Abstract
Spatial omics technologies decipher functional components of complex organs at cellular and subcellular resolutions. We introduce Spatial Graph Fourier Transform (SpaGFT) and apply graph signal processing to a wide range of spatial omics profiling platforms to generate their interpretable representations. This representation supports spatially variable gene identification and improves gene expression imputation, outperforming existing tools in analyzing human and mouse spatial transcriptomics data. SpaGFT can identify immunological regions for B cell maturation in human lymph nodes Visium data and characterize variations in secondary follicles using inhouse human tonsil CODEX data. Furthermore, it can be integrated seamlessly into other machine learning frameworks, enhancing accuracy in spatial domain identification, cell type annotation, and subcellular feature inference by up to 40%. Notably, SpaGFT detects rare subcellular organelles, such as Cajal bodies and Set1/COMPASS complexes, in highresolution spatial proteomics data. This approach provides an explainable graph representation method for exploring tissue biology and function.
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Introduction
Advancements in spatial omics offer a comprehensive view of the molecular landscape within the native tissue microenvironment, including genome, transcriptome, microbiome, T cell receptor (TCR)^{1}, epigenome, proteome, transcriptomeprotein markers coprofiling, and epigenome–transcriptome coprofiling^{2} (Fig. 1a and Supplementary Fig. 1). These approaches enable the investigation and elucidation of functional tissue units (FTUs)^{3}, which are defined as overrepresented multicellular functional regions with a unique physiologic function, with both cellcentric and genecentric approaches. Specifically, cellcentric approaches involve the identification of spatial domains with coherent gene expression and histology^{4}, studying cell composition and neighborhoods within specific domains^{5,6,7}, and understanding intercellular mechanisms. In parallel, genecentric approaches characterized FTUs by imputing gene expression^{8} and identifying spatially variable genes (SVG)^{9,10,11} in a highly complementary manner to cellcentric approaches.
Classic statistical methods, such as SPARK^{9}, SPARKX^{12}, and SpatialDE^{11}, have effectively modeled molecular variations and spatial relationships within a tissue. However, they did not fully explore the capacity to translate these relationships into understandable and analyzable features. In contrast, graphbased methods present a powerful alternative method that efficiently encodes and leverages spatial relationships within tissue in spatial omics data representation^{13}. We postulate that an FTU can be intuitively considered a graph; its nodes represent spots or cells, and edges connect spatially adjacent or functionally related nodes. Within this representation of FTUs, a binary graph signal (e.g., 0,1), representing discrete twostate information at each node, and cellular or subcellular composition or omics features (e.g., genes) constitute continuous graph signals, encoding a range of values across the graph’s nodes. These graph signals define the FTU’s characterization, connect cellcentric and genecentric analyses, and offer mutual interpretatibility^{14}, through the generation of a graph embedding that harmonizes the graph structure and signal magnitude. Furthermore, while graphbased machine learning methods are available to learn graph embeddings and carry out downstream tasks (e.g., graph classification), their learning progress is usually a “black box” and relies on an inductive bias (i.e., a hypothesis for a particular question) to train the model^{15}. The characteristics of the produced graph embeddings are specifically tailored to perform optimally in certain targeted downstream tasks. Therefore, there is a need for a generic graph signal representation framework with a solid mathematic foundation to reveal intricate relations between molecular signatures and FTUs across multiple resolutions of spatial omics data.
To achieve this, we present the Spatial Graph Fourier Transform (SpaGFT), an analytical feature representation approach to encode smooth graph signals for representing biological processes within tissues and cells. It bridges graph signal processing techniques and spatial omics data, enabling various downstream analyses and facilitating insightful biological findings. Computationally, SpaGFT outperformed other tools in identifying SVGs with hundredfold efficiency and gene expression imputation across human/mouse Visium data. Biologically, SpaGFT identified key immunological areas for B cell maturation processes from human lymph nodes Visium data and further illustrated secondary follicle cellular, morphological, and molecular diversity from exclusively inhouse human tonsils CODEX data. Moreover, SpaGFT can be seamlessly integrated into other machine learning frameworks regarding domain identification (e.g., SpaGCN^{4}), annotation transfer from cell types to spots (e.g., TACCO^{6}), the celltospot alignments (e.g., Tangram^{7}), and subcellular hallmark inference (e.g., CAMPA^{16}). Notably, enhanced CAMPA has enabled the discovery of rare subcellular structures like the Cajal body and Set1/COMPASS complex based on iterative indirect immunofluorescence image (4i) data^{17}, enhancing our understanding of cellular function using spatial omics technologies.
Results
SpaGFT reliably represents the smooth signal of spatial omics data
We summarize current spatially resolved omics as three types of spatial graphs related to the granularity of nodes, ranging from subcellular level (i.e., pixellevel) to broader cellular (i.e., celllevel) and multicellular scales (i.e., spotlevel) based on the spatial resolutions (Fig. 1b–k). This granularity can range from subcellular levels to broader cellular and multicellular scales. For example, based on the spatial graph of a spatially resolved transcriptomics (SRT) dataset, the transcriptomic profile of a specific gene is a graph signal and can be represented by the linear combination of its Fourier modes (FMs, Terminology Box). To elaborate, a lowfrequency FM contributes to a low and smooth graph signal variation, representing a spatially organized pattern, while a highfrequency FM contributes to rapid graph signal variation and usually refers to noises in spatial omics data^{18}. For example, if a gene exhibits a spatially organized pattern in SRT data, the Fourier coefficients (FCs) of corresponding lowfrequency FMs are more dominant than FCs of highfrequency FMs in the graph Fourier representation. Notably, FMs are associated with graph structure and do not assume any predefined patterns^{18}, ensuring flexibility in representing both welldefined and irregular spatial signal patterns. Thus, regardless of single (Fig. 1b–d, f, g, and i), multimodalities (Fig. 1e and h), or augmented features (Fig. 1j and k), the spatial omics can be analytically transformed into FCs to quantify the contribution of FMs in the frequency domain^{19}, a feature space for enhancing the interpretability and generalizability in downstream analyses.
SpaGFT identifies spatially variable genes and enhances gene and protein signals
Using the representation framework of SpaGFT (Fig. 2a), the mathematical formulation of SVG identification can be derived as a kbandlimited signal recognition problem, which determines the first k lowfrequency FMs to best approximate the original graph signal (Fig. 2b, Supplementary Fig. 2, and S1 of Supplementary Note 1). This formulation can overcome three main limitations of SVG identification methods: (i) no preassumption of regular patterns in model design (e.g., radial hotspot, curve belt, or gradient streak)^{9}; (ii) interpretable representation of SVG patterns^{20} with spatial context; and (iii) high computational efficiency^{12} when processing largescale datasets. Essentially, we defined and implemented a GFTscore for each gene to quantify the contribution of lowfrequency FMs by determining the first k lowfrequency FMs, weighting, and summing corresponding FCs (S2 of Supplementary Note 1). Based on the definition, a gene is identified as an SVG if (i) its GFTscore is greater than the inflexion point based on the distribution of all genes’ GFTscore and (ii) its FCs of the first k lowfrequency FMs are significantly higher than FCs of highfrequency FMs (S3 of Supplementary Note 1). Consequently, we evaluated the performance of SVG identification using 31 public SRT datasets from human and mouse brains (Supplementary Data 1)^{21,22,23,24}. As no goldenstandard SVG database was available, we collected 849 SVG candidates from five existing studies^{24,25,26,27,28}, and 458 of them were used as curated benchmarking SVGs based on crossvalidation with the in situ hybridization (ISH) database of Allen Brain Atlas^{29} (Supplementary Data 2 and 3, see the “Methods” section). The SVG prediction performance of SpaGFT was compared with SPARK^{9}, SPARKX^{12}, MERINGUE^{30}, SpatialDE^{11}, SpaGCN^{4}, and scGCO^{31} in terms of six referencebased and two referencefree metrics (Supplementary Note 2). The grid search of parameter combinations was conducted on three highquality brain datasets to evaluate each tool’s performance, in which SpaGFT showed the highest median and peak scores (Fig. 2c and Supplementary Data 4). In addition, the computational speed of SpaGFT was twofold faster than that of SPARKX and hundredsfold faster than those of the other four tools on the two Visium datasets (Supplementary Data 5). Although SpaGFT was slower than SPARKX on the SlideseqV2 dataset, it showed a remarkably enhanced SVG prediction performance compared to SPARKX. We then performed an independent test on 28 independent datasets using the parameter combination with the highest median Jaccard Index among three datasets from the above gridsearch test. The results revealed that SpaGFT promised supreme performance among the investigated tools based on the evaluation metrics (Fig. 2d, Supplementary Fig. 3a–d, and Supplementary Data 6). Within the top 500 SVGs from each of the above six tools, SpaGFT identified SVGs shared with other tools and also unique SVGs that were validated as the ground truth (Supplementary Fig. 3e and Supplementary Data 7). For example, Nsmf and Tbr1 were identified by all six tools and showed clear structures of the hippocampus, cortical region, and cerebral cortex. On the other hand, Cartpt, Cbln2, Ttr, and Pmch were uniquely identified by SpaGFT and showed key functions in the brain, such as Cartpt participating in dopamine metabolism^{29} (Fig. 2e, Supplementary Fig. 4, and Annotation 1 of Supplementary Note 3). These benchmarking results suggested that SpaGFT is capable of leveraging upon the FM representation of gene expression for robust and accurate identification of SVGs from SRT data. SpaGFT takes advantage of FM representation of gene expression patterns in SVG identification, and the SVGs identified by SpaGFT were distinguishably separated from nonSVGs on the FMbased UMAP with a clear boundary, whereas SVGs were irregularly distributed on the principal componentbased gene UMAP (Fig. 2f).
In addition, the distorted graph signal correction can be used as the mathematical formulation to impute a lowexpressed gene or denoise a highintensity but noisy protein in SpaGFT. Essentially, FCs are shifted towards a specific bandwidth by implementing a lowpass filter and are inversely transformed to an enhanced graph signal using an inverse graph Fourier transform (iGFT)^{32}. To enhance the main signal and mitigate noise, a lowpassing filter is employed to weigh and shift all FCs toward the lowfrequency bandwidth (see the “Methods” section). In the end, these weighted FCs are transformed back to a corrected graph signal via iGFT (Supplementary Fig. 5a). In assessing the performance of gene expression correction, we used 16 human brains SRT datasets with wellannotated spatial domains^{23,24} and utilized adjusted rand index (ARI) to measure the accuracy of predicting spatial domains using corrected gene expression. As a result, SpaGFT outperformed other gene enhancement tools in terms of ARI, including Sprod^{8}, SAVERX, scVI, netNMFsc, MAGIC, and DCA^{33,34} (Fig. 2g, Supplementary Fig. 5b, and Supplementary Data 8). For example, SpaGFT enhanced the lowintensity spatial omics signal broadly across different technologies and species, such as gene TNFRSF13C for human lymph node, gene Ano2 for mouse brain^{29} (Supplementary Fig. 5c), cell density for human prostate tumor (from data of Fig. 1k), protein IA, and corresponding gene H2ab1 for mouse breast tumor. Similarly, the noisy background can also be removed, such as protein LY6A/E and corresponding gene Ly6a and protein CD19 (Fig. 2h, i and Annotation 2 of Supplementary Note 3).
SpaGFT identifies the germinal center, T cell zone, B cell zone, and crosstalking regions in the human lymph node
As lowfrequency FC can represent smooth spatially variable patterns, they can be used for SVG clustering, and gene clusters can correlate with distinct FTUs from a gene perspective (Supplementary Fig. 6a). To demonstrate the application, we implemented SpaGFT in the publicly available Visium data of human lymph nodes, which, as secondary lymphoid organs contain wellknown recurrent functional regions, such as T cell zones, B cell zones, and germinal center (GC)^{20}. First, SpaGFT identified 1,346 SVGs and characterized nine SVG clusters (Fig. 3a and Supplementary Data 9). To recognize the FTUs of the T cell zone, B cell zone, and GC, we first used cell2location^{35,36} to determine the cell proportion (Supplementary Fig. 6b and Supplementary Data 10) for the nine SVG clusters and investigate function enrichment (Supplementary Fig. 6c–e) for three selected FTUs. Based on the molecular, cellular, and functional signatures of three regions^{35}, we found that SVG clusters 3, 5, and 7 (Fig. 3b) were associated with the T cell zone, GC, and B cell zone, respectively (Annotation 3 of Supplementary Note 3).
