural networks which will determine in an image which voxels belong to spine surfaces. Further algorithmic tools will be developed to track every spine of a dendrite individually over time. The same filtering tools will be used in a different application domain, the generation of illumination neutral images, in real-time. This will help the fast, high throughput removal of the effects of uneven illumination in images inhibiting the detection, by software or the naked eye of contours associated with shapes or textures in a scene. The investigators will develop the mathematical theory of illumination neutralization using concepts from fractal and microlocal analysis. The illumination neutralization algorithm is envisioned to work for real-time video analysis and in conjunction with face verification algorithms with the potential to be used in face recognition, laser microscopy and remote sensing applications.