A family of Mexican hat wavelet transforms associated with an isometry in the heat equation

Abstract

In this paper, Green's function of the heat equation is used to obtain one of the forms in a family of Mexican hat wavelet transforms. By implementing the theory of reproducing kernels, an isometry in the heat equation is also investigated. Further, a natural inversion is established for the Mexican hat wavelet transform. Several related recent investigations are also considered.

Characterizations of Continuous Fractional Bessel Wavelet Transforms

“…Proof. We can easily obtain the result (37), which is asserted by Theorem 7, from [6] by using the equations ( 21) to (23).…”

“…The theory that we have developed in this article is potentially useful for a variety of applications of the fractional Bessel wavelet transform in signal processing, image processing, quantum mechanics and other areas of engineering and applied sciences. Some instances of applications have been presented in Section 3 (see also several recent developments involving continuous and discrete wavelet transforms in [5,[18][19][20][21][22][23][24][25][26][27], each of which will presumably motivate further researches involving the continuous and discrete fractional Bessel wavelet transforms).…”

Non-Separable Linear Canonical Wavelet Transform

“…The most prompt ones are the fractional wavelet transform [10], linear canonical wavelet transform [11,12], special affine wavelet transform [13,14], quaternion linear canonical wavelet transform [15], and quadratic-phase wavelet transform [16]. Unfortunately, all these transforms only perform well at representing point singularities and are incompetent at handling the distributed singularities, such as curves or edges in higher-dimensional signals [17][18][19][20]. The intuitive reason for this inadequacy is that wavelets are isotropic entities generated by isotropically dilating the mother wavelet, and as such, they ignore the geometric properties of the structures to be analyzed.…”