In contrast to spatial domain detection tools, SpaGFT is not restricted to a rigid boundary for tissuelevel identification of microenvironments^{5}. Instead, SpaGFT allows overlapping regions to infer the functional coherence and collaboration among different FTUs. We therefore projected three FTUs represented by SVG clusters 3, 5, and 7 on the spatial map for visual inspection, and identified their close spatial proximity (Fig. 3c). These results are highly indicative of tissue regions of polyfunctionality amongst these three TFUs (four representative subregions are shown in Fig. 3c). To further investigate the crosstalk amongst these three TFUs, we projected spots (assigned to all three regions) to the Barycentric coordinates (the equilateral triangle in Fig. 3d), which displayed relations and abundance of the unique and overlapped regions regarding cell type components^{37}. We identified 614 spots overlapped with B cell zone and GC, 158 spots overlapped with GC and T zone, 93 spots overlapped with T zone and B cell zone, and 26 spots overlapped across three FTUs (Supplementary Data 11), in support of the complex interactions within these three TFUs. We next hypothesized that the spots from the overlapped region would vary in functions and cell components to support the polyfunctionality of these regions. We thus investigated the changes in enriched functions (Supplementary Data 12) and cell types (Supplementary Fig. 7) across seven regions (i.e., GC, GC–B, B, B–T, T, T–GC, and T–B–GC). Our results identified lymph noderelevant pathways and cell types, such as B and T cell activity and functions, as significantly varied across those regions (Fig. 3e, f, Annotation 4 of Supplementary Note 3), in support of our hypothesis.
SpaGFT reveals secondary follicle variability based on CODEX data
The results of Visium in Fig. 3 showcased the ability of SpaGFT to identify FTUs via SVG clustering. Given that the current resolution of Visium (~50 μm per pixel) limited our ability to interpret the variability of finer follicle structures and their corresponding functions at the cellular level, we next performed singlecell level spatial proteomics on a human tonsil using a 49plex CODEX panel at a ~0.37 μm per pixel resolution (Fig. 4a) to better characterize and interpret the follicle variability we observed and inferred using SpaGFT on the Visium data. Based on the anatomical patterns highlighted by B (e.g., CD20) and T cell (e.g., CD4) lineage markers, we selected fields of view (FOV) that would allow for a good representation of the complex tissue structures present in the tonsil (i.e., GC and interfollicular space^{38}) while still highlighting the variability in follicle structure^{39}. We first performed cell segmentation with DeepCell^{40}, followed by clustering with FlowSOM^{41} and Marker Enrichment modeling^{42} to identify the diverse cell phenotypes present in the data (Fig. 4b). Interestingly, we observed that the clear arrangement of T and B cell patterns (e.g., A3, A5, and A6) informed identifiable GC regions within the follicular structure, compared to others (e.g., A4) without clear T and B cell spatial organization (Fig. 4b). We, therefore, postulated that A4 is comprised of multiple follicles, unlike A5 and A6, to represent a more spatially complex FOV.
We investigated this further by directly using the raw CODEX images as inputs to identify FTUs formed from spatially variable protein (SVP) clusters within the tissue environment^{43}. To verify whether downsampling the CODEX image (Supplementary Fig. 8a) would result in a loss in the power of characterizing FTUs, we first used FOV 6 to generate three images across different resolutions (with downsampling), resulting in a (1) 1000by1000 pixel image (~0.8 μm per pixel size), (2) 500by500 pixel image (~1.6 μm per pixel size), and 3) 200by200 pixel image. Our results show that despite the generation of diverse low and highfrequency FMs from three pixellevel images (as illustrated in Supplementary Fig. 8b), SpaGFT was stable to resolution changes, characterizing FTUs across different resolutions with consistent patterns (Supplementary Fig. 8c). We subsequently calculated the structural similarity score (SSIM) to quantitatively evaluate pattern similarity among identified FTUs. Each gradient pixel size image identified six FTUs, and those patterns of FTUs showed pairwise consistency (Fig. 4c and Supplementary Fig. 8d), suggesting that 200by200 pixel downsampled images (an approximate factor of 105fold from the original pixel size) were sufficient in characterizing FTUs to balance between computational efficiency and biological insights.
We next implemented SpaGFT to characterize FTUs for the six FOVs with 200by200 pixel images and annotated follicles for each FOV based on cell components (Supplementary Fig. 9) and protein signatures (Supplementary Data 13; Supplementary Figs. 10a, b). Specifically, FTUs represented by SVP cluster 1 of A1 and SVP cluster 1 of A2 displayed morphological features akin to that of a mantle zone (MZ). Molecularly, we uncovered that the B cellspecific marker^{44} (CD20) and antiapoptotic factor (BCL2)^{45} were SVPs for these two FTUs of A1 and A2 (Fig. 4d, e and Supplementary Fig. 10c). Our results confirmed the presence of CD20 in delineating the MZ structure, and additionally suggest that the presence of BCL2 as an additional feature of MZ structures^{46}. In another case, FTUs represented SVP cluster 4 of A3, SVP cluster 9 of A4, and SVP cluster 4 of A5 displayed GCspecific T cell signatures (Fig. 4f, g and Supplementary Fig. 10d) and corresponding molecular features, including PD1^{47} and CD57^{48}, indicating the presence of wellcharacterized GCspecific T follicular helper cells^{49}. For FTUs represented by SVP cluster 2 of A6, we observed a complex molecular environment, where Podoplanin, CD11c, and CD11b were SVPs, thus showcasing the existence of follicular dendritic cell (FDC)^{50} and GCcentric macrophages^{51} networks (Fig. 4h). In addition to molecular heterogeneity, we further captured their variability in terms of lengthscale and morphology (Fig. 4i), cell type (Fig. 4j and k), cell–cell distance (Fig. 4l), and cellcell interactions (Fig. 4m). For example, from the tissue morphology perspective, A3–A6 captured clear oval shape patterns with different lengthscales, but A1 and A2 captured multiple partial MZ patterns (Fig. 4i). Although visual inspection was unable to distinguish between the morphological patterns of GCs in A4 (Fig. 4b), SpaGFT was able to determine three small lengthscale GC patterns at the molecular level (Fig. 4i).
Regarding cellular characteristics, six FTUs (i.e., two MZ from A1 and A2; four GCs from A3 to A6) were dominated by B and CD4 T cells with varying proportions (Fig. 4jk; Supplementary Data 14). Specifically, MZs from A1 and A2 showed an average composition of 58% B and 10% CD4 T cells. GC from A3 and A5 with similar lengthscale showed an average of 54% B and 32% CD4 T cells. A4 captured three lengthscale GCs and showcased 43% B and 46% CD4 T cells, while the largescale GC from A6 contained 70% B and 12% T cells, indicating B and T cell proportions varying in different lengthscale GC. We could also infer cell–cell interaction based on distance (Fig. 4l, m). In general, MZ from A1 and A2 show that the observed B–B distance was smaller than the expected distance, which suggests the homogeneous biological process of the significant B–B interaction in the GC region. In addition, cellcell interaction also shows heterogeneity for two MZ. The interactions between CD4 T cells and B cells were observed in two MZ from A1 and A2, showcasing the infiltration of CD4 T cells into the B cell right mantle zone^{52}. DCB and CD4 TB cell interactions in A3 and A4 suggest light zone functions for B cell selection^{53,54}. MacrophageB cell interactions in GC in A6 potentially indicated macrophage regulation on B cells (e.g., B cells that failed to trigger the cell proliferation signals during the B cell selection process underwent apoptosis and were subsequently engulfed by macrophages^{55}). Our results demonstrate the applicability of SpaGFT at an initial subsampled lower resolution from highplex spatial proteomics, thus efficiently identifying and characterizing highattention tissue regions, including secondary follicles, to uncover cellular and molecular variability that can be further confirmed at the original singlecell resolution. We also affirmed that FTUs identified by SpaGFT were not simply regions of cell aggregation but reflected both the cellular and regional activity and cell–cell interactions based on spatially orchestrated molecular signatures.
SpaGFT can generate new features and be implemented as an explainable regularizer for machinelearning algorithms
SpaGFT can also be beneficial to enhance the performance of existing methods as an explainable regularizer through feature or objective engineering. To elucidate its applicative power, we exemplified three representative analyses of SRT as follows (Supplementary Fig. 11 and see the “Methods” section).
First, we showcase how spot clustering can identify spatial domains spatially coherently in both gene expression and histology. Here, we selected SpaGCN^{4} as the demonstration to showcase the implementation of FC from the feature engineering aspect (Fig. 5a). To illustrate FCs being a feature, we extended the spatial expression matrix by concatenating a spotbyFC matrix derived from the spotspot similarity. Subsequently, the new feature matrix was input into the original SpaGCN model and predicted spatial domains. Same as the SpaGCN study, we utilized 12 datasets^{24} of human brain SRT data for training (two datasets from the same tissue section) the number of new features and testing for improving SpaGCN on 10 datasets. The results indicated improvements in eight out of ten datasets (Supplementary Data 15) in identifying the spatial domains of the dorsolateral prefrontal cortex. Notably, the top five datasets exhibited enhancements between 7.8% and 42.6%.
Second, annotation transfer will solve the challenge of insufficient data labeling for the increasing emergence of SRT. We used TACCO^{6} as an annotation transfer example tool to showcase the application of FC as a regularizer for the optimal transport (OT) method, which is a machine learning method that aimed to find the most efficient way (i.e., minimizing the overall cost associated with the movement) to move a probability distribution from one configuration to another. Specifically, TACCO allowed the transfer of phenotypelevel annotation labels (e.g., cell type) from scRNAseq to SRT using such an OT framework. Although TACCO has demonstrated algorithm effectiveness in consideration of cell similarity over all genes, we hypothesized that projecting cell similarity to the frequency domain and strengthening a topological regularization in OT’s objective function will be a potential avenue for performance enhancement. In our modification, we integrated a topological regularization term into the original cost matrix to derive a new cost matrix (Fig. 5b and see the “Methods” section). Leveraging the evaluation metrics of the original TACCO study, our tests underscored an 8.7–14.9% L2error decrease across five simulated bead sizes in terms of transferring annotated labels from scRNAseq to unannotated SRT mouse brain data (Supplementary Data 16).
Third, aligning singlecell data (e.g., scRNAseq) to low/highresolution SRT data was important to mutually benefit each other regarding spatial resolution and molecular diversity. We selected Tangram^{7} as an alignment tool to demonstrate the topological regulation of genes and spots in the frequency domain. Tangram optimized the celltospot mapping matrix through the gradientbased method, aiming to ensure the similarity between the reconstructed SRT based on scRNAseq and the original SRT. The objective function of Tangram is to measure cell density, genelevel similarity, and celllevel similarity in the vertex domain, respectively. In alignment with the hypothesis proposed in Fig. 5b, we constrained the similarity at both the gene and celllevel in the frequency domain (Fig. 5c). As a result, our tests illustrated 7.4–15.9% Pearson correlation coefficient increase improvement regarding aligning scRNAseq on simulated STARmap^{56} mouse brain SRT data (Supplementary Data 17).
SpaGFT introduces an inductive bias to regularize the deep learning method and identify rare subcellular organelles
We applied SpaGFT to obtain an interpretable spreading entropy regulation for a conditional variational autoencoder framework, CAMPA, to identify conserved subcellular organelles across multiple perturbed conditions on pixellevel 4i data (165 nm/pixel)^{16,17}. To modify the model, we introduced an entropy term in the original reconstruction loss of CAMPA to regularize the spreading of graph signals^{19}. Specifically, we constrained the entropy within the first k bandwidth and provided an inductive assumption for CAMPA to learn embeddings that represented kbandlimited signals (Supplementary Fig. 12a). Consequently, compared to the validation loss calculated from validation datasets (see the “Methods” section), the loss curve from the modified model showed a reduction and entered a stable state earlier (Fig. 6a). We observed that, by introducing the entropy term as a regularizer, the model enhanced the training efficacy in capturing and minimizing the reconstruction error and promoted faster convergence of the model.
Furthermore, we validated that the modified model significantly (pvalue = 0.035) improved the baseline model regarding batch effect removal (Supplementary Fig. 12b–e) using kBET testing^{57}, indicating that the learned embeddings retained conserved structures of subcellular organelles across multiple perturbations. Next, compared with the baseline (Fig. 6b–d), the modified model additionally identified two rare clusters (Supplementary Data 18), including cluster 5 (with an average of 0.16% pixels per cell) and cluster 6 (with an average of 0.10% pixels per cell). Notably, the pixels assigned to these two clusters are very stable (not random signals computationally) regardless of different resolution parameters of the Leiden clustering algorithm (Supplementary Data 19 and Supplementary Fig. 12f). Subsequently, clusters 5 and 6 were annotated as Cajal bodies^{58} and set1/COMPASS^{59}, respectively (Fig. 6e). Cluster 6 and its corresponding protein signature, SETD1A (Fig. 6f–h), displaying a highly concentrated pattern (with an average of 0.16% pixels per cell), were strongly shown as a kbandlimited signal in the frequency domain. Furthermore, we also observed similar characteristics of cluster 5 and the corresponding marker protein, COIL (Fig. 6i–k). Therefore, by integrating the regularization of lowfrequency signals from SpaGFT, the CAMPA model’s stability was enhanced in learning embeddings that represented subcellular organelles with kbandlimited characteristics. This approach, which we term “explainable regularization,” refines the detection and characterization of finer structures exhibiting spatially organized patterns.
