On the spectrums of frame multiresolution analyses

We first give conditions for a univariate square integrable function to be a scaling function of a frame multiresolution analysis (FMRA) by generalizing the corresponding conditions for a scaling function of a multiresolution analysis (MRA). We also characterize the spectrum of the ‘central space’ of an FMRA, and then give a new condition for an FMRA to admit a single frame wavelet solely in terms of the spectrum of the central space of an FMRA. This improves the results previously obtained by Benedetto and Treiber and by some of the authors. Our methods and results are applied to the problem of the ‘containments’ of FMRAs in MRAs. We first prove that an FMRA is always contained in an MRA, and then we characterize those MRAs that contain ‘genuine’ FMRAs in terms of the unique low-pass filters of the MRAs and the spectrums of the central spaces of the FMRAs to be contained. This characterization shows, in particular, that if the low-pass filter of an MRA is almost everywhere zero-free, as is the case of the MRAs of Daubechies, then the MRA contains no FMRAs other than itself.     

Classification of Nonexpansive Symmetric Extension Transforms for Multirate Filter Banks

This paper describes and classifies a family of invertible discrete-time signal transforms, referred to assymmetric extension transforms(SET's), for finite-length signals. SET's are algorithms for applying perfect reconstruction multirate filter banks to symmetric extensions of finite-length signals, thereby avoiding the boundary artifacts introduced by simple periodic extension. A key point is when such symmetric decompositions can be formed with no increase in data storage requirements (“nonexpansive decompositions”). Transforms based on three types of symmetric extension and four classes of linear phase filters are analyzed in terms of their memory requirements for generalM-channel perfect reconstruction filter banks. The classification is shown to be complete in the sense that it contains all possible nonexpansive SET's. Completeness is then used to deduce design constraints on the construction of nonexpansiveM-channel SET's, including new obstructions to the existence of certain classes of filter banks. This paper also forms the principal technical reference on the SET algorithms incorporated in the Federal Bureau of Investigation's digital fingerprint image coding standard.     

Results on Biorthogonal Filter Banks

For a maximally decimated nonuniform filter bank, the perfect reconstruction (PR) property is equivalent to biorthogonality. We start from this result and derive a number of properties of PR filter banks. For example, no two integer decimators in a biorthogonal system can be coprime; moreover, if all analysis and synthesis filters have unit energy, then perfect reconstruction is equivalent to orthonormality. We also generalize the Nyquist and power complementary properties of orthonormal filter banks, for the biorthogonal case. We then show that whenever the decimation ratios are such that biorthogonality is possible with rational filters, it is, in particular, possible to obtain orthonormality with rational filters. This is done by developing an orthonormalization procedure. While reminiscent of the Gram–Schmidt approach, the procedure converges in a finite number of steps and furthermore preserves the filter-bank-like form of the basis functions. We also show how this technique can be applied for the decorrelation of subband signals. We will consider the problem of alias cancellation and obtain a generalization of a previously known necessary condition called compatibility.     

Shannon multiresolution analysis on the Heisenberg group

We present a notion of frame multiresolution analysis on the Heisenberg group, abbreviated by FMRA, and study its properties. Using the irreducible representations of this group, we shall define a sinc-type function which is our starting point for obtaining the scaling function. Further, we shall give a concrete example of a wavelet FMRA on the Heisenberg group which is analogous to the Shannon MRA on R.     

Frames, modular functions for shift-invariant subspaces and FMRA wavelet frames

We introduce the concept of the modular function for a shift-invariant 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 and the multiplicity functions for general multiresolution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frame scaling functions and the low-pass filters.

     

Some Results in the Theory Modulated Filter Banks and Modulated Wavelet Tight Frames

