This paper presents a novel design method of a biorthogonal lapped transform that consists of long (overlapping) and short (nonoverlapping) basis functions (VLLBT), which can reduce the annoying blocking artifacts and ringing. We formulate the VLLBT by extending conventional lapped transforms. Then, we provide the theory of the Karhunen-Loeve transform in a subspace (SKLT). Using the theory of the SKLT, we show that given biorthogonal long basis functions of the VLLBT, the optimal short basis functions in the energy compaction sense are derived by solving an eigenvalue problem without iterative searching techniques. This leads to a desirable feature from a parameter optimization of view since the degree of freedom for the VLLBT can be theoretically reduced by means of the SKLT. Moreover, the SKLT enables us to easily construct a two-dimensional (2-D) VLLBT with nonseparable short basis functions. Experimental results show that compared to the case where all parameters are optimized, the reduction of free parameters by using the SKLT cause no decline in coding gain for the AR(1) process, and the proposed transform provides promising performance in coding efficiency of images.
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