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# Internal Clustering Validation on Riemannian manifolds for PolSAR Data Analysis

Gabriel Vasile, Marco Congedo

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Gabriel Vasile, Marco Congedo. Internal Clustering Validation on Riemannian manifolds for PolSAR Data Analysis. IGARSS 2025 - IEEE International Geoscience and Remote Sensing Symposium, Aug 2025, Brisbane, Australia. pp.4. [hal-05357968](https://hal.science/hal-05357968v1)

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INTERNAL CLUSTERING VALIDATION ON
RIEMANNIAN MANIFOLDS FOR POLSAR
DATA ANALYSIS

Gabriel Vasile

Univ. Grenoble Alpes, CNRS

GIPSA-lab, Grenoble, France

gabriel.vasile@grenoble-inp.fr

Marco Congedo

Univ. Grenoble Alpes, CNRS

GIPSA-lab, Grenoble, France

marco.congedo@grenoble-inp.fr

*Abstract*—Unsupervised clustering methods play a vital role in extracting information from PolSAR data, both for analyzing polarimetric signatures and identifying geospatial scenes of interest. This paper introduces three Riemannian extensions of internal clustering validation indexes for PolSAR data analysis. Combined with a fourth index presented in [1], the proposed analysis is illustrated using two airborne PolSAR datasets and the PolBaRi+Riemannian k-means clustering method to determine the optimal number of clusters. Considering the affine invariant Riemannian metric, these indexes can be directly applied to PolSAR coherency/covariance matrices or PolSAR Hermitian factors derived from polar decomposition.

*Index Terms*—Unsupervised clustering, PolSAR data, internal validation indexes, Riemannian manifold.

I. INTRODUCTION

One important application in PolSAR data analysis is classification. While both supervised and unsupervised methods are popular and have various levels of accuracy, we focus here only on the case of unsupervised classification (clustering).

Clustering methods can be categorized into several types, such as partitional, hierarchical, density-based, grid-based, or model-based approaches. Among these, centroid-based methods within the partitional category are the most commonly used for PolSAR unsupervised classification.

For example, the introduction of the Wishart classifier [2, 3] marked a significant milestone in PolSAR clustering. It has been demonstrated to function as an optimal Bayesian classifier, assuming that scattering vectors are modeled as zero-mean complex circular Gaussian vectors, fully described by their covariance matrix. The associated similarity measure is referred to as the Wishart “distance.”

The centroids are initialized using the $H-\alpha$ decomposition, which serves as a preliminary step [4]. This provides an initial estimate for each class centroid in the $H-\alpha$ plane, setting the number of clusters to $K = 8$. Following each iteration, the centroids are updated by complex multilooking the redistributed matrices.

With the increase in spatial resolution, various non-Gaussian clustering strategies have been adopted for PolSAR data analysis. Building on the conventional product model, two primary approaches to incorporating heterogeneity can be

identified. These involve either using non-Gaussian target vectors (e.g., K-, Kummer-U, or $G_0$-distributed clutter models) or employing compound covariance /coherency models (e.g., scale mixtures, $G_0$- or K-Wishart models), leading to the development of various classification algorithms [5–10].

Despite these advancements, the Wishart classifier remains one of the most widely used PolSAR clustering methods in practical applications due to its simplicity and effectiveness. Its significance has been demonstrated in numerous studies, which have proposed new or improved versions by modifying one or more stages of the general framework. Key enhancements include adjustments to initialization and the number of clusters [11–13], class assignment logic [12], and the distance metric [14].

The PolBaRi+Riemannian k-means clustering method introduced in [14, 15] presents a novel coherent approach that operates directly on the original scattering matrix using the polar decomposition. Unlike incoherent decomposition methods, which require data to be vectorized, this technique retains the matrix format. The polar decomposition is applicable to both symmetric and asymmetric scattering matrices, with the Hermitian factor being uniquely defined.

The proposed PolSAR processing method, termed “Riemannian multilooking,” shares similarities with incoherent techniques by incorporating spatial averaging on the Riemannian manifold to compute the centroids of Hermitian factors. This enables the method to handle distributed targets within a scene effectively. Computations are performed within the native Riemannian manifold of positive-definite matrices, leveraging the Affine Invariant Riemannian Metric (AIRM) to assess intercluster separation.

Class centers are initialized using k-means++ seeding based on the AIRM distance. Subsequently, each barycenter matrix is assigned to one of the $K$ clusters, and the cluster centers are updated iteratively. This process continues until the interclass transfer falls below a predefined threshold.

Given the geometrical nature of the PolBaRi+Riemannian k-means, an objective criterion for data-driven evaluation of the clustering result has been introduced [1] by adjusting the Calinski-Harabasz (variance ratio) index with the AIRM
metric. This index has been employed in order to retrieve the optimal number of classes for a subset of the PolSAR dataset.

Together with the Calinski-Harabasz index, this paper introduces the Riemannian extensions of three additional indexes for internal clustering validation with PolSAR data (either covariance / coherency matrix or Hermitian factor). Two airborne PolSAR datasets are then used to illustrate which index can be effectively used to select the optimum number of classes.

