Construction of Nonseparable Multiwavelets for Nonlinear Image Compression
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onstruction of Nonseparable Multiwavelets for Nonlinear Image Compression Ana M. C. Ruedin Departamento de Computación, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina Email: [email protected] Received 31 July 2001 and in revised form 2 November 2001 A procedure for the construction of balanced orthogonal nonseparable quincunx multiwavelets, having filters with good lowpass properties, is introduced. The matrix filter bank is viewed as the polyphase matrix of other filters, upon which the lowpass condition is imposed. The multiscaling functions obtained are plotted by means of the cascade algorithm. The process of transforming an image with these wavelets is outlined: formulae for analysis and synthesis are given, the first steps are illustrated with images, and the decomposition of the original image into two input images is addressed. Compression is achieved in a nonlinear process. Experimental results show that (i) the constructed multiwavelets having lowpass properties perform better than other nonseparable multiwavelets, (ii) the energy compaction in the fine detail subbands is greater for the multiwavelets than for the one-dimensional wavelets tried. Keywords and phrases: multiwavelets, nonseparable, quincunx, polyphase, balanced.
1. INTRODUCTION In the last 15 years wavelets have been an expanding research field, with many applications such as image compression, image denoising, and pattern recognition, among others. They have proved very efficient in image compression: they have good time-frequency localization, they decorrelate the data and give a sparse representation of the image. Multiwavelets are a generalization of the wavelet theory. They exploit the spatial correlations between various input images, such as multitemporal images. They can be designed to have several suitable properties simultaneously, such as orthogonality, polynomial approximation, short support and symmetry, see [1, 2, 3, 4, 5]. They have given good results for signal compression, see [6, 7, 8, 9]. In order to apply multiwavelets, either the input data must be prefiltered (see [10, 11, 12, 13, 14]) or the multiwavelets themselves must be balanced (see [15, 16]). Nonseparable wavelets have been introduced and investigated, and examples have been given in [17, 18, 19]. This is a more general setting than the classical one, nonseparable filters are used, and decimation is achieved with a dilation matrix. In this way the errors of thresholding the transformed coefficients do not lie mainly in the horizontal and vertical directions—which does not agree with our visual system. In an attempt to unify advances made in both directions: multiwavelets, and nonseparable bidimensional wavelets, examples were built of continuous, nonseparable, orthogonal multiscaling functions in [20]. They are compactly supported, have quincunx decimation, and have polynomial
approximation orders (i.e., accuracy) 2 and 3. Their corresponding multiwavelets were also found. In [21] other examples were given, with the additional property of be
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