B. Horn and E. Weldon. Direct methods for recovering motion. Int'l J. Computer Vision, pages 51--76, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....all analyses are based on the two step approach, analyzing the estimation of 3D motion from noisecontaminated optic flow or correspondence. However, as has been shown in previous work, the estimation of 3D motion does not necessarily require the prior computation of exact correspondence [11, 12, 13, 20, 29]. Flow measurements, or even their signs, along some direction in the image, such as for example the one provided by the spatial gradient, are sufficient for recovering 3D motion [3] Such measurements can be computed by even the simplest systems biological or artificial using, for ....

....though probably with some error. On the basis of this information, the best one can do to estimate the remaining translation is to assume that the flow field obtained by subtracting the estimated rotation is purely translational and apply a simple algorithm designed for only translation [2, 20, 29, 34]. Such algorithms, if based only on the constraint that the depth is positive, are formulated basically as constrained minimization problems. The underlying idea is illustrated in Figure 4. Assuming the observer is approaching the scene, the exact 2D motion vector at every point is away from the ....

B. Horn and E. Weldon, Jr. Direct method for recovering motion. International Journal of Computer Vision, 2:51--76, 1988.

....obtained without errors. As a remedy to this problem, a number of studies have recently been conducted addressing the motion recovery problem in a direct way, that is by employing as input to the recovery process only normal flow measurements the components of flow measurements along gradients [9, 10, 16, 21]. In this paper the general problem of self calibration, using normal flow as input, is considered. It is shown how image displacement fields originating from a sequence of images are related to the structure, motion and calibration parameters and how from this relationship the calibration ....

....(7) contains f , the flow r is independent of f , as can be easily seen by expanding the expressions. A calibrated camera (with focal length f) can be described as a special case of an uncalibrated camera, in which case K = f I, where I is the identity matrix. Equation (7) then becomes the usual [16] r c = Gamma (z Theta (t Theta r) z Theta (r Theta ( Theta r) 8) As can be seen by comparing equations (7) and (8) the first component of r is the same as the translational flow generated by a calibrated camera moving with translational velocity Kt=f . We thus call Kt (or ....

B. K. P. Horn and E. J. Weldon, Jr. Direct method for recovering motion. International Journal of Computer Vision, 2:51--76, 1988.

....of several motions in one large region. While computing motion on large regions, thus reducing noise and ambiguity, the segmentation assures that only relevant points will participate in motion computation. Analysis of camera motion relative to a static scene has been extensively treated [5, 7, 8, 9, 12]. Recovering this uniform motion does not require the computation of optical flow at every image point. Negahdaripour and Horn [12] present a method for recovering the observer s motion relative to g planar surface directly from the spatial and temporaI derivatives of the image brightness. Horn ....

....this uniform motion does not require the computation of optical flow at every image point. Negahdaripour and Horn [12] present a method for recovering the observer s motion relative to g planar surface directly from the spatial and temporaI derivatives of the image brightness. Horn and Weldon [7] follow ;his approach, and using a least squares method determine the observeFs motion in the special cases of known depth, pure rotation, or known rotational motion component. We use a method based on [7] for recovering uniform motion of a set of pixels to perform motion based segmentation, in ....

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B.K.P. Horn nd E.J. Weldon. Direct methods for recovering motion. International Journal of Computer Vision, 2(1):51-76, June 1988.

....by Chen and Medioni [6] and Besl and McKay [3] has been used extensively in the graphics literature to merge 3D laser range scans. In the vision literature much progress has been made on gradient based parametric motion estimation techniques which aggregate pointwise normal flow constraints [4, 20, 24]. ICP finds corresponding points between two 3D point clouds and tries to minimize the error (usually the euclidian distance) between the matched points. Chen and Medioni minimize this error based on a point to plane distance, while Besl and McKay minimize the di rect euclidian distance between ....

B.K.P. Horn and E.J. Weldon, Jr. Direct methods for recovering motion. IJCV, 2(1):51 76, June 1988.

....squares fit. This a#ne transformation is used to initialize a direct method which yields a projective transformation between the images. The direct method operates in an iterative manner using the LevenbergMarquardt method, a well known algorithm that is described in various papers (e.g. 5] [8]) It is also possible to use the Levenberg Marquardt method directly, although use of the KLT tracker speeds up the registration. Our algorithm provides a method for mosaicing that is robust to the presence of moving objects. First, as with several traditional mosaic techniques, we register each ....

