Peculiarities of the surface relief determination by the photoclinometric method in the millimeter range

Abstract

In the paper [l], a method of determination of the relief of a planet surface region by a series of its images is described. It is based on the fact that at given directions of illumination and observation the brightness of a surface element depends on its orientation. Knowing this, one can determine the field of slopes t(x,y), from which it is possible then to reconstruct the relief H ( x , y ) using the relation VH(x,y) =t(x,y)-(1) An attempt to determine H(x, y ) by solving the equation (1) directly leads to the incorrect problem due to accidental errors of calculation of t(x,y). Instead, as shown in [I, 21, one should formulate a statistical problem of determination of the most probable relief at the given field of slopes. In the simplest case this results in the Poisson's equation for the relief (2) with the boundary condition depending on the character of a priori data concerning the relief. Initially this method has been developed for optical wavelength range. However, steady advancement of radio technical tools towards more and more short waves makes it urgent to extend this method to radio wave range. The present work is a step in this direction.The present paper describes the case when the initial images have been received using a side-looking radar (SLR), with the focus on the information density of the received set of images. For definiteness sake, we shall assume that the surface dissipates radiation according to the Lambert law. Let the angle of the apparatus location and the surface element slope t are small, then the deviation Ii of the element brightness from that one averaged over the region when observed from i-th point of the path is linearly expressed through the vector t:where Ci and Si are the components of a unit vector n directed from the i-th point of the path to the observed element ( Fig.1). Each equation in the set of equations (3) corresponds to one position of the apparatus on the flight path, from which the survey of the region is performed. The given set of equations determines the slope components t , and t y of the given surface element. It is over determined and, if noise is available, incompatible; therefore it needs that some regularization procedures be performed. Application of the statistical approach appears to be the most natural here: it is proposed to determine the most probable value of the vector t at a given set of values Ii . Here the statistical features of errors in brightness <i are known and we will assume that they are distributed normally and independently of each other with the dispersions s i . Figure 1 Geometry of the experiment on receiving a series of images of a planet surface region by means of SLR. .The use of the least-squares method in the system (3) results in getting a normal set of equations for the most probable value of t 0-7803-841 1-3/04/$20.0002004 IEEE 905