A.D. Jepson and D.J. Heeger. A fast subspace algorithm for recovering rigid motion. In IEEE Workshop on Visual Motion, pages 124--131, Princeton, NJ, October 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....rotation Omega Gamma translation T , and scene depth kPk. The ego motion problem is to estimate Omega and T from a set of velocity vectors U i measured at points P i . Three well known algorithms for estimating motion are due to Bruss and Horn [2] Zhuang et al. 26] and Jepson and Heeger [5]. Although designed for a planar perspective projection, these algorithms are simple to adapt to spherical perspective projection. Bruss and Horn: In [2] Bruss and Horn derive a depth independent constraint from the planar motion field equation and then provide a least squares solution to this ....

....; 2l 6 ; k 1 ; k 2 ; k 3 ) The l i terms are functions of rotation and translation and (k 1 ; k 2 ; k 3 ) is collinear to T . Therefore the k i provide an estimate of the direction of translation. Note that for the case of planar perspective projection z = 1 and z = 0. Jepson and Heeger: In [5] and [19] Jepson and Heeger describe another linear algorithm for estimating motion. They ask the question, can a linear combination of motion vectors be found which is independent of rotation and orthogonal to translation Formally, given a set of n motion vectors at points P k , k = 1; ....

A.D. Jepson and D.J. Heeger. "A fast subspace algorithm for recovering rigid motion," Proc. of IEEE Workshop on Visual Motion, pp. 124-131, 1991.

....comparison of several flow generation algorithms. Heeger and Jepson [7] showed how the estimation of translation, rotation, and depth from flow data could be decoupled into separate sequential problems. They later reformulated the original nonlinear translation recovery method as a linear problem [8] to reduce computational expense. However, the resulting translation direction estimate was biased towards the optical axis. In [9] Jepson and Heeger performed a preliminary analysis and proposed a solution for eliminating the bias, as did MacLean and Jepson at a later date [12] Earnshaw and ....

....tested were 5 and 10 . 14 The first data set used a translational velocity of T = cm s) normalized to T = 00:4472 0 0:8944 and a rotational velocity of Omega 00:05 0:0 00:1 (rad s) Table 1 contains the mean translation estimates for the five methods: no bias compensation [8], unbiasing [3] dithering [9] whitening [12] and optimizing (with the unbiasing eigenvectors defining the search space) The error column contains the distances between the tips of the estimated and ideal translation direction vectors. As expected, the biased estimates are noticeably biased ....

A. D. Jepson and D. J. Heeger. "A fast subspace algorithm for recovering rigid motion". In IEEE Workshop on Visual Motion (1991.

....an appropriate basis for projective space and on the construction of two sets of linear constraints corresponding to certain combinations of the original epipolar constraints. This construction has been inspired by Jepson s and Heeger s linear subspace method for infinitesimal motion estimation [ Jepson and Heeger, 1991, Heeger and Jepson, 1992, Jepson and Heeger, 1992 ] and it is related to the linearized weak calibration method of Lawn and Cipolla [ Lawn and Cipolla, 1996 ] We will assume throughout this presentation that the point correspondences used as input to the weak calibration process have been ....

.... of Luong et al. 1993, 1995 ] Hartley s normalized eight point technique [ Hartley, 1995 ] and the virtual parallax algorithm of Boufama and Mohr [ 1995 ] This section also clarifies the relationship between Jepson s and Heeger s linear subspace approach to infinitesimal motion analysis [ Jepson and Heeger, 1991, Heeger and Jepson, 1992, Jepson and Heeger, 1992 ] and the Longuet Higgins characterization of epipolar geometry [ Longuet Higgins, 1981 ] Our approach to weak calibration is presented in Section 3. Its implementation is described in Section 4, where experimental results using both synthetic ....

[Article contains additional citation context not shown here]

A.D. Jepson and D.J. Heeger. A fast subspace algorithm for recovering rigid motion. In IEEE Workshop on Visual Motion, pages 124--131, Princeton, NJ, 1991.

....an appropriate basis for projective space and on the construction of two sets of linear constraints corresponding to certain combinations of the original epipolar constraints. This construction has been inspired by Jepson s and Heeger s linear subspace method for infinitesimal motion estimation [ Jepson and Heeger, 1991, Heeger and Jepson, 1992, Jepson and Heeger, 1992 ] We will assume throughout this presentation that the point correspondences used as input to the weak calibration process have been correctly established by a separate matching algorithm. Let us just mention that there is a vast literature on ....

