HDCluster: High-Degree Graph Clustering for Robust Analysis of Single Molecule Localization Microscopy

Abstract

Clustering is a fundamental task in data analysis: grouping similar objects together into distinguishable subsets. Here, we introduce HDCluster, a novel high-degree graph-based clustering algorithm designed to effectively and rapidly handle various real-world clustering applications, particularly in the context of super-resolution single molecule localization microscopy (SMLM). HDCluster efficiently handles datasets with large and variable numbers of clusters, without requiring prior knowledge of the cluster count, relying on only one parameter. The high speed and efficiency of HDCluster allow it to handle large SMLM datasets with millions of localizations. A comprehensive quantitative comparison against state-of-the-art clustering methods using simulated, public, and real-world datasets demonstrates that HDCluster outperforms other clustering algorithms in terms of time efficiency and clustering performance measures, such as ARI and AMI. HDCluster is particularly robust to noise, making it a promising and effective tool for various clustering tasks in big-data settings, such as SMLM.

INTRODUCTION

Clustering is an unsupervised machine learning task where data samples are grouped according to similarities, ensuring that similar samples fall into the same group while dissimilar ones are separated. Hence, grouping is formed based on intrinsic similarity patterns across the data¹. Data clustering is valuable in domains where ground truth labels are not available or hard to be collected, such as single molecule localization microscopy (SMLM).

SMLM comprises a family of super-resolution imaging techniques, such as Photoactivated Localization Microscopy (PALM)² and Stochastic Optical Reconstruction Microscopy (STORM)³, that circumvent the Abbe diffraction limit of conventional light microscopy via stochastic excitation of fluorophores and fitting of the point spread function (PSF) to localize single fluorophores at precisions of tens of nanometers⁴. DNA points accumulation for imaging in nanoscale topography (DNA-PAINT) utilizes the transient binding of fluorescent DNA strands to achieve both high precision and multiplexing⁵. Minimal photon fluxes (MINFLUX) combines spatially structured illumination with localization analysis⁶ and Resolution Enhancement by Sequential Imaging (RESI) combines DNA-PAINT with sequential imaging of sparse labeling sites⁷ achieving nanometer and sub-nanometer resolution. SMLM data are represented as a point cloud of 2D/3D coordinates of molecular localizations⁸. SMLM data cluster analysis enables quantitative study of nucleic acid organization, protein aggregates, and other cellular structures at the nanoscale^9–11. However, SMLM data exhibit challenges such as noise, variations in density and size, large dataset volumes, and the presence of a high and often unknown number of clusters¹².

Several algorithms have been proposed for data clustering in general, and specifically for SMLM data analysis, based on diverse principles, including distance, density, graph/spectral properties, or other similarity-driven approaches¹². Every clustering algorithm has underlying assumptions and parameters that restrict its universality across applications and data types. For example, the k-means algorithm requires the number of clusters to be specified ahead of time and assumes spherical clusters of similar sizes^1,13. Other clustering methods are sensitive to noise and outliers (e.g., hierarchical clustering), lack scalability to very large datasets in terms of time and space (i.e., memory), and struggle with clusters of varying shape and density^1,13. Here, we introduce HDCluster, a similarity graph-based clustering algorithm, and show that it outperforms competing clustering algorithms and is particularly robust to the challenges of noisy, heterogeneous point cloud-based SMLM data. HDCluster realizes various computational tasks aimed at SMLM data analysis, including reconstructing binding sites, identifying nanoclusters and macromolecular structures, and denoising the data.

RESULTS

1. HDCluster Framework and Computational Tasks

HDCluster is a spatial data clustering method that can be leveraged to find clusters in SMLM data. The HDCluster algorithm has one parameter, merging threshold ($mTh$), used to construct the similarity graph and then get the node degree as a feature to start iterating and merging the nodes to get the clusters. The setting of the $mTh$ parameter is related to the physical characteristics of the clusters, i.e., the size/scale of the sought clusters (Figure 1A). HDCluster finds the clusters after the convergence condition is achieved and returns the per-localization class label, including labels for noisy localizations. $\beta$ is an optional parameter to control noise removal with a default setting that can be changed if needed. See the Methods Section for more details about the algorithm.

