SPoCA: Spatial Possibilistic Clustering

• SPoCA is an advanced image segmentation algorithm designed to identify active regions and coronal holes in EUV solar images using spatial regularization.
• It extends classical possibilistic clustering with techniques like histogram acceleration and noise stabilization, ensuring efficient and accurate region detection.
• Implemented in the SPOCA-suite, it enables near real-time segmentation and provides reliable annotated catalogs to the Heliophysics Event Knowledgebase.

The Spatial Possibilistic Clustering Algorithm (SPoCA) is an advanced image segmentation methodology developed for delineating and tracking Active Regions (AR) and Coronal Holes (CH) in extreme ultraviolet (EUV) solar images. SPoCA extends classical possibilistic clustering with spatial regularization, enabling robust, automated region identification in the presence of photon noise and instrument artifacts. As implemented in the SPOCA-suite, this framework operates in near real-time on high-resolution data streams, notably from instruments such as SDO-AIA, supplying annotated region catalogs for the Heliophysics Event Knowledgebase (HEK) at a four-hour cadence (Delouille et al., 2012). SPoCA systematically augments Possibilistic C-Means to enforce local spatial coherence, incorporates algorithmic remedies for coincident clustering, and exploits histogram-based acceleration to efficiently segment multi-megapixel solar images.

1. Mathematical Formulation

SPoCA is grounded in a fusion of the classical Fuzzy C-Means (FCM) and Possibilistic C-Means (PCM) clustering paradigms, subsequently regularized to enforce neighborhood coherence. For an image $X = \{x_1, \ldots, x_N\} \subset \mathbb{R}^p$—where each $x_j$ denotes a pixel (intensity or vector)—the algorithm seeks $C$ clusters with prototypes $b_1, \ldots, b_C$ and a fuzzy-possibilistic partition matrix $U = [u_{ij}]$, with $u_{ij} \in [0,1]$ representing the membership of pixel $x_j$ to cluster $i$.

1.1 Fuzzy C-Means Objective

$$J_{\mathrm{FCM}}(B, U) = \sum_{i=1}^C \sum_{j=1}^N u_{ij}^m \|x_j - b_i\|^2$$

subject to $\sum_{i=1}^C u_{ij} = 1$ for all $j$, with $m>1$ as the fuzzifier (in practice, $m=2$).

1.2 Possibilistic C-Means Objective

$$J_{\mathrm{PCM}}(B, U) = \sum_{i=1}^C \left[ \sum_{j=1}^N u_{ij}^m \|x_j - b_i\|^2 + \eta_i \sum_{j=1}^N (1-u_{ij})^m \right]$$

for scale parameters $\eta_i > 0$, with the only constraint that $\max_i u_{ij} > 0$ for each $j$.

1.3 Spatial Regularization (SPoCA Objective)

SPoCA augments $J_{\mathrm{PCM}}$ with a spatial term:

$$J_{\mathrm{SPoCA}}(B,U) = \sum_{i=1}^C \sum_{j=1}^N u_{ij}^m \|x_j - b_i\|^2 + \sum_{i=1}^C \eta_i \sum_{j=1}^N (1-u_{ij})^m + \lambda \sum_{i=1}^C \sum_{j=1}^N \sum_{k \in N_j} w_{jk} |u_{ij} - u_{ik}|^m$$

where $N_j$ is the neighborhood of pixel $j$, $w_{jk}$ are normalized non-negative weights, and $\lambda \geq 0$ controls spatial smoothing.

Key Properties

• $\lambda=0$ recovers PCM.
• Large $\lambda$ enforces spatially smooth memberships across neighboring pixels.

2. Optimization and Update Scheme

SPoCA employs an alternating minimization of $J_{\mathrm{SPoCA}}$ over $U$, $B$, and (optionally) $\eta$.

2.1 Membership Update

The stationarity conditions yield (for $m=2$):

$$u_{ij} = \left[ 1 + \frac{ \|x_j-b_i\|^2 + \lambda \sum_{k \in N_j} w_{jk} (u_{ij} - u_{ik}) }{ \eta_i } \right]^{-1}$$

In practice, the iterative update at each step uses the previous U for the spatial term. For general $m$, expansion incorporates the full nonlinearity of the objective.

2.2 Prototype (Cluster Center) Update

$$b_i = \frac{ \sum_{j=1}^N u_{ij}^m x_j }{ \sum_{j=1}^N u_{ij}^m }$$

2.3 Scale Parameter Update

$$\eta_i = \frac{ \sum_j u_{ij}^m \|x_j - b_i\|^2 }{ \sum_j u_{ij}^m }$$

SPOCA-suite freezes $\eta_i$ as soon as it varies by more than a factor $\alpha$ (default $\alpha = 100$), stabilizing scale assignment.

