High Order Reconstruction Using Spectral Methods

研究人员开发工具和技术，以促进从压缩图像中准确地构建光谱数据。这种重建通常旨在从它们的傅里叶系数或光谱系数重建灰度函数，并且必须既需要图像跳跃不连续位置的精确信息，也需要通常通过投影进行适当的转换，以便可以在平滑区域中观察图像。目前用于边缘检测和投影重建的技术在避免Gibbs振荡方面取得了一定的成功，而不会损害边缘周围图像的完整性(即涂抹)。它们还可以检索在许多科学应用中出现的小尺度特征的信息。然而，成功似乎是依赖于函数的，并且很难选择在所有情况下都稳健有效的参数，特别是当给定的数据场中存在噪声时。此外，它们很难扩展到更高的维度。研究人员研究了改进边缘检测和投影方法的新方法，并最终创建了一种自动化的、健壮的、用户友好的、计算高效的、固有的多维图像重建方法。图像重建在许多科学领域都具有重要意义，高阶重建技术的发展既需要严格的数学证明来证明理论结果，也需要计算的稳健性来确保实际应用。在一个日益以图像为导向的社会中，当小尺度特征非常重要时，所提出的活动旨在高效和高精度地获取图像。例如，这些方法可以增强医学成像应用中的诊断能力。它们还可以用来更好地识别太阳活动中的小尺度特征，例如与洛克希德马丁公司的太阳成像套件有关的特征。这项研究对天气预报模型、地震和海啸预报以及任何其他可视化至关重要的领域也很有用。最后，这项研究将显著提高压缩数据的能力，因为当提出特定的压缩要求时，它将有可能优化重建参数。

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.