Figure 1. Overview of the HDCluster framework to cluster, denoise, and reconstruct the emitters/binding sites in simulated and DNA-PAINT data.

(A) Illustration of the HDCluster algorithm to obtain data clusters as well as the centroids (the consensus localizations that approximate the true emitter positions by computing the mean of means) of the extracted clusters. HDCluster is an iterative algorithm that starts from the similarity graph constructed based on the $mTh$ parameter. The degrees of the graph nodes are calculated. The nodes are sorted in descending order of node degree. The algorithm starts iterating from the node with the highest degree. All the nodes that share edges with this high-degree node are merged; the mean (in terms of average coordinates for the nodes’ spatial locations) of these merged nodes is calculated and all these merged nodes are marked as visited nodes. After that, the algorithm moves to the next highest degree node in the list, with at least one node that is marked as not visited, and so on until all the nodes are marked as visited. After concluding this first round of iterations, HDCluster enters another round by constructing an updated graph based on the mean nodes that were produced from the node merging of the first round of iterations. The algorithm will start iterating again from the mean node with the highest degree, as it did in the first round, and keep iterating and producing the mean of means until the convergence condition is reached. The convergence condition is reached when the mean nodes have no edges (i.e., the degree of the top-level mean nodes equals zero). Every one of the high-level mean nodes will be considered as a top-level cluster ID and will be used to backtrack the cluster labels to their means and so on down to the nodes that the means come from. The algorithm keeps propagating/backtracking the labels from the top-level mean node down to the nodes, and all the nodes will get an ID or label based on the top-level mean node. The algorithm does so for the rest of the top-level mean nodes. At the end, all the nodes get labeled and have membership in one and only one cluster with a unique ID. The top-level mean nodes are the emitters (consensus localizations that approximate the true emitter positions) of the nanoclusters in SMLM data. HDCluster is capable of filtering out the noisy localizations in addition to data clustering. (B) A general framework of SMLM data analysis and the computational tasks that HDCluster can perform for SMLM data analysis. (C) Quantitative analysis (ARI, AMI, #clusters) of clustering methods evaluated on the benchmark shape dataset¹⁹ presented in Supp. Fig. 1. The best parameter(s) for each method are selected based on the AMI measure.

Figure 1B illustrates HDCluster capabilities, encompassing various computational tasks for SMLM data analysis, such as (1) emitter reconstruction (or binding sites reconstruction¹⁴), (2) finding true biological clusters and nanoclusters in the SMLM data, and (3) denoising SMLM data.

2. HDCluster Performance Comparison with Other Clustering Methods

To evaluate clustering performance and tune the algorithm’s parameters, we used the adjusted Rand index (ARI)¹⁵ and the adjusted mutual information (AMI)¹⁶ as performance metrics. ARI measures similarity between predicted and ground truth labels and AMI captures the amount of shared information between clusters, corrected for chance. AMI is a more robust measure when the number of clusters varies and accounts for bias toward certain cluster size distributions^17,18.

We applied HDCluster and other commonly used clustering methods to the shape benchmarking dataset¹⁹, widely used for assessing clustering methods’ performance. This data set of 2D points and their corresponding membership as ground truth (GT) labels includes blob-like and spherical clusters similar to SMLM data of biological clusters: Aggregation with 7 clusters, Flame with 2 clusters, R14 with 14 clusters, and D31 with 31 clusters (Supp. Figure 1A). We tested state-of-the-art clustering methods, including Border Peeling (BP)²⁰, Quasi-Cluster Centers (QCC)²¹, Robust Continuous Clustering (RCC)²², Density Peaks (DP)²³, mean-shift (MS)²⁴, and Density-Based Spatial Clustering of Applications with Noise (DBSCAN)²⁵, as well as HDCluster, on these benchmark datasets (Supp. Table 1 shows the number of parameters and other information for the studied methods). We tuned each method’s parameters based on AMI. Supp. Figure 1A shows the qualitative results and Figure 1C summarizes the quantitative results when applying the clustering methods to the shape benchmarking dataset. Only HDCluster and DP retrieve the correct number of clusters across all tested datasets. However, DP shows poor performance on the Flame dataset (ARI of 33% and AMI of 41%), where HDCluster achieves much higher accuracy (92% ARI and 86% AMI). Notably, HDCluster outperforms DBSCAN on the D31 and R15 datasets, both of which mimic nanocluster structures observed in SMLM data.

