High Order Reconstruction Using Spectral Methods

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The investigators develop tools and techniques to facilitate the accuratereconstruction of spectral data from compressed images. Such reconstructiontypically aims at recreating gray-scale functions from their Fourier orspectral coefficients, and necessarily requires both precise informationof the location of the jump discontinuities of the images, as wellas an appropriate conversion, usually via projection, so that the imagecan be viewed in the regions of smoothness. Current techniques for edgedetection and projection reconstruction are somewhat successful in avoidingGibbs oscillations without compromising the integrity of the the imagesaround the edges (i.e. smearing). They also can retrieve information of smallscale features which arise in many scientific applications. However, thesuccess is seemingly function dependent, and it is difficult to chooseparameters that are robust to be effective in all cases, especiallywhen noise is present in the given data fields. Furthermore, they aredifficult to extend to higher dimensions. The investigators study newways to improve both edge detection and projection methods and ultimatelycreate an automated robust, user-friendly, computationally efficient, andinherently multi-dimensional image reconstruction method. The study teststhe resulting procedures against current image reconstruction methods on arange of applications.Image reconstruction is of critical importance in many scientific fields.The development of high order reconstruction techniques requires bothmathematical rigor to prove theoretical results and computationalrobustness to ensure practical usage. The proposed activities addresshow to obtain images efficiently and with high accuracy when small scalefeatures are of extreme interest in an increasingly image--oriented society.These methods can, for example, enhance the diagnostic ability in medicalimaging applications. They can also be used to better identify the smallscale features in solar activity such as those in connection with LockheedMartin's Solar Imaging Suite. This research is also useful for weatherforecasting models, earthquake and tsunami prediction, and any otherfields in which visualization is critical. Finally, this study stands tosignificantly enhance the ability to compress data, as it it will bepossible to optimize the reconstruction parameters when particularcompression requirements are proposed.