The remainder of this paper is organized as follows. Section II briefly defines the polar decomposition and the associated AIRM distance. Section III introduces three additional indexes for internal clustering validation on a Riemannian manifold. Section IV analyses the experimental results. Finally, the conclusions are discussed in Section V.

## II. RIEMANNIAN MANIFOLD EMBEDDING WITH POLSAR DATA

Any complex square matrix $S \in \mathbb{C}^{n \times n}$ can be decomposed using the polar decomposition as the product of two factors: a unitary matrix ($U, UU^H = I, U \in \mathbb{C}^{n \times n}$) and a Hermitian matrix ($H, H^H = H, H \in \mathbb{C}^{n \times n}$). $H$ is positive semi-definite (PSD), thus we have $v^H H v \ge 0$ for any nonzero column with complex elements vector, $v \in \mathbb{C}$. In any unitarily invariant norm, $U$ is the nearest unitary matrix to $S$ [16]. The impact of the two factors can be understood as follows: the unitary factor induces a rotation, while the Hermitian factor applies stretching or deformation.

In PolSAR, the polar decomposition is commonly employed as a coherent technique for feature extraction from the scattering matrix, $S \in \mathbb{C}^{2 \times 2}$. With no restrictions on its application, this factorization can be utilized for both symmetric and asymmetric, as well as monostatic and bistatic scattering matrices.

In the remainder of this paper, we utilize the polar decomposition of the scattering matrix but focus the clustering process exclusively on the Hermitian components (lying in a PSD manifold). In this approach, the unitary factor is excluded prior to PolSAR clustering. Considering the interpretation of a unitary matrix as a complex rotation matrix (comprising both normal rotation and phase terms), removing this rotation from the original scattering matrix may offer a significant advantage in terms of homogeneity [14, 15].

The best-known metric [17] used for the PSD manifold is the affine invariant Riemannian metric (AIRM). For any two positive definite matrices $A$ and $B$, AIRM yields a closed-form distance measure

$$d_{geod, \mathbb{P}(n)}(A, B) = \| \log(A^{-1/2}BA^{-1/2}) \|_{F}, \quad (1)$$

which can be interpreted as a similarity/dissimilarity criterion. Operator $log(X)$ represents the matrix logarithm. For positive-definite matrices it is usually computed using the eigenvalue decomposition: $X = VDV^H$, $D = diag(\lambda_1, \lambda_2, ..., \lambda_n)$ and the usual logarithm function. The operator $diag(\cdot)$ returns a diagonal matrix having the elements

inside parenthesis on the main diagonal. Then, $D_{log} = diag(\log(\lambda_1), \log(\lambda_2), ..., \log(\lambda_n))$ and

$$\log(\mathbf{X}) = \mathbf{V} \cdot \mathbf{D}_{log} \cdot \mathbf{V}^{\mathrm{H}}. \quad (2)$$

For real PolSAR data, the presence of noise (thermal or speckle) ensures that the Hermitian factors of the observed scattering matrices are always positive definite. These factors reside on a three-dimensional positive semi-definite (PSD) Riemannian manifold.

For $m$ positive definite matrices $\{H_1, H_2, ..., H_m\}, m > 2$, the Riemannian barycenter, i.e., geometric center of mass or geometric mean, is a point $H_0$ which attains the minimum value of [18]

$$\arg \min_{H_0} \sum_{i=1}^{m} d_{geod, \mathbb{P}(n)}(H_0, H_i)^2. \quad (3)$$

In the Riemannian manifolds of positive-definite matrices, it has been demonstrated that the solution to the minimization problem in (3) always exists and is unique [19]. Although a closed-form solution is not generally available, iterative minimization methods, such as gradient descent, yield convergent results [19].

Evaluating PolSAR barycenters is analogous to estimating the $N$-look geometric center of mass in the manifold of Hermitian polar factors. The barycenters are computed using an iterative method applied within square, local, sliding neighborhoods of a fixed size. This process is referred to in [14] by the acronym PolBaRi (POLar decomposition BArycenters estimation on the Riemannian manifold).

## III. INTERNAL EVALUATION INDEXES ON A RIEMANNIAN MANIFOLD

Internal clustering validation relies on the internal information of the clustering process to assess the quality of the clustering structure without referencing external data. It is also used to estimate the optimal number of clusters or the most suitable clustering algorithm. This is usually performed by jointly evaluating the compactness, connection and separation of the derived cluster partitions through internal evaluation indexes. We propose here to adjust three conventional indexes with the AIRM metric corresponding to the Riemannian manifold of PolSAR data (in either Hermitian factor or covariance / coherency matrix representation).

Firstly, the Davies-Bouldin index (DBI) internal index is defined by [20]:

$$DBI = \frac{1}{K} \sum_{k=1}^{K} \max_{k' \neq k} \frac{S_k + S_{k'}}{M_{k,k'}}, \quad (4)$$

with $S_k, S_{k'}$ being the average similarities of the cluster $k$ and the cluster $k'$ defined generally as

$$S_k = \frac{1}{N_k} \sum_{j=1}^{N_k} d_{geod, \mathbb{P}(n)}(H_j, H_k). \quad (5)$$
$\mathbf{H}_j$ are the matrices in $k$ and the $\mathbf{H}_k$ is the centroid of the same cluster $k$. The distance between the centroids of the cluster $k$ and the cluster $k'$ is:’

$$M_{k,k'} = d_{geod,\mathbb{P}(n)}(\mathbf{H}_k, \mathbf{H}_{k'}). \quad (6)$$

The minimum DBI will yield the optimal number of classes.