Discussion
SpaGFT provides a reliable feature representation through graph Fourier transform that enhances our biological understanding of complex tissues. This method aligns with the advanced analytical capabilities required to dissect the intricate spatial components of tissue biology, from subcellular to multicellular scales. It eliminates the need for predefined expression patterns and significantly improves computational efficiency, as demonstrated in the benchmarking across 31 human/mouse Visium and Slideseq V2 datasets. In addition, we also highlight our manually curated 458 mouse and human brain genes as closetooptimization standard SVGs. This will bring an alternative evaluation metric based on realistic human/mouse data, which is complementary to simulationbased evaluation methods, such as BSP^{60}, SPARKX, SpatialDE, SPARK, scGCO, and other benchmarking work^{61}. Furthermore, implementing a lowpass filter and inverse GFT effectively impute lowexpressed gene expression and denoise highnoisy protein intensity, leading to more precise spatial domain predictions, as showcased in the human dorsolateral prefrontal cortex. Notably, SpaGFT advances the interpretation of spatial omics data by enabling more accurate machine learning predictions. It has notably improved the performance of existing frameworks by 8–40% in terms of the accuracy of spatial domain identification, lower error of annotation transfer from cell types to spots, the correctness of the celltospot alignments, and the validation loss of subcellular hallmark inference, respectively.
From a computational standpoint, SpaGFT and scGCO are two graph representation methods, among others, for spatial omics data analysis, with the former focusing on omic feature representation and the latter focusing on SVG detection. scGCO employs a graphcut method to segment the tissue and compare the consistency between segmentations and gene expressions in support of SVG detection. SpaGFT uses the graph Fourier transform to find a novel latent space to represent gene expression and achieve various downstream tasks, including but not limited to, SVG identification, gene expression enhancement, and functional tissue unit inference.
In addition, there is a good potential for implementing SpaGFT into existing explainable spatial multimodalities frameworks^{2}, such as UnitedNet^{62}, MUSE^{63}, and modalitiesautoencoder^{64}. Considering UnitedNet^{62} as an example, it incorporates explainable machine learning techniques to dissect the trained network and quantify the relevance of features across different modalities, specifically looking at celltypespecific relationships. To bring more spatial insight into UnitedNet, SpaGFT can provide (1) augmented features (e.g., modified SpaGCN in Fig. 5a) and (2) an explainable regularizer (e.g., modified CAMPA in Fig. 6). To generate the augmented spatial omics features, SpaGFT can first calculate cellcell relations (e.g., calculation from H&E features, gene expression, or protein intensity) in the vertex domain and transform the relations to FCs, The FCs encode and quantify cell–cell variation patterns, which can be regarded as one of the inputs for UnitedNet. Regarding implementing SpaGFT as an explainable regularizer, the spreading entropy can be introduced into UnitedNet’s reconstruction loss function, as UnitedNet has an encoderdecoder structure. By regularizing the entropy of encoded and decoded spatial omics features on the Fourier domain, the UnitedNet may be guided to learn spatially organized regions that present a lowfrequency signal (e.g., one functional tissue unit with a specific pattern and function). These enhancements are pivotal in characterizing complex biological structures using explainable regularization for deep learning framework, including identifying rare subcellular organelles, thus providing deeper insights into the cellular machinery.
Regarding the biological implications, SpaGFT offers alternative perspectives on spatial biology questions. Specifically, by grouping SVGs identified by SpaGFT, we can uncover distinct FTUs within organs. This has led to the identification of critical immunological regions in the human lymph node Visium data, enhancing our knowledge of B cell maturation and the polyfunctional areas it encompasses, such as the B cell zone, T cell zone, GC, B–T zone, GC–B zone, T–GC, and trizone Additionally, exclusive inhouse CODEX data, SpaGFT has revealed secondary follicle differences in the morphology, molecular signatures, and cellular interactions in the human tonsil, offering a more nuanced understanding of B cell maturation. Additionally, SpaGFT introduces kbandlimited signal entropy within the CAMPA framework. This has led to the groundbreaking identification of rare subcellular organelles, which are the Cajal body and the Set1/COMPASS complex. The former is integral to the regulation of gene expression, while the latter plays a critical role in epigenetic modifications. By enabling the investigation of these organelles with unprecedented detail, SpaGFT propels us closer to a comprehensive understanding of the spatial dynamics of gene expression and the epigenetic landscape within cells.
However, there is still room for improving prediction performance and understanding the FTU mechanism. First, SpaGFT discusses lowfrequency signals in the frequency domain, but there is a lack of discussion on medium and highfrequency signals. Although a previous study^{65} described that most functionally related biological signals are presented in the lowfrequency region, certain special signals are also found in the medium and highfrequency region. For instance, in the human brain fMRI (functional magnetic resonance imaging, a technique that measures brain activity by detecting changes associated with blood flow), lowfrequency FMs capture the global variation signals (e.g., daydreaming and retrieving memories). Mediumfrequency FMs capture brain networks with less global variation but more rapid processing (e.g., working memory or executive functions). Highfrequency FMs capture responses to new or complex stimuli that involve local connections between close brain regions (e.g., acute, localized brain activities). Analogous to spatial omics data, we assume that medium and highfrequency signals may also have corresponding special biological signals with more local and less global variation (e.g., regions stimulation from the environment), complementing the current kbandlimited signal approach of representing smooth global variation. Therefore, in future studies, we might focus more on multifrequency signal interpretation. Second, although the SpaGFT computation speed is very competitive, it can be further enhanced by reducing the computational complexity from \(O({n}^{2})\) to \(O(n\times \log (n))\) using fast Fourier transform algorithms^{66}. Third, the alteration of the spot graph and FTU topology represents a potential challenge in identifying FTUs across spatial samples from different tissues or experiments, which results in diverse FM spaces and renders the FCs incomparable. This is similar to the “batch effect” issue in multiple singlecell RNA sequencing (scRNAseq) integration analyses. One possible solution to this challenge is to embed and align spatial data points to a fixed topological space using machine learning frameworks, such as optimal transport. Another possibility is to use H&E images as a common reference for all to make the embedding tissueaware. Fourth, SpaGFT implementation on the CODEX image relies on experts’ knowledge to preselect functional regions. The future direction of analyzing multiplexed images is to develop a topological learning framework to automatically detect and segment functional objects based on SpaGFT feature representation. Overall, we believe the value of our study is to bring an alternative view for explainable artificial intelligence in spatial omics modeling, including multiresolution spatial omics data integration and pattern analysis across spatiotemporal data^{13}.
Methods
We introduce Spatial Graph Fourier Transform (SpaGFT) to represent spatial omics features. The core concept of SpaGFT is to transform spatial omics features into Fourier coefficients (FC) for downstream analyses, such as SVG identification, expression signal enhancement, and topological regularization for other machine algorithms. SpaGFT framework provides graph signal transform and seven downstream tasks: SVG identification, gene expression imputation, protein signal denoising, spatial domain characterization, cell type annotation, cellspot alignment, and subcellular landmark inference. The detailed theoretical foundation of kbandlimited signal recognition can be found in Supplementary Note 1.
Graph signal transform
Knearest neighbor (KNN) Graph construction
Given a gene expression matrix containing n spots, including their spatial coordinates and m genes, SpaGFT calculates the Euclidean distances between each pair of spots based on spatial coordinates first. In the following, an undirected graph \(G=\left(V,\,E\right)\) will be constructed, where \(V=\{{v}_{1},\,{v}_{2},\ldots,\,{v}_{n}\}\) is the node set corresponding to n spots; E is the edge set while there exists an edge \({e}_{{ij}}\) between \({v}_{i}\) and \({v}_{j}\) in \(E\) if and only if \({v}_{i}\) is the KNN of v_{j} or v_{j} is the KNN of \({v}_{i}\) based on Euclidean distance, where \(i,\,{j}=1,\,2,\,\ldots,{n}\); and \(i\,\ne\, j\). Based on the benchmarking results in Supplementary Data 4, the default K is defined as 1*\(\sqrt{n}\) among 0.5*\(\sqrt{n,\,}\) 1*\(\sqrt{n}\), 1.5*\(\sqrt{n}\), and 2*\(\sqrt{n}\). Note that all the notations of matrices and vectors are bolded, and all the vectors are treated as column vectors in the following description. An adjacent binary matrix \({{{\bf{A}}}}=({a}_{{ij}})\) with rows and columns as n spots is defined as:
A diagonal matrix \({{{\bf{D}}}}={{{\rm{diag}}}}(d_{1},\,{d}_{2},\,\ldots,\,{d}_{n})\), where \({d}_{i}={\sum}_{j=1}^{n}{a}_{ij}\) represents the degree of \({v}_{i}\).
Fourier mode calculation
Using matrices \({{{\bf{A}}}}\) and \({{{\bf{D}}}}\), a Laplacian matrix \({{{\bf{L}}}}\) can be obtained by
\({{{\bf{L}}}}\) can be decomposed using spectral decomposition:
where the diagonal elements of \({{{\bf{\Lambda }}}}\) are the eigenvalues of \({{{\bf{L}}}}\) with \({\lambda }_{1}\le {\lambda }_{2}\le \ldots \le {\lambda }_{n},\) where \({\lambda }_{1}\) is always equal to 0 regardless of graph topology. Thus, \({\lambda }_{1}\) is excluded from the following analysis. The columns of \({{{\bf{U}}}}\) are the unit eigenvector of \({{{\bf{L}}}}\). μ_{k} is the kth Fourier mode (FM), \({{{{\boldsymbol{\mu }}}}}_{{k}}\in {{\mathbb{R}}}^{n},\) \(k=1,\,2,\,\ldots,{n}\), and the set {μ_{1}, μ_{2}, ..., μ_{k}} is an orthogonal basis for the linear space. For \({{{{\boldsymbol{\mu }}}}}_{{k}}=\left({\mu }_{k}^{1},\,{\mu }_{k}^{2},\,\ldots,\,{\mu }_{k}^{n}\right)\), where \({\mu }_{k}^{i}\) indicates the value of the kth FM on node \({v}_{i}\), the smoothness of μ_{k} reflects the total variation of the kth FM in all mutual adjacent spots, which can be formulated as
The form can be derived by matrix multiplication as
where \({{{{\mathbf{\mu }}}}}_{{{{\bf{k}}}}}^{{{{\rm{T}}}}}\) is the transpose of μ_{k}. According to the definition of smoothness, if an eigenvector corresponds to a small eigenvalue, it indicates the variation of FM values on adjacent nodes is low. The increasing trend of eigenvalues corresponds to an increasing trend of oscillations of eigenvectors; hence, the eigenvalues and eigenvectors of L are used as frequencies and FMs in our SpaGFT, respectively. Intuitively, a small eigenvalue corresponds to a lowfrequency FM, while a large eigenvalue corresponds to a highfrequency FM.
Graph Fourier transform
The graph signal of a gene g is defined as \({{{{\bf{f}}}}}_{{{{\bf{g}}}}}=\left({f}_{{g}}^{1},\,{f}_{{g}}^{2},\,\ldots,\,{f}_{{g}}^{n}\right)\in {{\mathbb{R}}}^{n},\) which is a ndimensional vector and represents the gene expression values across n spots. The graph signal f_{g} is transformed into a Fourier coefficient \({\hat{{{{\bf{f}}}}}}_{{{{\bf{g}}}}}\) by
In such a way, \({{\hat{f}}_{{g}}^{{k}}}\) is the projection of f_{g} on FM μ_{k}, representing the contribution of FM μ_{k} to graph signal f_{g}, k is the index of f_{g} (e.g., \(k=1,\,2,\,\ldots,{n}\)). This Fourier transform harmonizes gene expression and its spatial distribution to represent gene g in the frequency domain. The details of SVG identification using \({\hat{{{{\bf{f}}}}}}_{{{{\bf{g}}}}}\) can be found below.
