RECOVERING HEIGHT INFORMATION FROM SAR IMAGES OF TERRAIN Adrian G. Bors and Edwin R. Hancock Department of Computer Science, University of York, York YO10 5DD, U.K. ABSTRACT In this paper we suggest a new approach for recovering 3-D depth from a field of surface normals. The surface normals are extracted from Synthetic Aperture Radar (SAR) images using the radar re- flectivity function and local statistical estimations. Reconstructing the 3-D surface from the flow of surface normals has an inexact solution due to missing information. A two stage algorithm is pro- posed for calculating the height information. In the first stage a site of unknown height and one of known height are chosen ac- cording to the orientation of the surface normal component in the horizontal plane. In the second stage a gradient updating algorithm is used for calculating the unknown height. We apply the proposed algorithm on SAR images of terrain. 1. INTRODUCTION Shape-from-shading has been employed for extracting surface ori- entation and height information, from Lambertian objects, by solv- ing the image irradiance equation. The problem is invariably cou- ched in a variational framework, where data-closeness and smooth- ness penalties are minimized subject to constraints imposed by boundary conditions [1, 2, 3]. The problem of applying shape from shading in synthetic aperture radar (SAR) images is further restricted by the statistical and non-Lambertian characteristics of these images [4]. In [5] we have suggested a maximum a posteriori criterion for estimating the field of surface normals from the statistics of SAR images. A log-likelihood edge estimator is used to find terrain fea- tures such as ravines and ridges by employing a Rayleigh-Bessel statistical model for radar signal [6]. The parameters of the statis- tical model are estimated using robust statistics. The surface nor- mals are constrained, by the terrain features and by the local radar image statistics, to lay on a range of cones. Various algorithms such as the marginal median, the vector median and a curvature consistency operator [7], have been employed for smoothing the surface normal field. The result of this robust processing module is a smooth field of surface normals. However, in order to have a well-defined object representation we should estimate the 3D ob- ject depth. The problem of extracting the height information from the smoothed vector field is under-constrained. The same vector field can be produced by various 3-D height maps. Let us consider known a boundary condition representing a known height for a certain site. We start from the sites of known depth and update their neighbours heights. We use a two step updating algorithm. In the first step we choose the sites of known height which are the closest to the slopes originating in the sites of unknown heights. At the second step the height is updated using the known height. The estimation of the 3-D information is propagated for all the sites in the map. The height reconstruction algorithm is applied for mod- elling 3-D topography from smoothed local surface normals. 2. ESTIMATING SURFACE NORMALS IN SAR IMAGES Let us denote by the a posteriori probability of jointly recovering the field of surface normals , and the edge map , from the 3D shape denoted by and its observed projection on the 2D plane, denoted by . The 3-D height information can be expressed by means of the Bayes theorem in : (1) where the probability density models the recov- ery of the 3D shape, from the given 2D projection, its correspond- ing field of surface normals , and the edge map . The condi- tional probability can be further factorized : (2) where represents the probability of recovering the edges from the available image data. The surface normals can be es- timated using shape-from-shading in the case of Lambertian sur- faces [1, 2, 3, 7]. In the case of SAR images we have to interpret their statistics and to model their characteristic reflectance model [4, 6]. The probabilities from (1) and (2) are described by means of energy functions modelling the characteristics of SAR images. We have proposed a Rayleigh-Bessel statistical model for SAR images in [6]. The reflectivity function models the non-linear de- pendency of the surface normals to the image intensity and it is de- rived by using the knowledge of a digital elevation map [5]. We de- rive the local surface normals by considering the Rayleigh-Bessel distribution for modelling SAR image signal. Sharp changes in radar statistics are modelled as edges and extracted using a maxi- mum log-likelihood ratio derived from the Rayleigh-Bessel statis- tical model. Edges are classified in ravines and ridges according to a simple statistical test. Ridges are used to enforce the source of divergent vector fields while ravines are employed as sinks of convergent vector fields. The geometrical locus for each surface normal is on a range of cones driven by the local statistics [5, 7]. Due to the uncertainty in the statistical estimation, the surface nor- mals may not be entirely consistent with the global model for the terrain. We apply local smoothers onto the surface normal field in order to enforce shape consistency. The probabilities from (2) are expressed as energy functions which take into account the statistical model for SAR images. The maximization of the probabilities corresponds to the minimization of their corresponding energy functions : (3) where represents the image size, is the inverse of the reflectance function, is the surface normal at location ,