Preconditioning of frames

Tight frames are extremely useful in applications. A scalable frame was recently introduced as a frame with the property of generating a tight frame by rescaling its frame vectors. In this paper, we consider piecewise scalable frames. We obtain some characterizations for them, and demonstrate that scalability is stable under unitary operators and isomorphisms between two Hilbert spaces. We further obtain a relation between the piecewise scalable frames in Hilbert spaces, and their tensor product

Irreducible cyclic codes have been an interesting subject of study for a long time. Their weight distribution is known in only a few cases. In this paper, the weight distribution of the irreducible cyclic codes in a number of other cases is determined. The number of nonzero weights in the codes dealt with in this paper varies between one and four.

A tight frame in Rn is a redundant system which has a reconstruction formula similar to that of an orthonormal basis. For a unit-norm frame F = {f i }k i=1 , a scaling is a vector c = (c(l),…, c(k)) e Rk≥0 such that {c(i)f i }k i=1 is a tight frame in Rn. If a scaling c exists, we say that F is a scalable frame. A scaling c is a minimal scaling if {fi : c{i) > 0} has no proper scalable subframes. In this paper, we present the uniqueness of the orthogonal partitioning property of any set of minimal scalings and provide a construction of scalable frames by extending the standard orthonormal basis of Rn.

The unitary extension principle by Ron and Shen is one of the cornerstones of wavelet frame theory; it leads to tight frames for L 2 ( R ) L^{2}(\mathbb {R}) and associated expansions of functions f ∈ L 2 ( R ) f\in L^{2}(\mathbb {R}) of similar type as those for orthonormal wavelet bases. In this paper, the unitary extension principle is extended to the setting of a locally compact abelian group, equipped with a collection of nested co-compact subgroups. Unlike all previously known generalizations of the unitary extension principle, the current one is taking place within the setting of continuous frames, which means that the resulting decompositions of functions in the underlying Hilbert space in general are given in terms of integral representations rather than discrete sums. The frame elements themselves appear by letting a collection of modulation operators act on a countable family of basic functions.

So far there has not been paid attention to frames that are balanced, i.e. those frames which sum is zero. In this paper we consider balanced frames, and in particular balanced unit norm tight frames, in finite dimensional Hilbert spaces. Here we discover various advantages of balanced unit norm tight frames in signal processing. They give an exact reconstruction in the presence of systematic errors in the transmitted coefficients, and are optimal when these coefficients are corrupted with noises that can have non-zero mean. Moreover, using balanced frames we can know that the transmitted coefficients were perturbed, and we also have an indication of the source of the error. We analyze several properties of these types of frames. We define an equivalence relation in the set of the dual frames of a balanced frame, and use it to show that we can obtain all the duals from the balanced ones. We study the problem of finding the nearest balanced frame to a given frame, characterizing completely its existence and giving its expression. We introduce and study a concept of complement for balanced frames. Finally, we present many examples and methods for constructing balanced unit norm tight frames.

An efficient algorithm for designing projection matrix in compressive sensing based on alternating optimization

We give a comprehensive introduction to a general modular frame construction in Hilbert C*-modules and to related linear operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that admit an orthonormal modular Riesz basis. Interrelations and applications to classical frame theory are indicated. Resorting to frames in Hilbert spaces we discuss some measures for pairs of frames to be close to one another. In particular, the existence and uniqueness of the closest (normalized) tight frame to a given frame is investigated. For Riesz bases with certain restrictions the set of closest tight frames often contains a multiple of its symmetric orthogonalization.

A complex equiangular tight frame (ETF) is a tight frame consisting of N unit vectors in C d whose absolute inner products are identical. One may view complex ETFs as a natural geometric generalization of an orthonormal basis. Numerical evidence suggests that these objects do not arise for most pairs ( d , N). The goal of this paper is to develop conditions on ( d , N) under which complex ETFs can exist. In particular, this work concentrates on the class of harmonic ETFs, in which the components of the frame vectors are roots of unity. In this case, it is possible to leverage field theory to obtain stringent restrictions on the possible values for ( d , N).

The sparsity of signals and images in a certain dictionary domain has been explored in the past several decades. Recently, a newly developed transform model is proved to be powerful. We propose a new framework for overcomplete transform learning, searching the optimal transform within tight frame set, which is much simpler than the existing one. A closed form solution is derived to update such a transform. Experiments demonstrate that the proposed algorithm outperforms the existing ones in terms of recovery of reference transform for synthetic data and denoised PSNR for image denoising.

In this paper, the set of all complete multi-normalized tight frame vectors NFr(U) with multiplicity r and the set of all complete multi-frame vectors Fr(U) with multiplicity r for a system U of unitary operators acting on a separable Hilbert space are characterized in terms of co-isometries and surjective operators in \(\mathcal{C}_{\Psi ^r } \) (U), the set of all operators which locally commute with U at Ψr, a fixed complete wandering r-tuple for U. Then we study the linear combinations of multi-frame vectors for U and establish some conditions under which these combinations are still the same type of multi-frame vectors for U. Finally, we establish some interesting properties for multi-frame vectors when U is a unitary group. All these results have potential applications in the theory of multi-Gabor systems and multi-wavelet systems.

We give a physical interpretation for finite tight frames along the lines of Columb's Law in Physics. This allows us to use results from classical mechanics to anticipate results in frame theory. As a consequence, we are able to classify those frames for an <i>N</i>-dimensional Hilbert space which are the closest to being tight (in the sense of minimizing potential energy) while having the norms of the frame vectors prescribed in advance. This also yields a <i>fundamental inequality </i>that all finite tight frames must satisfy.