2015/05/20 by Milana Gatarić, Gataric, Milana, Clarice Poon +1
Computer Science · Engineering · Medicine · #Advanced X-ray and CT Imaging #FOS: Mathematics #Image and Signal Denoising Methods #Medical Imaging Techniques and Applications #Numerical Analysis (math.NA) #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1505.05308
openalex publication_date 2015/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a series of recent papers (Adcock, Hansen and Poon, 2013, Appl. Comput.\nHarm. Anal. 45(5):3132-3167), (Adcock, Gataric and Hansen, 2014, SIAM J.\nImaging Sci. 7(3):1690-1723) and (Adcock, Hansen, Kutyniok and Ma, 2015, SIAM\nJ. Math. Anal. 47(2):1196-1233), it was shown that one can optimally recover\nthe wavelet coefficients of an unknown compactly supported function from\npointwise evaluations of its Fourier transform via the method of generalized\nsampling. While these papers focused on the optimality of generalized sampling\nin terms of its stability and error bounds, the current paper explains how this\noptimal method can be implemented to yield a computationally efficient\nalgorithm. In particular, we show that generalized sampling has a computational\ncomplexity of \O(M(N)\log N) when recovering the first N\nboundary-corrected wavelet coefficients of an unknown compactly supported\nfunction from M(N) Fourier samples. Therefore, due to the linear\ncorrespondences between the number of samples M and number of coefficients\nN shown previously, generalized sampling offers a computationally optimal way\nof recovering wavelet coefficients from Fourier data.\n