Muhlich, M.; Mester, R.: The Role of Total Least Squares in Motion Analysis. Proc. ECCV'98, pp. 305-321.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Fondation (DFG) through the research unit Image Sequence Processing to Investigate Dynamic Processes . lead to biases in the estimates, as all gradients are generally obscured by noise [14] Under these circumstances the use of a total least squares (TLS) method [20] is the estimator of choice [16]. The local constraints of gradient based optical flow techniques do generally not incorporate brightness changes. This can of course only be a first approximation, as brightness changes due to inhomogeneous or fluctuating illumination prevail in real world scenes. Moreover, in scientific ....

M. Mhlich and R. Mester. The role of total least squares in motion analysis. In ECCV, pages 305--321, Freiburg, Germany, 1998.

....axes X Z. Translation parallel to rotation axis: Figures 6 and 7 show the experiments performed for our algorithm when translation is parallel to the axis of rotation. The non isotropic normalization procedure proposed by Hartley [2] and statistically justified by Muhlich and Mester [12] was used to Although in this paper we do not outline the algorithm, it should be clear from Section 3.2. For specifying the Rotation Translation axes, we simply use symbols such as XY Y Y ZZ which means: for the 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 5 10 15 20 25 30 ....

....of noise in the estimation of the calibration matrix for T=R = 1 and a rotation of = 20 between consecutive frames. It can be seen that the normalization procedure improves the estimation of the calibration matrix, but the improvement is not significant. This result is consistent with that of [12], since the effect of normalization is more important for large noise levels. On the other hand, the performance of the algorithm is not as good as that of the pure rotation case, but still an error of 5 is reached for a noise level of 2 pixels. Figure 7 shows the effect of the angle of rotation ....

M. Muhlich and R. Mester. The role of total least squares in motion analysis. In Proceedings of European Conference on Computer Vision, pages 305--321, 1998.

.... extended, in particular by the application of geometric invariance principles [see for example 12 and 13] and development of the tri and quadrifocal tensors [14 16] but it appears that a subtle point, first noticed recently in computer vision in the context of estimating the fundamental matrix [17], the way image measurement errors are treated has not been addressed. The aim of this paper is to show how, for the approximate linear morphing relationships most frequently used in practice, a correct error analysis may be carried out and to evaluate the resulting improvements in the geometry ....

....containing the constants a 5 and b 5 , and these errors may be correlated. The technique capable of dealing with all such problems is the generalised total least squares method [22] introduced recently in computer vision research to deal with similar problems in calculating the fundamental matrix [17]. Its application is similar to that of the total least squares method used here, except that it leads to a generalised eigenproblem [23] whereas the total least squares requires only a conventional eigenproblem or singular value decomposition. Since the total least squares method is the more ....

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Mhlich M. and Mester R. (1998) "The Role of Total Least Squares in Motion Analysis", In Proceedings of 5th European conference on Computer Vision, ECCV'98, Springer Verlag, pp 305-321.

....to observe the performance of our algorithm and illustrate the reliability evaluation process. 1. Introduction Computing the fundamental matrix from corresponding points over two images is one of the basic elements of computer vision. Already, many algorithms have been presented for this purpose [1, 3, 6, 7, 13, 18, 22, 23, 24, 25]. These are roughly classified into two approaches: the bundleadjustment and the linear algorithm. The bundle adjustment is based on maximal likelihood estimation after reprojection: we assume all the parameter values, predict where the image data should ideally be observed, and determine the ....

M. Muhlich and R. Mester, The role of total least squares in motion analysis, Proc. 5th Euro. Conf. Comput. Vision , June 1998, Freiburg, Germany, pp. 305-- 321.

....problems in computer vision may be couched in terms of parameter estimation. Accordingly, much effort has gone into the development of sophisticated techniques for generating estimates of parameters. Some of these techniques utilise covariance information characterising uncertainty in the data [1, 2]. This paper is concerned with the application of a recently introduced covariance based method [3] to the problem of estimating the fundamental matrix (see also [1, 4 10] However, we assume here that, as is often the case, covariance information is unavailable. A 3D point in a scene ....

....generating estimates of parameters. Some of these techniques utilise covariance information characterising uncertainty in the data [1, 2] This paper is concerned with the application of a recently introduced covariance based method [3] to the problem of estimating the fundamental matrix (see also [1, 4 10]) However, we assume here that, as is often the case, covariance information is unavailable. A 3D point in a scene perspectively projected onto the image plane of a camera gives rise to an image point represented by a pair (m 1 ; m 2 ) of coordinates, or equivalently, by the vector m = m 1 ; m ....

