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As imaging, computing, and data storage technologies improve, there is an increasing opportunity for multiscale analysis of three-dimensional datasets (3-D). Such analysis enables, for example, microscale elements of multiple macroscale specimens to be compared throughout the entire macroscale specimen. Spatial comparisons require bringing datasets into co-alignment. One approach for co-alignment involves elastic deformations of data in addition to rigid alignments. The elastic deformations distort space, and if not accounted for, can distort the information at the microscale. The algorithms developed in this work address this issue by allowing multiple data points to be encoded into a single image pixel, appropriately tracking each data point to ensure lossless data mapping during elastic spatial deformation. This approach was developed and implemented for both 2-D and 3D registration of images. Lossless reconstruction and registration was applied to semi-quantitative cellular gene expression data in the mouse brain, enabling comparison of multiple spatially registered 3-D datasets without any augmentation of the cellular data. Standard reconstruction and registration without the lossless approach resulted in errors in cellular quantities of ~ 8%.