Spherical Object Tracking using Von Mises-Fisher Distribution in Catadioptric System

Corresponding Author: Khald Anisse LRIT, Rabat IT Center Faculty of Sciences, Mohamed V University, B.P.1014 RP, Rabat, Morocco Email: khald.anisse@gmail.com Abstract: Tracking objects on video sequences is a very challenging task in computer vision applications. However, there is few articles that deal with this topic in catadioptric vision. This paper describes a new approach of omnidirectional images (gray level) processing based on inverse stereographic projection in the image plane. Our work is based on minimizing the distance between two models. The model named von Mises-Fisher distribution have as input the gabor phase and the measure used is Kullback– Leibler Divergence (KLD). In one hand, this model matching respect the deformed geometry of omnidirectional images due to using the spherical neighbourhood. In the other hand, the simulation results show that our approach gives as better performance in terms of overlapping estimation.


Introduction
In the context of monitoring system, we describe a method for images processing and analysing. Indeed, Object Tracking is an important task in computer vision applications including video surveillance, Radar, robot navigation.
This work defines this process in a omnidirectional images sequence. This is a challenging task because these images contain significant distortions because of mirror geometry. This is why; we must take them into account during processing.
This work is focused on two main goals: The one including detection and matching features. After that, the adapted process according to spherical coordinates (, ) over the image plane in order to make possible the nonrigid objects tracking.
The omnidirectional image provided by a camera with a Single View Point [(SVP) (Geyer and Daniilidis, 2000;Barreto and Araujo, 2001;Chiuso and Picci, 1998)] is modelled by a spherical image (Fig. 1). This model called unified projection model and was introduced in (Geyer and Daniilidis, 2000) and used in (Baker and Nayar, 1998). Also (Mei and Rives, 2007) make using this model for camera calibration.
Following the catadioptric mirror shape used (parabolic in our case), the projection process onto the sphere and the catadioptric plan is shown in Fig. 2 The stereographic projection of Ps from the sphere to the catadioptric plane can be expressed on Cartesian coordinates in Equation (2):  (2000) Using Equation (1) and (2), we obtain the image point P(u, v) expressed on spherical coordinates as Equation (3): where,  is the latitude varying between 0 and  and  is the longitude varying between 0 and 2. The localization of a spherical point according to spherical coordinates system is defined by (, ).
In the context of tracking, catadioptric cameras give us the advantage of making possible to find objects in surrounding scene. According to many works, we get constraints either when omnidirectional image is presented by a set of perspective images or as a panoramic image without taking into account existing distortions. That is why; we must adapt the algorithm according to omnidirectional image geometry. This method must takes into account the specific geometry of the calibrated camera by representing the image onto the unit sphere. The object motion, changing illumination and background, are issues to solve. In this study, we address the problem related to changing size and shape of object by using model matching method.
In our work, we perform the tracking as a Model Matching Algorithm where the underlying Model is a distribution on the spherical space, named von Mises-Fisher distribution (Chiuso and Picci, 1998). After computing models that have as input the Gabor phase, we minimize the distance (KLD) between two image block and this algorithm is repeated in the rest of sequence in order to track the object block.
The paper is organized as follows. Section II presents the general mathematical background and formulas for tracking onto the sphere with the von Mises-Fisher distribution. Section III describes the proposed object tracking approach based on the von MisesFisher distribution. Section IV presents the results and   discussion of the synthetic data experiments, while Section V concludes the paper.

Related Works
Object tracking in a complex environment needs a powerful algorithm. Consequently, many methods have been developed in literature and can be divided into tree groups: Tracking based on kernel, based on point and based on edges. These methods have been successfully employed in various application domains. They cannot be directly applied to images acquired by catadioptric cameras like the work's (Kumar, 2015). When the author attempts to estimate self-position of intelligent vehicles by fusion several techniques. Also in (Aggarwal, 2015), the same author use GPS information for estimation self-position of mobile robots. Mei et al. (2006), the author presents a homography-based approach for tracking multiple planar templates. First, the adaptation of conventional particle filter to the catadioptric geometry was purposed in (Lucas and Kanade, 1981). A few methods have been developed in the literature in this context. Consequently, the must adapted methods are based on statistic estimation. We find the adaptation of conventional particle filter to the catadioptric geometry that was purposed in (Ikoma, 2008). This is done by adapting the window used to define the object appearance on the unitary sphere. Also, authors in (Hurych et al., 2011) propose a new method to display tracking result from weighted particles obtained from the estimation process by Sequential Monte Carlo (SMC) implementation of Probability Hypothesis Density (PHD) filter. Also Radgui et al. (2011) using spherical adaptation of the KLD method makes flow estimation in omnidirectional images.
In this study, we use methods based on kernel. Specially, based probability distributions whither give us information about area of block without knowing size or shape. In addition, in context of computer vision, there are many sets of probability distributions on the sphere in several fields.
Early uses of spherical distributions was found in geology field (Bingham, 1974).
In recent years, these approaches have found their way into many domains like estimation and filtering not just to descriptive statistics. For example, the task of tracking objects using omnidirectional cameras (Marković et al., 2014a) can naturally be mapped to a spherical process problem by using the probability estimation. Thus, it even becomes possible to consider multiple objects tracking onto the unit sphere (Marković et al., 2014b).
In the same context, we consider the Gabor phase information as estimator input. This is because it can take very different values even if it is sampled at image locations only a few pixels apart (Struc and Pavesic, 2010).

