ANTONINI et aL.: IMAGE CODING USING WAVELET TRANSFORM TABLE I FILTER COEFFICIENTS FOR THE SPLINE VARIANT WITH LESS DISSIMILAR ± 2-1/h,0.6029490.266864-0.078223-0.0168640.02674 2-120.5575430.295636-0.028 0.045636 d etsψ. ngths: /=4-kk=4).(a) Scaling function o.(b)Scaling function o(c)Wavelet 4. (d) Wavelet 2 various extensions of the one-dimensional wavelet FILTER COEFFICIENTS FOR EXAMPLE 3. THE ENTRIES ARE RATIONAL, AND transform to higher dimensions. We follow Mallat [27 LAPLACIAN PYRAMID FILTER PROPOSED IN [9]. IN THIS CASE and use a two-dimensional wavelet transform in which =2=k,E= horizontal and vertical orientations are considered pre 0 ±4 esential In two-dimensional wavelet analysis one introduces like o in the one-dimensional case, a scaling function o(x. y) 17/28 such that o(x, v)= o(x)o( v) (11) The two biorthogonal filters in this example are both close to an orthonormal wavelet filter of length 6 con- where o(r) is a one-dimensional scaling function structed in [17], where it was called a:coiflet. Being Let v(x)be the one-dimensional wavelet associated with an orthonormal wavelet filter, the coiflet is nonsymme- the scaling function (x). Then, the three two-dimer tric. The filters in this example are shorter than in exam- sional wavelets are defined as ples I and 2, but k is also smaller. The next example in this family corresponds to k= 4(and /=4); the filters h ψ"(x,y)=o(x)ψ(y) and h then have length 9 and 15: they are both close to a coiflet of length 12 (x,y)=ψ(x)q(y) 5) Extension to the Two-Dimensional Case: There ex ψ(x,y)=ψ(x)(y) (12)