Lapped Transforms

The idea of a lapped transform (LT, for short) maintaining orthogonality and nonexpansion of the samples was developed in the early 80’s 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 mantain orthogonality. Cassereau [1] introduced the lapped orthogonal transform (LOT), and Malvar [5],[6],[7] gave the LOT its design strategy and a fast algorithm. It was later pointed by Malvar [9] the equivalence between an LOT and a multirate filter bank. 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 were 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 algorihtms based on special factorizations od the basis functions. We assume a one-dimensional input sequence x(n) which is transformed into several coefficients yi(n), where yi(n) would belong to the i-th subband. We also will use the discrete cosine transform [23] and another cosine transform variation, which we abbreviate as DCT and DCT-IV (DCT type 4), respectively [23].