A New Approach for Target Detection iIn Radar Images Based on Geometric Properties of Covariance Matrices’ Spaces. Une Nouvelle Approche pour la Détection de Cibles Dans les Images Radar Basée sur des Distances et Moyennes dans des Espaces de Matrices de

The following paper aims at presenting new theoretical and algorithmic developments to the problem of target detection in radar imaging. This concept has been explored very recently by the authors as a possible alternative to the standard Fourier transform-based algorithm called TFAC. Indeed, TFAC proves inaccurate in cases where the number of impulsesper emission salvo in the radar image becomes small. To avoid lack of resolution of the Fourier transform, we propose an entirely different approach that involves covariance structures of the data instead of raw datas themselves. Working with covariance matrices, as defined by (1), of the complex vectors that represents the radar data has already appeared useful in some other issues related to radar image processing, which motivated their use in this particular problem. The basic idea of the algorithm is identical to other detection schemes : we want to compare a given location on the radar image with its neighbourhood and detect a target as a point where radar data is highly different from the average of its closest neighbours. The difference in our setting comes from the fact that we try to evaluate it through these covariance matrices defined at each point. Assuming that we can derive them from the data, the problem becomes one of de fining and computing adapted distances and averages of such objects. This involves recent results in differential geometric studies of hermitian positive definite matrices'space and is developed in section 2 and 3. The metric on HPD_n(C) needs to satisfy some invariances due to the special structure of covariance matrices, in particular invariance by basis changes which imposes condition (2). This excludes all the usual distances we know like for instance the Frobenius norm. A pleasant framework to derive a good distance is the one of Riemannian geometry. Considering the manifold HPD_n(C), we define a metric on the identity tangent space which we transport to the whole space according to the invariances we want (4). Taking the most simple metric on the identity tangent space, ie the usual euclidean metric, we have defined a riemannian metric satisfying the conditions. Another good point comes from the fact that we have an explicit parametrisation of the geodesics (9) and therefore, integrating along them, the expression of the distance as in (7) and (8). It is very remarkable that the resulting distance is actually the Siegel distance in HPD_n(C), which is also Rao's distance measure between zero-mean Gaussian distributions in information geometry.

Once the metric is defined, there is a canonical way to express the mean of several covariance matrices so that we preserve the same invariances. It is given by equation (10). Finding the mean then requires the resolution of a minimization problem to which we cannot express the solution in the general case. This is solved by a gradient descent scheme made possible both by the expression of the geodesics and the gradient. Computation steps are given by the recursive relations (11). Eventually, the defined mean on HPDn(C) has many interesting properties summarized in theorem 3.1, which makes it more appropriate to our applications than other definitions that can be found in literature. In section 4, we give a first set of results on real images provided by a coastal radar. Starting with raw data, we need to estimate covariance matrices on each radar location. This is done classically by considering auto-regressive models. Since it is not the point of the article, we simply refer to the papers cited about the subject ([7], [10]). Once calculated, we can apply all the previous framework on the covariance matrices. For detection puposes, we compute at each location the mean of the adjacent matrices and compare it to the current covariance matrix through the Siegel distance. In figure 1, this distance is represented function to the point's distance to the radar. Several peaks corresponding to special objects appear on the graph and can be compared to the equivalent peaks obtained by the usual TFAC scheme (figure 2), clearly showing a far better detection efficiency. This result is highlighted when artificial targets are inserted on the initial image (cf. figures 3 and 4).

Finally, in section 5, we describe an improved version which was considered in order to reduce computational cost of the algorithm. The strategy consists in the use of a more compact representation of the data given directly by reflexion coeffcients of the auto-regressive model. Again, a metric and a mean are defined on the new space following the ideas of information geometry, leading to equations (13) and (14). As shown on figures 6 and 7, detection quality remains similar while computation time is notably improved by a factor 4 within this new scheme. Although, as can be imagined, results are still perfectible, we have presented several new ideas that, in our sense, could pave the way for an interesting approach to radar image processing in general.

Résumé

La présente étude vise à introduire une toute nouvelle classe d’algorithmes en vue de l’application à la détection automatique des cibles sur les images radar dans lesquelles on dispose d’un faible nombre d’échantillons par salve d’émission. Contrairement aux méthodes déjà mises en oeuvre (filtrage TFAC, etc...) qui agissent directement sur les données radar, celle exposée ici consiste à calculer des matrices de covariance à partir de ces données puis de chercher en quels points une matrice de covariance diffère de la matrice «moyenne» calculée dans un voisinnage. On espère ainsi obtenir des pics de détection plus prononcés au niveau des cibles qu’avec les algorithmes standards. Ceci impose dans un premier temps l’exploitation de résultats déjà mis au point en géométrie différentielle pour définir distances et moyennes dans de tels espaces (cf. [5], [11], [16], [18]) ce que nous récapitulerons dans les sections 2 et 3. L’algorithme à proprement dit ainsi que quelques résultats de détection par cette nouvelle approche sont ensuite présentés dans la section 4 et comparés aux algorithmes classiques. Enfin, dans la section 5, on verra une amélioration possible des méthodes basées sur les matrices de covariance notamment en terme de temps de calcul, grâce à l’utilisation des processus auto-régressifs ([7], [10]).