SVG identification
GFTscore definition
We designed a GFTscore to quantitatively measure the randomness of gene expressions distributed in the spatial domain, defined as
where \({\lambda }_{k}\) is the precalculated eigenvalue of L, and \({{\rm {e}}}^{{\lambda }_{k}}\) is used to weigh the \({\widetilde{f}}_{g}^{k}\) to further enhance the smoothness of the spatial omics variation signal and reduce its noisy components (Supplementary Note 1S2.3)^{18,67}. The normalized Fourier coefficient \({\widetilde{f}}_{g}^{k}\) is defined as
The gene with a high GFTscore tends to be SVG and vice versa. Therefore, all m genes are decreasingly ranked based on their GFTscore from high to low and denote these GFTscore values as \({y}_{1}\ge {y}_{2}\ge \ldots \ge {y}_{m}\). In order to determine the cutoff y_{z} to distinguish SVG and nonSVGs based on GFTscore, we applied the Kneedle algorithm^{68} to search for the inflection point of a GFTscore curve described in Supplementary Note 1.
Wilcoxon ranksum test implementation for determining SVGs
Although the above GFTscore is an indicator to rank and evaluate the potential SVGs, a rigorous statistical test is needed to calculate the pvalue for SVGs and control type I error. First, SpaGFT determines lowfrequency FM and highfrequency FMs and corresponding FCs by applying the Kneedle algorithm to the eigenvalues of L. The inflection points are used for determining the lowfrequency FMs and highfrequency FMs when the direction parameters are ‘increasing’ and ‘decreasing’, respectively. Second, the Wilcoxon ranksum test is utilized to test the differences between lowfrequency FCs and highfrequency FCs to obtain statistical significance. If a gene has a high GFTscore and significantly adjusted pvalue, the gene can be regarded as an SVG. We use \(f=({f}_{1},\,{f}_{2},\ldots,{f}_{n})\) to represent the expression of a random signal on n spots. If the gene corresponding to the graph signal is a nonSVG, the gene expressions on neighboring spots are independent. Otherwise, it will exhibit spatial dependence. Hence, we can assume that \(({f}_{1},\ldots,\,{f}_{n}) \sim N({\mu }_{f},\,{\sigma }_{f}^{2}I)\), similar in SpatialDE^{11}, where \({\mu }_{f}\), \({\sigma }_{f}^{2}\) and I are the mean, variance, and identity matrix, respectively. In this case, each \({f}_{i}\) follows a Gaussian distribution, which is independent and identically distributed. By implementing GFT on \(({f}_{1},\,{f}_{2},\ldots,\,{f}_{n})\), we obtain Fourier coefficients \({{F{C}}}_{1},\,{{{F}{C}}}_{2},\,\cdots,{{{F}{C}}}_{p}\), where \(p\) is the number of lowfrequency FCs and reflects the contributions from lowfrequency FMs. We also obtain the \({{F{C}}}_{p+1},{{{F}{C}}}_{p+2},\,\cdots,{{{F}{C}}}_{p+q}\), where \(q\) is the number of highfrequency FCs and reflects the contributions from noise. Hence, we form the null hypothesis that no difference exists between lowfrequency FCs and highfrequency FCs (Proof can be found in S3 of Supplementary Note 1). Accordingly, a nonparametrical test (i.e., Wilcoxon ranksum test) is used for testing the difference between median values of lowfrequency FCs and highfrequency FCs. Especially, the null hypothesis is that the median of lowfrequency FCs of an SVG is equal to or lower than the median of highfrequency FCs. The alternative hypothesis is that the median of lowfrequency FCs of an SVG is higher than the median of highfrequency FCs. The pvalue of each gene is calculated based on the Wilcoxon onesided ranksum test and then adjusted using the false discovery rate (FDR) method. Eventually, a gene with GFTscore higher than \({y}_{z}\) and adjusted pvalue less than 0.05 is considered an SVG.
Benchmarking data setup
Dataset description
Thirtytwo spatial transcriptome datasets were collected from the public domain, including 30 10X Visium datasets (18 human brain data, 11 mouse brain data, and one human lymph node data) and two SlideseqV2 datasets (mouse brain). More details can be found in Supplementary Data 1. Those samples were sequenced by two different SRT technologies: 10X Visium measures ~55 μm diameter per spot, and SlideseqV2 measures ~10 μm diameter per spot. Three datasets were selected as the training sets for gridsearch parameter optimization in SpaGFT, including two highest readdepth datasets in Visium (HEcoronal) and SlideseqV2 (Puck20011508), one signature dataset in Maynard’s study^{24}. The remaining 28 datasets (excluding lymph node data) were used as independent test datasets.
Data preprocessing
For all 32 datasets, we adopt the same preprocessing steps based on squidpy (version 1.2.1), including filtering genes that have expression values in <10 spots, normalizing the raw count matrix by counts per million reads method, and implementing logtransformation to the normalized count matrix. No specific preprocessing step was performed on the spatial location data.
Benchmarking SVG collection
We collected SVG candidates from five publications^{24,25,26,27,28}, with data from either human or mouse brain subregions. (i) A total of 130 layer signature genes were collected from Maynard’s study^{24}. These genes are potential multiplelayer markers validated in the human dorsolateral prefrontal cortex region. (ii) A total of 397 celltypespecific (CTS) genes in the adult mouse cortex were collected from Tasic’s study (2016 version)^{28}. The authors performed scRNAseq on the dissected target region, identified 49 cell types, and constructed a cellular taxonomy of the primary visual cortex in the adult mouse. (iii) A total of 182 CTS genes in mouse neocortex were collected from Tasic’s study^{27}. Altogether, 133 cell types were identified from multiple cortical areas at singlecell resolution. (iv) A total of 260 signature genes across different major regions of the adult mouse brain were collected from Ortiz’s study^{25}. The authors’ utilized spatial transcriptomics data to systematically profile subregions and delivered the subregional genes using consecutive coronal tissue sections. (v) A total of 86 signature genes in the cortical region shared by humans and mice were collected from Hodge’s study^{26}. Collectively, a total of 849 genes were obtained, among which 153 genes were documented by multiple papers. More details, such as gene names, targeted regions, and sources, can be found in Supplementary Data 2.
Next, the above 849 genes were manually validated on the insitu hybridization (ISH) database deployed on the Allen Brain Atlas (https://mouse.brainmap.org/). The ISH database provided ISH mouse brain data across 12 anatomical structures (i.e., Isocortex, Olfactory area, Hippocampal formation, Cortical subplate, Striatum, Pallidum, Thalamus, Hypothalamus, Midbrain, Pons, Medulla, and Cerebellum). We filtered the 849 genes as follows: (i) If a gene is showcased in multiple anatomical plane experiments (i.e., coronal plane and sagittal plane), it will be counted multiple times with different expressions in the corresponding experiments, such that 1327 genes were archived (Supplementary Data 3). (ii) All 1327 genes were first filtered by low gene expressions (cutoff is 1.0), and the FindVariableFeatures function (“vst” method) in the Seurat (v4.0.5) was used for identifying highly variable genes across twelve anatomical structures. Eventually, 458 genes were kept and considered as curated benchmarking SVGs. The evaluation criteria can be found in Supplementary Note 2.
Statistics and reproducibility
In our benchmarking experiment, we implemented a twosided Wilcoxonrank sum test to conduct a significance test. No data were excluded from the analyses. The experiments were not randomized. Randomization is not relevant to this study since each data was analyzed separately. We then computed the key evaluation metrics, including the Jaccard index, odds ratio, precision, recall, F1 score, Tversky index, Moran’s Index, and Geary’s C.
SpaGFT implementation and grid search of parameter optimization
A gridsearch was set to test for six parameters, including ratio_neighbors (0.5, 1, 1.5, 2) for KNN selection and S (4, 5, 6, 8) for the inflection point coefficient, resulting in 16 parameter combinations. We set \(K=\sqrt{n\,}\) as the default parameter for constructing the KNN graphs in SpaGFT. SVGs were determined by genes with high GFTscore via the KneeLocator function (curve=’convex’, direction=’deceasing’, and S = 6) in the kneed package (version 0.7.0) and FDR (cutoff is less than 0.05).
Parameter setting of other tools
(i) SpatialDE (version 1.1.3) is a method for identifying and describing SVGs based on Gaussian process regression used in geostatistics. SpatialDE consists of four steps, establishing the SpatialDE model, predicting statistical significance, selecting the model, and expressing histology automatically. We selected two key parameters, design_formula (‘0’ and ‘1’) in the NaiveDE.regress_out function and kernel_space (“{‘SE’:[5.,25.,50.],‘const’:0}”, “{‘SE’:[6.,16.,36.],‘const’:0}”, “{‘SE’:[7.,47.,57.],‘const’:0}”, “{‘SE’:[4.,34.,64.],‘const’:0}”, “{‘PER’:[5.,25.,50.],‘const’:0}”, “{‘PER’:[6.,16.,36.],‘const’:0}”, “{‘PER’:[7.,47.,57.],‘const’:0}”, “{‘PER’:[4.,34.,64.],‘const’:0}”, and “{‘linear’:0,‘const’:0}”) in the SpatialDE.run function for parameter tunning, resulting in 18 parameter combinations.
(ii) SPARK (version 1.1.1) is a statistical method for spatial count data analysis through generalized linear spatial models. Relying on statistical hypothesis testing, SPARX identifies SVGs via predefined kernels. First, raw count and spatial coordinates of spots were used to create the SPARK object via filtering lowquality spots (controlled by min_total_counts) or genes (controlled by percentage). Then the object was followed by fitting the countbased spatial model to estimate the parameters via spark.vc function, which is affected by the number of iterations (fit.maxiter) and models (fit.model). Lastly, ran spark.test function to test multiple kernel matrices and obtain the results. We selected four key parameters, percentage (0.05, 0.1, 0.15), min_total_counts (10, 100, 500) in CreateSPARKObject function, fit.maxiter (300, 500, 700), and fit.model (“poisson”, “gaussian”) in spark.vc function for parameter tuning, resulting in 54 parameter combinations.
(iii) SPARKX (version 1.1.1) is a nonparametric method that tests whether the expression level of the gene displays any spatial expression pattern via a general class of covariance tests. We selected three key parameters, percentage (0.05, 0.1, 0.15), min_total_counts (10, 100, 500) in the CreateSPARKObject function, and option (“single”, “mixture”) in the sparkx function for parameter tuning, resulting in 18 parameter combinations.
(iv) SpaGCN (version 1.2.0) is a graph convolutional network approach that integrates gene expression, spatial location, and histology in spatial transcriptomics data analysis. SpaGCN consisted of four steps, integrating data into a chart, setting the graph convolutional layer, detecting spatial domains by clustering, and identifying SVGs in spatial domains. We selected two parameters, the value of the ratio (1/3, 1/2, 2/3, and 5/6) in the find_neighbor_cluster function and res (0.8, 0.9, 1.0, 1.1, and 1.2) in the SpaGCN.train function for parameter tuning, resulting in 20 parameter combinations.
(v) MERINGUE (version 1.0) is a computational framework based on spatial autocorrelation and crosscorrelation analysis. It is composed of three major steps to identify SVGs. Firstly, Voronoi tessellation was utilized to partition the graph to reflect the length scale of cellular density. Secondly, the adjacency matrix is defined using geodesic distance and the partitioned graph. Finally, genewise autocorrelation (e.g., Moran’s I) is conducted, and a permutation test is performed for significance calculation. We selected min.read (100, 500, 1000), min.lib.size (100, 500, 1000) in the cleanCounts function and filterDist (1.5, 2.5, 3.5, 7.5, 12.5, 15.5) in the getSpatialNeighbors function for parameter tuning, resulting in 54 parameter combinations.
(vi) scGCO (version 1.1.2) is a graphcut approach that integrates gene expression and spatial location in spatial transcriptomics data analysis. scGCO consists of four steps: representing a gene’s spatial expression with hidden Markov random field (HMRF), optimizing HMRF with graph cuts with varying hyperparameters, identifying best graph cuts, and calculating the significance of putative SVGs. We selected three parameters, the value of unary_scale_factor (50, 100, and 150) and smooth_factor (5, 10, and 15) in the identify_spatial_genes function for parameter tuning and fdr_cutoff (0.025, 0.05, and 0.075) in the final pipeline for identification of SVG, resulting in 27 parameter combinations.
Visualization of frequency signal of SVGs in PCA and UMAP
Mouse brain (i.e., HE coronal sample) with 2702 spots was used for demonstrating FCs on distinguishing SVG and nonSVG in the 2D UMAP space. SpaGFT determined 207 lowfrequency FMs using the Kneedle Algorithm and computed corresponding FCs. PCA was also used for producing lowdimension representation. The transposed and normalized expression matrix was decomposed by using the sc.tl.pca function from the scanpy package (version 1.9.1). The first 207 principal components (PC) were selected for UMAP dimension reduction and visualization. The function sc.tl.umap was applied to conduct UMAP dimension reduction for FCs and PCs.
