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... The simplest function to model an "ON-OFF" signal is the Haar function, defined by the following relations (Meyer 1994): ...
... Consequently, we hope to benefit from this to consider the wavelet theory. A function is a basic wavelet if it verifies several main conditions (Daubechies 1992;Meyer 1994;Holschneider 1995). ...
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A luminous stimulus which penetrates in a retina is converted to a nerve message. Ganglion cells give a response that may be approximated by a wavelet. We determine a function PSI which is associated with the propagation of nerve impulses along an axon. Each kind of channel (inward and outward) may be open or closed, depending on the transmembrane potential. The transition between these states is a random event. Using quantum relations, we estimate the number of channels susceptible to switch between the closed and open states. Our quantum approach was first to calculate the energy level distribution in a channel. We obtain, for each kind of channel, the empty level density and the filled level density of the open and closed conformations. The joint density of levels provides the transition number between the closed and open conformations. The algebraic sum of inward and outward open channels is a function PSI of the normalized energy E. The function PSI verifies the major properties of a wavelet. We calculate the functional dependence of the axon membrane conductance with the transmembrane energy.
... We will make it into a GMRA by tensoring it with the dilation by 2 Haar GMRA. It is known (as in [18]) that the tensor product of two GMRAs gives a GMRA. By tensoring our almost GMRA with an actual one, we will preserve all the properties of the almost GMRA, and eliminate the non-trivial intersection. ...
We discuss how generalized multiresolution analyses (GMRAs), both classical and those defined on abstract Hilbert spaces, can be classified by their multiplicity functions m and matrix-valued filter functions H. Given a natural number valued function m and a system of functions encoded in a matrix H satisfying certain conditions, a construction procedure is described that produces an abstract GMRA with multiplicity function m and filter system H. An equivalence relation on GMRAs is defined and described in terms of their associated pairs (m,H). This classification system is applied to MRAs and other classical examples in L2(Rd) as well as to previously studied abstract examples.
We construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity. The order of regularity increases linearly with the support width. We start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction. The construction then follows from a synthesis of these different approaches.
L’analyse par ondelettes. Pour la science
  • Y S Jaffard
  • O Meyer