Weak nonhomogeneous wavelet dual frames for Walsh reducing subspace of L2(ℝ+)

In wavelet analysis, refinable functions are the bases of extension principles for constructing (weak) dual wavelet frames for [Formula: see text] and its reducing subspaces. This paper addresses refinable function-based dual wavelet frames construction in Walsh reducing subspaces of [Formula: see text]. We obtain a Walsh–Fourier transform domain characterization for weak [Formula: see text]-adic nonhomogeneous dual wavelet frames; and present a mixed oblique extension principle for constructing weak [Formula: see text]-adic nonhomogeneous dual wavelet frames in Walsh reducing subspaces of [Formula: see text].

Optical multiple-image encryption in discrete multiple-parameter fractional Fourier transform scheme using complex encoding, theta modulation and spectral fusion

We present a novel encryption method for multiple images in a discrete multiple-parameter fractional Fourier transform scheme, using complex encoding, theta modulation and spectral fusion. All pairs of original images are encoded separately into a complex signal. The spectrum of each complex signal can then be scattered into various positions in the spectral plane and multiplexed into one spectral image with a combination of theta modulation and spectral fusion. After Fourier transforming back to the spatial domain, the multiplexed signal is encrypted in the discrete multiple-parameter fractional Fourier transform domain. Information about the original images can only be successfully decrypted given the possession of all correct keys. The parameters of chaotic pixel scrambling for the proposed method enlarge the key space. Moreover, the proposed method solves the crosstalk problem of multiple images and improves the multiplexing capacity. Numerical simulations demonstrate the effectiveness of the proposed method.

The N-Dimensional Uncertainty Principle for the Free Metaplectic Transformation

The free metaplectic transformation is an N-dimensional linear canonical transformation. This transformation operator is useful, especially for signal processing applications. In this paper, in order to characterize simultaneously local analysis of a function (or signal) and its free metaplectic transformation, we extend some different uncertainty principles (UP) from quantum mechanics including Classical Heisenberg’s uncertainty principle, Nazarov’s UP, Donoho and Stark’s UP, Hardy’s UP, Beurling’s UP, Logarithmic UP, and Entropic UP, which have already been well studied in the Fourier transform domain.