SVG signal enhancement
An SVG may suffer from low expression or dropout issues due to technical bias^{8}. To solve this problem, SpaGFT implemented the lowpass filter to enhance the SVG expressions. For an SVG with an observed expression value \({{{{\bf{f}}}}}_{{{{\bf{g}}}}}\in {{\mathbb{R}}}^{n}\), we define \({\bar{{{{\bf{f}}}}}}_{{{{\bf{g}}}}}\in {{\mathbb{R}}}^{n}\) as the expected gene expression value of this SVG, and \({{{{\bf{f}}}}}_{{{{\bf{g}}}}}{{{\boldsymbol{=}}}}{\bar{{{{\bf{f}}}}}}_{{{{\bf{g}}}}}{{{\boldsymbol{+}}}}{{{{\boldsymbol{\epsilon }}}}}_{{{{\bf{g}}}}}\), where \({{{{\boldsymbol{\epsilon }}}}}_{{{{\boldsymbol{g}}}}}\in {{\mathbb{R}}}^{n}\) represents noises. SpaGFT estimates an approximated FCs \({{{{\bf{f}}}}}_{{{{\bf{g}}}}}^{{{{\boldsymbol{\star }}}}}\) to expected gene expression \({\bar{{{{\bf{f}}}}}}_{{{{\bf{g}}}}}\) in the following way, resisting the noise \({{{{\boldsymbol{\epsilon }}}}}_{{{{\boldsymbol{g}}}}}\). The approximation has two requirements (i) the expected gene expression after enhancement should be similar to the originally measured gene expression, and (ii) keep low variation within estimated gene expression to prevent inducing noises. Therefore, the following optimization problem is proposed to find an optimal solution \({{{{\bf{f}}}}}_{{{{\bf{g}}}}}^{{{{\boldsymbol{\star }}}}}\) for \({\bar{{{{\bf{f}}}}}}_{{{{\bf{g}}}}}\)
where  ∙  is the \(L2\)norm, \({{{\bf{f}}}}=\left({f}^{1},\,{f}^{2},\,\ldots,\,{f}^{n}\right)\in {{\mathbb{R}}}^{n}\) is the variable in solution space, and \(i,\,{j}=1,\,2,\,\ldots,{n}\). \(c\) is a coefficient to determine the importance of variation of the estimated signals, and \(c\, > \,0\). According to convex optimization, the optimal solution \({{{{\bf{f}}}}}_{{{{\bf{g}}}}}^{\star}\) can be formulated as
where \({{{\mathbf{\Lambda }}}}={{{\rm{diag}}}}\left({\lambda }_{1},\,{\lambda }_{2},\,\ldots,\,{\lambda }_{n}\right)\), and I is an identity matrix. \({\left({{{\bf{I}}}}+c{{{\mathbf{\Lambda }}}}\right)}^{1}\) is the lowpass filter and \({\left({{{\bf{I}}}}+c{{{\mathbf{\Lambda }}}}\right)}^{1}{\hat{{{{\bf{f}}}}}}_{{{{\bf{g}}}}}\) is the enhanced FCs. \({{{{\bf{f}}}}}_{{{{\bf{g}}}}}^{\star}={{{{\bf{U}}}}\left({{{\bf{I}}}}+c{{{\mathbf{\Lambda }}}}\right)}^{1}{\hat{{{{\bf{f}}}}}}_{{{{\bf{g}}}}}\) represents the enhanced SVG expression using inverse graph Fourier transform. Specifically, in HEcoronal mouse brain data analysis, we selected 1300 (\(=25\sqrt{n},{n}=2702\)) lowfrequency FCs for enhancing signal and recovering spatial pattern by using iGFT with \(c=0.0001\).
Data preprocessing on the human prostate cancer Visium data
Cell segmentation
The Visium image of human prostate cancer (adenocarcinoma with invasive carcinoma) from the 10X official website was cropped into patches according to spot center coordinates and diameter length. Each patch is processed by Cellpose for nuclei segmentation using the default parameter. Cell density in each patch is determined using the number of segmented cells.
Microbial alignment
Following the tutorial^{69}, the corresponding bam files were processed via Kraken packages by (1) removing host sequences and retaining microbial reads, (2) assigning microbial reads to a taxonomic category (e.g., species and genus), and (3) computing the relative abundance of different species in each spot.
SVG signal enhancement benchmarking
Sixteen human brain datasets with wellannotated labels were used for enhancement benchmarking^{23,24}. Samples 151510, 151672, and 151673 were used for grid search. Other 13 datasets were used for independent tests. SpaGFT can transform graph signals to FCs, and apply correspondence preprocessing in the frequency domain to realize signal enhancement of genes. Briefly, it is composed of three major steps. Firstly, SpaGFT is implemented to obtain FCs. Secondly, a lowpass filter is applied to weigh and recalculate FCs. Lastly, SpaGFT implements iGFT to recover the enhanced FCs to graph signals. We select c (0.003, 0.005, 0.007) and ratio_fms (13, 15, 17) in the low_pass_enhancement function, resulting in 9 parameter combinations. c = 0.005 and ratio_fms = 15 were selected for the independent test. For the parameters used for other computational tools, the details can be found as follows.

(i)
SAVERX (version 1.0.2) is designed to improve data quality, which extracts genegene relationships by adopting a deep autoencoder and a Bayesian model simultaneously. SAVERX is composed of three major steps roughly. Firstly, train the target data with an autoencoder without a chosen pretraining model. Secondly, filter unpredictable genes using crossvalidation. Lastly, estimate the final denoised values with empirical Bayesian shrinkage. Two parameters were considered to explore the performance as well as the robustness of SAVERX, including batch_size (32, 64, 128) in the saverx function and fold (4, 6, 8) in the autoFilterCV function, resulting in 9 parameter combinations.

(ii)
Sprod (version 1.0) is a computational framework based on latent graph learning of matched location and imaging data by leveraging information from the physical locations of sequencing to impute accurate SRT gene expression. The framework of Sprod can be divided into two major steps roughly, which are building a graph and optimizing objective function for such a graph to obtain the denoised gene expression matrix. To validate its robustness, two parameters were adjusted, including sprod_R (0.1, 0.5) and sprod_latent_dim (8, 10, 12), to generate nine parameter combinations.

(iii)
DCA (version 0.3.1) is a deep count autoencoder network with specialized loss functions targeted to denoise scRNAseq datasets. It uses the autoencoder framework to estimate three parameters \(({{{\rm{\mu }}}},{{{\rm{\theta }}}},{{{\rm{\pi }}}})\) of zeroinflated negative binomial distribution conditioned on the input data for each gene. In particular, the autoencoder gives three output layers, representing for each gene the three parameters that make up the genespecific loss function to compare to the original input of this gene. Finally, the mean \(({{{\rm{\mu }}}})\) of the negative binomial distribution represents denoised data as the main output. We set neurons of all hidden layers except for the bottleneck to (48, 64, 80) and neurons of bottleneck to (24, 32, 40) for parameter tuning, resulting in 9 parameter combinations.

(iv)
MAGIC (version 3.0.0) is a method that shares information across similar cells via data diffusion to denoise the cell count matrix and fill in missing transcripts. It is composed of two major steps. Firstly, it builds its affinity matrix in four steps which include a data preprocessing step, converting distances to affinities using an adaptive Gaussian Kernel, converting the affinity matrix A into a Markov transition matrix M, and data diffusion through exponentiation of M. Once the affinity matrix is constructed, the imputation step of MAGIC involves sharing information between cells in the resulting neighborhoods through matrix multiplication. We applied the knn settings (3, 5, 7) and the level of diffusion (2, 3, 4) in the MAGIC initialization function for parameter tuning, resulting in 9 parameter combinations.

(v)
scVI (version 0.17.3) is a hierarchical Bayesian model based on a deep neural network, which is used for probabilistic representation and analysis of singlecell gene expression. It consists of two major steps. Firstly, the gene expression is compressed into a lowdimensional hidden space by the encoder, and then the hidden space is mapped to the posterior estimation of the gene expression distribution parameters by the neural network of the decoder. It uses random optimization and deep neural networks to gather information on similar cells and genes, approximates the distribution of observed expression values, and considers the batch effect and limited sensitivity for batch correction, visualization, clustering, and differential expression. We selected n_hidden (64, 128, 256) and gene_likelihood (‘zinb’, ‘nb’, ‘poisson’) in the model.SCVI function for parameter tuning, resulting in 9 parameter combinations.

(vi)
netNMFsc (version 0.0.1) is a nonnegative matrix decomposition method for network regularization, which is designed for the imputation and dimensionality reduction of scRNAseq analysis. It uses a priori gene network to obtain a more meaningful lowdimensional representation of genes, and network regularization uses a priori knowledge of gene–gene interaction to encourage gene pairs with known interactions to approach each other in lowdimensional representation. We selected d (8, 10, 12) and alpha (80, 100, 120) in the netNMFGD function for parameter tuning, resulting in 9 parameter combinations.
SVG clustering and FTU identification
The pipeline is visualized in Supplementary Fig. 6a. As the pattern of one SVG cluster can demonstrate specific functions of one FTU, the FTU may not necessarily display a clear boundary to its neighbor FTUs. On the contrary, the existence of overlapped regions showing polyfunctional regions is allowed. Computationally, the process of FTU identification is to optimize the resolution parameter of the Louvain algorithm for obtaining a certain number of biologyinformed FTUs, which minimizes the overlapped area. Denote G' as the set of SVGs identified by SpaGFT. For each resolution parameter \({{{\rm{res}}}}\, > \,0\), G' can be partitioned to {\({G}_{1}^{{\prime} },\,{G}_{2}^{{\prime} },\,\ldots,\,{G}_{{n}_{{{\rm {res}}}}}^{{\prime} }\left.\right\}\) (i.e., \({\bigcup }_{k}{G}_{i}^{{\prime} }={G}^{{\prime} }\) and \({G}_{k}^{{\prime} }\bigcap {G}_{l}^{{\prime} }\,=\, \varnothing,\,\forall k\,\ne\,l.\)) by applying the Louvain algorithm on FCs, and the resolution will be optimized by the loss function below. Denote \(X=({x}_{s,g})\in {{\mathbb{R}}}^{\leftS\right\times \left{G}^{{\prime} }\right}\) as the gene expression matrix, where \(S\) is the set of all spots. In the following, for each SVG group \({G}_{k}^{{\prime} }\), \({{{\rm{pseudo}}}}({s}_{s,{G}_{k}^{{\prime} }})={\sum }_{g\in {G}_{k}^{{\prime} }}\log ({x}_{s,g})\) represents the pseudo expression value^{4} for spot \(i\). Apply kmeans algorithms with k = 2 on \(\{{{{\rm{pseudo}}}}\left({s}_{1,{G}_{k}^{{\prime} }}\right),{{{\rm{pseudo}}}}\left({s}_{2,{G}_{k}^{{\prime} }}\right),\,\ldots,\,{{{\rm{pseudo}}}}\left({s}_{\leftS\right,{G}_{k}^{{\prime} }}\right)\}\) to pick out one spot cluster whose spots highly express genes in SVG group \({G}_{k}^{{\prime} }\) and such spot cluster is identified as a FTU, denoted as \({S}_{i}\in S\). Our objective function aims to find the best partition of \({G}^{{\prime} }\) such that the average overlap between any two \({S}_{i},\,{S}_{j}\) is minimized:
\({{{{\rm{argmin}}}}}_{{{{res}}} > 0}\frac{2\times {\sum}_{k\ne l}\left{S}_{k}\cap {S}_{l}\right}{{n}_{{{{res}}}}\times ({n}_{{{{res}}}}1)}\)
SpaGFT implementation on the lymph node Visium data and interpretation
Lymph node SVG cluster identification and FTU interpretation
SVGs were identified on the human lymph node data (Visium) with the default setting of SpaGFT. To demonstrate the relations between cell composition and annotated FTUs, cell2location^{35} was implemented to deconvolute spot and resolve finegrained cell types in spatial transcriptomic data. Cell2location was first used to generate the spotcell type proportion matrix as described above, resulting in a cell proportion of 34 cell types. Then, pseudoexpression values across all spots for one FTU were computed using the method from the FTU identification section. Then, an element of the FTUcell type correlation matrix was calculated by computing the Pearson correlation coefficient between the proportion of a cell type and the pseudoexpression of an FTU across all the spots. Subsequently, the FTUcell type correlation matrix was obtained by calculating all elements as described above, with rows representing FTUs and columns representing cell types. Lastly, the FTUcell type matrix was generated and visualized on a heatmap, and three major FTUs in the lymph node were annotated, i.e., the T cell zone, GC, and B follicle.
