Two-dimensional geometric lifting

Wavelets provide a sparse representation for piecewise smooth signals in 1-D; however, separable extensions of wavelets to multiple dimensions do not achieve the same level of sparseness. Recently proposed directional lifting offers transforms sensitive to edges that are not aligned with the coordinate axes, yet the concatenation of separate 1-D slices implicitly assumes independent directional slices and could create large or isotropic support. True 2-D filters and lifting schemes will avoid both of these problems. By aligning the support of the filters with the expected edge, the filters will create fewer non-zero coefficients. Because these filters correspond to interpolation, the theory of Neville filters can automatically determine the coefficients. For images that consist of two bilinear functions divided by a line, geometric lifting demonstrates a 2-4 times reduction of the number of non-zero coefficients compared with the Daubechies order 2 wavelet. In addition, there is a gain of 2.4 dB in nonlinear approximation.