Perfect reconstruction (PR) FIR filter banks, obtained by modulation of a linear-phase, lowpass, prototype filter and of length 2Mm are well known. Recently, PR modulated filter banks (MFBs) with the analysis and synthesis banks obtained from different prototypes have been reported. This paper describes a general form of modulation that includes modulations used in the literature. This modulation depends on an integer parameter, the modulation phase. The PR property is characterized for MFBs with finite and infinite impulse response filters. The MFB PR problem reduces to roughly M/2 two-channel PR problems. A natural dichotomy in the PR conditions leads us to the concepts of Type 1 and Type 2 MFBs. Unitary MFBs are characterized by the M/2 two-channel PR filter banks also being unitary (for FIR filters of length N = 2Mm, these results are given in (Malvar, Electr Lett. 26, June 1990, 906–907; Koilpillai and Vaidyanathan, IEEE Trans. SP40, No. 4, Apr. 1992, 770–783)). We also give a necessary and sufficient condition for a large class (including FIR) unitary MFB prototypes to have symmetric (even or odd) prototype filters, and exhibit unitary MFBs without symmetric prototypes. A parameterization of all FIR unitary MFBs is also given. An efficient design procedure for FIR unitary MFBs is developed. It turns out that MFBs can be implemented efficiently using Type III and Type IV DCTs. Compactly supported modulated wavelet tight frames are shown to exist and completely parameterized. K-regular modulated WTFs are designed numerically and analytically by solving a set of non-linear equations over the parameters. Design of optimal modulated WTFs for the representation of any given signal is described with examples, and this is used to design smooth modulated WTFs.     

The characterization of a class of subspace pseudoframes with arbitrary real number translations

In this article, the notion of generalized multiresolution structure is introduced. The concept of subspace pseudoframes with arbitrary real number translations is proposed. A new method for constructing a generalized multiresolution structure in Paley–Wiener subspace of L²(R) is presented. A pyramid decomposition scheme is established based on such a generalized multiresolution structure. Finally, affine frames of space L²(R) with arbitrary real number translations are obtained by virtue of the subspace pseudoframes and the pyramid decomposition scheme. Relation to some physical theories such as quarks confinement is also investigated.     

Theory,implementation and applications of nonstationary Gabor frames

Signal analysis with classical Gabor frames leads to a fixed time–frequency resolution over the whole time–frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time–frequency resolution changing over time or frequency. We describe the construction of the resulting nonstationary Gabor frames and give the explicit formula for the canonical dual frame for a particular case, the painless case. We show that wavelet transforms, constant-Q transforms and more general filter banks may be modeled in the framework of nonstationary Gabor frames. Further, we present the results in the finite-dimensional case, which provides a method for implementing the above-mentioned transforms with perfect reconstruction. Finally, we elaborate on two applications of nonstationary Gabor frames in audio signal processing, namely a method for automatic adaptation to transients and an algorithm for an invertible constant-Q transform.     

Theory, implementation and applications of nonstationary Gabor frames

Signal analysis with classical Gabor frames leads to a fixed time-frequency resolution over the whole time-frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time-frequency resolution changing over time or frequency. We describe the construction of the resulting nonstationary Gabor frames and give the explicit formula for the canonical dual frame for a particular case, the painless case. We show that wavelet transforms, constant-Q transforms and more general filter banks may be modeled in the framework of nonstationary Gabor frames. Further, we present the results in the finite-dimensional case, which provides a method for implementing the above-mentioned transforms with perfect reconstruction. Finally, we elaborate on two applications of nonstationary Gabor frames in audio signal processing, namely a method for automatic adaptation to transients and an algorithm for an invertible constant-Q transform.     

Biorthogonal wavelets with 4-fold axial symmetry for quadrilateral surface multiresolution processing

Surface multiresolution processing is an important subject in CAGD. It also poses many challenging problems including the design of multiresolution algorithms. Unlike images which are in general sampled on a regular square or hexagonal lattice, the meshes in surfaces processing could have an arbitrary topology, namely, they consist of not only regular vertices but also extraordinary vertices, which requires the multiresolution algorithms have high symmetry. With the idea of lifting scheme, Bertram (Computing 72(1–2):29–39, 2004) introduces a novel triangle surface multiresolution algorithm which works for both regular and extraordinary vertices. This method is also successfully used to develop multiresolution algorithms for quad surface and \(\sqrt 3\) triangle surface processing in Wang et al. (Vis Comput 22(9–11):874–884, 2006; IEEE Trans Vis Comput Graph 13(5):914–925, 2007) respectively. When considering the biorthogonality, these papers do not use the conventional \(L^2({{\rm I}\kern-.2em{\rm R}}^2)\) inner product, and they do not consider the corresponding lowpass filter, highpass filters, scaling function and wavelets. Hence, some basic properties such as smoothness and approximation power of the scaling functions and wavelets for regular vertices are unclear. On the other hand, the symmetry of subdivision masks (namely, the lowpass filters of filter banks) for surface subdivision is well studied, while the symmetry of the highpass filters for surface processing is rarely considered in the literature. In this paper we introduce the notion of 4-fold symmetry for biorthogonal filter banks. We demonstrate that 4-fold symmetric filter banks result in multiresolution algorithms with the required symmetry for quad surface processing. In addition, we provide 4-fold symmetric biorthogonal FIR filter banks and construct the associated wavelets, with both the dyadic and \(\sqrt 2\) refinements. Furthermore, we show that some filter banks constructed in this paper result in very simple multiresolution decomposition and reconstruction algorithms as those in Bertram (Computing 72(1–2):29–39, 2004) and Wang et al. (Vis Comput 22(9–11):874–884, 2006; IEEE Trans Vis Comput Graph 13(5):914–925, 2007). Our method can provide the filter banks corresponding to the multiresolution algorithms in Wang et al. (Vis Comput 22(9–11):874–884, 2006) for dyadic multiresolution quad surface processing. Therefore, the properties of the scaling functions and wavelets corresponding to those algorithms can be obtained by analyzing the corresponding filter banks.     