Visualization of GC, T cell zone, and B follicles in the Barycentric coordinate system
Spotcell proportion matrix was used to select and merge signature cell types of GC, T cell zone, and B follicles for generating a merged spotcell type proportion matrix (an Nby3 matrix, N is equal to the number of spots). For GC, B_Cycling, B_GC_DZ, B_GC_LZ, B_GC_prePB, FDC, and T_CD4_TfH_GC were selected as signature cell types. For T cell zone, T_CD4, T_CD4_TfH, T_TfR, T_Treg, T_CD4_naive, and T_CD8_naive were selected as signature cell types. For B follicle, B_mem, B_naive, and B_preGC were regarded as signature cell types. The merged spotcell type proportion matrix was calculated by summing up the proportion of signature cell types for GC, T cell zone, and B follicle, respectively. Finally, annotated spots (spot assignment in Supplementary Data 11) were selected from the merged spotcell type proportion matrix for visualization. The subset spots from the merged matrix were projected on an equilateral triangle via Barycentric coordinate project methods^{37}. The projected spots were colored by FTU assignment results. Unique and overlapped spots across seven regions (i.e., GC, GC–B, B, B–T, T, T–GC, and T–GC–B) from three FTUs were assigned and visualized on the spatial map. Gene module scores were calculated using the AddModuleScore function from the Seurat (v4.0.5) package. Calculated gene module score and cell type proportion were then grouped by seven regions and visualized on the line plot (Fig. 3e, f). Oneway ANOVA using function aov in R environment was conducted to test the difference among the means of seven regions regarding gene module scores and cell type proportions, respectively.
CODEX tonsil tissue staining
An FFPE human tonsil tissue (provided by Dr. Scott Rodig, Brigham and Women’s Hospital Department of Pathology) was sectioned onto a No. 1 glass coverslip (22x22mm) pretreated with Vectabond (SP18007, Vector Labs). The tissue was deparaffinized by heating at 70 °C for 1 h and soaking in xylene 2× for 15 min each. The tissue was then rehydrated by incubating in the following sequence for 3 min each with gentle rocking: 100% EtOH twice, 95% EtOH twice, 80% EtOH once, 70% EtOH once, ddH_{2}O thrice. To prepare for heatinduced antigen retrieval (HIER), a PT module (A80400012, Thermo Fisher) was filled with 1X PBS, with a coverslip jar containing 1X Dako pH 9 antigen retrieval buffer (S2375, Agilent) within. The PT module was then prewarmed to 75 °C. After rehydration, the tissue was placed in the prewarmed coverslip jar, then the PT module was heated to 97 °C for 20 min and cooled to 65 °C. The coverslip jar was then removed from the PT module and cooled for ~15–20 min at room temperature. The tissue was then washed in rehydration buffer (232105, Akoya Biosciences) twice for 2 min each then incubated in CODEX staining buffer (232106, Akoya Biosciences) for 20 min while gently rocking. A hydrophobic barrier was then drawn on the perimeter of the coverslip with an ImmEdge Hydrophobic Barrier pen (310018, Vector Labs). The tissue was then transferred to a humidity chamber. The humidity chamber was made by filling an empty pipette tip box with paper towels and ddH_{2}O, stacking the tip box on a cool box (432021, Corning) containing a −20 °C ice block, then replacing the tip box lid with a sixwell plate lid. The tissue was then blocked with 200 μL of blocking buffer.
The blocking buffer was made with 180 μL BBDG block, 10 μL oligo block, and 10 μL sheared salmon sperm DNA; the BBDG block was prepared with 5% donkey serum, 0.1% Triton X100, and 0.05% sodium azide prepared with 1X TBS IHC Wash buffer with Tween 20 (935B09, Cell Marque); the oligo block was prepared by mixing 57 different custom oligos (IDT) in ddH_{2}O with a final concentration of 0.5 μM per oligo; the sheared salmon sperm DNA was added from its 10 mg/ml stock (AM9680, Thermo Fisher). The tissue was blocked while photobleaching with a custom LED array for 2 h. The LED array was set up by inclining two Happy Lights (6460231, Best Buy) against both sides of the cool box and positioning an LED Grow Light (B07C68N7PC, Amazon) above. The temperature was monitored to ensure that it remained under 35 °C. The staining antibodies were then prepared during the 2h block.
DNAconjugated antibodies at appropriate concentrations were added to 100 μL of CODEX staining buffer, loaded into a 50kDa centrifugal filter (UFC505096, Millipore) prewetted with CODEX staining buffer, and centrifuged at 12,500×g for 8 min. Concentrated antibodies were then transferred to a 0.1 μm centrifugal filter (UFC30VV00, Millipore) prewetted with CODEX staining buffer, filled with extra CODEX staining buffer to a total volume of 181 μL, added with 4.75 μL of each Akoya blockers N (232108, Akoya), G (232109, Akoya), J (232110, Akoya), and S (232111, Akoya) to a total volume of 200 μL, then centrifuged for 2 min at 12,500×g to remove antibody aggregates. The antibody flow through (99 μL) was used to stain the tissue overnight at 4 °C in a humidity chamber covered with a foilwrapped lid.
After the overnight antibody stain, the tissue was washed in CODEX staining buffer twice for 2 min each before fixing in 1.6% paraformaldehyde (PFA) for 10 min while gently rocking. The 1.6% PFA was prepared by diluting 16% PFA in CODEX storage buffer (232107, Akoya). After 1.6% PFA fixation, the tissue was rinsed in 1X PBS twice and washed in 1X PBS for 2 min while gently rocking. The tissue was then incubated in the cold (−20 °C) with 100% methanol on ice for 5 min without rocking for further fixation and then washed thrice in 1X PBS as before while gently rocking. The final fixation solution was then prepared by mixing 20 μL of CODEX final fixative (232112, Akoya) in 1000 μL of 1x PBS. The tissue was then fixed with 200 μL of the final fixative solution at room temperature for 20 min in a humidity chamber. The tissue was then rinsed in 1X PBS and stored in 1X PBS at 4 °C prior to CODEX imaging.
A black flat bottom 96well plate (07200762, Corning) was used to store the reporter oligonucleotides, with each well corresponding to an imaging cycle. Each well contained two fluorescent oligonucleotides (Cy3 and Cy5, 5 μL each) added to 240 μL of plate master mix containing DAPI nuclear stain (1:600) (7000003, Akoya) and CODEX assay reagent (0.5 mg/mL) (7000002, Akoya). For the first and last blank cycles, an additional plate buffer was used to substitute for each fluorescent oligonucleotide. The 96well plate was securely sealed with aluminum film (14222342, Thermo Fisher) and kept at 4 °C prior to CODEX imaging.
CODEX antibody panel
The following antibodies, clones, and suppliers were used in this study:
BCL2 (124, Novus Biologicals, 1:50), CCR6 (polyclonal, Novus Biologicals, 1:25), CD11b (EPR1344, Abcam, 1:50), CD11c (EP1347Y, Abcam, 1:50), CD15 (MMA, BD Biosciences, 1:200), CD16 (D1N9L, Cell Signaling Technology, 1:100), CD162 (HECA452, Novus Biologicals, 1:200), CD163 (EDHu1, Novus Biologicals, 1:200), CD2 (RPA2.10, Biolegend, 1:25), CD20 (rIGEL/773, Novus Biologicals, 1:200), CD206 (polyclonal, R&D Systems, 1:100), CD25 (4C9, Cell Marque, 1:100), CD30 (BerH2, Cell Marque, 1:25), CD31 (C31.3 + C31.7 + C31.10, Novus Biologicals, 1:200), CD4 (EPR6855, Abcam, 1:100), CD44 (IM7, Biolegend, 1:100), CD45 (B11 + PD7/26, Novus Biologicals, 1:400), CD45RA (HI100, Biolegend, 1:50), CD45RO (UCHL1, Biolegend, 1:100), CD5 (UCHT2, Biolegend, 1:50), CD56 (MRQ42, Cell Marque, 1:50), CD57 (HCD57, Biolegend, 1:200), CD68 (KP1, Biolegend, 1:100), CD69 (polyclonal, R&D Systems, 1:200), CD7 (MRQ56, Cell Marque, 1:100), CD8 (C8/144B, Novus Biologicals, 1:50), collagen IV (polyclonal, Abcam, 1:200), cytokeratin (C11, Biolegend, 1:200), EGFR (D38B1, Cell Signaling Technology, 1:25), FoxP3 (236A/E7, Abcam, 1:100), granzyme B (EPR20129217, Abcam, 1:200), HLADR (EPR3692, Abcam, 1:200), IDO1 (D5J4E, Cell Signaling Technology, 1:25), LAG3 (D2G4O, Cell Signaling Technology, 1:25), mast cell tryptase (AA1, Abcam, 1:200), MMP9 (L51/82, Biolegend, 1:200), MUC1 (955, Novus Biologicals, 1:100), PD1 (D4W2J, Cell Signaling Technology, 1:50), PDL1 (E1L3N, Cell Signaling Technology, 1:50), podoplanin (D240, Biolegend, 1:200), Tbet (D6N8B, Cell Signaling Technology, 1:100), TCR β (G11, Santa Cruz Biotechnology, 1:100), TCRγ/δ (H41, Santa Cruz Biotechnology, 1:100), Tim3 (polyclonal, Novus Biologicals, 1:50), Vimentin (RV202, BD Biosciences, 1:200), VISTA (D1L2G, Cell Signaling Technology, 1:50), αSMA (polyclonal, Abcam, 1:200), and βcatenin (14, BD Biosciences, 1:50). Readers of interest are referred to publication^{70} for more details on the antibody clones, conjugated fluorophores, exposure, and titers.
CODEX tonsil tissue imaging
The tonsil tissue coverslip and reporter plate were equilibrated to room temperature and placed on the CODEX microfluidics instrument. All buffer bottles were refilled (ddH_{2}O, DMSO, 1X CODEX buffer (7000001, Akoya)), and the waste bottle was emptied before the run. To facilitate the setting up of imaging areas and z planes, the tissue was stained with 750 μL of nuclear stain solution (1 μL of DAPI nuclear stain in 1500 μL of 1X CODEX buffer) for 3 min, then washed with the CODEX fluidics device. For each imaging cycle, three images that corresponded to the DAPI, Cy3, and Cy5 channels were captured. The first and last blank imaging cycles did not contain any Cy3 or Cy5 oligos, and thus are used for background correction.
The CODEX imaging was operated using a ×20/0.75 objective (CFI Plan Apo λ, Nikon) mounted to an inverted fluorescence microscope (BZX810, Keyence) which was connected to a CODEX microfluidics instrument and CODEX driver software (Akoya Biosciences). The acquired multiplexed images were stitched, and background corrected using the SINGER CODEX Processing Software (Akoya Biosciences). For this study, six independent 2048 × 2048 fieldofviews (FOV) were cropped from the original 20,744 × 20,592 image. The FOVs were selected to include key cell types and tissue structures in tonsils, such as tonsillar crypts or lymphoid nodules.
Cell segmentation
Custom ImageJ macros were used to normalize and cap nuclear and surface image signals at the 99.7th percentile to facilitate cell segmentation. Cell segmentation was performed using a local implementation of Mesmer from the DeepCell library (deepcelltf 0.11.0)^{40}, where the multiplex_segmentation.py script was modified to adjust the segmentation resolution (microns per pixel, mpp). model_mpp = 0.5 generated satisfactory segmentation results for this study. Singlecell features based on the cell segmentation mask were then scaled to cell size and extracted as FCS files.
Cell clustering and annotation
Singlecell features were normalized to each FOV’s median DAPI signal to account for FOV signal variation, arcsinh transformed with cofactor = 150, capped between 1st–99th percentile, and rescaled to 0–1. Sixteen markers (cytokeratin, podoplanin, CD31, αSMA, collagen IV, CD11b, CD11c, CD68, CD163, CD206, CD7, CD4, CD8, FoxP3, CD20, CD15) were used for unsupervised clustering using FlowSOM^{41} (66 output clusters). The cell type for each cluster was annotated based on its relative feature expression, as determined via Marker Enrichment Modeling^{42}, and annotated clusters were visually compared to the original images to ensure accuracy and specificity. Cells belonging to indeterminable clusters were further clustered (20 output clusters) and annotated as above.
SpaGFT implementation on tonsil CODEX data and interpretation
Resize CODEX images and SpaGFT implementation
As each FOV consisted of 2048 by 2048 pixels (~0.4 μm per pixel size), the CODEX image needed to be scaled down to 200 by 200 pixels (~3.2 μm per pixel size) to reduce the high computational burden (Supplementary Fig. 8a). Therefore, original CODEX images (2048 by 2048 pixels) were resized to 200 by 200 images by implementing function “resize” and selecting cubic interpolation from the imager package (v.42) in R environments. SpaGFT was then applied to the resized images by following default parameters.
