Splines and multiresolution are two independent concepts, which – considered together – yield a vast variety of bases for image processing and image analysis. The idea of a multiresolution analysis is to construct a ladder of nested spaces that operate as some sort of mathematical looking glass. It allows to separate coarse parts in a signal or in an image from the details of various sizes. Spl...

Abstract: A wavelet basis is an orthonormal basis for L(R), the space of square-integrable functions on the real line, of the form {gnk}n,k∈Z, where gnk(t) = 2 n/2 g(2t − k) and g is a single fixed function, the wavelet. Each multiresolution analysis for L(R) determines such a basis. To find a multiresolution analysis, one can begin with a dilation equation f(t) = ∑ ck f(2t− k). If the solution...

The contribution of this work is to generalize the real and complex wavelet transforms and to derive for the first time a quaternionic wavelet pyramid for multi-resolution analysis using three phases. The paper can be very useful for researchers and practitioners interested in understanding and applications of the quaternion wavelet transform.

The LULU operators, well known in the nonlinear multiresolution analysis of sequences, are extended to functions defined on continuous domain, namely, a real interval Ω ⊆ R. Similar to their discrete counterparts, for a given δ > 0 the operators Lδ and Uδ form a fully ordered semi-group of four elements. It is shown that the compositions Lδ ◦ Uδ and Uδ ◦ Lδ provide locally δ-monotone approximat...