M. Muhlich and R. Mester, "The role of total least squares in motion analysis," in Proc. 5th European Conference on Computer Vision, Freiburg, Germany. 1998, vol. 1407 of Lecture Notes in Computer Science, pp. 305--321, Springer-Verlag, Berlin.

....and as such may fail to have a non zero solution. In this situation, we are concerned with finding that best fits the data in some sense. The form of this vision problem involving (known) covariance information was first studied in detail by Kanatani [12] and later by various others (see, e.g. [4, 13,14,20, 21]) Conic fitting is one problem of this kind [2, 23] Two other conformant problems are estimating the coefficients of the epipolar equation [6] and estimating the coefficients of the differential epipolar equation [3, 22] Each of these problems involves an 2 ancillary cubic constraint. The ....

M. Muhlich and R. Mester, The role of total least squares in motion analysis, Computer Vision---ECCV'98, Fifth European Conference on Computer Vision (Freiburg, Germany, June 2--6, 1998) (H. Burkhardt and B. Neumann, eds.), Lecture Notes in Computer Science, vol. 1407, Springer, Berlin, 1998, pp. 305--321.

....the covariance matrices by a common scalar factor. As a consequence, the fundamental numerical scheme is elegant, computationally efficient and accurate. 5 Experimental results The fundamental numerical scheme and other algorithms were tested on the problem of estimating the fundamental matrix [6, 8 10, 13, 15, 18, 20 23]. Tests reported here are synthetic as these permit precise control of the conditions under which performance can be evaluated. Results obtained with real images are presented in later work. It should be noted that, in estimating the fundamental matrix, no special knowledge of the domain was ....

M. Muhlich and R. Mester, The role of total least squares in motion analysis, Computer Vision---ECCV'98 (Fifth European Conference on Computer Vision, Freiburg, Germany, June 2--6, 1998) (H. Burkhardt and B. Neumann, eds.), Lecture Notes in Computer Science, vol. 1407, Springer, Berlin, 1998, pp. 305--321. 11

....axes X Z. Translation parallel to rotation axis: Figures 6 and 7 show the experiments performed for our algorithm 8 when translation is parallel to the axis of rotation. 9 The non isotropic normalization procedure proposed by Hartley [2] and statistically justified by Muhlich and Mester [12] was used to 8 Although in this paper we do not outline the algorithm, it should be clear from Section 3.2. 9 For specifying the Rotation Translation axes, we simply use symbols such as XY Y Y ZZ which means: for the 12 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 5 10 15 20 25 30 35 ....

....noise in the estimation of the calibration matrix for T=R = 1 and a rotation of = 20 o between consecutive frames. It can be seen that the normalization procedure improves the estimation of the calibration matrix, but the improvement is not significant. This result is consistent with that of [12], since the effect of normalization is more important for large noise levels. On the other hand, the performance of the algorithm is not as good as that of the pure rotation case, but still an error of 5 is reached for a noise level of 2 pixels. Figure 7 shows the effect of the angle of rotation ....

M. Muhlich and R. Mester. The role of total least squares in motion analysis. In Proceedings of European Conference on Computer Vision, pages 305--321, 1998.

....Figure 5: Rotation axes X Y , oe = 2. Translation parallel to rotation axis: Figures 6 and 7 show the experiments performed when translation is parallel to the axis of rotation. 10 The non isotropic normalization procedure proposed by Hartley [7] and statistically justified by Muhlich and Mester [23] was used to estimate the fundamental matrix. Figure 6 shows the effect of noise in the estimation of the calibration matrix for T=R = 1 and a rotation of = 20 o between consecutive frames. It can be seen that the normalization procedure improves the estimation of the calibration matrix, but ....

....noise in the estimation of the calibration matrix for T=R = 1 and a rotation of = 20 o between consecutive frames. It can be seen that the normalization procedure improves the estimation of the calibration matrix, but the improvement is not significant. This result is consistent with that of [23], since the effect of normalization is more important for large noise levels. On the other hand, the performance of the algorithm is not as good as that of the pure rotation case, but still an error of 5 is reached for a noise level of 2 pixels. Figure 7 shows the effect of the angle of rotation ....

M. Muhlich and R. Mester. The role of total least squares in motion analysis. In Proceedings of European Conference on Computer Vision, pages 305--321, 1998.

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Muhlich, M.; Mester, R.: The Role of Total Least Squares in Motion Analysis. Proc. ECCV'98, pp. 305-321.

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Muhlich, M.; Mester, R.: The Role of Total Least Squares in Motion Analysis. Proc. European Conference on Computer Vision (ECCV) 1998.

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Muhlich, M., Mester, R.: The role of total least squares in motion analysis. In: Proc. Europ. Conf. Comp. Vision. (1998)

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Muhlich, M., Mester, R.: The role of total least squares in motion analysis. In: Proc. Europ. Conf. Comp. Vision. (1998)

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M. Muhlich and R. Mester. The role of total least squares in motion analysis. In Proc. Europ. Conf. Comp. Vision, 1998.

