Unsupervised Clustering of Depth Images using Watson Mixture Model Md. Abul Hasnat, Olivier Alata and Alain Tr´ emeau Universit´ e Jean Monnet, Saint Etienne, France. Email: {mdabul.hasnat, olivier.alata, alain.tremeau}@univ-st-etienne.fr Abstract—In this paper, we propose an unsupervised cluster- ing method for axially symmetric directional unit vectors. Our method exploits the Watson distribution and Bregman Divergence within a Model Based Clustering framework. The main objectives of our method are: (a) provide efficient solution to estimate the parameters of a Watson Mixture Model (WMM); (b) generate a set of WMMs and (b) select the optimal model. To this aim, we develop: (a) an efficient soft clustering method; (b) a hierarchical clustering approach in parameter space and (c) a model selection strategy by exploiting information criteria and an evaluation graph. We empirically validate the proposed method using synthetic data. Next, we apply the method for clustering image normals and demonstrate that the proposed method is a potential tool for analyzing the depth image. Keywords—Unsupervised Clustering, Model Based Clustering, Watson Distribution, Mixture Model, Depth Image Analysis. I. I NTRODUCTION Model Based Clustering (MBC) methods [1], [2] have been widely used in the literature for data analysis and statistics. In general, these methods assume that the data are generated from a statistical Mixture Model [2]. Hence, in the context of an unsupervised clustering with MBC method, the goal is to find an appropriate mixture model for a given dataset. To this aim, we adopt a MBC method which first generates a set of candidate models and then selects the best model that satisfies the pre-specified objective function. Our method focuses on four prominent issues: (a) what type of models to generate? (b) how to generate? (c) how many models? and (d) which objective function to satisfy for selecting the best model? Directional distributions [3] model directional data sam- ples which are defined by a unit vector. Multivariate Watson Distribution (mWD) is a fundamental distribution that models axially symmetric directional data (i.e., unit vectors where ±x is equivalent). Watson Mixture Model (WMM) is a generative model, which assumes that the data samples are issued from a mixture of mWDs [4]. Our clustering method considers WMM as the core model for the data (issue (a)). Bregman Soft Clustering (BSC) is a partitional and para- metric clustering method, which arises by a special choice of Bregman divergence [5]. It employs Expectation Maximization based method to perform Maximum Likelihood Estimate of parameters (subject to a given initialization). Compare to other soft-clustering methods, BSC employs a computationally efficient Maximization Step. Moreover, it is applicable for any distribution belonging to the exponential family. The multivari- ate Watson distribution (mWD) belongs to such family. This motivates us to develop Bregman Soft Clustering for mWD. Clustering a mixture model generates a simplified model with smaller number of components. Therefore, it can be an efficient approach to generate a set of mixture models. In general, a hierarchical agglomerative clustering (HAC) in the model parameter space is used to cluster a mixture model [6], [7]. The main element of such clustering is the measure of a distance among the distributions. For example, to simplify the Gaussian Mixture Model, Goldberger et al. [6] and Zhong et al. [1] used KL Divergence and Garcia et al. used [7] Bregman Divergence (BD). Similarly, BD is used to construct a hierarchy of von Mises-Fisher mixture models [8]. In our method, we use HAC with BD to generate the desired set of models (issue (b)). To deal with the next issue (issue (c)), we generate a set of models within a certain bound (e.g. k min to k max ) of number of components. Clustering methods with this type of bounds are called deterministic method [9]. In general, an objective function (to select the best model) is defined based on minimizing certain model selection criteria [2], [9], [10], [11]. A different approach evaluates a graph/plot [12], [13] for model selection. The objective is to fit two lines in the plot and identify the point (called knee/elbow) that mini- mizes the line fitting error. We examine both approaches (issue (d)) and select one of them based on empirical justification. We develop a clustering method by combining the answers for all the issues. Based on the literature [1] it belongs to the category of hybrid Model Based Clustering method. This category of methods is more effective than using a single complex model for clustering (see chapter 6 of [1] for details). In depth images, normals are 3D unit vectors that describe the orientation of the pixels. The most common method uses plane fitting method to compute normals [14]. However, it generates ambiguous sign of the normal. Due to the axially symmetric property, the Watson distribution can be effectively applied to overcome such ambiguity. Indeed, this motivates us to employ our method to analyze a depth image via clustering. In this paper, we propose a novel unsupervised clustering method for axially symmetric directional data. Our contribu- tions are: (a) a mathematical formulation to compute Bregman Divergence among multivariate Watson distributions (mWD); (b) an efficient method to construct a set of Watson Mixture Models (WMM) and (c) an empirical model selection strategy. To the best of our knowledge, till date no unsupervised Model Based Clustering method is proposed with mWD. Using syn- thetic data, we experimented and compared our method with the state of the art methods. Numerical evaluations confirm that our method performs better in terms of clustering accuracy and