Wavelets with Frame Multiresolution Analysis

A frame multiresolution (FMRA for short) orthogonalwavelet is a single-function orthogonal wavelet such that theassociated scaling space V0 admits a normalized tight frame(under translations). In this article, we prove that for anyexpansive matrix A with integer entries, there existA-dilation FMRA orthogonal wavelets. FMRA orthogonal waveletsfor some other expansive matrix with non integer entries are also discussed.     

Compactly Supported Tight Frames Associated with Refinable Functions

It is well known that in applied and computational mathematics, cardinal B-splines play an important role in geometric modeling (in computer-aided geometric design), statistical data representation (or modeling), solution of differential equations (in numerical analysis), and so forth. More recently, in the development of wavelet analysis, cardinal B-splines also serve as a canonical example of scaling functions that generate multiresolution analyses of L²(−∞,∞). However, although cardinal B-splines have compact support, their corresponding orthonormal wavelets (of Battle and Lemarie) have infinite duration. To preserve such properties as self-duality while requiring compact support, the notion of tight frames is probably the only replacement of that of orthonormal wavelets. In this paper, we study compactly supported tight frames Ψ={ψ¹,…,ψ^N} for L²(−∞,∞) that correspond to some refinable functions with compact support, give a precise existence criterion of Ψ in terms of an inequality condition on the Laurent polynomial symbols of the refinable functions, show that this condition is not always satisfied (implying the nonexistence of tight frames via the matrix extension approach), and give a constructive proof that when Ψ does exist, two functions with compact support are sufficient to constitute Ψ, while three guarantee symmetry/anti-symmetry, when the given refinable function is symmetric.     

Directional Haar wavelet frames on triangles

Traditional wavelets are not very effective in dealing with images that contain orientated discontinuities (edges). To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. In recent years several approaches like curvelets and shearlets have been studied providing essentially optimal approximation properties for images that are piecewise smooth and have discontinuities along C²-curves. While curvelets and shearlets have compact support in frequency domain, we construct directional wavelet frames generated by functions with compact support in time domain. Our Haar wavelet constructions can be seen as special composite dilation wavelets, being based on a generalized multiresolution analysis (MRA) associated with a dilation matrix and a finite collection of ‘shear’ matrices. The complete system of constructed wavelet functions forms a Parseval frame. Based on this MRA structure we provide an efficient filter bank algorithm. The freedom obtained by the redundancy of the applied Haar functions will be used for an efficient sparse representation of piecewise constant images as well as for image denoising.     

A Characterization of Schauder Frames Which Are Near-Schauder Bases

A basic problem of interest in connection with the study of Schauder frames in Banach spaces is that of characterizing those Schauder frames which can essentially be regarded as Schauder bases. In this paper, we give a solution to this problem using the notion of the minimal-associated sequence spaces and the minimal-associated reconstruction operators for Schauder frames. We prove that a Schauder frame is a near-Schauder basis if and only if the kernel of the minimal-associated reconstruction operator contains no copy of c ₀. In particular, a Schauder frame of a Banach space with no copy of c ₀ is a near-Schauder basis if and only if the minimal-associated sequence space contains no copy of c ₀. In these cases, the minimal-associated reconstruction operator has a finite dimensional kernel and the dimension of the kernel is exactly the excess of the near-Schauder basis. Using these results, we make related applications on Besselian frames and near-Riesz bases.     

