An Efficient Method for Generating Points Uniformly Distributed in Hyperellipsoids Jean Dezert and Christian Musso ONERA/DTIM, 29 Av. Division Leclerc 92320 Chˆ atillon, France ABSTRACT In this paper, we consider the problem of generating efficiently random points uniformly distributed in hyperel- lipsoid. Such kind of problem arises frequently in Monte Carlo simulations for the performance evaluation of multitarget tracking algorithm in cluttered environment since the false alarms are usually assumed to be uniformly distributed in the (ellipsoidal) validation gate. We present here a more efficient algorithm for solving this problem which outperforms all existing methods both in term of reduction of computational loads and in term of quality of uniformity obtained. The efficiency and the feasability of our new method is demonstrated by several simulation results and comparisons. Moreover, the Matlab TM source code of algorithm is also provided for convenience. 1. INTRODUCTION In the Monte Carlo simulations for the study and design of multitarget tracking algorithms [2,3], one needs fre- quently to generate false alarms (FA) in target validation gates defined by hyperellipsoids in measurement space computed from predicted target measurement and covariance of measurement innovation. False alarms are usually supposed to be independent and uniformly distributed in validation gates. During many years, the only inefficient method for generating such random points [1] was to generate points in the minimal hypercube containing hyperel- lipsoid, and then sort and keep points which have been drawn in the hyperellipsoid based on a Mahalanobis distance test. This method which can be used whenever measurement space dimension and spatial density of false alarms are low, become very inefficient with the growth of FA spatial density and measurement space dimension because of the exponential rejection ratio which will be presented in section 2.3. To overcome this major drawback, X.R. Li has been the first one (to the knowledge of the authors) to propose in 1992 [12] a new algorithm, for generating points uniformly distributed in hyperellipsoids. In 1999, T.J. Ho and M. Farooq have however pointed out in [10] an obstacle in the practical use of Li’s approach. They have then proposed an improved approach (referred here as HF algorithm; HF standing for initials of authors) based on the orthogonal factorization of covariance matrix which avoids the indefinite number of iterations occuring within Li’s algorithm. It is worthwhile to note that both approaches are based on computation of eigenvalues of matrix . This requirement is time consuming (high computation burden) when measurement space dimension becomes high. In recent tracking developments, authors have tested intensively the HF algorithm and have discovered the poor performances of this algorithm in term of spatial uniformity of random points generated in validation gates. A presentation of HF algorithm results will be detailed in the sequel. To overcome this major drawback, we propose a new fast, efficient and reliable algorithm for generating directly random points really uniformly distributed in hyperellipsoid which has the following two important properties : its complexity is ( being the measurement space dimension), and it does not require the computation of eigenvalues of matrix and itself as in previous existing methods. The new method proposed in this paper follows exactly the same assumptions as in [12,10]: 1) the number of false measurements to be generated can be described by a suitable Poisson model; 2) the false measurements are uniformly distributed in validation gate and are independent from scan to scan. Authors can be contacted at following E-mail addresses: Jean.Dezert@onera.fr or Christian.Musso@onera.fr Matlab TM is a trademark of MathWorks, Inc.