We investigate the problem of reconstructing a 2D piecewise smooth function from its bandlimited Fourier measurements. This is a well known and well studied problem with many real world implications, in particular in medical imaging. While many techniques have been proposed over the years to solve the problem, very few consider the accurate reconstruction of the discontinuities themselves. In this work we develop an algebraic reconstruction technique for two-dimensional functions consisting of two continuity pieces with a smooth discontinuity curve. By extending our earlier one-dimensional method, we show that both the discontinuity curve and the function itself can be reconstructed with high accuracy from a finite number of Fourier measurements. The accuracy is commensurate with the smoothness of the pieces and the discontinuity curve. We also provide a numerical implementation of the method and demonstrate its performance on synthetic data.

翻译：本文研究从带限傅里叶测量数据重建二维分段光滑函数的问题。这是一个具有重要现实意义（尤其在医学成像领域）且被广泛研究的经典问题。尽管多年来已提出多种求解技术，但极少有方法关注间断点本身的高精度重建。本研究针对由两个连续片段及一条光滑间断曲线构成的二维函数，发展了一种代数重建技术。通过扩展我们先前的一维方法，我们证明从有限个傅里叶测量值可以高精度地重建间断曲线及函数本身，其精度与各片段及间断曲线的光滑度相匹配。我们还提供了该方法的数值实现，并在合成数据上验证了其性能。