Wavelet Gabor phase
Gabor kernels or also called Gabor wavelets or filters have known to be a robust tool for feature detection. They represent an optimal localization in the spatial and frequency domains. They extract, multi-scale invariant, spatially local features of a suitable frequency band. In the spatial domain (Struc and Pavesic, 2010), the family of Gabor filters is defined as follows: where, x = xcos(v + ysinv and y = -sin(v) + ycos (v) denote axes that are rotated by v = where,  is a constant.
In Equation (6), G(u,v)(x,y) represents the complex convolution which can be decomposed into real R(u,v)(x,y) and imaginary parts I(u,v)(x,y) as follows: 1232 where, I(x, y) is the input image. the local magnitude M and the phase  of the output are given by: The gabor phase is represented in Fig. 3. We apply the Gabor filter over each block image. This block in perspective image corresponds to the different shape block in omnidirectional image. In our algorithm, we defined the neighborhood by size block given by  and  as follows: N et M are the neighborhood orders. We extract the histogram phase of each block and it will be the input of simulator in order to create the model.

Von Mises-Fisher Distribution
The probability distribution f (, ) onto the unit sphere defined by density function (pdf) firstly in (Mardia and Jupp, 1999) (Wood, 1994) at order p = 2 d -1. The density in Equation (Ikoma, 2008) invariant by rotation around the mean direction. Also it is equivalent to the multivariate Gaussian distribution. This distribution is characterized by the maximum entropy Principe in order to maximize the Boltzmann-Shannon entropy according to the directional mean. Von Mises-Fisher distributions presents an exponential family (Rényi, 1961) defined by parameters  =   R d and the log-normalizing function given by: The vMF distribution is generally determined by the directional (angular) mean: where, Ad is the ratio of the following Bessel functions: In order to simulate the VMF distribution, we obtain a random vector from the VMF knowing  and . To get the desired direction, we only need to apply the suitable rotation. An example of a von Mises-Fisher distribution on the unit sphere with different mean directions and concentration parameters is shown in Fig. 4.

KL-Divergence for the VMF Distribution
In this study, we use a method that requires to compute the distance between the current model and others in their vicinity. We propose the following procedure based on statistical distances (Diethe, 2015).
Given that, we derive the Renyi -divergence (Rényi, 1961) for the vMF distribution as a distance that will be used to determine the measurement.
We chose this divergence because is a statistical class of generalized distances including the Kullback-Leibler (KL) distance and the Bhattacharyya distance.
The Renyi -divergence is given by the following expression: And it is characterised by the real parameter a when in the limit  

Algorithm
The tracking algorithm of our approach is presented beside.

Evaluation Methodology
In this study, we use the Spatial Overlapping estimation. Overlapping estimation. This metric is the bounding box overlap as shown in Fig. 5. Given the tracked bounding box rt and the ground truth box ra, the overlap measure is defined as and  represent the intersection and union of two regions, respectively and || denotes the number of pixels in the region. To measure the performance on a sequence of frames, we count the number of successful frames whose overlap S is larger than the given threshold equal to 0.2.  In Table 1, we compared the adapted Mean Shift algorithm (Rameau et al., 2011) with our approach based on adapted model matching in omnidirectional images in term of the Spatial Overlapping.
The results in Fig. 6 show that the directional model matching in spherical domain gives better performances.

Conclusion
In this study, we present a tracking method in catadioptric system. It is based on an adapted Model Matching. In one hand, we the adapted process done by image projection onto the sphere. In the other hand, the model was defined by Von mises Fisher Distribution parameters. In tracking, we optimize the distance between models for finding the moving object block. Experimental results taking into account the change size and shape of block show the good performance of our approach. A direction of future work would be an extension for multiple objects tracking with real time processing.