The investigators develop tools and techniques to facilitate the accurate reconstruction of spectral data from compressed images. Such reconstruction typically aims at recreating gray-scale functions from their Fourier or spectral coefficients, and necessarily requires both precise information of the location of the jump discontinuities of the images, as well as an appropriate conversion, usually via projection, so that the image can be viewed in the regions of smoothness. Current techniques for edge detection and projection reconstruction are somewhat successful in avoiding Gibbs oscillations without compromising the integrity of the the images around the edges (i.e. smearing). They also can retrieve information of small scale features which arise in many scientific applications. However, the success is seemingly function dependent, and it is difficult to choose parameters that are robust to be effective in all cases, especially when noise is present in the given data fields. Furthermore, they are difficult to extend to higher dimensions. The investigators study new ways to improve both edge detection and projection methods and ultimately create an automated robust, user-friendly, computationally efficient, and inherently multi-dimensional image reconstruction method. The study tests the resulting procedures against current image reconstruction methods on a range of applications.
Image reconstruction is of critical importance in many scientific fields. The development of high order reconstruction techniques requires both mathematical rigor to prove theoretical results and computational robustness to ensure practical usage. The proposed activities address how to obtain images efficiently and with high accuracy when small scale features are of extreme interest in an increasingly image--oriented society. These methods can, for example, enhance the diagnostic ability in medical imaging applications. They can also be used to better identify the small scale features in solar activity such as those in connection with Lockheed Martin's Solar Imaging Suite. This research is also useful for weather forecasting models, earthquake and tsunami prediction, and any other fields in which visualization is critical. Finally, this study stands to significantly enhance the ability to compress data, as it it will be possible to optimize the reconstruction parameters when particular compression requirements are proposed.