Secondly, the Krzanowski-Lai index (KL) is using the within cluster sum of squares and it is expressed for two consecutive number of clusters $k$ and $k + 1$ as [21]:

$$KL = \frac{\mathrm{diff}(k)}{\mathrm{diff}(k + 1)}, \quad (7)$$

with $\mathrm{diff}(k) = (K - 1)^{2/p}S_{k-1} - K^{2/p}S_k$ and $p = 3$ is the number of variables in the PolSAR Hermitian factor. The maximum value of $KL$ is considered to be the optimal number of classes. While $K^{2/p}S_k$ is not monotone with $K$, the $KL$ index is.

Thirdly, the mean Silhouette index (SI) depends on the difference between the average distance within the cluster and the minimum distance between the clusters [22]:

$$SI = \frac{1}{N} \sum_{i=1}^{N} \frac{b(i) - a_k(i)}{\max\{a_k(i), b(i)\}}, \quad (8)$$

where $N$ is the total number of points, $a_k(i)$ is the average Riemannian distance of sample $i$ to other samples within the cluster $k$

$$a_k(i) = \frac{1}{N_k - 1} \sum_{j=1}^{N_k - 1} d_{geod,\mathbb{P}(n)}(\mathbf{H}_i, \mathbf{H}_j), \quad (9)$$

and $b_k(i)$ represents the minimum Riemannian distance between all the clusters:

$$b(i) = \min_{k} d_{geod,\mathbb{P}(n)}(\mathbf{H}_i, \mathbf{H}_k). \quad (10)$$

High values of $SI$ are obtained when the partition is good, and thus the corresponding $K$ value is appropriate. Note $SI$ is not monotone in $K$.

To these three internal indexes, we add the Riemannian Calinski-Harabasz index (CHI) defined in [1].

## IV. RESULTS AND DISCUSSION

Two PolSAR datasets are used for illustration purposes.

The first C-band dataset was acquired in by the SAR ElectroMagnetic Institute Synthetic Aperture Radar (EMISAR) instrument over the Foulum area, Denmark. It shows a mixture of vegetation areas (different crop fields, forest) and a small urban area (illustrated in Fig. 1-(a)).

The second dataset was acquired by the French Aerospace Lab (ONERA) over Brétigny, France (shown in Fig. 1-(b)). The mean incidence angle is $30^\circ$. It represents a fully polarimetric (monostatic) X-band acquisition with a spatial resolution of approximately 1.5 m in range and azimuth.

Multiple PolBaRi+Riemannian k-means runs, with an increasing $K$, are operated on a 300×300 pixels sub-image from the Foulum dataset and from the Brétigny dataset. Figs. 2 and 3 show the results obtained with the four previously discussed

<figure><img src="image_2.png" /><figcaption>Fig. 1. PolSAR datasets, span of the Hermitian factor [dB]: the Foulum area (a) and the Brétigny area (b).</figcaption></figure>

internal evaluation indexes on the Riemannian manifold of the PolSAR Hermitian factors. For each $K$, a total number of 9 trials, with different k-means++ initializations have been performed.

For the Foulum C-band data, one can observe that both the CHI and KL indexes give the same optimal number of classes at $K = 7$. The SIL follows with a maximum at $K = 5$. The DBI index does not converge to a clear minimum (except for the first $K = 4$).

The Brétigny X-band dataset shows a similar behavior for the CHI and the KL indexes. The SIL index is quasi-similar for $K = 12$ and $K = 5$. The DBI index does not converge,
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<figure><img src="image_6.png" /><figcaption>(d)</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(a)</figcaption></figure>

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<caption>Fig. 3. Brétigny dataset - internal evaluation indexes on the Riemannian manifold of the PolSAR Hermitian factors (normalized display): (a) Davies-Bouldin, (b) Krzanowski-Lai, (c) Silhouette and (d) Calinski-Harabasz.</caption>

except for the first *K* = 4 value. Globally, *CHI* (and partly *KL*) internal index is exhibiting the smoothest trend.

V. CONCLUSIONS

The Riemannian extensions of three internal clustering validation indexes for PolSAR data — Davies-Bouldin, Krzanowski-Lai, and mean Silhouette — have been introduced. Alongside the previously developed Riemannian Calinski-Harabasz index, these tools can provide valuable insights into the optimal number of clusters and/or the quality of two or

more partitions obtained using different clustering algorithms with the same number of classes. By leveraging the affine invariant Riemannian metric, these indexes can be directly applied to PolSAR coherency/covariance matrices or to PolSAR Hermitian factors derived from the polar decomposition.

Moreover, these indexes offer insights into the data driven relevance of expert-generated or physically informed ground truths, aiming to align terrain features with the information content of polarimetric data.
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