9.9 Lapped Transforms

The lapped orthogonal transforms (LOT) were originally proposed to be block transforms whose basis functions extended beyond the block boundaries, that is, the basis functions from neighboring blocks overlapped with each other. Its main goal was to reduce blocking effects, usually present under quantization of the block transform coefficients, as is the case of the DCT. Blocking effects are discontinuities that appear across the block boundaries. They occur because each block is transformed and quantized independently of the others, and this type of distortion is particularly annoying in images (Malvar, 1992). If the transform basis functions are allowed to overlap, the blocking effects are greatly reduced.

Since block transforms are equivalent to M-band perfect reconstruction filter banks, one idea would be to replace the DCT by a cosine-modulated filter bank. However, the cosine-modulated filter banks discussed in the previous section have the nonlinear phase of the analysis filters, an undesirable feature in applications such as image coding. LOT-based filter banks are very attractive because they lead to linear-phase analysis filters and have fast implementation. The LOT-based filter banks are members of the family of paraunitary FIR perfect reconstruction filter banks with linear phase. Although there are a number of possible designs for linear-phase filter banks with perfect reconstruction, the LOT-based design is simple to derive and to implement. The term LOT applies to the cases where the analysis filters have length 2M. Generalizations of the LOT to longer analysis and synthesis filters (length LM)...