Mathematical Sciences: Wavelets and their Applications to Neural Network Theory, Vision, and Image Processing

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9410859 Kon The investigator and his colleagues study wavelets and their applications to neural network theory and processing of visual images. It has been shown recently that wavelets are useful as activation functions in neural networks, permitting faster adaptation to required tasks than do sigmoidal activation functions, in common use up to now. The success of wavelet methods has already been impressively demonstrated in important optimization problems using radial basis functions. The work of this project develops better theoretical underpinnings for the construction of neural nets that learn efficiently, and studies minimal complexities of neural nets that emulate "intelligent" tasks, e.g., recognition of visual images. The investigators also study the construction of wavelet-based algorithms for image compression. They examine the optimization of signal compression, aiming at an improved theoretical understanding of how such signal compression works from the standpoint of the mapping between functions and their wavelet transforms. Applications of the mathematics proposed here are closely related to the creation of "intelligent" systems using neural network technology. This technology in many ways emulates the operation of biological nervous systems. The present work is based on indications that the actions of biological neurons have in some cases been too well emulated in artificial neural systems. It has been shown in the work of Girosi and Poggio and others that attaining desired behavior in neural networks can be better achieved with networks that have "localized activation functions," i.e., ones for which output eventually decreases with very large inputs. An additional issue related to effecting proper input-output behavior using neural networks is knowledge of the basic complexity of certain desired tasks, and how neural networks can best achieve this level of complexity. Such tasks (e.g., the visual recognition of object s) can already be performed by biological neural networks, and have a complexity that can be defined and studied in a relatively precise way, to determine how complicated artificial neural networks need to be in order to emulate the behaviors of the biological networks. In addition, the work on signal compression using wavelets has a two-fold impact. The first involves the improvement of techniques for compression, e.g., storage of large quantities of video, audio or other data on compact media (such as compact discs, hard disks, or random access memory). The second is related to the fact that if such data are compressed to smaller sizes, certain kinds of information may be extracted from them more easily. For example, it is easier to find evidence of an irregular heartbeat (or precursors to arhythmia) using a computer if the cardiological data have been compressed in a way that allows them to be more easily manipulated. These and other applications have made the data compression issues that are studied here an area with high priority in mathematical signal processing.