.... of Luong et al. 1993, 1995 ] Hartley s normalized eight point technique [ Hartley, 1995 ] and the virtual parallax algorithm of Boufama and Mohr [ 1995 ] This section also clarifies the relationship between Jepson s and Heeger s linear subspace approach to infinitesimal motion analysis [ Jepson and Heeger, 1991, Heeger and Jepson, 1992, Jepson and Heeger, 1992 ] and the Longuet Higgins characterization of epipolar geometry [ Longuet Higgins, 1981 ] Our approach to weak calibration is presented in Section 3. Its implementation is described in Section 4, where experimental results using both synthetic ....

[Article contains additional citation context not shown here]

A.D. Jepson and D.J. Heeger. A fast subspace algorithm for recovering rigid motion. In IEEE Workshop on Visual Motion, pages 124--131, Princeton, NJ, 1991.

....initial estimates of the translation and rotation. The rst is the standard linear 8 point algorithm [20] as improved by Hartley [7] It can deal with motions of any size but works best for large motions [32] The second is an improved version [28] 21] of the linear subspace technique of [14][15], which essentially eliminates the earlier techniques s bias toward recovering e within the FOV. It can deal with translations of any size but requires small to moderate rotations. For small motions, it in e ect minimizes the exact projective least squares error and, thus, should have less bias ....

A.D. Jepson and D.J. Heeger, \A fast subspace algorithm for recovering rigid motion," Motion Workshop , Princeton, N.J., 124-131, 1991.

....temporal coherence complement one another in many cases, but not always. The temporal coherence does however help ll in small holes where aperture problem occurs in textured patches. This may be important for egomotion methods that require dense ow (e.g. the convolution form of subspace methods [15]) Finally, Figure 5 shows the ow elds computed from the synthetic sequences using a third order pre lter with 1 1 = 1:25, the quantitative results for which are given in the second row of Table 3. Figure 6 shows results of the same lters on two real image sequences that were also used by ....

Jepson, A. and Heeger, D. (1991) A fast subspace algorithm for recovering rigid motion. Proc. IEEE Workshop on Visual Motion, Princeton, pp. 124-131

....rotation Omega Gamma translation T , and scene depth kPk. The ego motion problem is to estimate Omega and T from a set of velocity vectors U i measured at points P i . Three well known algorithms for estimating motion are due to Bruss and Horn [2] Zhuang et al. 26] and Jepson and Heeger [5]. Although designed for a planar perspective projection, these algorithms are simple to adapt to spherical perspective projection. Bruss and Horn: In [2] Bruss and Horn derive a depth independent constraint from the planar motion field equation and then provide a least squares solution to this ....

....; k 1 ; k 2 ; k 3 ) T : The l i terms are functions of rotation and translation and (k 1 ; k 2 ; k 3 ) T is collinear to T . Therefore the k i provide an estimate of the direction of translation. Note that for the case of planar perspective projection z = 1 and z = 0. Jepson and Heeger: In [5] and [19] Jepson and Heeger describe another linear algorithm for estimating motion. They ask the question, can a linear combination of motion vectors be found which is independent of rotation and orthogonal to translation Formally, given a set of n motion vectors at points P k , k = 1; ....

A.D. Jepson and D.J. Heeger. "A fast subspace algorithm for recovering rigid motion," Proc. of IEEE Workshop on Visual Motion, pp. 124-131, 1991.

....of an appropriate basis for projective space and on the construction of two sets of linear constraints corresponding to certain combinations of the original epipolar constraints. This construction has been inspired by Jepson s and Heeger s linear subspace method for infinitesimal motion estimation (Jepson and Heeger 1991, Heeger and Jepson 1992, Jepson and Heeger 1992) and it is related to the linearized weak calibration method of Lawn and Cipolla (1996) We will assume throughout this presentation that the point correspondences used as input to the weak calibration process have been correctly established by a ....