B.K.P. Horn and E.J. Weldon. Direct methods for recovering motion. IJCV, 2(1):51-76, June 1988.

....explain how sampling techniques can be appled to represent the three probability distributions introduced above and to derive an estimator of x k . 2.3 Integrating Tracking and SFM Feature tracking and SFM are often treated as separate problems with some notable exceptions. Direct approaches [2] [9] recover the camera motion without explicit feature correspondences. However direct methods assume small camera motion between frames which is not always true in robot navigation tasks. Torr [21] estimates the fundamental matrix with RANSAC and uses the recovered fundamental matrix to guide the ....

B.K.P Horn and E. Weldon. Direct methods for recovering motion. International Journal on Computer Vision, (1), 1988.

....can be classified into two categories: estimation of depth and use of low level visual cues. There have been several attempts to achieve robot navigation using depth information from the environment. Although some of these attempts use visual cues such as focusing [54] and depth from motion [30], binocular stereovision has been the favorite method [39, 47] Ratler and Nomad at CMU, and Robby and HMMWV at JPL are examples of mobile robots that use a stereovision system as the main sensor for navigation. In these systems the goal has been a geometrical reconstruction of the depth structure ....

B. Horn and E. Weldon. Direct methods for recovering motion. International Journal of Computer Vision, 2(1), 1988.

....produce better subjective results using simple wireframe models for typical head and shoulder scenes whilst requiring less computation. 2. ALGORITHMIC DESCRIPTION The motion of each modelled object in a scene is approximated by computational means. A number of algorithms have been proposed [3] 4][10], but more typically the global motion of an object is calculated using parametric analysis of a block matched displacement field (see figure 1) This can be summarised as: If a point [ Z Y X P , is a point on a rigid object at time t and moves to a point Z Y X P , at time ....

B K P Horn and E J Weldon. Direct Methods for Recovering Motion. International Journal of Computer Vision pp 51-76, 1988

....in addition to the resampling filter, and the resulting predicted gradients are of high quality. Note that the calculation of gradients is an important and often neglected element of all gradient based motion estimation algorithms, and the performance of an implementation often depends on it [8, 2]. Equation (16) can now be interpreted as follows: the Jacobian images, representing the partial derivatives of the image with respect to one pose variable, can be obtained as a linear combination of the two Figure 3: Vector field induced by a change in yaw . gradient images. The coefficients ....

B. k P Horn and E. J. Weldon. Direct methods for recovering motion. International Journal of Computer Vision, 1(2):51-- 76, 1988.

....oriented entity with certain semantic attributes at a di erent position more likely. Here we want to stress that another important regularity in visual data, with quite distinct properties compared to collinearity and parallelism, is motion, most importantly rigid body motion (RBM) see, e.g. [16, 31]) Knowing the RBM between two frames, deterministic predictions between frames can be made (see, e.g. 15, 64] The occurrence of an event in the rst frame makes the occurrence of a certain event in the second frame mandatory (except in the special case of occlusion) While RBM leads to ....

B.K.P. Horn and Weldon E, J. Direct methods for recovering motion. International Journal of Computer Vision, 2:51-76, 1988.

....from eq. 1) If we also use the Longuett Higgins and Prazdny[12] small motion assumptions: z Z y x f 1 x y f 1 (10) where f is the focal length, and w x #w y are the rotations around the X and Y axes. Thus for the first motion weget: which first appeared in [8]. And similarly for the second motion: 4 Eliminating k =1=z from equations (11) and (12) we obtain the 15 parameter model based brightness constraint: I s [t ; t ]v =0 (13) where s# v are defined below: s = xI x ; yI y v = I y ; y(xI x yI ....

B.K.P. Horn and E.J. Weldon. Direct methods for recovering motion. International Journal of Computer Vision, 2:51--76, 1988.

....slope. The computation of the direction of motion in a steady environment is found to require the knowledge of at least seven variables related to the egomotion [45] Recently, the computational expense of the optical flow has given rise to direct methods for calculating the velocity components [53,54]. Thus motion vision can calculate, from time varying images, the distance from objects in the scene if the camera motion is known. The objects in the scene can be stationary or can be moving. Given moving objects in a scene, one can easily appreciate the added difficulty in tracking objects and ....

B.K.P. Horn and E.J. Weldon Jr., "Direct Methods for Recovering Motion," International Journal of Computer Vision, Vol. 2, 1988.

