xIV-LDDMM Toolkit: A Suite of Image-Varifold Based Technologies for Representing and Mapping 3D Imaging and Spatial-omics Data Simultaneously Across Scales.

Advancements in imaging and molecular techniques enable the collection of subcellular-scale data. Diversity in measured features, resolution, and physical scope of capture across technologies and experimental protocols pose numerous challenges to integrating data with reference coordinate systems and across scales. This resource paper describes a collection of technologies that we have developed for cross-modality 3D mapping for the alignment of transcriptomics at the micron scales of genes and cells to the anatomical tissue scales. Our collection of technologies include (i) an explicit censored data representation for the partial matching problem mapping whole brains to subsampled subvolumes, (ii) image-varifold measure norms for supporting nearly universal crossing of modality, (iii) a multi, scale-space optimization technology for generating resampling grids optimized to represent spatial geometry at fixed complexities, and (iv) mutual-information based functional feature selection. Collectively, these methods afford efficient representations of peta-scale imagery providing the algorithms for mapping from the nano to millimeter scales which we term cross-modality image-varifold LDDMM (xIV-LDDMM).