.... calibration of Luong et al. 1993, 1996) Hartley s normalized eight point technique (Hartley 1995) and the virtual parallax algorithm of Boufama and Mohr (1995) This section also clarifies the relationship between Jepson s and Heeger s linear subspace approach to infinitesimal motion analysis (Jepson and Heeger 1991, Heeger and Jepson 1992, Jepson and Heeger 1992) and the Longuet Higgins characterization of epipolar geometry (Longuet Higgins 1981) Our approach to weak calibration is presented in Section 3. Its implementation is described in Section 4, where experimental results using both synthetic and real ....

[Article contains additional citation context not shown here]

Jepson, A. and Heeger, D.: 1991, A fast subspace algorithm for recovering rigid motion, IEEE Workshop on Visual Motion, Princeton, NJ, pp. 124--131.

....an appropriate basis for projective space and on the construction of two sets of linear constraints corresponding to certain combinations of the original epipolar constraints. This construction has been inspired by Jepson s and Heeger s linear subspace method for infinitesimal motion estimation [11, 13]. 2 The Approach In [11, 13] Jepson and Heeger propose a linear subspace method for infinitesimal motion estimation, which can be summarized as follows: suppose that a moving perspective camera with unit focal length observes some scene. The motion field is u v = Gamma1 0 u 0 ....

....space and on the construction of two sets of linear constraints corresponding to certain combinations of the original epipolar constraints. This construction has been inspired by Jepson s and Heeger s linear subspace method for infinitesimal motion estimation [11, 13] 2 The Approach In [11, 13], Jepson and Heeger propose a linear subspace method for infinitesimal motion estimation, which can be summarized as follows: suppose that a moving perspective camera with unit focal length observes some scene. The motion field is u v = Gamma1 0 u 0 Gamma1 v ( 1 z t ....

A.D. Jepson and D.J. Heeger. A fast subspace algorithm for recovering rigid motion. In IEEE Workshop on Visual Motion, pages 124--131, Princeton, NJ, 1991.

....the projective structure itself than does a Euclidean estimate (i.e. an estimate based on an assumed calibration) derived for the incorrect calibration parameters. Our approach for the constant translation direction case has some connection with previous work on optical flow by Heeger and Jepson [3, 5, 7, 4]. The new features of our work include a generalization of their technique for canceling first order rotation and the analysis of the finite rotation case [11] More importantly, this paper includes the extension of their work to non infinitesimal translational motion, the recovery not just of ....

.... has the familiar optical flow form: f(R h ; p h i ) h y ; Gamma h x ) t ( Gammay i ; x i ) t h z Gamma b i ( h x y i Gamma h y x i ) 8) The second stage of our algorithm determines the translation direction using a generalization of an earlier optical flow technique [3, 5, 7] to cancel the first order residual rotation (8) Note that though [3, 5, 7] specifically deal with infinitesimal motion, we can recover the direction of translation direction here for finite motion with no further approximation. Multiply (2) for each feature point by i j T z ( Gammay i ; x i ....

[Article contains additional citation context not shown here]

A.D. Jepson and D.J. Heeger, "A fast subspace algorithm for recovering rigid motion," Motion Workshop , Princeton, N.J., 124-131, 1991.

....defined by the first n Gamma 1 rows of H an is a rank n Gamma 1 matrix annihilating n. Matrices annihilating multiple vectors can be computed by taking appropriate products of matrices derived in this way. The linear elimination of rotation dependence was first proposed by Heeger and Jepson [9, 7, 11] in the quite different context of recovering motion from optical flow. However, their technique did not extend simply to annihilating general vectors or to sparse flows[10] whereas our Householder based method does. Denote the (2M Gamma 3) Theta 2M matrix annihilating V x ; V y ; V z ....

A.D. Jepson and D.J. Heeger, "A fast subspace algorithm for recovering rigid motion," Motion Workshop , Princeton, N.J., 124-131, 1991.

....the observed optical flow data. Heeger and Jepson [4] illustrated how the estimation of translation, rotation, and depth could be decoupled into separate problems which could be solved in order. They later reformulated the original iterative translation recovery method as a noniterative problem [6]. However, the resulting translation direction estimates tended to be biased towards the optical axis. Jepson and Heeger [7] performed a preliminary bias analysis and proposed a solution for eliminating it. This paper presents a detailed error analysis of the bias which includes alternative ....

A. D. Jepson and D. J. Heeger. "A fast subspace algorithm for recovering rigid motion". In IEEE Workshop on Visual Motion (1991), pages 124--131, 1991.

