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This correspondence reviews the logical transform and pre-sents new theoretical and empirical developments. Novel extensions of the logical transform into spaces of three and four dimensions are de-scribed using the hexahedral and octahedral tessellations. These new tessellations are important in that they yield sufficiently compact neighborhoods to permit multidimensional cellular logic transforms to be carried out in real time by table lookup. Applications of logical transforms in these tessellations to data and image analysis are provided.