Fast image reconstruction techniques are becoming important with the increasing number of scientific cases in high resolution micro and nano tomography. The processing of the large scale 3D data demands new mathematical tools for the tomographic reconstruction. Due to the high computational complexity of most current algorithms, big data sizes demands powerful hardware and more sophisticated numerical techniques. Several reconstruction algorithms are dependent on a mathematical tool called backprojection (a transposition process). A conventional implementation of the backprojection operator has cubic computational complexity. In the present manuscript we propose a new fast backprojection operator for the processing of tomographic data, providing a low-cost algorithm for this task. We compare our formula against other fast transposition techniques, using real and simulated large data sets.

Recently, specially crafted unidimensional optimization has been successfully used as line search to accelerate the overrelaxed and monotone fast iterative shrinkage-threshold algorithm (OMFISTA) for computed tomography. In this paper, we extend the use of fast line search to the monotone fast iterative shrinkage-threshold algorithm (MFISTA) and some of its variants. Line search can accelerate the FISTA family considering typical synthesis priors, such as the ℓ 1 -norm of wavelet coefficients, as well as analysis priors, such as anisotropic total variation. This paper describes these new MFISTA and OMFISTA with line search, and also shows through numerical results that line search improves their performance for tomographic high-resolution image reconstruction.

This paper proposes an accelerating process through the use of a fast and exact line search for the over-relaxed monotone fast iterative shrinkage-threshold algorithms (OMFISTA). This algorithm is applied to high resolution tomographic image reconstruction using data from the Brazilian Synchrotron Light Source (LNLS). The LNLS can capture lots of data from the scanned objects in each angle, but in order to reduce acquisition time, only few angles are captured. Due to the reduced angles, the l\ penalty is used as a prior to provide a proper reconstruction. Algorithms, such as the fast iterative shrinkage-threshold algorithms (FISTA) and the OMFISTA can be used with some success. However, due to the large computational time at each iteration, even with fast projection/backprojection operators, any reduction in the number of iterations is welcome. In this paper, the fast and exact unidimensional optimization for íi-í\ is used as line search to accelerate the OMFISTA. The results in this paper illustrate that this line search accelerated OMFISTA is faster than FISTA, MFISTA and the OMFISTA.

Internet users are very familiar with the results of a search query displayed as a ranked list of snippets. Each textual snippet shows a content summary of the referred document (or webpage) and a link to it. This display has many advantages, for example, it affords easy navigation and is straightforward to interpret. Nonetheless, any user of search engines could possibly report some experience of disappointment with this metaphor. Indeed, it has limitations in particular situations, as it fails to provide an overview of the document collection retrieved. Moreover, depending on the nature of the query--for example, it may be too general, or ambiguous, or ill expressed--the desired information may be poorly ranked, or results may contemplate varied topics. Several search tasks would be easier if users were shown an overview of the returned documents, organized so as to reflect how related they are, content wise. We propose a visualization technique to display the results of web queries aimed at overcoming such limitations. It combines the neighborhood preservation capability of multidimensional projections with the familiar snippet-based representation by employing a multidimensional projection to derive two-dimensional layouts of the query search results that preserve text similarity relations, or neighborhoods. Similarity is computed by applying the cosine similarity over a "bag-of-words" vector representation of collection built from the snippets. If the snippets are displayed directly according to the derived layout, they will overlap considerably, producing a poor visualization. We overcome this problem by defining an energy functional that considers both the overlapping among snippets and the preservation of the neighborhood structure as given in the projected layout. Minimizing this energy functional provides a neighborhood preserving two-dimensional arrangement of the textual snippets with minimum overlap. The resulting visualization conveys both a global view of the query results and visual groupings that reflect related results, as illustrated in several examples shown.

Abstract In this work we propose a new discretization method for the Laplace–Beltrami operator defined on point‐based surfaces. In contrast to the existing point‐based discretization techniques, our approach does not rely on any triangle mesh structure, turning out truly mesh‐free. Based on a combination of Smoothed Particle Hydrodynamics and an optimization procedure to estimate area elements, our discretization method results in accurate solutions while still being robust when facing abrupt changes in the density of points. Moreover, the proposed scheme results in numerically stable discrete operators. The effectiveness of the proposed technique is brought to bear in many practical applications. In particular, we use the eigenstructure of the discrete operator for filtering and shape segmentation. Point‐based surface deformation is another application that can be easily carried out from the proposed discretization method.