This paper considers two nonlocal regularizations for image recovery, which exploit the spatial interactions in images. We get superior results using preprocessed data as input for the weighted functionals. Applications discussed include image deconvolution and tomographic reconstruction. The numerical results show our method outperforms some previous ones.

This work develops a material reconstruction method for spectral CT, namely Total Image Constrained Material Reconstruction (TICMR), to maximize the utility of projection data in terms of both spectral information and high signal-to-noise ratio (SNR). This is motivated by the following fact: when viewed as a spectrally-integrated measurement, the projection data can be used to reconstruct a total image without spectral information, which however has a relatively high SNR; when viewed as a spectrally-resolved measurement, the projection data can be utilized to reconstruct the material composition, which however has a relatively low SNR. The material reconstruction synergizes material decomposition and image reconstruction, i.e., the direct reconstruction of material compositions instead of a two-step procedure that first reconstructs images and then decomposes images. For material reconstruction with high SNR, we propose TICMR with nonlocal total variation (NLTV) regularization. That is, first we reconstruct a total image using spectrally-integrated measurement without spectral binning, and build the NLTV weights from this image that characterize nonlocal image features; then the NLTV weights are incorporated into a NLTV-based iterative material reconstruction scheme using spectrally-binned projection data, so that these weights serve as a high-SNR reference to regularize material reconstruction. Note that the nonlocal property of NLTV is essential for material reconstruction, since material compositions may have significant local intensity variations although their structural information is often similar. In terms of solution algorithm, TICMR is formulated as an iterative reconstruction method with the NLTV regularization, in which the nonlocal divergence is utilized based on the adjoint relationship. The alternating direction method of multipliers is developed to solve this sparsity optimization problem. The proposed TICMR method was validated using both simulated and experimental data. In comparison with FBP and total-variation-based iterative method, TICMR had improved image quality, e.g., contrast-to-noise ratio and spatial resolution.