....image brightness information directly to recover the motion and shape. Previous work in direct motion vision has used the Brightness Change Constraint Equation (BCCE) for rigid body motion [8] s t Et v Z 0 (1) to solve special cases such as known depth [5] pure translation or known rotation [6], pure rotation [6] and planar world [8] All these direct methods are restricted in the types of the motion or shape that they can handle. Recently, we introduced a direct method called fization for solving the motion vision problem in the general case without placing restrictions on the motion ....

....directly to recover the motion and shape. Previous work in direct motion vision has used the Brightness Change Constraint Equation (BCCE) for rigid body motion [8] s t Et v Z 0 (1) to solve special cases such as known depth [5] pure translation or known rotation [6] pure rotation [6], and planar world [8] All these direct methods are restricted in the types of the motion or shape that they can handle. Recently, we introduced a direct method called fization for solving the motion vision problem in the general case without placing restrictions on the motion or the shape [12, ....

B. K. P. Horn and E. J. Weldon Jr. Direct methods for recovering motion. Int'l Journal of Computer Vision, 2:51-76, 1988.

....the visual motions, because visual motion depends only on camera motion and the 3 D structure. But reliably extracting optical flow is usually very difficult in real images because of the well known aperture problem , occlusion, noise and absence of obvious intensity variation. In [NH87] [HW88], SB88] and [WL91] direct methods of shape from motion were used to determine depth based on the constraint equation. These gradientbased techniques do not compute optical flow as a necessary intermediate step, nor establish correspondence between points in two images. But the results are still ....

B.K.P.Horn and E.J.Weldon Jr., Direct methods for recovering motion, Int. Journal of Computer Vision, Vol.2, 1988, pp. 51-76

....this approach to make hand eye (extrinsic) calibration. Our method for hand eye calibration boils down to recovering the motion of the camera, using only image derivatives, when we have knowledge about the motion of the robot hand. This work has a lot in common with the work by Horn and Weldon [6] and Negahdaripour and Horn [9] where the same kind of intensity constraints are developed. We use, however, an active approach which allows us to choose the type of motions so that, for example, the unknown depth parameter Z can be e#ectively eliminated. The equations are developed with the goal ....

B. K. P. Horn and E. J. Weldon Jr. Direct methods for recovering motion. Int. Journal of Computer Vision, 2(1):51--76, June 1988.

....and object tracking. Rigid and affine models for direct parametric motion estimation have been extensively explored in the past decade. Horn and Weldon provided an early and comprehensive description of the brightness constraints implied by egomotion or the rigid motion of an object in the world [7]. They observed that motion estimation was in general quite difficult to solve with unknown scene depth, although it is possible in several restricted cases. Bergen et al. 1] were among the first to demonstrate image stabilization and object tracking using an affine model with direct image ....

....y; t# I t #x; y; t# (2) where I x #x; y; t#, I y #x; y; t#,andI t #x; y; t# are image intensity gradients with respect to x, y,andt, as a function of space and time. Cancellation of the I#x; y; t# terms and rearrangement into matrix form yields the commonly used gradient formulation of the BCCE [7]: I t = # I x I y # # v x v y # (3) This equation constrains image plane velocities, but we are interested in solving for 3 D world velocities. For a perspective camera with focal length f , the relationship between the two sets of velocities may be derived from the perspective camera ....

B.K.P. Horn and E.J. Weldon, "Direct Methods for recovering Motion ", International Journal of Computer Vision, 2:51-76, 1988

....of distinguished features (points, lines, contours) extracted from successive frames [24] 12] 8] Both approaches depend on the accuracy of the feature detection, which can not always be assured. Methods for computing the ego motion directly from image intensities were also suggested [10] [13]. Camera rotations and translations can induce similar image motions [2] 9] causing ambiguities in their interpretation. The problem of recovering the 3D camera motion from a flow field is therefore an ill conditioned problem, since small errors in the 2D flow field usually result in large ....

B.K.P. Horn and E.J. Weldon, "Direct Methods for Recovering Motion," Int'l J. Computer Vision, vol. 2, no. 1, pp. 51--76, June 1988.

