Direct passive navigation: Analytical solution for planes

1. I n t r o d u c t i o n The problem of determining rigid body motion and surface structure from image data has been the topic of many research papers in the area of machine vision 11-22]. Many approaches based on tracking feature points [5,11,19,20] or contours 191, using optical flow 11,3,4,10,12,16,17,21,22], texture 121, or image intensity gradients j14,15] have been proposed in the literat,ure. In the feature point matching schemes, information about a finite number of well-separated points is used to recover the motion (general 8-point 2-frame algorithms of Longuet-Higgins i l l ] , Tsai and Huang [20), Buxton et al. [ ~ j , and the algorithm of Tsai, Huang and Zhu 1191 for planar surfaces). These methods require identifying and matching feature points in a sequence of images. The minimum number of points required depends on the number of image frames. With 2 frames, in most cases, a minimum of 5 points results in a unique solution from a set of nonlinear equations. However, using 8 points, as in algorithms cited above, one only solves linear equations. Here, it is assumed that the more difficult problem of establishing point correspondence has already been solved. In general, this involves determining corners along contours using iterative searches. For images of smooth objects, it is difficult to find good features or corners. For the smooth surfaces, Longuet-Higgins and Prazdny j l l j suggested a method that uses the optical flow and its first and second derivatives at a single point. Later, Waxman and Ullman j21! developed this into an algorithm for recovering the structure and motion parameters from a set of nonlinear equations. Subbarao and Waxman 1171 recently found a closed form solution to the original formulation in 1211 for planar surfaces. These methods, while mathematically elegant, are very sensitive t o errors in the optical flow data since second order derivatives of noisy da ta are used. At the expense of more computation, more robrlst algorithms have been suggested using the optical flow a t every image point 11:3,4]. Longuet-Higgins 1121 has presented a closed form solution for planar surfaces, very similar to ours, using the coefficients of the second order optical flow equations. However, it is assumed that both components of the flow field have already been computed for a minimum of 5 image points. By representing a planar surface in the form of a closed contour, Kanatani 191 has shown that the surface and motion parameters can be computed by measuring "diameters" of the contour using line and surface integrals. Here, no point correspondence is required. Assuming that the planar surface has a uniform texture density, Aloimonos and Chou j2] have presented a procedure for computing the motion and surface orientation from texture. In much of the research work in recovering surface structure and motion from the optical flow field, it is assumed that a reasonable estimate of the full optical flow field is available. In general, the computation of the local flow field exploits a constraint equation between the local intensity ch%nges and the two components of the optical flow. However, this only gives the component of the flow in the direction of the intensity gradient. To compute the full flow field, one needs additional constraints such as the heuristic assumption that the flow field is locally smooth [7,8]. This, in many cases, leads to optical flow fields that are not consistent with the true motion field. In an earlier paper, we presented an ilerative scheme for recovering the motion of an observer relative to a planar surface directly from the image brightness derivatives, without the need to compute the local flow field 114,151. Further, using a compact vector notation, we showed that, a t most, two interpretations are possible for planar surfaces and derived the relationship between them. Here, we present a closed form solution to the same problem. We first solve a linear matrix equation for the elements of a 3x3 matrix using intensity derivatives a t a minimum of 8 points. The special structure of this matrix allows us t o compute the motion and structure parameters very easily. CH2282-2/86/0000/1157$01.00 O 1986 IEEE 2. P r e l i m i n a r i e s We first recall some details about perspective projection, the motion field, the brightness change constraint equation, rigid body motion and planar surfaces. This we do using vector notation in order to keep the resulting equations as compact as possible. 2.1. Perspec t ive P r o j e c t i o n Let the center of projection be a t the origin of a Cartesian coordinate system. Without loss of generality we assume that the effective focal length is unity. The image is formed on the plane a = 1, parallel t o the zy-plane, that is, the optical axis lies along the z-axis. Let R be a point in the scene. Its projection in the image is r , where The z-component of r is clearly equal to one, i.e., r .i = 1. 2.2. M o t i o n F ie ld a n d O p t i c a l F low The motion field is the vector field induced in the image plane by the relative motion of the observer with respect t o the environment. The optical flow is the apparent motion of brightness patterns. Under favourable circumstances the optical flow is identical t o the motion field (Moving shadows or uniform objects in motion could create discrepancies between the motion field and the optical flow. Here, we assume that the motion flow field and the optical flow are the same). The velocity of the image r of a point R is given by For convenience, we introduce the notation rf and Rt for the time derivatives of r and R, respectively. We then have that can also be written in the compact form since a x ( b x c ) = (c . a ) b ( a . b)c. The vector rr lies in the image plane, and so (rt . i ) = 0. Further, rl = 0 , if Rf ( 1 R, as expected. Finally, noting that R = ( R . i ) r , we obtain 1 rt = -(i x ( R t x r ) ) . R . 2 2.3. R i g i d B o d y M o t i o n In the case of the observer moving relative to a rigid environment with translational velocity t and rotational velocity w, we find that the motion of a point in the environment relative to the observer is given by Since R = ( R . i ) r , we can write this as Substituting for RL in the formula derived above for r t , we obtain I t is important t o remember that there is an inherent ambiguity here, since the same motion field results when distance and the translational velocity are multiplied by an arbitrary constant. This can be seen easily from the above equation since the same image plane velocity is obtained if one multiplies both R and t by some constant. 2.4. B r i g h t n e s s C h a n g e E q u a t i o n The brightness of the image of a particular patch of a surface depends on many factors. It may for example vary with the orientation of the patch. In many cases, however, it remains a t least approximately constant as the surface moves in the environment. lf we assume that the image brightness of a patch remains constant, we have where aEjar = ( a E j a z , a ~ j a ~ , ~ ) ~ is the image brightness gradient. It is convenient to use the notation E, for this quantity and Et for the time derivative of the brightness. Then we can write the brightness change equation in the simple form E, . rt + El = 0. Substituting for rl we obtain