Structural similarity (SSIM) calculation
The Structural Similarity (SSIM) score was a measurement for locally evaluating the similarity between two images regardless of image size^{71}. The SSIM score ranged from 0 to 1; a higher score means more similarity between two images. It was defined as follows:
x and y were windows with 8 by 8 pixels; \(l\left(x,\,y\right)=\frac{2{\mu }_{x}{\mu }_{y}+{C}_{1}}{{\mu }_{x}^{2}+{\mu }_{x}^{2}+{C}_{1}}\) was the luminance comparison function for comparing the average brightness of the two images regarding pixels x and \(y\). \({C}_{1}\) is constant, and \(\alpha\) is the weight factor of luminance comparison. \(c\left(x,\,y\right)=\frac{2{\sigma }_{x}{\sigma }_{y}+{C}_{1}}{{\sigma }_{x}^{2}+{\sigma }_{x}^{2}+{C}_{2}}\) was the contrast comparison function for measuring the standard deviation of two images. \({C}_{2}\) is constant, and \(\beta\) is the weight factor of contrast comparison. \(s\left(x,\,y\right)=\frac{{\sigma }_{{xy}}+{C}_{3}}{{\sigma }_{x}{\sigma }_{y}+{C}_{3}}\) was the structure comparison by calculating the covariance between the two images. \({C}_{3}\) is constant, and \(\gamma\) is the weight factor of structure comparison.
Cell–cell distance and interaction analysis
To compute cell–cell distance within one FTU, we first select cells assigned to each FTU. An undirected cell graph was then constructed, where the cell was a node and edge connected by every two cells defined by the Delaunay triangulation using the deldir function from the deldir package (v.1.06). Subsequently, the edge represented the observed distance between the connected two cells, and Euclidean distance was used for calculating the distance^{72}. Lastly, the average distance among different cell types was computed by taking the average of the observed cell–cell distance to generate the network plot. Regarding the determination of the cell–cell interaction, the spatial location of cells assigned in each FTU was permutated and recalculated cell–cell distance as expected distance. If the cell–cell distance is lower than 15 μm^{73} (~5 pixels in the 200 by 200pixel image), the cells will contact and interact with each other. Wilcoxon ranksum test was used for the computed pvalue for expected distance and observed distance. If the expected distance was significantly smaller than the observed distance, it suggested that cells would interact with each other.
SpaGFT implementation in SpaGCN
Let X_{spa} be the SRT gene expression matrix with the dimension \({n}_{{{\rm {spot}}}}\times {n}_{{{\rm {gene}}}}\), in which \({n}_{{{\rm {spot}}}}\) and \({n}_{{{\rm {gene}}}}\) represent the numbers of spots and genes, respectively. Upon normalization, the spot cosine similarity matrix \({{{{\boldsymbol{X}}}}}_{{{{\boldsymbol{s}}}}}\) is computed by the formula \({{{{\bf{X}}}}}_{{{{\bf{s}}}}}{{{\boldsymbol{=}}}}{{{{\bf{X}}}}}_{{{{\bf{spa}}}}}{{{{\bf{X}}}}}_{{{{\bf{spa}}}}}^{{{{\rm{T}}}}}\), yielding a matrix with dimension \({n}_{{{\rm {spot}}}}\times {n}_{{{\rm {spot}}}}\). Denote \({{{\bf{U}}}}=({{{{\bf{\mu }}}}}_{{{{\bf{1}}}}}{{,}}\,{{{{\bf{\mu }}}}}_{{{{\bf{2}}}}}{{{\boldsymbol{,}}}}\,{{\ldots }}{{,}}\,{{{{\bf{\mu }}}}}_{{{{{\bf{n}}}}}_{{{{\bf{FC}}}}}})\), where each \({{{{\bf{\mu }}}}}_{{{{\bf{l}}}}}\) is the lth eigenvector of the Laplacian matrix of the spatial graph and \({n}_{{{\rm {FC}}}}\) is the number of Fourier coefficients. Hence, graph Fourier transform is implemented to transform \({{{{\bf{X}}}}}_{{{{\bf{s}}}}}\) into the frequency domain by:
Subsequently, the newly augmented spotbyfeature matrix is obtained by concatenating SRT gene expression matrix \({{{{\bf{X}}}}}_{{{{\bf{spa}}}}}\) and transformed signal matrix \({\hat{{{{\bf{X}}}}}}_{{{{\bf{s}}}}}\):
Finally, the matrix X_{new} is inputted into SpaGCN as a replacement for the original gene expression matrix to predict the spatial domain cluster labels across all spots.
To evaluate the performance of such modification, 12 human dorsolateral prefrontal cortex of 10x Visium datasets were applied in benchmarking based on annotations from the initial study of SpaGCN^{4}. The adjusted Rand index (ARI) was selected as the evaluation metric to measure the consistency between the predicted spot clusters and manually annotated spatial domain. The parameter num_fcs, which controlled the count of FCs, was determined by utilizing a grid search methodology executed on datasets 151508 and 151670. The search spanned a range of values from 600 to 1400, sampled per 100 steps. Upon analysis, the optimal parameter value was established at 1000 (Supplementary Data 15), while the other parameters were set to the default in SpaGCN. Next, the performance was compared on the 10 remaining datasets for the independent test.
SpaGFT implementation in TACCO
SpaGFT was implemented to improve the performance of TACCO, which leveraged optimal transport (OT) to transfer annotation labels from scRNAseq to spatial transcriptomics data. The core objective function of TACCO is denoted by a cost matrix \({{{{\bf{C}}}}}=(c_{{tb}})\) and a proportion matrix \({{{\bf{\Gamma }}}}=({\gamma }_{{tb}})\):
Specifically, \({c}_{{tb}}\) quantifies the cost that transports an object \(b\) to an annotation \(t\). In TACCO, principal component analysis (PCA) was used to reduce the dimension of scRNAseq and spatial transcriptomics gene expression matrices to the PC matrices by keeping the first 100 PCs, respectively. Subsequently, \({{{\bf{C}}}}\) is computed by calculating the Bhattacharyya coefficients between cell typeaveraged scRNAseq and spatial transcriptomics PC matrices. Finally, the OT’s optimization is solved by using the Sinkhorn–Knopp matrix scaling algorithm to yield a ‘good’ proportion matrix \({{{\bf{\Gamma }}}}\).
For finding \({{{\bf{\Gamma }}}}\), the cost matrix \({{{\bf{C}}}}\) plays the most important role in the OT’s optimization process. Based on the originally calculated \({{{\bf{C}}}}\), an updated cost matrix \({{{{\bf{C}}}}}^{{{{\bf{update}}}}}\) considering spatial topology information is fused. To incorporate this topology information from the spatial data, the coordinates of spatial spots are used to construct a spatial graph, which is as the input with gene expression and initial TACCOcalculated mapping \({{{\bf{\Gamma }}}}\), which represent celltype proportions into SpaGFT for calculating FCs of genes and cell types (CT). Subsequently, these gene FCs matrices were weighted and averaged by spot expression value to obtain the spots’ FCs for obtaining spot level constraints. The cosine distance is calculated between the FCs of spatial spots and the FCs of cell types to create the updated CTspot cost matrix \({{{{\bf{C}}}}}^{{{{\bf{update}}}}}\). The \({{{{\bf{C}}}}}^{{{{\prime} }}}\) is a united cost matrix fused by \({{{\bf{C}}}}\) and \({{{{\bf{C}}}}}^{{{{\bf{update}}}}}\) with a balancing parameter \(\beta\) as
This updated \({{{{\mathbf{C}}}}}^{{\prime} }\) is then fed back into TACCO’s OT algorithm to predict revised cell type proportions for the spatial data. In addition, we used a simulated validation dataset with the setting of \({{{\rm{bead\; size}}}}=5\) to conduct a grid search on the input parameters \(S\), the sensitivity in the Kneedle algorithm from SpaGFT, and \(\beta\) for determining these hyperparameters. While maintaining computational efficiency, we ascertained that the updated TACCO with \(\beta=0.8\) and \(S=24\) can achieve the best performance. Our experiments reveal that the updated TACCO, enriched with SpaGFT features, outperforms the baseline TACCO model in the simulated independent test dataset with the setting of \({{{\rm{bead\; size}}}}\in [10,\,20,\,30,\,40,\,50]\).
SpaGFT implementation in Tangram
Denote \({{{{\bf{X}}}}}_{{{{\bf{sc}}}}}\) as the gene expression matrix of scRNAseq with the dimension \({n}_{{{\rm {cell}}}}\times {n}_{{{\rm {gene}}}},\) in which \({n}_{{{\rm {cell}}}}\) and \({n}_{{{\rm {gene}}}}\) represent the numbers of cells and genes, respectively. \({{{{\bf{X}}}}}_{{{{\bf{spa}}}}}\) is the SRT gene expression matrix with dimension \({n}_{{{\rm {spot}}}}\times \,{n}_{{{\rm {gene}}}}\), and \({n}_{{{\rm {spot}}}}\) represents the number of spots. Tangram aims to find a mapping matrix \({{{\bf{M}}}}={\left({m}_{{ij}}\right)}_{{n}_{{{\rm {cell}}}}\times {n}_{{{\rm {spot}}}}}\), where \(0\le {m}_{{ij}}\le 1\), \({\sum }_{i}^{{n}_{{{\rm {spot}}}}}{m}_{{ij}}=1\) and \({m}_{{ij}}\) reflects the probability of cell \(i\) mapping to spot \(j\). Hence, \({{{{\bf{M}}}}}^{{{{\rm{T}}}}}{{{{\bf{X}}}}}_{{{{\bf{sc}}}}}\) can be treated as the reconstructed SRT gene expression matrix using scRNAseq. Let \({{{{{\bf{X}}}}}_{{{{\bf{re}}}}}{{{\boldsymbol{=}}}}{{{\bf{M}}}}}^{{{{\rm{T}}}}}{{{{\bf{X}}}}}_{{{{\bf{sc}}}}}\). The regularization part of the original objective function of Tangram is as follows:
where the first term describes the cosine similarity of gene \(k\) across all spots in reconstructed SRT gene expression matrix and real SRT gene expression matrix, weighted by \({w}_{1}\); and the second term describes the cosine similarity of spot \(j\) across all genes in reconstructed SRT gene expression matrix and real SRT gene expression matrix, weighted by \({w}_{2}\). By maximizing the objective function, the optimal mapping matrix \({{{{\bf{M}}}}}^{{{{\boldsymbol{*}}}}}\) can be obtained.
Denote \({{{\bf{U}}}}=\left({{{{\bf{\mu }}}}}_{{{{\bf{1}}}}}{{{\boldsymbol{,}}}}\,{{{{\bf{\mu }}}}}_{{{{\bf{2}}}}}{{,}}\,{{\ldots }}{{,}}\,{{{{\bf{\mu }}}}}_{{{{{\bf{n}}}}}_{{{{\bf{FC}}}}}}\right)\), where each \({{{{\boldsymbol{\mu }}}}}_{{{{\bf{l}}}}}\) is the lth eigenvector of the Laplacian matrix of the spatial graph and \({n}_{{{\rm {FC}}}}\) is the number of Fourier coefficients. Hence, we can implement graph Fourier transform for genes by
Therefore, both \({\hat{{{{\bf{X}}}}}}_{{{{\bf{spa}}}}}\) and \({\hat{{{{\bf{X}}}}}}_{{{{\bf{re}}}}}\) are the representations of genes in the frequency domain with the dimension \({n}_{{{\rm {FC}}}}\times {n}_{{{\rm {gene}}}}\). In addition, \({{{{\bf{X}}}}}_{{{{\bf{spa}}}}}^{{{{\prime} }}}{{=}}{{{{\bf{X}}}}}_{{{{\bf{spa}}}}}{{{{\bf{X}}}}}_{{{{\bf{spa}}}}}^{{{{\rm{T}}}}}\) can be considered as the spot similarity matrix calculated by gene expression from real SRT data with dimension is \({n}_{{{\rm {spot}}}}\times {n}_{{{\rm {spot}}}}\). Similarly, \({{{{\bf{X}}}}}_{{{{\bf{re}}}}}^{{{{\prime} }}}=({{{{\bf{M}}}}}^{{{{\rm{T}}}}}{{{{\bf{X}}}}}_{{{{\rm{sc}}}}}){({{{{\bf{M}}}}}^{{{{\rm{T}}}}}{{{{\bf{X}}}}}_{{{{\bf{sc}}}}})}^{{{{\rm{T}}}}}\) represents the spot similarity matrix calculated by gene expression in reconstructed SRT data. In this way, we can implement graph Fourier transform for spots by:
Therefore, both \({\widetilde{{{{\boldsymbol{X}}}}}}_{{{{\boldsymbol{spa}}}}}\) and \({\widetilde{{{{\boldsymbol{X}}}}}}_{{{{\boldsymbol{re}}}}}\) are the new representations of spots in the frequency domain with the dimension \({n}_{{{\rm {FC}}}}\times {n}_{{{\rm {spot}}}}\). Therefore, we improved the objective function of Tangram by adding the similarity measurements of genes and spots in the frequency domain. The new objective function is
where w_{1}weights similarities of genes in the vertex domain; \({w}_{2}\) weights similarities of spots in the vertex domain; \({w}_{3}\) weights the similarities of genes in the frequency domain and \({w}_{4}\) weights the similarities of spots in the frequency domain.
