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A wavelet theory for local fields and related groups
 J. GEOM. ANAL
, 2004
"... Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G = Qp, the field of padic rational numbers (as a group under addition), which has compact open subgroup H = Zp, the ring of padic integers. Classical wavelet theories, which require a ..."
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Cited by 35 (1 self)
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Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G = Qp, the field of padic rational numbers (as a group under addition), which has compact open subgroup H = Zp, the ring of padic integers. Classical wavelet theories, which require a non trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. A wavelet theory is developed on G using coset representatives of the discrete quotient ̂G/H ⊥ to circumvent this limitation. Wavelet bases are constructed by means of an iterative method giving rise to socalled wavelet sets in the dual group ̂G. Although the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, it is observed that their analogues for G are equivalent.
A Wreath Product Group Approach to Signal and Image Processing: Part II  Convolution, Correlation, and Applications
 IEEE Trans. on Sig. Proc
, 1999
"... This paper continues the investigation of the use of spectral analysis on certain noncommutative #nite groupswreath product groupsin digital signal processing. We describe here the generalization of discrete cyclic convolution to convolution over these groups and showhow it reduces to multip ..."
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Cited by 22 (3 self)
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This paper continues the investigation of the use of spectral analysis on certain noncommutative #nite groupswreath product groupsin digital signal processing. We describe here the generalization of discrete cyclic convolution to convolution over these groups and showhow it reduces to multiplication in the spectral domain. Finite groupbased convolution is de#ned in both the spatial and spectral domains and its properties established. Wepay particular attention to wreath product cyclic groups and further describe convolution properties from a geometric view point, in terms of operations with speci#c signals and #lters. Groupbased correlation is de#ned in a natural way and its properties follow from those of convolution. We #nally consider an application of convolution: the detection of similarity of perceptually similar signals, and an application of correlation: the detection of similarity of group transformed signals. Several examples using images are included to demo...
ELA WAVE PACKET TRANSFORMS OVER FINITE FIELDS∗
"... Abstract. This article introduces the abstract notion of finite wave packet groups over finite fields as the finite group of dilations, translations, and modulations. Then it presents a unified theoretical linear algebra approach to the theory of wave packet transforms (WPT) over finite fields. It i ..."
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Abstract. This article introduces the abstract notion of finite wave packet groups over finite fields as the finite group of dilations, translations, and modulations. Then it presents a unified theoretical linear algebra approach to the theory of wave packet transforms (WPT) over finite fields. It is shown that each vector defined over a finite field can be represented as a finite coherent sum of wave packet coefficients as well.