The Improvement of Discrete Wavelet Transform

Discrete wavelet transform and discrete periodic wavelet transform have been widely used in image compression and data approximation. Due to discontinuity on the boundary of original data, the decay rate of the obtained wavelet coefficients is slow. In this study, we use the combination of polynomial interpolation and one-dimensional/two-dimensional discrete periodic wavelet transforms to mitigate boundary effects. The decay rate of the obtained wavelet coefficients in our improved algorithm is faster than that of traditional two-dimensional discrete wavelet transform. Moreover, our improved algorithm can be extended naturally to the higher-dimensional case.

Construction of nonuniform periodic wavelet frames on non-Archimedean fields

In real life applications not all signals are obtained by uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool. Gabardo and Nashed, and Gabardo and Yu filled this gap by the concept of nonuniform multiresolution analysis and nonuniform wavelets based on the theory of spectral pairs for which the associated translation set \(\Lambda= \{0,r/N\}+2\mathbb{Z}\) is no longer a discrete subgroup of \(\mathbb{R}\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we introduce a notion of nonuniform periodic wavelet frame on non-Archimedean field. Using the Fourier transform technique and the unitary extension principle, we propose an approach for the construction of nonuniform periodic wavelet frames on non-Archimedean fields.

On sufficient frame conditions for periodic wavelet systems

Sufficient frame conditions for periodic dual wavelet are established. These conditions are given in terms of the Fourier coefficients. A way of constructing dual wavelet Riesz bases starting with trigonometric polynomials is provided.

Periodic wavelet descriptor of plant leaf and its application in botany

Leaf is one of the most important organs of plant. Leaf contour or outline, usually a closed curve, is a fundamental morphological feature of leaf in botanical research. In this paper, a novel shape descriptor based on periodic wavelet series and leaf contour is presented, which we name as Periodic Wavelet Descriptor (PWD). The PWD of a leaf actually expresses the leaf contour in a vector form. Consequently, the PWD of a leaf has a wide range in practical applications, such as leaf modeling, plant species identification and classification, etc. In this work, the plant species identification and the leaf contour reconstruction, as two practical applications, are discussed to elaborate how to employ the PWD of a plant leaf in botanical research.