Nieves et al.²⁶ introduced a standard benchmarking framework for comparing clustering algorithms on SMLM point cloud data. Their work includes generating simulated ground truth point patterns with realistic blinking and localization uncertainty. The dataset comprises 10 distinct scenarios/conditions (variations in density, cluster size, blinking behavior, background noise, etc.) with and without multiple blinking artifacts. Each scenario contains 50 different point cloud files. They tested and compared the performance of several clustering methods, which include: DBSCAN²⁵, topological mode analysis tool (ToMATo)²⁷, kernel density estimation (KDE)²⁸, fast optimized clustering algorithm for localizations (FOCAL)²⁹, cluster analysis by machine learning (CAML)³⁰, ClusterViSu³¹, and SR-Tesseler³². The methods’ parameter spaces are scanned systematically under each condition. In their results, DBSCAN, ToMATo, and KDE generally perform well across multiple conditions and are relatively robust, with DBSCAN showing superiority. The other methods, ClusterViSu, SR-Tesseler, and FOCAL, show weaker performance in many settings. All methods suffer performance drops under multiple blinking localization conditions. We assessed HDCluster using the benchmarking SMLM datasets in accordance with the methodology outlined by Nieves et al.²⁶. All experiments were performed with the denoising feature activated. We tuned the key parameters of HDCluster, mTh and the optional filtering parameter $\beta$, to maximize the average AMI of the 50 files of each scenario.

For performance comparison, we selected DBSCAN, the most effective method identified by Nieves et al.²⁶, and employed their reported parameters to ensure fairness. The AMI and ARI values for HDCluster and DBSCAN are presented in Supp. Table 2. Across all simulated scenarios, HDCluster outperforms or matches DBSCAN in both ideal (i.e., no added blinking) and multiple blinking scenarios. In the ideal scenarios, both methods achieve comparable results, with HDCluster showing slightly higher ARI and AMI values in most cases (e.g., Scenario 3: 0.90 ± 0.017 vs. 0.84 ± 0.017 for ARI). However, when multiple blinking is introduced, HDCluster demonstrates greater robustness, maintaining high average clustering accuracy (average ARI ≈ 0.57 and AMI ≈ 0.68) compared to DBSCAN, whose performance drops noticeably (average ARI ≈ 0.51 and AMI ≈ 0.62). Under multiple-blinking (the harder, SMLM-like condition), HDCluster shows a clear advantage of about +0.06 absolute difference in both ARI and AMI (≈ 9.7–11.8% relative improvement). This highlights HDCluster’s ability to efficiently handle emitter counting and multiple blinking artifacts typical in SMLM data.

3. HDCluster Robustness to Noise and Scalability

To test the ability of the different clustering methods to handle noisy data, we generated datasets that comprise clusters with various shapes and densities with additive noise points as described in Methods Section 4. Figure 2A shows one experiment for the generated clusters concatenated with a noise level of 20% of the total clustered points drawn from a uniform distribution. DBSCAN and HDCluster efficiently handle the noisy data and extract clusters, while QCC, DP, RCC and MS are unable to accurately segment noise from clusters. We varied the noise level from 0% (clusters only) to 100% with a step size of 10% and compared DBSCAN, BP and HDCluster at each noise level (Figure 2B). The parameter(s) for DBSCAN and HDCluster are selected based on ground truth data and the AMI score, as shown in Figure 2C, which shows a single case where the noise level is 100%. DBSCAN and HDCluster are robust to high noise levels; for BP, AMI and ARI are progressively reduced as the noise level exceeds 40% (Figure 2B).

Figure 2. Noise robustness analysis of various clustering methods at different noise levels. Runtime comparison of the clustering methods based on the simulated and real SMLM dataset.