....pairs. In almost all cases, either optical flow or feature points correspondence are used as the initial measurements. In the first case, some inherent problems (aperture, large motions, etc. related to optical flow computation, suggests that errors can never be lowered to a negligible level (see [1, 2, 3, 4]) Even methods using the intensity derivatives directly or normal flow, as in [11, 12, 8, 4, 5, 6, 7] suffer from high noise sensitivity. For feature based methods, the reliable selection and tracking of meaningful feature points is generally very difficult, see [8, 9, 10] All prior methods of ....

A. D. Jepson and D. J. Heeger. A fast subspace algorithm for recovering rigid motion. In Proc. IEEE Workshop on Visual Motion, pages 124--131, Princeton, NJ, 1991.

....images that have much more than one pixel displacement [36] This chapter is divided into two parts. In the first part we study analytically motion and geometry configurations that yield noise sensitive 3D information. We present our results and compare them with a state of the art approach [19, 37]. In the second part, we try to overcome some of the instabilities of the first part by paying special attention on the statistical terms of bias and variance. Then, we unify our approach by treating both point and line correspondences in the same way. Last, we analyze the case of recovering the ....

....the Focus of Expansion. 12 Kostas Daniilidis and Minas E. Spetsakis 2.2. The bias towards the viewing direction In this section we are going to prove the bias in the estimated translation direction towards the viewing direction if the field of view is small. This bias was already observed in [47, 48, 49, 37]. We will show here using the arguments in [39, 11] that this bias can be eliminated if the error metric is derived by the statistical analysis in the first part of the chapter. The bias affected error metrics are those derived directly from the epipolar constraint in its discrete form x 2 T (T ....

[Article contains additional citation context not shown here]

A. Jepson and D.J. Heeger. A fast subspace algorithm for recovering rigid motion. In Proc. IEEE Workshop on Visual Motion, pages 124--131, Princeton, NJ, Oct. 7-9, 1991. 28 Kostas Daniilidis and Minas E. Spetsakis

.... estimation, on the other hand, is a numerically stable problem, because the 2D problem is highly overdetermined (only six unknowns in the affine model, eight unknowns in the projective model) Previous works on 3D motion estimation use the optical or normal flow field derived between two frames [1, 2, 8, 17, 20, 21], or the correspondence of previously extracted distinguished features (points, lines, contours) 10, 22] Methods for computing the ego motion directly from image intensities were also suggested [9, 11, 23] but each method has its limitations. In this section we propose the following scheme in ....

A.D. Jepson and D.J. Heeger. A fast subspace algorithm for recovering rigid motion. In IEEE Workshop on Visual Motion, pages 124--131, Princeton, NJ, October 1991.

....(MFSFM) for point features described here may be seen as a generalization of the earlier work by Tomasi [16] to the case of full perspective for instance, our approach works well on the Rocket Field sequence, where perspective effects are crucial. It also relates to the work of Heeger and Jepson [6, 4, 7, 5, 15] on recovering translational motion from optical flow, yielding a simple algorithm for recovering translation from sparse as well as dense optical flow. Our motivation is to develop methods for MFSFM that are approximate but make effective use of the information available and are fast. They are ....

....of zeros. Since V x ; V y ; V z are known, we design a measurement matrix from D which is rank 2 with right singular vectors given by just (7) This is done by postmultiplying D by a rank 2M Gamma3 matrix annihilating V x ; V y ; V z . Heeger and Jepson used this basic trick [6, 4, 7] in the context of recovering motion from optical flow, but their technique did not extend to annihilating general vectors or to sparse flows. Our extension is based on Householder matrices. The Householder matrix [3] H ab is an orthogonal matrix that takes a to b by a reflection. With n j (0; 0 ....

A.D. Jepson and D.J. Heeger, "A fast subspace algorithm for recovering rigid motion, " Motion Workshop , Princeton, N.J., 124-131, 1991.

....uses the observed optical flow data. Heeger and Jepson [4] illustrated how the estimation of translation, rotation, and depth could be decoupled into separate problems which could be solved in order. They later reformulated the original nonlinear translation recovery method as a linear problem [5]. However, the resulting translation direction estimates tended to be biased towards the optical axis. Jepson and Heeger [6] performed a preliminary bias analysis and proposed a solution for eliminating it. Earnshaw and Blostein [1] 2] provided an alternate explanation and bias compensation ....

