This paper extends the theory of the one-dimensional oversampled linear-phase perfect reconstruction filter banks (OLPPRFBs) developed by Gan et al. to multidimensional (MD) cases and proposes MD nonseparable oversampled lapped transforms (NSOLTs). NSOLTs allow us to achieve an overcomplete analysis-synthesis system with nonseparable, symmetric, realvalued, overlapping, and compact-supported filters. The proposed systems are based on lattice structures and the redundancy is flexibly controlled by the number of channels and downsampling ratio. The proposed structure is shown to be capable of constructing Parseval tight frames in any number of dimensions. The number of design parameters are examined under the Parseval tight frame constraint. In order to design NSOLTs specified for sparse approximation of image or volume data, an example-based design procedure is introduced. The effectiveness of this method is verified by examining design samples and evaluating their sparse approximation performance using the iterative hard thresholding algorithm for a natural image and MRI volume data patch.