(A) Simulated 3D data of 80 Gaussian clusters with various densities and ellipsoidal shapes is generated with various noise levels. In this simulation, we show the data with a 20% noise level of the total number of localizations. The various clustering methods have been applied to find the clusters and label the noisy localizations. For the parametric methods, the method’s parameter(s) are tuned based on the AMI. (B) Comparison of the clustering methods with the denoising capability for various noise levels. Some methods can handle noise but are not robust to high noise values in the data. (C) Parameter selection for the DBSCAN and HDCluster methods when applied to data with a 100% noise level. Both ARI and AMI measures were plotted for both methods at various parameters. The parameters with high AMI value were selected to report the results presented in (B). (D) Calibrate the runtime for the various clustering methods when applied to the simulated dataset presented in (A). All the methods were run on the same computer and the same dataset for fair comparison. Not all the methods can be used on large datasets. (E) Runtime analysis for the methods that can be run on large datasets, such as SMLM data. HDCluster, DBSCAN, and mean-shift (MS) algorithms were run on a dataset of Caveolin-1 localizations³³ presented in the Supp. Figure 2. We start running the clustering methods on 10% of the whole data and calibrate the runtime, then we increase the localizations by 10% until we reach 100% of the total localizations. Not all the methods scale with the dataset size. HDCluster* refers to the version of the algorithm with the noise-handling option enabled.

To assess the speed of the clustering algorithm for SMLM data, we applied the clustering methods to the same datasets and on the same hardware (Intel(R) Core(TM) i7-4790 CPU @ 3.6 GHz. Memory 32 GB, without other processes/applications running during testing). We first calibrated the running time for each algorithm on the small dataset from Figure 2A. Some of the algorithms are implemented in Python, while others are implemented in MATLAB. In our comparison, we used the latest optimized implementation of DBSCAN in MATLAB 2024b. HDCluster is the fastest algorithm, retrieving clusters in ~0.046 seconds, with DBSCAN second followed by MS, DP, RCC, and QCC, and finally the BP algorithm (Figure 2D). We then applied the algorithms to a real SMLM dataset of Caveolin-1 labeled HeLa cells of more than 1.5 million localizations³³ (Supp. Figure 2).

Only HDCluster, DBSCAN and MS were able to handle the data on our testing workstation (Intel(R) Xeon(R) CPU E5–2630 v3 @ 2.40 GHz. Memory 94 GB) with the other algorithms crashing the computer and failing to complete. We calculated the runtime for MS, DBSCAN, and HDCluster starting with 10% of total localizations, progressively increasing the sample size to the total number of localizations (Figure 2E). The runtime for MS grew exponentially with dataset size, and we stopped the analysis after reaching 30% of the 1.5 million localizations. For DBSCAN and HDCluster runtimes were calculated for increments up to the full data size (i.e., 100%, ~1.5 million localizations). At 100% of the total data, HDCluster is >55 times faster than DBSCAN, and we expect the trend to continue as the dataset becomes larger (Figure 2E).

4. HDCluster Binding Site Reconstruction Outperforms DBSCAN

Binding site and emitter reconstruction and protein counting are important tasks in SMLM data analysis^14,34. To test whether the representative (or consensus) localizations of HDCluster iterative merging approximate true emitter positions, we analyzed SMLM data from the Gatta-PAINT 80nm nanoruler that has three binding sites (emitters) organized linearly with a spacing of 80nm (Supp. Figure 3A). Calculation of the 1st–5th nearest neighbor (NN) distances calculated for HDCluster reconstructed emitters, shows that the majority of emitters are within ~80nm. HDCluster was then applied to the DNA origami nanostructure dataset published recently for the SPINNA method that compares experimental single-protein NN distances with simulated data generated from user-defined protein oligomerization models³⁴. HDCluster was used to reconstruct the binding sites for linear and triangular DNA origami nanostructures with three binding sites spaced ~15nm (Supp. Figure 3B). We used $mTh=0.04\times \mathit{\text{pixelSize}}$ (i.e., $mTh=5.2nm$), where the pixel size is 130nm. For linear origami, HDCluster effectively reconstructs the binding sites for the two origami structures and calculated NN distances, where the first peak is at ~15nm and a second peak at ~30nm for linear origami and ~15nm for triangular nanostructures.

