We study a functional with variable exponent, 1 ≤ p(x) ≤ 2, which provides a model for image denoising, enhancement, and restoration. The diffusion resulting from the proposed model is a combination of total variation (TV)-based regularization and Gaussian smoothing. The existence, uniqueness, and long-time behavior of the proposed model are established. Experimental results illustrate the effectiveness of the model in image restoration.

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Modeling of trickle-bed reactors requires the knowledge of liquid flow texture. A dye-adsorption method was employed to study the liquid flow texture in the bed. Pictures of cross sections of the dismantled bed at different depths were taken. The texture in prewetted and nonprewetted beds of glass beads (nonporous) and alumina particles (porous) of different sizes was studied. To supplement the visual observation, exit liquid distribution and pressure drop were measured. The liquid flow texture was deduced from the visual observations and exit liquid distribution. The present study revealed several unexpected features. The behavior of a bed of porous particles was strikingly different from that of nonporous particles. The interpretation of the previous work on the hysteresis of pressure drop, gas−liquid and liquid−solid interfacial areas, and the implications of the observed liquid flow texture on modeling of trickle-bed reactors are discussed.

Structural heterogeneity of particles can be investigated by their three-dimensional principal components. This paper addresses the question of whether, and with what algorithm, the three-dimensional principal components can be directly recovered from cryo-EM images. The first part of the paper extends the Fourier slice theorem to covariance functions showing that the three-dimensional covariance, and hence the principal components, of a heterogeneous particle can indeed be recovered from two-dimensional cryo-EM images. The second part of the paper proposes a practical algorithm for reconstructing the principal components directly from cryo-EM images without the intermediate step of calculating covariances. This algorithm is based on maximizing the (posterior) likelihood using the Expectation-Maximization algorithm. The last part of the paper applies this algorithm to simulated data and to two real cryo-EM data sets: a data set of the 70S ribosome with and without Elongation Factor-G (EF-G), and a data set of the inluenza virus RNA dependent RNA Polymerase (RdRP). The first principal component of the 70S ribosome data set reveals the expected conformational changes of the ribosome as the EF-G binds and unbinds. The first principal component of the RdRP data set reveals a conformational change in the two dimers of the RdRP.

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