3. Parameterization and Initialization

Parameter configuration reflects both instrument noise and empirical constraints:

• $m$ fixed at 2 for all runs.
• $(b_i, \eta_i)$ are initialized via a single-pass FCM applied to the image histogram.
• $\lambda$ is adjusted according to image noise: $\lambda \in [0.1, 1.0]$; higher for noisier instruments (e.g., SOHO-EIT), lower for high-SNR data (e.g., SDO-AIA 193 Å).
• $w_{jk}$ are uniform over either the 4- or 8-connected neighborhood.
• Initial cluster centers $b_i^0$ derive from FCM analysis; $\eta_i^0$ from the above ratio.
• For subsequent frames, the median of the prior 10 $b_i$ estimations is used, enhancing temporal smoothness.

4. Algorithmic Enhancements in the SPOCA-suite

Several engineering improvements ensure robust operation in the solar imaging context:

• PCM2 Exponent: To mitigate "coincident clustering" (prototype collapse), the exponent $1/(m-1)$ in the PCM update is replaced by $2/(m-1)$, yielding more compact memberships.
• $\eta$-Constraints: Empirically enforced inequalities such as $\eta_{QS}/\eta_{CH} < b_{QS}/b_{CH}$ prevent dark CH pixels erroneously aggregating into AR classes.
• Histogram Acceleration: Given computational intractability for full $4096 \times 4096$ AIA frames, clustering is performed on the image histogram (bin size $\Delta I=0.01$). Cluster parameters are recovered on the histogram, then memberships are reconstructed over the full image.
• Limb-brightening Correction and Variance Stabilization: Preprocessing includes exposure normalization, Anscombe (square-root) transform for Poisson-noise stabilization, and smooth limb-brightening correction ($I_{smooth}(x,y) = (1-f(\rho))I + f(\rho)I_{corr}$ with $f(\rho)$ ramped from 70–112% $R_{\odot}$), followed by normalization to the on-disk median.

A high-level pseudocode as described in Delouille et al. is:

1. Pre-process image $I(x, y) \to I'(x, y)$.
2. Build histogram $H$ of $I'$.
3. Apply FCM to $H \to$ initial $(b_i, \eta_i)$.
4. Refine with PCM2 + $\lambda$-regularization on $H$.
5. Expand $U$ to full image using closed-form membership.
6. Optionally iterate 4–5 until convergence.
7. Hard assignment of classes ($\arg\max$ of $u_{ij}$ or a custom threshold rule for AR).
8. Post-process segmentation.
9. Region tracking over time (directed graph and overlap).

5. Application to EUV Solar Imaging

SPoCA's principle application domain is segmentation of SDO-AIA EUV images for AR and CH cataloging.

• CH Detection (SPOCA-CH): Employs single-channel (193 Å) FCM on histogram, followed by class assignment via maximum membership. Pixel intensity partitioning results in clustering of dark regions (CH), mid-tones (QS), and high-intensity (AR).
• AR Detection (SPOCA-AR): Utilizes two-channel (171 Å, 193 Å) feature-vector clustering within the PCM2 + regularization framework ($\lambda \approx 0.2$). AR regions are isolated by a QS-membership threshold (e.g., $I_j > b_{QS}$ and $u_{QS,j} < 10^{-4} \implies$ AR).

Image Processing Workflow

• Preprocessing: exposure correction, Anscombe transform, limb-brightening correction (Eq 8, Delouille), histogram normalization.

• Post-processing steps:

  1. Sinusoidal/Lambert projection for foreshortening correction.
  2. Morphological erosion (6 arcsec) to eliminate speckles.
  3. Morphological closing (32 arcsec) to aggregate adjacent regions.
  4. Inverse projection.
  5. Area thresholding: $<3000$ arcsec² (CH), $<1500$ arcsec² (AR) are discarded.
  6. Boundary coding: chain code extraction ($\leq 100$ vertices) via recursive Douglas–Peucker.
  7. Temporal tracking: overlap and split/merge linkage in directed graphs; CH persisting $>72$ h are retained.

6. Performance and Validation

Histogram-based acceleration enables the SPOCA-suite to process full $4096 \times 4096$ images (preprocessing, clustering, post-processing) in under 15 seconds on a modern 8-core CPU. The operational cadence is every 4 hours for SDO/AIA, with catalog updates provided to HEK within 5 minutes of image acquisition.

Empirical validation against manually-drawn CH masks over one year yields an $F_1$-score of approximately $0.80$–$0.85$ for CH boundaries; AR detection exhibits $90\%$ completeness with a $10\%$ false-alarm rate. Example outputs include overlays of AR and CH boundaries on AIA frames and time series of CH filling factors, demonstrating precise spatial adherence and consistent temporal region tracking.

7. Summary and Context

SPoCA represents an overview of possibilistic clustering and spatial regularization, augmented with a series of optimizations specific to EUV solar imaging. The approach leverages the noise robustness of PCM2, enforces local spatial smoothness, and incorporates practical engineering considerations such as $\eta$-constraints, histogram-based scalability, and solar-specific image corrections. This methodology underpins the automated, stable, and fast extraction of active regions and coronal holes from solar data streams, as evidenced by its deployment within the operational SPOCA-suite and integration with the Heliophysics Event Knowledgebase.