We then compared the binding site reconstruction performance of HDCluster and DBSCAN using a dataset of 3 × 4 origami grids arranged at 20nm spacing (Figure 3A)³⁵. HDCluster was applied with $mTh=0.06\times \mathit{\text{pixelSize}}$ (i.e., $mTh=7.8nm$ for a pixel size of 130nm) and evaluated against the results reported by Stein et al.³⁵, who analyzed NN of DBSCAN cluster centers using the same parameters reported in their study: $ϵ=2\times {\sigma }_{\mathit{\text{NeNA}}}=2\times 4.2=8.4nm$ and $\mathit{\text{MinPts}}=6$. Calculations of the 1st–5th NN distance show that the peaks are sharper and clearer for HDCluster relative to DBSCAN, with the highest peak at 20nm (peak (a) in Figure 3B), the second highest peak at 28nm (peak (b)) and so on (Figure 3B). The zoomed-in regions show superior clustering performance when using HDCluster over DBSCAN. With DBSCAN, some of the binding sites are filtered out, some merged and some averaged, affecting counting and the subsequent analysis of NN values. Indeed, the frequency of occurrences is more representative for HDCluster than DBSCAN; the expected 17 occurrences of peak (a) vs 8 occurrences of peak (d) in the 3 × 4 origami grids are accurately reported by HDCluster but not DBSCAN, which shows a similar frequency of events for peaks (a) and (d). Overall, HDCluster outperforms DBSCAN at various benchmarking clustering tasks and is significantly faster.

Figure 3. Qualitative and quantitative comparison of binding sites reconstruction and quality of clustering when using DBSCAN and HDCluster when applied to 3 × 4 DNA origami dataset³⁵.

(A) The quality of clusters impacts the binding sites’ subsequent analysis, as in the nearest neighbor (NN) analysis. We adopt the same DBSCAN parameters used in Stein et al.³⁵ to find the clusters. DBSCAN can find the clusters, but when the clusters overlap, it might get them wrong. On the other hand, HDCluster can easily be adopted with one parameter to find more accurate clusters and consequently the binding sites. (B) Quantitative analysis of the reconstructed binding sites from both DBSCAN and HDCluster methods. The NN analysis for the binding sites from the 3 × 4 DNA 20 nm origami grids shows that the histogram peaks correspond to the geometrical grid of sites that distant 20 nm apart for both DBSCAN and HDCluster methods. However, the HDCluster is more realistic, while DBSCAN has limitations for the overlapped clusters and is sensitive to noisy points.

5. Conclusion

HDCluster is a simple, scalable, one-parameter clustering method that is very fast and robust to noise. By combining density-based clustering with graph-based techniques, HDCluster provides robust and accurate cluster identification in SMLM datasets, even with very large datasets of hundreds of thousands or millions of DNA-PAINT localizations, which allows for precise emitter reconstruction.

METHODS

1. HDCluster Algorithm

HDCluster is an unsupervised algorithm for spatial data clustering. It is an iterative similarity graph-based clustering method that discovers clusters and estimates their centroids, which are referred to as consensus localizations that approximate the true emitter positions. Figure 1A illustrates the algorithm based on 2D data points and details the underlying methodology. However, the algorithm works for 3D spatial data as well. The one parameter that the user needs to choose to run the algorithm is the “merging threshold ($mTh$)”. The algorithm begins by constructing a similarity graph based on the spatial proximity parameter $mTh$ and calculating the node degree. Nodes are sorted based on the degree in descending order, so the algorithm iteratively merges each high-degree node with its connected neighbors, computing their mean spatial coordinates. The algorithm keeps iterating until all nodes are visited and produces a set of mean nodes that represent preliminary cluster centers. In the next stage, HDCluster reconstructs a new graph using the mean nodes and repeats the merging process across multiple iterations until the convergence condition is reached, where no edges remain between mean nodes, which is an indication that clusters are fully separated. Each top-level mean node is then assigned as a unique cluster ID, and labels are backtracked from these centroids to the original nodes such that every localization/point belongs to one cluster only. The resulting top-level mean nodes represent the emitters of the extracted nanocluster/cluster. Additionally, HDCluster can incorporate a denoising step to filter out low-degree nodes, effectively removing noisy/background localizations (see Methods Section 3 for a more extensive discussion of filtering options).

