Two-stage adaptive random Fourier sampling method for image reconstruction. (September 2021) - Yun, Joo Dong; Kim, Yunho
Highlights: We discover a column-wise maximum coherent structure in the Fourier-Haar interplay. A few examples of exact recovery of the Haar wavelet coefficients of an image are provided in a constrained ℓ 1 minimization with a small number of high magnitude Fourier samples based on the maximum coherent structure. A two-stage adaptive Fourier sampling scheme is proposed to acquire such a small number of meaningful Fourier samples supported by the exact recovery examples. The proposed sampling scheme is confirmed to reveal high frequency patterns better than other well-known efficient sampling schemes, showing its numerical superiority in the ℓ 1 minimization. Abstract: We propose a random Fourier sampling scheme to enhance the accuracy of the high frequency pattern estimation for image reconstruction. This method is designed to work in a constrained ℓ 1 minimization based on the Fourier-Haar interplay revealing a column-wise maximum coherent structure that we provide. Essential in the scheme is to generate a data-driven density function by a small percentage of Fourier samples. The density function governs a random sampling procedure to acquire high frequency information, resulting in better reconstruction of the Haar wavelet coefficients. We also discuss a few examples of exact recovery of the Haar wavelet coefficients from which the proposed sampling scheme has emerged. Numerical experiments confirm superiority of the proposed sampling scheme to other conventional sampling schemes in the ℓ 1 framework.