To evaluate the performance of such modification. We adopted the evaluation scheme from Bin Li et al. study. In addition, we simulated this SRT dataset by ‘gridding’ a dataset (STARmap) using various window sizes (400, 450, …, 1200). In addition, simulated datasets of window sizes 400 and 1200 were used for grid search to determine the hyperparameters. In this way, \({w}_{3}\) and \({w}_{4}\) were set to 11 and 1, respectively, and other parameters (including \({w}_{1}\) and \({w}_{2}\)) were the default parameters of Tangram. Our experiments reveal that the updated Tangram, enriched with SpaGFT features, outperforms the baseline Tangram model.
SpaGFT implementation in CAMPA
Overall, the CAMPA framework, a conditional variational autoencoder for identifying conserved subcellular organelle on pixellevel iterative indirect immunofluorescence, was modified by adding an entropy term on its loss function to regularize graph signal (e.g., protein intensity) spreading or concentration. Specifically, compared with the baseline CAMPA loss function, which computed the mean squared error (MSE) loss for each pixel, the modified loss function additionally considered protein global spreading at the cell level.
Data preparation for model training, testing, and validation
Following the baseline CAMPA paper and guidelines^{16}, 292,548 (0.05% of full data) pixels datasets were downsampled from processed cell nuclei of I09 (normal), I10 (Triptolide treatment), I11 (normal), and I16 (TSA treatment) wells based on 184A1 cell line. The training, testing, and validation data were set to 70%, 10%, and 20%, respectively.
Entropy regularization
For cell \(i\in I\), where \(I\) was the complete set of all cells in the downsampled data, the corresponding original protein signatures in each cell were denoted as \({{{{\bf{X}}}}}^{{{{\rm{i}}}}}\) with the dimension \({n}_{{{{pixel}}}}\times {n}_{{{{channel}}}}\), where \({n}_{{{{pixel}}}}\) and \({n}_{{{{channel}}}}\) represented the number of pixels in one cell and the number of proteins, respectively. Similarly, \({\hat{{{{\bf{X}}}}}}^{{{{\rm{i}}}}}\) was denoted as reconstructed protein signatures for cell \(i\). To measure the spreading of reconstructed protein signatures in the frequency domain, \({\hat{{{{\bf{X}}}}}}^{{{{\rm{i}}}}}\) and the coordinates of pixels were input into SpaGFT for computing the FC \({\hat{{{{\bf{F}}}}}}^{{{{\rm{i}}}}}\) with the dimension, in which \({n}_{{{{FC}}}}\) was the number of FC. Denote \({{{\bf{U}}}}=({{{{\bf{\mu }}}}}_{{{{\bf{1}}}}}{{,}}\,{{{{\bf{\mu }}}}}_{{{{\bf{2}}}}}{{,}}\,{{\ldots }}{{,}}\,{{{{\bf{\mu }}}}}_{{{{{\bf{n}}}}}_{{{{\bf{FC}}}}}})\), where each \({{{{\bf{\mu }}}}}_{{{{\bf{k}}}}}\) was the kth eigenvector of the Laplacian matrix of the spatial neighboring graph for cell \(i\). Hence, FCs of reconstructed protein signatures for cell \(i\) was calculated by
Subsequently, \({\hat{{{{\bf{F}}}}}}^{{{{\rm{i}}}}}=({\hat{{{{\bf{f}}}}}}_{{{{\bf{1}}}}}^{{{{\bf{i}}}}}{{{\boldsymbol{,}}}}\,{\hat{{{{\bf{f}}}}}}_{{{{\bf{2}}}}}^{{{{\bf{i}}}}}{{{\boldsymbol{,}}}}\,{{{\boldsymbol{\ldots }}}}{{{\boldsymbol{,}}}}\,{\hat{{{{\bf{f}}}}}}_{{{{{\bf{n}}}}}_{{{{\bf{FC}}}}}}^{{{{\bf{i}}}}})\) was used to calculate entropy by the entropy function, which regularized a concentrated graph signal^{19,74}
where \({\parallel \cdot \parallel }_{2}\) presents \({L}^{2}\)norm.
In addition, the \(\eta\) parameter was used as a weighting term to balance the initial loss function and the entropydecreasing loss function, assigned with 0.3 as default. The formula of the modified loss function \({{{{\rm{L}}}}}_{{{{\rm{modified}}}}}\) was as follows:
where D is a constant, which was used the same as the baseline mode (D = 0.5). The initial decoder loss function was a part of the objective function in CAMPA, which used an analytical solution from \(\sigma\)VAE^{75} to learn the variance of the decoder. The MSE and the logarithm of the variance were minimized through \(\sigma\), which was a weighting parameter between the MSE reconstruction term and the KLdivergence term in the CAMPA objective function. There was an analytic solution to compute the value of \(\sigma\):
\({\sigma }^{*2}\) was estimated value for \({\sigma }^{2}\) and \({{{{\boldsymbol{\nu }}}}}^{{{{\bf{i}}}}}\) presented the estimated latent mean for \({{{{\bf{X}}}}}^{{{{\bf{i}}}}}\).
Regarding the implementation, the training and testing datasets were selected to build the modified and baseline models, respectively. Subsequently, to fairly compare the two models’ training efficiency, the same validation dataset and initial loss were implemented to evaluate the convergence of validation loss.
To interpret the modified CAMPA training efficiency improvement regarding biological perspective, batch effect removal and prediction accuracy were evaluated. Regarding batch effect removal, a proportion of 1% of pixels were subsampled from prepared data. First, UMAP embeddings calculated from the CAMPA latent representations were generated to visualize the mixture of three perturbation conditions. To quantitatively compare the batch effect removal between the baseline and modified model, the kBET^{57} score was computed using the CAMPA latent representations across perturbation conditions. Following the kBET suggestion, 0.5% pixels (~1500 pixels) were iteratively selected for calculating the kBET score (a higher rejection rate suggested a better batch effect removal result) 10000 times using 1–100 neighbors.
Subsequently, the CAMPA latent representations were clustered utilizing the Leiden algorithm^{16} at resolutions of 0.2, 0.4, 0.6, 0.8, 1.2, 1.6, and 2.0. To understand the identity of each cluster predicted by the modified CAMPA under the resolution of 0.2, the protein intensities in each pixel cluster were visualized in the heatmap. Each pixel’s channel values were averaged at the cluster level and scaled by channel (columnlevel) zscore. Clusters were annotated based on the highest expressed markers and human protein atlas.
To evaluate the conserveness and homogeneity of the predicted cluster across different resolutions, we implemented highlabel entropy to quantify the trend of diverging from one cluster into two clusters^{76}. For example, at the resolution of 0.2, all pixels of cluster 6 predicted by the modified model were used to calculate entropy via a probability vector with two lengths. The first element of this vector was a percentage of pixels at the current resolution (i.e., 0.2), which tended to be the largest cluster at the next resolution (e.g., 0.4). The second element of this vector was the percentage of the rest of the pixels at the current resolution, which tended to be other clusters at the next resolution. The highlabel entropy was repeatedly calculated on the same pixels of one cluster within/across baseline and modified model across gradient resolutions (i.e., 0.2, 0.4, 0.6, 0.8, 1.2, 1.6, and 2.0). To visualize intact cells and summarize the relation between pixel and cell in Supplementary Data 19, seven clusters predicted by the modified model based on resolution 0.2 were transferred to all pixels from fullsize data via function project_cluster_data in the CAMPA package. The illustrated examples (id: 367420 and 224081) were extracted to calculate the FC of COIL and SETD1A and visualize.
Reporting summary
Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.
Data availability
All datasets from 10x Visium can be accessed from https://www.10xgenomics.com/products/spatialgeneexpression. SlideDNAseq data is available as accession code SCP1278 in the Single Cell Portal. SlideTCRseq data is available as accession code SCP1348 in the Single Cell Portal. The GSM5519054_Visium_MouseBrain data can be accessed via the GEO database with an accession code GSM5519054. Regarding the human brain dataset, twelve samples can be accessed via endpoint “jhpce#HumanPilot10x” on Globus data transfer platform at http://research.libd.org/globus/. The other six human brain datasets (23AD_Visium_HumanBrain, 28AD_Visium_HumanBrain, T4857AD_Visium_HumanBrain, 25_Visium_HumanBrain, 1864_Visium_HumanBrain, and 11_Visium_HumanBrain) can be accessed via the GEO database with an accession code GSE220442 and https://bmbls.bmi.osumc.edu/scread/stofad2. The two SlideseqV2 datasets are available as accession code SCP815 in the Single Cell Portal. MERFISH data (Slice1_Replicate1Vizgen_MouseBrainReceptor) can be accessed from https://console.cloud.google.com/marketplace/product/gcppublicdatavizgen/vizgenmousebrainmap?pli=1&project=vizgengcpshare. Xenium data (Rep1Cancer_Xenium_HumanBreast) is downloaded from https://www.10xgenomics.com/products/xeniuminsitu/humanbreastdatasetexplorer. SpatialCITEseq data can be accessed via the GEO database with an accession number of GSE213264. Spatial epigenome–transcriptome coprofiling data (spatial_ATAC_RNA_MouseE13) can be accessed via the GEO database with an accession code GSE205055. The 184A1 datasets used to train modified CAMPA reported in this manuscript can be found at https://doi.org/10.5281/zenodo.7299516. SPOT data can be accessed via the GEO database with an accession number of GSE198353. The CODEX tonsil data generated in this study have been deposited in the Zenodo database under accession code 10433896. Source data are provided in this paper. Source data are provided with this paper.
Code availability
SpaGFT is a Python package for modeling and analyzing spatial transcriptomics data. The SpaGFT source code and the analysis scripts for generating results and figures in this paper are available at https://github.com/OSUBMBL/SpaGFT. The source code is also available on Zenodo^{77} with link https://doi.org/10.5281/zenodo.12595086.
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Acknowledgements
This work was part of the PhD thesis of Y.C., who was comentored by Z.L. and Q.M. and was supported by research grants P01CA278732 (A.M. and Z.L.), P01AI177687 (A.M., Y.J., and D.H.B.), R21HG012482 (Ma), U54AG075931 (A.M.), R01DK138504 (A.M.), NIH DP2AI171139 (Y.J.), and R01AI149672 (Y.J.) from the National Institutes of Health. This work was supported by Gilead’s Research Scholars Program in Hematologic Malignancies (Y.J.), Sanofi iAward (Y.J.), the Bill & Melinda Gates Foundation INV002704 (Y.J.), the Dye Family Foundation (Y.J.), and the Bridge Project, a partnership between the Koch Institute for Integrative Cancer Research at MIT and the DanaFarber/Harvard Cancer Center (Y.J.). This work was also supported by the Pelotonia Institute of ImmunoOncology (PIIO). Figure 1a and Supplementary Fig. 1, created with BioRender.com, were released under a Creative Commons AttributionNonCommercialNoDerivs 4.0 International license.
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Conceptualization: Q.M.; methodology: J.L., Y.C., B.L., Q.M.; software coding: J.L., Y.J., and Y.C.; data collection and investigation: Y.C., Q.G., and M. M.; experiment and interpretation: Z.L., D.X., Y.Y.Y., S.J., S.R., G.N., and D.B.; data analysis and visualization: Y.C., Y.J. and J.L.; case study design and interpretation: Y.C., J.L., S.J., J.E.K. and A.M.; software testing and tutorial: J.L., Y.J., and Y.C.; graphic demonstration: Y.C., Y.J., and A.M.; manuscript writing, review, and editing: all the authors.
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S.J. is a cofounder of Elucidate Bio Inc., has received speaking honorariums from Cell Signaling Technology, and has received research support from Roche unrelated to this work. The other authors declare no competing interests.
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Chang, Y., Liu, J., Jiang, Y. et al. Graph Fourier transform for spatial omics representation and analyses of complex organs. Nat Commun 15, 7467 (2024). https://doi.org/10.1038/s41467024515905
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DOI: https://doi.org/10.1038/s41467024515905
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