The $mTh$ parameter selection is based on the physical characteristics of the underlying clusters, and one of the heuristics to set this parameter is based on Ripley’s H-function (see Methods Section 2 for a more extensive discussion of $mTh$). However, it can be selected based on the application and the spread of the clusters, as illustrated in Supp. Figure 4 and Equation (2). $\beta$ is an optional parameter to control the noise removal with a default setting that can be changed if needed (see Methods Section 3 for a more extensive discussion of $\beta$).

2. HDCluster Merging threshold (mTh) and Evaluation Measures

Merging threshold ($mTh$) in HDCluster is a spatial proximity parameter that governs the similarity graph construction and guides the iterative merging of nodes toward the convergence of cluster means of consensus localization. Selection of $mTh$ depends on the intrinsic properties of the dataset and the physical characteristics of the clusters, such as their spatial extent or point dispersion. Also, it can be determined by H-Ripley’s function or empirically through a data-driven heuristic to balance cluster separation and cohesion.

To demonstrate how we can set the $mTh$ parameter, we simulated various scenarios of neighboring emitters (Supp. Figure 4A) with varying separation distance, defined as the space between the centers of the clusters. $\sigma$ is the standard deviation of the Gaussian that quantifies the spread of the localizations/points around each cluster center. We derived the overlapping distance as a function of $\sigma$ and separation distance to quantify the extent of overlap between the clusters, with negative values reflecting the separation between the boundaries of non-overlapping clusters (Equation (1)). The parameter $mTh$ can be set according to Equation (2). We also define three measures of error: counting error ($\mathit{\text{error\_C}}$), localization error ($\mathit{\text{error\_L}}1$), and localization error ($\mathit{\text{error\_L}}2$) (Equations (3–5)).

To assess the applicability of the HDCluster in reconstructing the emitters, we simulate various scenarios that may be similar to real-world situations, where the molecules can be sparse or very dense. We simulated two Gaussian emitters and change two parameters, (1) the separation distance (i.e., the distance between the centers/means of the two Gaussians or the emitters, which we use to simulate the density of the molecule in larger grids) and (2) the standard deviation of the Gaussians $\sigma$, that simulates the localization precision, i.e., the uncertainty in detecting the position of that emitter across repeated localizations, as shown in Supp. Figure 4A. The overlapping distance between the two clusters is defined as the distance between the borders of the clusters. The overlap between the localizations of the emitters is calculated according to Equation (1). The overlapping distance is a function of $\sigma$ and separation distance. In the simulation, we can change the separation distance and $\sigma$ to compute the amount of overlap between the neighboring clusters. The overlap is negative when the localizations of the neighboring emitters are not overlapping, and the overlap is positive when the localizations of the neighboring emitters are overlapping. The overlap equals zero when the borders of the two clusters are touching each other (Supp. Figure 4A).

$$\mathit{\text{Overlap}}=-(\mathit{\text{Separation}}-2\sigma )$$ Eq(1)

Where $\sigma$ is the standard deviation of the Gaussian-generated data.

The HDCluster parameter, merging threshold ($mTh$), can be selected to reconstruct the emitters’ locations based on Equation (2).

$$mTh=\left\{\begin{array}{cc}2\sigma & :\mathit{\text{Overlap}}\le 0\\ \left|2\sigma -\left|\frac{\mathit{\text{Overlap}}}{2}\right|\right|& :\mathit{\text{Overlap}}>0\end{array}\right.$$ Eq(2)

In Equation (2), when the neighboring clusters are well separated or their borders touch each other (i.e., $\mathit{\text{Overlap}}\le 0$), the parameter $mTh=2\sigma$ ensures appropriate merging within each cluster. However, as clusters become closer and their borders overlap (i.e., $\mathit{\text{Overlap}}>0$), $mTh$ should be reduced to prevent unintended merging of adjacent overlapping clusters (i.e., $mTh$ can be safely adjusted to $mTh=\left|2\sigma -\left|\frac{\mathit{\text{overlap}}}{2}\right|\right|$). This approach aligns with adaptive thresholding strategies used in hierarchical and graph-based clustering methods, where thresholds decrease as clusters become denser or closer.

