Search results for "Zak transform"
showing 4 items of 4 documents
Non-periodic Discrete Splines
2015
Discrete Splines with different spans were introduced in Sect. 3.3.1. This chapter focuses on a special case of discrete splines whose spans are powers of 2. These splines are discussed in more detail. The Zak transform provides an integral representation of such splines. Discrete exponential splines are introduced. Generators of the discrete-spline spaces are described whose properties are similar to properties of polynomial-spline spaces generators. Interpolating discrete splines provide efficient tools for upsampling 1D and 2D signals. An algorithm for explicit computation of discrete splines is described.
Non-periodic Polynomial Splines
2015
In this chapter, we outline the essentials of the splines theory. By themselves, they are of interest for signal processing research. We use the Zak transform to derive an integral representation of polynomial splines on uniform grids. The integral representation facilitated design of different generators of spline spaces and their duals. It provides explicit expressions for interpolating and smoothing splines of any order. In forthcoming chapters, the integral representation of splines will be used for the constructions of efficient subdivision schemes and so also for the design spline-based wavelets and wavelet frames.
Polynomial Spline-Wavelets
2015
This chapter presents wavelets in the spaces of polynomial splines. The wavelets’ design is based on the Zak transform, which provides an integral representation of spline-wavelets. The exponential wavelets which participate in the integral representation are counterparts of the exponential splines that were introduced in Chap. 4. Fast algorithms for the wavelet transforms of splines are presented. Generators of spline-wavelet spaces are described, such as the B-wavelets and their duals and the Battle-Lemarie wavelets whose shifts form orthonormal bases of the spline-wavelet spaces.
Periodic Discrete Splines
2014
Periodic discrete splines with different periods and spans were introduced in Sect. 3.4. In this chapter, we discuss families of periodic discrete splines, whose periods and spans are powers of 2. As in the polynomial splines case, the Zak transform is extensively employed. It results in the Discrete Spline Harmonic Analysis (DSHA). Utilization of the Fast Fourier transform (FFT) enables us to implement all the computations in a fast explicit way.