It is a challenging task to reconstruct images from their noisy, blurry, and/or incomplete measurements, especially those with important details and features such as medical magnetic resonance (MR) and CT images. We propose a novel regularization model that integrates two recently developed regularization tools: total generalized variation (TGV) by Bredies, Kunisch, and Pock; and shearlet transform by Labate, Lim, Kutyniok, and Weiss. The proposed model recovers both edges and fine details of images much better than the existing regularization models based on the total variation (TV) and wavelets. Specifically, while TV preserves sharp edges but suffers from oil painting artifacts, TGV "selectively regularizes" different image regions at different levels and thus largely avoids oil painting artifacts. Unlike the wavelet transform, which represents isotropic image features much more sparsely than anisotropic ones, the shearlet transform can efficiently represent anisotropic features such as edges, curves, and so on. The proposed model based on TGV and the shearlet transform has been tested in the compressive sensing context and produced high-quality images using fewer measurements than the state-of-the-art methods. The proposed model is solved by splitting variables and applying the alternating direction method of multiplier (ADMM). For certain sensing operators, including the partial Fourier transform, all the ADMM subproblems have closed-form solutions. Convergence of the algorithm is briefly mentioned. The numerical simulations presented in this paper use the incomplete Fourier, discrete cosine, and discrete wavelet measurements of MR images and natural images. The experimental results demonstrate that the proposed regularizer preserves various image features (including edges and textures), much better than the TV/wavelet based methods.

Quantitative optical measurements of deep sub-wavelength, three-dimensional, nanometric structures with sensitivity to sub-nanometer details address an ubiquitous measurement challenge. A Fourier domain normalization approach is used in the Fourier optical imaging code to simulate the full three-dimensional scattered light field of nominally 15 nm sized structures, accurately replicating the light field as a function of the focus position. Using the full three-dimensional light field, nanometer scale details such as a 2 nm thin conformal oxide and nanometer topography are rigorously fitted for features less than 1/30th of the wavelength in size. The densely packed structures are positioned nearly an order of magnitude closer than the conventional Rayleigh resolution limit and can be measured with sub-nanometer parametric uncertainties. This approach enables a practical measurement sensitivity to size variations of only a few atoms in size using a high throughput optical configuration with broad application in measuring nanometric structures and nanoelectronic devices.

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In this paper, we propose a variational multiphase image segmentation model based on fuzzy membership functions and L1-norm fidelity. Then we apply the alternating direction method of multipliers to solve an equivalent problem. All the subproblems can be solved efficiently. Specifically, we propose a fast method to calculate the fuzzy median. Experimental results and comparisons show that the L1-norm based method is more robust to outliers such as impulse noise and keeps better contrast than its L2-norm counterpart. Theoretically, we prove the existence of the minimizer and analyze the convergence of the algorithm.

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There has been much recent work in developing advanced optical metrology methods that use imaging optics for critical dimension measurements and defect detection. Sensitivity to nanometer-scale changes has been observed when measuring critical dimensions of subwavelength 20 nm features or when imaging defects below 15 nm using angle-resolved and focus-resolved optical data. However, these methods inherently involve complex imaging optics and analysis of complicated three-dimensional electromagnetic fields. This paper develops a new approach to enable the rigorous analysis of three-dimensional, through-focus, or angle-resolved optical images. We use rigorous electromagnetic simulation with enhanced Fourier optical techniques, an approach to optical tool normalization, and statistical methods to evaluate sensitivities and uncertainties in the measurement of subwavelength three-dimensional structures.

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