Fig. 1: An overview illustrator of high correlated feature pairs combined with spatial morphological alignment(haCCA).

The procedure consist of 5 main steps: Data preparing (A, B): data is normalized into the construction form of \(\{d,f,l\}\). Where \(d\) represent the coordinates, \(f\) represent the feature matrix, \(l\) represents the cluster label. Gross alignment and Further alignment (C, D): Corresponding points were selected based on mutual spatial region(marked by square) with high similarity between \({m}_{a}\) and \({m}_{b}\). (C, upper) After gross alignment, some spots in \({Dat}{a}_{B}\) is not perfectly overlayed with\({Dat}{a}_{A}\). (red edge in C below). After further alignment, a maximum overlap of \({d}_{{m}_{b}}\) and \({d}_{{m}_{a}}\) can be achieved. (C, below) After gross alignment and further alignment the \({d}_{m}\) is noted as \({d}_{m}^{{\prime} }\). The spatial registration of \({haCCA}\) is now finished. Anchor spot pairs and high correlated feature pairs identification (E, F): Anchor spots were shown on \({m}_{a}\) and \({m}_{b}\)(E). After detection of anchor spot pairs, noted as \({anc}h{or}\; {spot}({m}_{a}^{{anc}h{or}},\,{m}_{b}^{{anc}h{or}})\), Pearson correlation matrix was calculated using the feature vector \(f\) among all anchor spot pairs. After Peason correlation matrix was calculated, the index of k-top-most R values are recorded as high-correlated feature pairs\({\{({f}_{{m}_{a}}^{i},\,{f}_{{m}_{b}}^{j})\}}_{k}\). Feature aid fine alignment. G, H \(({f}_{{m}_{a}}^{i},\,{f}_{{m}_{b}}^{j})\) undergo CCA transfer to generate a pair of high-correlated variate \(\{{f}_{{m}_{a}}^{{cca}},\,{f}_{{m}_{b}}^{{cca}}\}\) as a latent space shared by \({m}_{a}\) and \({m}_{b}\). With \(\{{f}_{{m}_{a}}^{{cca}},\,{f}_{{m}_{b}}^{{cca}}\}\) and \(\{{d}_{{m}_{a}}^{{\prime} },{d}_{{m}_{b}}^{{\prime} }\}\), Alignment between 2 assay is carried out by minimizing the \({los}{s}_{h{aCCA}}({{\rm{H}}})\). \(\alpha\) is set by default = 0.5. A 3-D visualization of \(m=\{{d}_{m}^{{\prime} },\,{f}_{m}^{{cca}}\}\) before and after Feature aid fine alignment was shown in G. integrating (I, J): An alignment of \(\{{m}_{{a}_{i}},{m}_{{b}_{j}}\}\) was found For each spot \({m}_{{a}_{i}}=\{{d}_{m{a}_{i}},\,{f}_{m{a}_{i}},\,{l}_{m{a}_{i}}\}\). For each spot \({m}_{{a}_{i}}=\{{d}_{m{a}_{i}},\,{f}_{m{a}_{i}},\,{l}_{m{a}_{i}}\}\), the data of its alignment spot \({m}_{{b}_{j}}\) in another assay is integrated in order to generate multi-modal data \({m}_{{ai}}^{{integrated}}=\{{d}_{m{a}_{i}},\,{f}_{m{a}_{i}},{f}_{m{b}_{j}},\,{l}_{m{a}_{i}}\}\). I shown the label distribution of \({m}_{b}\) in \({m}_{a}^{{integrated}}\).