Fast Spatial Combinative Lifting Algorithm of the Wavelet Transform with the 9/7 Filter for Image Blocks

the 9/7 Filter for Image Blocks Hongying Meng and Zhihua Wang A new fast spatial Combinative lifting algorithm (SCLA) of the wavelet transform with the 9/7 filter for image blocks is proposed. In comparison with its lifting-based implementation, the multiplication is reduced by a ratio of 5/12. The SCLA can speed up the implementation of the wavelet transform. Introduction: The popular 9/7 filter is one of the biorthogonal wavelet filters proposed by Cohen, Daubechies and Feauveau [1] in 1992. It has been used as the default filter in the irreversible wavelet transform of the upcoming new still image compression standard JPEG2000 [2]. The implementation of the wavelet transform with the 9/7 filter has two modes: a convolution-based [3] and a lifting-based implementation (LBI) [4]. The former has more computation cost than the latter. It is reported that the JPEG2000 encoder takes approximately 34 times longer than the JPEG encoder [2]. The reason is mainly due to the wavelet transform. This letter proposes a new fast spatial combinative lifting algorithm (SCLA) for the wavelet transform with the 9/7 filter in the image blocks compression. In comparison with the LBI, the multiplication is reduced by a ratio of 5/12. LBI with decomposition level one: For a 1-D signal   1 , , 1 , 0 | ) (    N k k x X  with length N (assume L N 2  and L is a natural number). If the symmetric extension method on the boundary is chosen, the LBI with decomposition level one by the 9/7 filter can have the following matrix representation: EDCBAX X T Y N N    (1) where   1 , , 1 , 0 | ) (    N k k y Y  is the obtained coefficient vector and N N T  is the associated wavelet transform operator matrix. A, B, C, D, E are the constant matrices associated with each step of the LBI.