Dual Gramian analysis is one of the fundamental tools developed in a series of papers [37, 40, 38, 39, 42] for studying frames. Using dual Gramian analysis, the frame operator can be represented as a family of matrices composed of the Fourier transforms of the generators of (generalized) shiftinvariant systems, which allows us to characterize most properties of frames and tight frames in terms of their generators. Such a characterization is applied in the above-mentioned papers to two widely used frame systems, namely Gabor and wavelet frame systems. Among many results, we mention here the discovery of the duality principle for Gabor frames [40] and the unitary extension principle for wavelet frames [38]. This paper aims at establishing the dual Gramian analysis for frames in a general Hilbert space and subsequently characterizing the frame properties of a given system using the dual Gramian matrix generated by its elements. Consequently, many interesting results can be obtained for frames in Hilbert spaces, e.g., estimates of the frame bounds in terms of the frame elements and the duality principle. Moreover, this new characterization provides new insights into the unitary extension principle in [38], e.g., the connection between the unitary extension principle and the duality principle in a weak sense. One application of such a connection is a simplification of the construction of multivariate tight wavelet frames from a given refinable mask. In contrast to the existing methods that require completing a unitary matrix with trigonometric polynomial entries from a given row, our method greatly simplifies the tight wavelet frame construction by converting it to a constant matrix completion problem. To illustrate its simplicity, the proposed construction scheme is used to construct a few examples of multivariate tight wavelet frames from box splines with certain desired properties, e.g., compact support, symmetry or anti-symmetry.

The subject of this article is the duality principle, which, well beyond its stand at the heart of Gabor analysis, is a universal principle in frame theory that gives insight into many phenomena. Its fiber matrix formulation for Gabor systems is the driving principle behind seemingly different results. We show how the classical duality identities, operator representations and constructions for dual Gabor frames are in fact aspects of the dual Gramian matrix fiberization and its sole duality principle, giving a unified view to all of them. We show that the same duality principle, via dual Gramian matrix analysis, holds for dual (or bi-) systems in abstract Hilbert spaces. The essence of the duality principle is the unitary equivalence of the frame operator and the Gramian of certain adjoint systems. An immediate consequence is, for example, that, even on this level of generality, dual frames are characterized in terms of biorthogonality relations of adjoint systems. We formulate the duality principle for irregular Gabor systems which have no structure whatsoever to the sampling of the shifts and modulations of the generating window. In case the shifts and modulations are sampled from lattices we show how the abstract matrices can be reduced to the simple structured fiber matrices of shift-invariant systems, thus arriving back in the well understood territory. Moreover, in the arena of multiresolution analysis J Fourier Anal Appl (MRA)-wavelet frames, the mixed unitary extension principle can be viewed as the duality principle in a sequence space. This perspective leads to a construction scheme for dual wavelet frames which is strikingly simple in the sense that it only needs the completion of an invertible constant matrix. Under minimal conditions on the MRA, our construction guarantees the existence and easy constructability of non-separable multivariate dual MRA-wavelet frames. The wavelets have compact support and we show examples for multivariate interpolatory refinable functions. Finally, we generalize the duality principle to the case of transforms that are no longer defined by discrete systems, but may have discrete adjoint systems.

In electron microscopy, three-dimensional (3D) reconstruction is one key component in many computerized techniques for solving 3D structures of large protein assemblies using electron microscopy images of particles. Main challenges in 3D reconstruction include very low signal-to-noise ratio and very large scale of data sets involved in the computation. Motivated by the recent advances of sparsity-based regularization in the wavelet frame domain for solving various linear inverse problems in imaging science, we proposed a wavelet tight frame based 3D reconstruction approach that exploits the sparsity of the 3D density map in a wavelet tight frame system. The proposed approach not only runs efficiently in terms of CPU time but also requires a much lower memory footprint than existing framelet-based regularization methods. The convergence of the proposed iterative scheme and the functional it minimizes is also examined, together with the connection to existing wavelet frame based regularizations. The numerical experiments showed good performance of the proposed method when it is used in two electron microscopy techniques: the single particle method and electron tomography. Introduction.In the past few decades, with the advances in specimen preparation methods and image processing techniques, electron microscopy (EM) techniques [14,17,23,24,50,29] have become indispensable tools for determining the three-dimensional (3D) structures of macromolecules, macromolecular complexes, and cells. Among all computerized techniques used in EM, the single particle method (SPM) and electron tomography (ET) are two of the most popular ones. The specimen preparation methods mainly include the negative staining method, in which specimens are stained in heavy metal salts, and the frozen-hydrated method, in which specimens are embedded in vitreous ice. The EM technique using ice-embedded specimens is often called cryo-EM. SPM includes a set of image processing techniques for building up the 3D structure of a particle by using two-dimensional (2D) digitized EM images (projections) of many identical particles. The main modules in the SPM include image alignment and classification; image filtering and contrast transfer function (CTF) correction; and 3D reconstruction (see [24,50,30,29] for more details). These modules are integrated in an iterative framework to iteratively refine the 3D result using the projection matching methods [41]. ET is a tomography technique to reconstruct a detailed 3D structure of the macromolecular complex or cell from 2D images collected from different tilted stages of the object. The main image

Transmission imaging, as an important imaging technique widely used in astronomy, medical diagnosis, and biology science, has been shown in [49] quite different from reflection imaging used in our everyday life. Understanding the structures of images (the prior information) is important for designing, testing, and choosing image processing methods, and good image processing methods are helpful for further uses of the image data, e.g., increasing the accuracy of the object reconstruction methods in transmission imaging applications. In reflection imaging, the images are usually modeled as discontinuous functions and even piecewise constant functions. In transmission imaging, it was shown very recently in [49] that almost all images are continuous functions. However, the author in [49] considered only the case of parallel beam geometry and used some too strong assumptions in the proof, which exclude some common cases such as cylindrical objects. In this paper, we consider more general beam geometries and simplify the assumptions by using totally different techniques.In particular, we will prove that almost all images in transmission imaging with both parallel and divergent beam geometries (two most typical beam geometries) are continuous functions, under much weaker assumptions than those in [49], which admit almost all practical cases. Besides, taking into accounts our analysis, we compare two image processing methods for Poisson noise (which is the most significant noise in transmission imaging) removal. Numerical experiments will be provided to demonstrate our analysis.