The idea of a lapped transform (LT) maintaining orthogonality and non-expansion of the samples was developed in the early 1980s at MIT by a group of researchers unhappy with the blocking artifacts so common in traditional block transform coding of images. The idea was to extend the basis function beyond the block boundaries, creating an overlap, in order to eliminate the blocking effect. This idea was not new, but the new ingredient to overlapping blocks would be the fact that | de Queiroz . Lapped Transforms Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton CRC Press LLC 1999 1999 by CRC Press LLC 38 Lapped Transforms Ricardo L. de Queiroz Advanced Color Imaging Xerox Corporation Introduction Orthogonal Block Transforms Orthogonal Lapped Transforms Useful Transforms ExtendedLapped Transform ELT GeneralizedLinear-Phase Lapped Orthogonal Transform GenLOT Remarks References Introduction The idea of a lapped transform LT maintaining orthogonality and non-expansion of the samples was developed in the early 1980s at MIT by a group of researchers unhappy with the blocking artifacts so common in traditional block transform coding of images. The idea was to extend the basis function beyond the block boundaries creating an overlap in order to eliminate the blocking effect. This idea was not new but the new ingredient to overlapping blocks would be the fact that the number of transform coefficients would be the same as if there was no overlap and that the transform would maintain orthogonality. Cassereau 1 introduced the lapped orthogonal transform LOT and Malvar 5 6 7 gave the LOT its design strategy and a fast algorithm. The equivalence between an LOT and a multirate filter bank was later pointed out by Malvar 9 . Based on cosine modulated filter banks 15 modulated lapped transforms were designed 8 25 . Modulated transforms were generalized for an arbitrary overlap later creating the class of extended lapped transforms ELT 10 13 . Recently a new class of LTs with symmetric bases was developed yielding the class of generalized LOTs GenLOT 17 19 20 . As we mentioned filter banks and LTs are the same although studied independently in the past. We however refer to LTs for paraunitary uniform FIR filter banks with fast implementation algorithms based on special factorizations of the basis functions. We assume a one-dimensional input sequence x n which is transformed into several .
