High Order Reconstruction Using Spectral Methods

研究人员开发了工具和技术，以方便从压缩图像中准确构建光谱数据。这样的重建通常旨在从它们的傅立叶或光谱系数中重建灰度函数，并且必然需要图像跳跃不连续位置的精确信息，以及通常通过投影进行适当的转换，以便在平滑区域中查看图像。当前的边缘检测和投影重建技术在避免吉布斯振荡而不损害边缘周围图像的完整性（即涂抹）方面取得了一定的成功。它们还可以检索在许多科学应用中出现的小尺度特征信息。然而，成功似乎是函数依赖的，很难选择在所有情况下都有效的鲁棒参数，特别是当给定数据域中存在噪声时。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、鲁棒的、用户友好的、计算效率高的、固有的多维图像重建方法。该研究测试了产生的程序与当前的图像重建方法的应用范围。图像重建在许多科学领域都是至关重要的。高阶重建技术的发展既需要数学上的严谨性来证明理论结果，又需要计算上的鲁棒性来确保实际应用。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，所提出的活动解决了如何高效、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。它们还可以用于更好地识别太阳活动的小尺度特征，例如与洛克希德马丁公司的太阳成像套件相关的特征。这项研究对天气预报模型、地震和海啸预测以及其他可视化至关重要的领域也很有用。最后，本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，它将有可能优化重构参数。

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.