....line on the unit sphere and that the ratio v x =v y as well as the component of the angular velocity parallel to (v x ; v y ; 0) T could be reliably estimated in the presence of noise. The influence on the error sensitivity of the translation direction has been emphasized by [Weng et al. 89b] Horn Weldon 88] by geometric arguments. The importance of the translation direction has been shown experimentally by [Mitiche et al. 87] Adiv 89] and [Weng et al. 89b] Weng et al. 89b] carried out an analysis of the errors in the estimated eigenvectors in order to compare it with the actual error. However, ....

B.K.P. Horn, Direct methods for recovering motion, Int. Journal of Computer Vision 2 (1988) 51-76.

....(u) L p (u) q (2.14) The model based optical flow constraint equation in the image can be found by rewriting (2.13) using (2.14) IL p (u) q I t = 0 (2.15) Formulations which are basically identical to (2. 15) although are often confined to rigid motion) can be found in [Adi85, BAHH92, CAHT94, HW88, LRF93, NH87, NS85] Negahdaripour and Horn [NH87] refers to a formulation such as this as a direct method for motion estimation. The discussion of (2.15) in [BAHH92, NH87, NS85] is specialized for rigid motion, and while still general, requires a lengthy derivation by hand. Using the modular ....

....representation for the rotations) There are a number of techniques available for solving (2.15) The most common is the iterative minimization of the quadratic error measure, summed over a set of pixels: min q i (I i L p (u i ) q I t i ) 2 (2. 16) which is the approach taken in [BAHH92, HW88, NH87, NS85] An alternative method solves the least squares problem using the pseudo inverse of the matrix formed by stacking a set of equations like (2.15) for a set of pixels [CAHT94, NS85, LRF93] B q I t = 0 (2.17) 18 which is solved as: q = B I t (2.18) where B is the ....

B. Horn and E. Weldon. Direct methods for recovering motion. International Journal of Computer Vision, 2(1):51--76, June 1988.

.... second, interpreting the motion from the point correspondences [1 6] or the optical flow field [7 11] Instead of calculating the point correspondences or the optical flow field as an intermediate result, some other methods estimate the motion directly from the spatial and temporal gradients [12,13]. In this paper, we concentrate on the estimation of the socalled ego motion of an observer moving in a static environment by using the optical flow fields. Ego motion provides useful information for human computer interaction and vehicle navigation [14 18] In the literature, Burger and Bhanu ....

Berthold K.P. Horn and E.J. Weldon JR., "Direct methods for recovering motion", International Journal of Computer Vision, vol. 2, no. 1, pp. 51--76, 1988.

....errors cannot be large. Motivated by these factors, as in [18] 26] 16] we disregard the correspondence problem and present an algorithm that reconstructs from pre tracked point features over a large number of images. See [21] 20] for extensions of our approach to a direct method ( 5][8]) that reconstructs directly from the image intensities as well as from pre tracked point and line data. For small displacements, most scene points are visible in all the images. This makes it possible to use fast factorization methods, as we do here [18] 26] 16] 31] 30] It also means that one ....

B.K.P. Horn, and E. J. Weldon, Jr. "Direct Methods for Recovering Motion," IJCV 2:1, 51--76, 1988.

....to noise is the dominant issue in the reconstruction problem. The problem itself is inherently sensitive, and no algorithm or numerical implementation will make this go away. As a consequence, reconstruction can only be applied when the field of view of the camera is sufficiently wide [KvD87] [HW88], motion is sufficiently extended around the object [TK93] and image measurements are both sufficiently accurate (low bias) and precise (low standard deviation) From a numerical standpoint, since the homogeneous F Theta 5 system (8) is expected to have exactly one affine shape solution with ....

B. K. P. Horn and E. J. Weldon Jr. Direct methods for recovering motion. IJCV, 2:51--76, 1988.

....be quite difficult, however, especially as the deviation between the model and data becomes large. Model based optical flow: Instead of computing an unconstrained flow field (a grid of arrows) a model based approach explains the optical flow information in terms of motion parameters of the model [1, 5, 9, 23, 28, 35, 36]. While the problem is non linear, these frameworks can use either a single step linear least squares solution [9, 28, 36] or an iterative least squares solution [1, 5, 23, 35] The motion model can be a 2D model of image motion [5, 6] or a 3D model (rigid or non rigid) of object motion [5, 9, ....