More formally, we simulated 3D grids of size 3 × 3 × 3 and we changed the separation distance and the standard deviation of the Gaussians $\sigma$. Moreover, we also changed the density of the localizations around everyone of the simulated emitters as shown in Supp. Figure 4B. HDCluster is used to reconstruct the emitters. The parameter $mTh$ is selected according to Equation (2). To assess the quality of emitter location reconstruction, we introduced three measures, as we have the ground truth location for the emitters. We quantitatively calculated three types of errors: $\mathit{\text{error\_C}}$ Equation (3), $\mathit{\text{error\_L}}1$ Equation (4), and $\mathit{\text{error\_L}}2$ Equation (5).

$$\mathit{\text{Error\_C}}=\frac{1}{G}\times (R-G)\times 100\%$$ Eq(3)

where $G$ is the number of ground truth (GT) emitters and $R$ is the reconstructed emitters or binding sites. In the $\mathit{\text{error\_C}}$, positive values mean overcounting, negative values mean undercounting, and zero means no counting error.

$$\mathit{\text{Error\_L}}1\left(nm\right)=\frac{1}{G}{\sum }_{i=1}D{1}_i$$ Eq(4)

where $D{1}_i$ is the distance between the GT emitter $i$ and the location of the nearest reconstructed emitter. $\mathit{\text{Error\_L}}1$ is normalized by the number of ground truth emitters $G$.

$$\mathit{\text{Error\_L}}2\left(nm\right)=\frac{1}{R}{\sum }_{i=1}D{2}_i$$ Eq(5)

where $D{2}_i$ is the distance between the reconstructed emitter $i$ and the location of the nearest GT emitter. $\mathit{\text{Error\_L}}2$ is normalized by the number of reconstructed emitters $R$.

We generated a larger 3D 3 × 3 × 3 grid of simulated emitters and localizations, where the localizations’ membership and the emitters’ location are considered as a ground truth (GT) for evaluation via the error measures (Supp. Figure 4B). We generated 20 grids for each pair of $\sigma$ and separation distance and computed the three types of counting (i.e., $\mathit{\text{error\_C}}$) and localization errors (i.e., $\mathit{\text{error\_L}}1$ and $\mathit{\text{error\_L}}2$). We started with a low density of 10 localizations per emitter and increased the density to 35 localizations per emitter. We notice that the counting error ($\mathit{\text{error\_C}}$) is improved (Supp Figure 4B, left vs right), indicating emitter reconstruction is much better for a higher density of localizations. As $\sigma$ decreases (i.e., more precise localizations), the 3 types of error become smaller (Supp. Figure 4B). For example, when using $\sigma =2$, for any separation distance ≥10nm, and for a number of localizations 35 per emitter, the $\mathit{\text{error\_C}}$ is zero, and both the $\mathit{\text{error\_L}}1$ and $\mathit{\text{error\_L}}2$ are almost zeros (i.e., the location of the reconstructed emitters is almost identical to the ground truth location). The error graphs show that the reconstruction errors remain relatively low, even under challenging conditions where nearby emitters exhibit substantial overlap. In the worst cases, $\mathit{\text{error\_L}}1$ and $\mathit{\text{error\_L}}2$ stay below ~30nm and 14nm, respectively. $\mathit{\text{Error\_L}}1$ and $\mathit{\text{error\_L}}2$ fluctuate due to overcounting or undercounting, specifically in challenging cases of highly overlapping or low-density nanoclusters.

