Fig. 5: SpaGFT implementation for three cell-centric tools and the figure consists of four columns, each corresponding to spatial omics analysis, computational formulation, implementation of FC in optimizing examples tools, and performance evaluation.

a Spot clustering can be formulated as a many-to-one mapping problem. Regarding the modified workflow of SpaGCN, we changed the original input of SpaGCN. A newly formed matrix was then placed into the frozen SpaGCN model for computation. The top 5 performance-increased samples are distinctly showcased, where the y-axis is the ARI value, and the x-axis is the sample number. b Annotation transfer is formulated as a many-to-many mapping problem. Regarding the modified workflow of TACCO, we modified the cost matrix for optimal transport. In the new cost matrix calculation method, we use weighted FCs as the feature to calculate the distance between CT and spots and then optimize the baseline mapping matrix (e.g., TACCO output). In the evaluation, we refer to TACCO methods to simulate spots with different bead sizes using scRNA-seq data and use L2 error to measure differences between predicted and known cell composition in each simulated spot. The y-axis is the bead size for a simulation data value, and the x-axis is the L2 error. Lower L2 error scores indicate better performance. c The cell-spot alignment can be formulated as a many-to-many mapping problem. Regarding the modified workflow of Tangram, we have added two additional constraint terms to its original objective function. The first constraint is designed from a gene-centric perspective, calculating the cosine similarity of the gene by FC matrix between the reconstructed and the original matrix. The second constraint is designed from a cell-centric perspective, calculating the cosine similarity on the spot by the FC matrix between the reconstructed and the original matrix. In the evaluation, we first simulate spatial gene expression data using different window sizes based on STARmap data. Subsequently, we measure the similarity between predicted and known cell proportions in each simulated spot using the Pearson correlation coefficient. A higher PPC indicates better performance (Source data are provided as a Source Data file).