Quantitative Dip Bounds for the Two-dimensional Discrete Wavelet Transform the Two-dimensional Discrete Wavelet Transform
نویسندگان
چکیده
An analysis of the discrete wavelet transform of dipping segments with a signal of given frequency band leads to a quantitative explanation of the known division of the two-dimensional wavelet transform into horizontal, vertical and diagonal emphasis panels. The results must be understood in a \fuzzy" sense: since wavelet mirror lters overlap, the results stated can be slightly violated with violation tending to increase with shortness of the wavelet chosen. The speciic angles that delimit the three wavelet panels can be stated in terms of the Nyquist frequency F t in the rst dimension and the Nyquist frequency F x in the second dimension. The angle HV = arctan(F t =F x) approximately separates the dips < HV that appear in the horizontal panel from those greater dips that appear in the vertical panel. Similarly, the angles arctan(F t =2F x), arctan(2F t =F x) approximately bound the dips appearing in the diagonal panel. These results are simple and probably have been observed by other researchers, but we haven't found a prior reference for them. The simplest way to obtain a wavelet basis for a two-dimensional space, say, t (time) and x (space), is to multiply the one-dimensional bases: jj 0 kk 0 (t; x) = jk (t) j 0 k 0 (x): (1) This is analogous to replacing the one-dimensional Fourier kernel e i!t with e i!t e ?ikx and does, indeed, provide a two-dimensional complete orthonormal basis in the wavelet case just as it does for Fourier transform. The disadvantage of this basis is the mixing of the scales j and j 0. It turns out that it is possible to construct a basis using only a single scale j at the expense of having three wavelets (basis functions) at each j level. These three functions are H jkk 0 (t; x) = jk (t) jk 0 (x); V jkk 0 (t; x) = jk (t) jk 0 (x); (2) D jkk 0 (t; x) = jk (t) jk 0 (x): 1
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