3. HDCluster Denosing Parameter - Spatial Filtering Option

HDCluster incorporates a denoising option to filter out noisy localizations (background, outlier points, etc.). The denoising capability/function in HDCluster can be enabled or disabled by the user. If enabled, the user can set a $\beta$ value or rely on the default $\beta$ value. The embedded filtering works by calculating the degree for all the nodes, then labeling all the nodes with a degree below a low degree with a 0 label (i.e., noisy localizations). The $\mathit{\text{low\_degree}}$ value is calculated as shown in Equation (6).

$$\mathit{\text{Low\_degree}}=\beta \times \mathit{\text{mean}}\left(\mathit{\text{degree}}\right),$$ Eq(6)

where mean(degree) is the average node degree that is calculated when the graph is constructed with the $mTh$ parameter, $\beta$ which is a scalar that controls the denoising process. As a rule-of-thumb, the default value $\beta =0.2$ is selected to label the non-clustered localizations as noisy. However, the user can select other $\beta$ values that might control the noise removal according to the data, where larger $\beta$ values employ harsh filtering. $\beta$ is a scaling factor that controls the minimum node degree relative to the average connectivity of the graph, which determines how aggressively HDCluster removes low-connectivity nodes (potential noise). Setting $\beta =0.2$ ensures that only nodes whose connectivity <20% of the mean graph degree are labeled as noise and generalizes well across different datasets without prior tuning. Setting $\beta$ balances between retaining true cluster points at the periphery and filtering isolated/background points that do not contribute meaningfully to the cluster structure. We provided the user with the ability to change $\beta$. Higher values (i.e., >0.2) can lead to harsher filtering, which can fragment clusters and remove localizations within low-density regions.

4. Generating Data with Various Shapes and Densities with Additive Noise

To test the robustness of the clustering methods to noise, we generated a dataset with $K=80$ Gaussian clusters that varied in density and were randomly positioned in 3D space. Each cluster has a minimum of 10 points and a maximum of 100 (for various-density clusters), and of various shapes, varying the covariance matrix diagonal of the Gaussians as given in Equation (7).

Let each cluster $i\in \{\mathrm{1,2},\dots ,K\}$ be defined by:

• Mean ${\mu }_i={\left[{\mu }_{ix},{\mu }_{iy},{\mu }_{iz}\right]}^T\in R^3$

  $${\mathbf{\Sigma }}_i=\left[\begin{array}{ccc}{\sigma }_{ix}^2& 0& 0\\ 0& {\sigma }_{iy}^2& 0\\ 0& 0& {\sigma }_{iz}^2\end{array}\right]$$
• covariance matrix (diagonal):

  $$\text{each cluster}i\text{contains}{n}_i\text{points}:x_{ij}~N\left({\mu }_i,{\Sigma }_i\right),j=1,\dots ,n_i$$ Eq(7)

Sampling procedure:

1. For each cluster $i\in \{\mathrm{1,2},\dots ,K\}$, randomly sample the mean:

   $${\mu }_i~\mathit{\text{Uniform}}\left(\left[x_{\mathit{\text{min}}},x_{\mathit{\text{max}}}\right]\times \left[y_{\mathit{\text{min}}},y_{\mathit{\text{max}}}\right]\times \left[z_{\mathit{\text{min}}},z_{\mathit{\text{max}}}\right]\right)$$
2. randomly sample diagonal variances:

   $${\sigma }_{ix}^2,{\sigma }_{iy}^2,{\sigma }_{iz}^2~\mathit{\text{Uniform}}\left({\sigma }_{min}^2,{\sigma }_{max}^2\right),\text{where}{\sigma }_{min}=1\text{and}{\sigma }_{max}=21$$

from the diagonal covariance matrix ${\Sigma }_i$ as shown above. Sample $n_i$ points from the 3D Gaussian distribution $x_{ij}~N\left({\mu }_i,{\Sigma }_i\right)$. Notice that the off-diagonal elements are all zero, so the clusters are axis-aligned ellipsoids. The variance values control the spread along each axis independently to get different shapes. We simulated the irregular cluster sizes by drawing $n_i$ randomly (i.e., uniform distribution) to vary the clusters’ densities. We then varied the noise by controlling the noise level, where the noise level can be selected for every experiment and drawn from a uniform distribution.