Woven Continuous Frames in Hilbert Spaces

Continuous framings for Banach spaces

Some equalities and inequalities for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>g</mml:mi></mml:math>-Bessel sequences in Hilbert spaces

Multiplication is an important fundamental function in arithmetic operations. Multiplication-based operations such as Multiply and Accumulate(MAC) and inner product are among some of the frequently used computation Intensive Arithmetic Functions(CIAF) currently implemented in many Digital Signal Processing (DSP) applications such as convolution, Fast Fourier Transform(FFT), filtering and in microprocessors m its arithmetic and logic unit. Since multiplication dominates the execution tune of most DSP algorithms, so there is a need of high speed multiplier. Currently, multiplication time is still the dominant factor in determining the instruction cycle time of a DSP chip. The demand for high speed processing has been increasing as a result of expanding computer and signal processing applications. Higher throughput arithmetic operations are important to achieve the desired performance in many real-time signal and image processing applications . One of the key arithmetic operations in such applications is multiplication and the development of fast multiplier circuit has been a subject of interest over decades. Multiplier based on Vedic Mathematics is one of the fast and low power multiplier. Employing this technique in the computation algorithms will reduce the complexity, execution time, power etc. This vedic based multiplier is compared with binary multiplier(partial products method).

The purpose of this paper is to propose a definition of continuous frames of rank n for Krein spaces and to study their basic properties. Similarly to the Hilbert space case, continuous frames are characterized by the analysis, the pre-frame and the frame operator, where the latter gives rise to a frame decomposition theorem. The paper includes a discussion of similar, dual and Parseval frames and of reproducing kernels. In addition, the importance of the fundamental symmetry in the formula for the frame operator in a Krein space is clarified. As prime examples, it is shown how to transfer continuous frames for Hilbert spaces to Krein spaces arising from a possibly non-regular Gram operator.

Numerically and spectrally optimal dual frames in Hilbert spaces

A note on some equalities for frames in Hilbert spaces

As a starting point we assume to have a continuous frame in a Hilbert space with respect to a measure space. This frame inherits a unitary structure from a unitary representation of a locally compact abelian group in the Hilbert space. In this setting we state a continuous sampling result for the range space of the associated analysis frame operator. The data sampling are functions also defined by using the underlying unitary structure. The result is illustrated by using continuous frames in Paley–Wiener and shift-invariant spaces generated by translates of fixed functions. A sampling strategy working only for discrete abelian groups is also discussed.

Dual frames are generalized Riesz bases which have potential applications in signal processing. In this paper, the construction of dual frames of matrix-valued wave packet systems in the matrix-valued function space [Formula: see text] from dual pairs of atomic wave packet frames in [Formula: see text] is studied. A class of matrix-valued dual generators from its associated dual pair of atomic wave packets has been obtained. We provide a characterization of matrix-valued dual window functions in terms of orthogonality of wave packet Bessel sequences. A perturbation result with respect to window functions for matrix-valued dual frames is given. It is well known that an orthogonal Parseval Hilbert frame for a Hilbert space turned out be an orthonormal basis for the space, however, this is not true for an orthogonal matrix-valued wave packet Parseval frame for the underlying matrix-valued function space. We give a type of matrix-valued orthonormal basis associated with an orthonormal basis of [Formula: see text].

Multi‐dimensional Discrete Fourier Transforms (DFTs) play an important role in signal and image processing applications. Image reconstruction is a key component in signal processing applications like medical imaging, computer vision, face recognition etc. Two dimensional fast Fourier Transform (2D FFT) and Inverse FFT plays vital role in reconstruction. In this paper we present a fast 64 × 64 point 2D FFT architecture based on radix‐43 algorithm using a parallel unrolled radix‐4 3 FFT as the basic block. Our radix‐4 3 architecture is a memory optimized parallel architecture which computes 64‐point FFT, with least execution time. Proposed architecture produces reordered output of both 64‐point one dimensional (1D) FFT and 64 × 64 point 2D FFT, without using any additional hardware for reordering. The proposed architecture has been implemented in UMC 40nm CMOS technology with clock frequency of 500 MHz and area of 0.841 mm 2 . The power consumption of proposed architecture is 358 mW at 500 MHz. Energy efficiency (FFTs computed per unit of energy) is 341 points/Joule. Computation time of 64 × 64 point FFT is 8.19 μs. ASIC implementation results shows better performance of proposed work in terms of computation time when compared with state‐of‐art implementation. Proposed architecture has also been implemented in Virtex‐7 FPGA which gives comparable area.

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.

In this paper, in terms with the analysis operator and bounded linear operators, we give some novel conclusions on the sum of [Formula: see text]-frames in Hilbert space, which completely improve and generalize the existing results of the sum of [Formula: see text]-frames, and construct some numerical examples to illustrate the validity and rationality of our conclusions.

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

K-stationarity and wavelets

The 16 papers in this special issue focus on adaptive sparse representation of data and applications in signal and image processing.

Diffraction and holography are fertile areas for application of signal theory and processing. Recent work on 3DTV displays has posed particularly challenging signal processing problems. Various procedures to compute Rayleigh-Sommerfeld, Fresnel and Fraunhofer diffraction exist in the literature. Diffraction between parallel planes and tilted planes can be efficiently computed. Discretization and quantization of diffraction fields yield interesting theoretical and practical results, and allow efficient schemes compared to commonly used Nyquist sampling. The literature on computer-generated holography provides a good resource for holographic 3DTV related issues. Fast algorithms to compute Fourier, Walsh-Hadamard, fractional Fourier, linear canonical, Fresnel, and wavelet transforms, as well as optimization-based techniques such as best orthogonal basis, matching pursuit, basis pursuit etc., are especially relevant signal processing techniques for wave propagation, diffraction, holography, and related problems. Atomic decompositions, multiresolution techniques, Gabor functions, and Wigner distributions are among the signal processing techniques which have or may be applied to problems in optics. Research aimed at solving such problems at the intersection of wave optics and signal processing promises not only to facilitate the development of 3DTV systems, but also to contribute to fundamental advances in optics and signal processing theory.