Image restoration is a computationally intensive problem as a large number of pixel values have to be determined. Since the pixel values of digital images can attain only a finite number of values (e.g., 8-bit images can have only 256 gray levels), one would like to recover an image within some dynamic range. This leads to the imposition of box constraints on the pixel values. The traditional gradient projection methods for constrained optimization can be used to impose box constraints, but they may suffer from either slow convergence or repeated searching for active sets in each iteration. In this paper, we develop a new box-constrained multiplicative iterative (BCMI) algorithm for box-constrained image restoration. The BCMI algorithm just requires pixelwise updates in each iteration, and there is no need to invert any matrices. We give the convergence proof of this algorithm and apply it to total variation image restoration problems, where the observed blurry images contain Poisson, Gaussian, or salt-and-pepper noises.

Since it was first recognised that eukaryotic genes are fragmented into coding segments (exons) separated by non-coding segments (introns), the reason for this phenomenon has been debated. There are two dominant theories: that the piecewise arrangement of genes allows functional protein domains, represented by exons, to recombine by shuffling to form novel proteins with combinations of functions; or that introns represent parasitic DNA that can infest the eukaryotic genome because it does not interfere grossly with the fitness of its host. Differing distributions of exon lengths are predicted by these two theories. In this paper we examine distributions of exon lengths for six different organisms and find that they offer empirical evidence that both theories may in part be correct.

This paper first develops a general multiplicative iterative (MI) algorithm for tomographic image reconstructions, where the objective function is only specified as a general function containing two components: a data mismatch component and a penalty component. This general algorithm is then applied to different objective functions deduced from different probability models for measurements in emission or transmission tomography, such as Poisson, Gaussian, or shifted Poisson models. Furthermore, an approximate line search step can be easily incorporated into the algorithm so that the objective function is guaranteed to increase during the iterations. This MI algorithm (with line search) is easy to implement, as it performs only one forward- and one or two back-projection in each iteration, and it respects the positivity constraint usually imposed on reconstructions.