.... a model based approach explains the optical flow information in terms of motion parameters of the model [1, 5, 9, 23, 28, 35, 36] While the problem is non linear, these frameworks can use either a single step linear least squares solution [9, 28, 36] or an iterative least squares solution [1, 5, 23, 35]. The motion model can be a 2D model of image motion [5, 6] or a 3D model (rigid or non rigid) of object motion [5, 9, 28] along with a camera model to relate to the images) It is also possible to compute an unconstrained optical flow field using standard techniques, and fit a parametric motion ....

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B. Horn and E. Weldon. Direct methods for recovering motion. International Journal of Computer Vision, 2(1):51--76, June 1988.

....hereafter referred to as a map. The specific issues that are addressed include imaging the surface, aligning all of the images in a common frame of reference, and reducing the amount of memory required to store the map. 2. Previous Work Navigating from imagery is a basic technique in robotics [3][4], but such techniques are often concerned with the much harder problem of a three dimensional scene, usually of unknown geometry. Mosaic based positioning deals with a two dimensional pose and assumes the scene geometry is flat, a fair assumption to make for floors. The idea of mosaicing, or ....

B.K.P. Horn and E.J. Weldon, "Direct Methods for Recovering Motion," Int. J. Computer Vision, Vol. 1, pp. 51-76, 1988.

.... 38] techniques that do the opposite [1, 23, 26, 31, 33, 39, 42] and techniques that estimate all motion parameters simultaneously [7, 8, 17, 36, 45] The positive depth constraint, which has been used for normal flow fields, is relatively new and is employed in the so called direct algorithms [8, 21, 27]. One has to search for the 3D motion that is consistent with the input and produces the minimum amount of negative depth. Put differently, in these approaches the function representing the amount of negative depth must be minimized. Finally, one may be able to use the positive depth constraint ....

....vectors t close to the center of gravity of the points r. This bias has been recognized [42] and alternatives have been proposed that reduce this bias, but without eliminating the confusion between rotation and translation. 5 Then estimation of 3D motion from normal flow amounts to minimizing [20, 21, 27] the function M nd = Z Z image I nd (r)dr: 8) Expressing r in terms of the real motion from (1) and (2) functions (6) and (8) can be expressed in terms of the actual and estimated motion parameters t, t and (or, equivalently, the actual motion parameters t; and the errors t ffl = t ....

B. K. P. Horn and E. J. Weldon, Jr. Direct methods for recovering motion. International Journal of Computer Vision, 2:51--76, 1988.

....and thus contributes false measurements for ego motion calculation; also rain drops, wiper moving in rain conditions, glare, and so forth, all contribute false measurements for egomotion recovery. To overcome these problems, first and foremost, we propose an approach based on a direct method [4,1,11,10] where each pixel contributes a measurement. These measurements are then combined in a global probability function for the parameters of the ego motion model. The direct approach has the advantage of avoiding the calculation of optic flow and in turn avoids the use of feature tracking. As a ....

....zero mean Gaussian noise we get: P #I#x#y# t# # I#x #udt#y# vdt# t #dt## #N #s 2 # 0# (13) and a maximum likelihood solution is sought. Using eq. 12) alone to find correspondences has proven to be difficult and computationaly expensive. To avoid this step we use the direct method approach of [4] (see also [1, 11, 10] Following this approach we compute the motion parameters directly from the images by combining the geometric constraints embodied in eq. 11) together with the photometric constraints (12) Given two consecutive images y#x#y# and y # #x#y#, our goal is to compute the ....

B. K. P. Horn and E. J. Weldon, Jr. Direct methods for recovering motion. International Journal of Computer Vision, 2:51--76, 1988.

.... 1a moving in a static environment with instantaneous translation t = U; V; W ) and instantaneous rotation = ff; fi; fl) measured in the coordinate system OXY Z) Then a scene point R moves with velocity (relative to the camera) R = Gammat Gamma Theta R (2) The image motion field is then [16] r = Gamma 1 (R Delta z) z Theta (t Theta r) 1 f z Theta (r Theta ( Theta r) 1 Z u tr (t) u rot ( 3) where Z is used to denote the scene depth (R Delta z) and u tr ; u rot the direction of the translational flow and the rotational flow respectively. Due to the ....

....estimates. Formally, if r is an image point, define the indicator function I nd (r) 8 : 1 for i u tr ( t) Delta n j ( r Gamma u rot ( 0 0 for i u tr ( t) Delta n j ( r Gamma u rot ( 0 : Then estimation of 3D motion from normal flow amounts to minimizing [9,10,16] the function M nd = Z Z image I nd (r)dr: 11) As before, we assume a probabilistic model for the scene with the depth values uniformly distributed between two arbitrary values Z min (or R min ) and Z max (or R max ) We further assume that the directions in which flow measurements are made ....

