Tight frames and rank-one quantum measurements are shown to be intimately related. In fact, the family of normalized tight frames for the space in which a quantum mechanical system lies is precisely the family of rank-one generalized quantum measurements (POVMs) on that space. Using this relationship, frame-theoretical analogues of various quantum-mechanical concepts and results are developed. ...

A Wilson orthonormal basis was constructed in 1991 by I. Daubechies, S. Jaffard and J.L. Journé from Gabor tight frame elements, when the redundancy of the Gabor system is 2. In 1994 P. Auscher gave a characterization of the atoms for which the Wilson system is an orthonormal basis. Afterwards, K. Gröchenig posed a question whether the construction of an orthonormal Wilson basis is possible for...

In this paper, we study the performance of the PCM scheme with linear quantization rule for quantizing finite unit-norm tight frame expansions for R and derive the PCM quantization error without the White Noise Hypothesis. We prove that for the class of unit norm tight frames derived from uniform frame paths the quantization error has an upper bound of O(δ) regardless of the frame redundancy. T...

Spline wavelet tight frames of [20] have been used widely in frame based image analysis and restorations (see, e.g. survey articles [15, 22]). However, except for the tight frame property and the approximation order of the truncated series (see [13,20]), there are few other properties of this family of spline wavelet tight frames to be known. This paper is to present a few new properties of thi...

Tight frames can be characterized as those frames which possess optimal numerical stability properties. In this paper, we consider the question of modifying a general frame to generate a tight frame by rescaling its frame vectors; a process which can also be regarded as perfect preconditioning of a frame by a diagonal operator. A frame is called scalable, if such a diagonal operator exists. We ...

This paper is to construct tight wavelet frame systems containing a set of canonical filters by applying the unitary extension principle of [20]. A set of filters are canonical if the filters in this set are generated by flipping, adding a conjugation with a proper sign adjusting from one filter. The simplest way to construct wavelets of s-variables is to use the 2−1 canonical filters generated...

We introduce the concept of the modular function for a shiftinvariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix A and...

In this article, we show that a finite dimensional Hilbert space can have an infinite Bessel sequence, but a normalized Bessel sequence in a finite dimensional Hilbert space must be of finite length. A relation between the dimension of a given finite dimensional Hilbert space and the bound of any finite normalized tight frame for the underlying space is obtained. Also some properties of the fra...

Sparse representation of images under certain transform domain has been playing a fundamental role in image restoration tasks. One such representative method is the widely used wavelet tight frame systems. Instead of adopting fixed filters for constructing a tight frame to sparsely model any input image, a data-driven tight frame was proposed for the sparse representation of images, and shown t...

Finite tight frames for polynomial subspaces are constructed using monic Hahn polynomials and Krawtchouk polynomials of several variables. Based on these polynomial frames, two methods for constructing tight frames for the Euclidean spaces are designed. With r(d, n) := ( n+d−1 n ) , the first method generates, for each m ≥ n, two families of tight frames in R with r(d + 1,m) elements. The secon...