Comparison between Orthogonal and Bi-Orthogonal Wavelets

Theoretical aspects and analysis of wavelets have applications in mathematical modeling, artificial neural networks, digital signal processing, and image processing and numerical methods. The term orthogonal deals with the mathematical part which covers a wide area of digital signal processing and image processing. An orthogonal wavelet generates the wavelet whose nature is orthogonal. This means an inverse or transpose wavelet transform is nothing but the adjoint of a wavelet transform. If this condition fails by missing orthogonality it may result in biorthogonal wavelets. Single scaling functions and single wavelets are generated but orthogonal wavelet filter bank. A biorthogonal wavelet associated with the wavelet transformation is invertible. There is no need that if it is invertible so it should be orthogonal. The biorthogonal wavelet allows maximum freedom in the case of designing the orthogonal wavelet. It also supports the construction of symmetric wavelet functions. In biorthogonal wavelets, as the name indicates, two scaling factors or functions are responsible for the generation of the various multi-resolutions on the basis of different wavelets. For the image and signal reconstruction purpose we need that wavelet. We get a better result in the presence of biorthogonal wavelets. In the present work, we analyzed the performance of orthogonal and biorthogonal wavelet filters for image processing. We test the image and observed that the filter coefficient and image quality for the orthogonal and biorthogonal wavelet. On the basis of performance analysis it is concluded that biorthogonal wavelets are better than orthogonal wavelets.

Construction of wavelets and framelets on a bounded interval

Many problems in applications are defined on a bounded interval. Therefore, wavelets and framelets on a bounded interval are of importance in both theory and application. There is a great deal of effort in the literature on constructing various wavelets on a bounded interval and exploring their applications in areas such as numerical mathematics and signal processing. However, many papers on this topic mainly deal with individual examples which often have many boundary wavelets with complicated structures. In this paper, we shall propose a method for constructing wavelets and framelets in [Formula: see text] from symmetric wavelets and framelets on the real line. The constructed wavelets and framelets in [Formula: see text] often have a few simple boundary wavelets/framelets with the additional flexibility to satisfy various desired boundary conditions. To illustrate our construction method, from several spline refinable vector functions, we present several examples of (bi)orthogonal wavelets and spline tight framelets in [Formula: see text] with very simple boundary wavelets/framelets.