A. D. Jepson and D. J. Heeger. "A fast subspace algorithm for recovering rigid motion". In IEEE Workshop on Visual Motion (1991), pages 124--131, 1991.

.... assuming a limited model of the world [Adi85] restricting the range of possible motions [HW88] or assuming some type of temporal motion constancy over a longer sequence [DF90] 3D motion is often estimated from the optical or normal flow field derived between two frames [Adi85, LR83, HJ90, GU91, JH91, Sun91, NL91, HA91, TS91, AD92, DRD93] or from the correspondence of distinguished features (points, lines, contours) previously extracted from successive frames [FLT87, Hor90, COK93] Both approaches depend on the accuracy of the pre processing stage, which can not always be assured [WHA89] ....

A.D. Jepson and D.J. Heeger. A fast subspace algorithm for recovering rigid motion. In IEEE Workshop on Visual Motion, pages 124--131, Princeton, NJ, October 1991.

....from point correspondences is called weak calibration. The synthesis of new views of a scene from a set of pre recorded pictures is called image based rendering. In this thesis, we investigate these two problems and develop algorithms to solve them. It has been shown by Jepson and Heeger in [43, 50] that motion estimation for calibrated cameras undergoing in nitesimal displacements can be reduced to a linear problem. A primary objective of this thesis was to answer the following question: is it possible to generalize Jepson s and Heeger s method to the nite motion case As we will see in ....

....are available (see Figure 1.2 for an example) 1.1 Thesis Organization In Chapter 2, the main elements of epipolar geometry are summarized and various approaches to weak calibration are reviewed. This chapter also clari es the relationship between Jepson s and Heeger s linear subspace approach [50, 43, 51] to in nitesimal motion analysis and the Longuet Higgins characterization of epipolar geometry. 3 ###### ### Example image synthesis: 30 images (top 6 rows) are used to obtain the synthesized image for a novel view (larger image) 4 In Chapter 3, weintroduce a linear algorithm for estimating ....

[Article contains additional citation context not shown here]

A.D. Jepson and D.J. Heeger. A fast subspace algorithm for recovering rigid motion. In IEEE Workshop on Visual Motion, pages 124-131, Princeton, NJ, 1991.

....# ## # # ## # denotes the unknown velocity vector in homogeneous coordinates. Uncertainty is caused by noise in measuring image derivatives and by violations of brightness constancy. Linear constraint equations also arise in subspace methods for estimating the 3D direction of camera translation [9, 10]. In this case local optical flow vectors are combined linearly to obtain a set of constraints, ## , that satisfy ## # T # # # (2) where T is the unknown camera translation. Because of uncertainty in optical flow, the subspace constraint measurements, ## , will be noisy. Previous approaches ....

....in optical flow affects all components of the 3D measurement vector, ## . In these cases, with linear constraints and noise in all components of the measurement vector, total least squares (TLS) 5] has been used as an estimator, both for optical flow [21, 1] and 3D translation direction [9, 15]. The TLS estimator is a ML estimator for independent and identically distributed (IID) additive Gaussian noise [4, 20] However, the TLS formulation does not provide a likelihood function. Previous approaches to obtain this likelihood function, like error in variables [4] introduced the true ....

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A. D. Jepson and D. J. Heeger. A fast subspace algorithm for recovering rigid motion. In Proceedings IEEE Workshop on Visual Motion, pages 124--131, Princenton, 1991.

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A.D. Jepson and D.J. Heeger. A fast subspace algorithm for recovering rigid motion. In IEEE Workshop on Visual Motion, pages 124--131, Princeton, NJ, October 1991.

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A.D. Jepson and D.J Heeger. A fast subspace algorithm for recovering rigid motion. In Proceedings of the IEEE Workshop on Visual Motion, volume ix+349, pages 124--31, October 1991.

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A. D. Jepson and D. J. Heeger. "A fast subspace algorithm for recovering rigid motion". In IEEE Workshop on Visual Motion (1991.

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A. D. Jepson and D. J. Heeger. "A fast subspace algorithm for recovering rigid motion". In IEEE Workshop on Visual Motion (1991.

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A.D. Jepson and D.J. Heeger, "A fast subspace algorithm for recovering rigid motion," Motion Workshop , Princeton, N.J., 124-131, 1991.

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