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The theory and design for a novel class of a variable-length lapped transform are presented. In this class, the synthesis filter bank is firstly defined with arbitrary combination of FIR filters, giving much freedom of design compared to previously presented variable-length lapped transforms. The analysis filter bank is then obtained as the inverse of the synthesis bank by the Neumann series expansion to achieve the perfect reconstruction. Moreover, by introducing a unitary transform, we improve the compression performance of the analysis bank. We provide several design examples and experimental results of image coding, which show that the proposed transform is promising and comparable with conventional subband transforms including wavelets.