Erratum to: Sobolev Duals in Frame Theory and Sigma-Delta Quantization

A new class of alternative dual frames is introduced in the setting of finite frames for ℝ ^d . These dual frames, called Sobolev duals, provide a high precision linear reconstruction procedure for Sigma-Delta (ΣΔ) quantization of finite frames. The main result is summarized as follows: reconstruction with Sobolev duals enables stable rth order Sigma-Delta schemes to achieve deterministic approximation error of order O(N^-r)\mathcal{O}(N^{-r}) for a wide class of finite frames of size N. This asymptotic order is generally not achievable with canonical dual frames. Moreover, Sobolev dual reconstruction leads to minimal mean squared error under the classical white noise assumption.     

Painless Reconstruction from Magnitudes of Frame Coefficients

The goal of this paper is to develop fast algorithms for signal reconstruction from magnitudes of frame coefficients. This problem is important to several areas of research in signal processing, especially speech recognition technology, as well as state tomography in quantum theory. We present linear reconstruction algorithms for tight frames associated with projective 2-designs in finite-dimensional real or complex Hilbert spaces. Examples of such frames are two-uniform frames and mutually unbiased bases, which include discrete chirps. The number of operations required for reconstruction with these frames grows at most as the cubic power of the dimension of the Hilbert space. Moreover, we present a very efficient algorithm which gives reconstruction on the order of d operations for a d-dimensional Hilbert space.     

Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear filter

In this work we study connections between various asymptotic properties of the nonlinear filter. It is assumed that the signal has a unique invariant probability measure. The key property of interest is expressed in terms of a relationship between the observation σ field and the tail σ field of the signal, in the stationary filtering problem. This property can be viewed as the permissibility of the interchange of the order of the operations of maximum and countable intersection for certain σ-fields. Under suitable conditions, it is shown that the above property is equivalent to various desirable properties of the filter such as

(a) uniqueness of invariant measure for the signal,

(b) uniqueness of invariant measure for the pair (signal, filter),

(c) a finite memory property of the filter,

(d) a property of finite time dependence between the signal and observation σ fields and

(e) asymptotic stability of the filter.

Previous works on the asymptotic stability of the filter for a variety of filtering models then identify a rich class of filtering problems for which the above equivalent properties hold.     

Nonparametric denoising of signals with unknown local structure, I: Oracle inequalities

We consider the problem of pointwise estimation of multi-dimensional signals s, from noisy observations (y_τ) on the regular grid . Our focus is on the adaptive estimation in the case when the signal can be well recovered using a (hypothetical) linear filter, which can depend on the unknown signal itself. The basic setting of the problem we address here can be summarized as follows: suppose that the signal s is “well-filtered”, i.e. there exists an adapted time-invariant linear filter with the coefficients which vanish outside the “cube” {0,…,T}^d which recovers s₀ from observations with small mean-squared error. We suppose that we do not know the filter q^*, although, we do know that such a filter exists. We give partial answers to the following questions:

– is it possible to construct an adaptive estimator of the value s₀, which relies upon observations and recovers s₀ with basically the same estimation error as the unknown filter ?

– how rich is the family of well-filtered (in the above sense) signals?

We show that the answer to the first question is affirmative and provide a numerically efficient construction of a nonlinear adaptive filter. Further, we establish a simple calculus of “well-filtered” signals, and show that their family is quite large: it contains, for instance, sampled smooth signals, sampled modulated smooth signals and sampled harmonic functions.

Existence of Matrix-Valued Multiresolution Analysis-Based Matrix-Valued Tight Wavelet Frames

In this article, we introduce and study the matrix-valued tight wavelet frames for analyzing matrix-valued signal based on matrix-valued multiresolution analysis (MMRA). We put our emphasis on the existence of the MMRA-based matrix-valued tight wavelet frames by establishing the correspondence with their the unitary extension principle (UEP). Here in particular we introduce the square brackets product and the quasi-interpolatory operator, which makes the certificating process for UEP become relatively simple. Some interesting byproducts, such as features on the quasi-interpolatory operator 𝒫_n in the matrix-valued function space case, are the critical foundation for our main work.