B. K. P. Horn and E. J. Weldon, Jr. Direct methods for recovering motion. International Journal of Computer Vision, 2:51--76, 1988.

....the use of a calibration object, using only normal flow is presented in [1] Our method uses a calibrated camera, but [1] shows that it should be possible to start with an uncalibrated camera. The first methods that used the normal flow for navigation and reconstruction were presented in [10] and [5], where the term direct methods for methods relying only on intensity derivatives was coined. As bi product of our method we will obtain a depth reconstruction of the current scene. Another method for this using the intensity gradients and also the notion of the trifocal tensor is described in ....

B. K. P. Horn and E. J. Weldon Jr. Direct methods for recovering motion. Int. Journal of Computer Vision, 2(1):51--76, June 1988.

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Horn, B.K.P. & E.J. Weldon Jr. (1988) "Direct Methods for Recovering Motion," International Journal of Computer Vision, Vol. 2, No. 1, pp. 51--76, June.

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B. Horn and E. Weldon. Direct methods for recovering motion. Int'l J. Computer Vision, pages 51--76, 1988.

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B.K.P. Horn, and E.J. Weldon, "Direct Methods for Recovering Motion," International Journal of Computer Vision, vol. 2, pp. 51-76, 1988.

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B. K. P. Horn and E. J. Weldon, "Direct Methods for Recovering Motion," International Journal of Computer Vision, vol. 2, no. 1, pp. 51-76, June 1988.

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B. K. P. Horn and E. J. Weldon, "Direct methods for recovering motion," Int. Journal of Computer Vision, vol. 2, no. 1, pp. 51--76, June 1988.

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B. K. P. Horn and E. J. Weldon, "Direct methods for recovering motion," Int. Journal of Computer Vision, vol. 2, no. 1, pp. 51--76, June 1988.

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B.K.P. Horn and E.J. Weldon. Direct methods for recovering motion. International Journal of Computer Vision, 2(1):51--76, June 1988.

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B.K.P. Horn and E. Weldon, "Direct methods for recovering motion," IJCV, pp. 51--76, 1988.

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B.K.P. Horn and Weldon E, J. Direct methods for recovering motion. International Journal of Computer Vision, 2:51-76, 1988.

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B.K.P. Horn and E.J. Weldon JR, Direct methods for recovering motion, Int. J. Computer Vision (1988).

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B.K.P. Horn and E.J. Weldon Jr., "Direct Methods for Recovering Motion," Kluwer Int'l J. Computer Vision, vol. 2, no. 1, pp. 51-76, June 1988.

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B. Horn and E. Weldon Jr. Direct methods for recovering motion. Int. Journal of Computer Vision, 2(1):51--76, 1988.

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B. K. P. Horn and E. J. Weldon, "Direct methods for recovering motion," Int. Journal of Computer Vision, vol. 2, no. 1, pp. 51--76, June 1988.

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B.K.P. Horn and Jr. Weldon, E.J. Direct methods for recovering motion. International Journal of Computer Vision, 2(1):51--77, June 1988.

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B. K. P. Horn and E. J. Weldon. Direct methods for recovering motion. International Journal of Computer Vision, 2(1):51--76, 1988.

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B. Horn and E. Weldon Jr. Direct methods for recovering motion. Int. Journal of Computer Vision, 2(1):51--76, 1988.

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B.K.P. Horn and E.J. Weldon. Direct methods for recovering motion. International Journal of Computer Vision, 2:51--76, 1988.

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B. K. P. Horn and E. J. Weldon, Direct Methods for Recovering Motion, Int. J. Computer Vision, vol 1, pp 51-76, 1988.

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B.K.P. Horn, E.J. Weldon Jr., Direct methods for recovering motion, Int. J. Comput. Vision 2 (1988) 51-77.

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B. K. P. Horn and E. J. Weldon, Jr., "Direct methods for recovering motion, " Int. J. Comput. Vis., 1988.

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B. K. Horn and E. J. Weldon. Direct methods for recovering motion. International Journal of Computer Vision, 2:51--76, 1988. REFERENCES 423

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B. K. Horn and E. J. Weldon. Direct methods for recovering motion. International Journal of Computer Vision, 2:51--76, 1988. REFERENCES 423

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