Increasingly detailed biomedical data must be integrated to better understand chronic disease. There is urgent need for better analysis of heterogeneous data and images acquired by different devices and modalities enabling quantitative tracking of changes over time. The rapid developments in applied harmonic analysis over the past years have the potential to overcome present barriers. Scientists with broad experience both on the mathematical side as well as long-standing experience in cooperation with medical teams are needed. Building upon my previous human retina project jointly with the National Eye Clinic at the NIH, my research is guided by clinical retinal imaging that serves as a testbed for wider applications. We propose to develop customized tools from applied harmonic analysis to better study symptoms and concomitants of age related macular degeneration (the leading cause of blindness among the elderly) and diabetic retinopathy. The Vienna Reading Center as part of the Department of Ophthalmology and Optometry of the Medical University of Vienna is the main partner institution providing medical images and designing a prospective study that incorporates multi-spectral retinal autofluorescence imaging as developed at the NIH within my former project. The primary medical goals are automated, hence unbiased and efficient, quantification of subretinal fluid, cysts, and microexudates, tracking their changes over time enabling the evaluation of treatment decisions, and the prediction of disease progression. The low signal in spectral domain optical coherence tomography (SD-OCT) requires the joint analysis with other imaging modalities. Thus, we must fuse images from several modalities with SD-OCT, segment and quantify fluid, cysts, and exudates in fused data, track changes over time, and predict progression.
The following scheme describes an iterative process, in which the Vienna Reading Center's continuous feedback supports refining the mathematical tools. Eventually, the developed methods shall be applied to reading processes by means of a software package at the Vienna Reading Center.
(Step 1) FUSION FRAMES and DIMENSION REDUCTION:
To fuse data from different modalities with SD-OCT, we use the concept of fusion frames, where each modality is modeled as a projector of a sufficient rank on a high-dimensional feature space. Distorted or missing measurements shall be robustly recovered by using semidefinite programming. The fused image data across modalities require dimension reduction to perform segmentation and volume quantification in the subsequent step. For both, fusion frame and dimension reduction, we shall derive certain optimal configurations of projectors with sufficient rank.
(Step 2) Variational segmentation using the p-LAPLACIAN, SPARSITY, and WAVELETS/SHEARLETS:
To quantify subretinal fluid, cysts, and microexudates, we shall develop a hybrid segmentation scheme based on the p-Laplacian with sparsity constraints in wavelet/shearlet frames enabling sufficient flexibility to incorporate characteristics of fused modality data and target features.
(Step 3) DIFFUSION GEOMETRY and TIME-FREQUENCY ANALYSIS on GRAPHS:
Statistical shape analysis combined with frame algorithms shall be adopted to track changes over time. Clustering yields large hypergraphs, whose function on the nodes represent likelihood of the progression of fluid, cyst, and exudate volume. To study this graph, we shall further develop optimal time-frequency dictionaries on hypgergraphs and use diffusion geometry techniques to predict disease progression.
(Add-on) We shall verify some of the proposed mathematical concepts' usefulness in other disciplines already within the project lifetime. Therefore, the Acoustics Research Institute acts as a secondary partner. We aim to reduce measurements of the head-related transfer function, the responses that characterize how an ear receives a sound, and aim to solve a reconstruction problem related to speech recognition.
Further modifications of the developed tools will have wider applications in multi-factorial disease and normal development offering new means to personalized medicine, where integration and analysis of large heterogeneous datasets are the key challenges.