....i x # i x # i = NI. After the computation of the trifocal tensor T or the essential matrix E, this normalization transformation has to be reversed to get the real motion parameters. This heuristically introduced transformation has been justified and improved in two papers by Muhlich and Mester [5, 4]. These methods are not really linear as they are based on the calculation of eigenvectors which is certainly no linear operation. Another two view method involves the non linear minimization of the sum of squared distances between image points and their corresponding epipolar lines. The ....

....R12 instead of R12 . Our requirement that all three translation vectors must lie in the same plane is equivalent to the statement that they are linearly dependent. This means that the matrix M defined as M = t 12 , t , t 13 ) 14) must be singular. The Eckard Young Mirsky theorem (see e.g. [5]) solves this problem using the singular value decomposition (SVD) If we assume that the error distribution in all vectors is equal and if we neglect the e#ect of the transformation of t 23 , all three vectors should be scaled to the same length, e.g. to unit vectors. The columns of the ....

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M. Muhlich and R. Mester. The Role of Total Least Squares in Motion Analysis. in Proc. ECCV'98, Freiburg, pages 305--321, 1998.

....to be the eigenvector of C g corresponding to the smallest eigenvalue. This type of problem appears in other orientation estimation schemes (cf. JAE95] the degree of orientedness can be derived from the eigenvalues of C g . Additionally, we haveastrongrelationtothetotal least squares problem [MUEH98], where a detailed analysis of the error structure of the considered measurement matrix (here: C g ) is crucial for obtaining precise and unbiased results [MUEH00] 3.2 Estimation in the Fourier spectrum Using Parseval s theorem, one can easily show that criterion (3) is equivalent to the ....

Muhlich, M.; Mester, R.: The Role of Total Least Squares in Motion Analysis. Proc. ECCV'98, pp. 305-321.

....matrices W L and W R . We can use the equilibration technique to exploit non i.i.d. error structures because equilibration adjusts the error metric. The TLS estimate #x # of the equilibrated matrix A # has to be multiplied by W R in order to get the equilibrated TLS solution: #x = W R #x # [6]. The matrices W R and W L canbeusedtohandle di#erent variances and covariances in the columns or 1 The TLS problem is often formulated as A#x # # b. The vector # b can always be added to A as a further column vector when the element 1 is appended to #x. Then we obtain our homogeneous ....

....A T A, but the SVD is numerically more stable. 3 This case is also handled by Demmel in [2] rows resp. of the error matrix D. Note that equilibrating with a diagonal matrix containing arbitrary small entries # is a convenient way to handle constant (i.e. error free) rows or columns. In [6] it has been shown that the requirement E D T D = c I with an arbitrary constant c must hold to ensure unbiased estimates for the right singular values, i.e. an unbiased TLS solution. 4 Aright multiplication with W R is su#cient to achieve this. With the transformation D # = DW R we ....

Muhlich, M.; Mester, R.: The Role of Total Least Squares in Motion Analysis. Proc. ECCV'98, Springer, 1998.

....out that many computer vision problems are in fact Total Least Squares (TLS) problems. For this reason we give a short overview on TLS here. A very comprehensive and detailed description can be found in [VHU91] and a special treatment of TLS in the context of vision problems can be found in [MUE98]. Readers who are familiar with the basic TLS concept may skip this section without any loss. Definition 1 (Total Least Squares Problem) Given an overdetermined set of N equations #a T #x =0with M unknowns x i , compiled to a matrix equation A#x = # 0, our task is to estimate #x when all ....

....distance #A#x# 2 (for a proof of this equivalence, see [MUE99a] It is given by the right hand singular vector of A corresponding to the smallest singular value. 3 Muhlich extensively discusses the requirement to have an i.i.d. error structure in matrix A for obtaining unbiased TLS estimates [MUE98]. 4 Equilibration If the error matrix of the TLS problem has no i.i.d. structure, for instance if the errors are correlated or have di#erent variances, an improved estimate of vector #x can be obtained using the technique of equilibration which essentially consists in replacing the metric #A ....

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Muhlich, M.; Mester, R.: The role of total least squares in motion analysis. Proc. ECCV'98, Springer, 1998, pp.305-321.

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M. Muhlich, R. Mester, The role of total least squares in motion analysis, Proceedings of European Conference on Computer Vision, Freiburg, Germany (1998) 305 -- 321.

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Muhlich, M., Mester, R.: The role of total least squares in motion analysis. In: ECCV, Freiburg, Germany (1998) 305--321

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M. Muhlich and R. Mester, "The role of total least squares in motion analysis," in Computer Vision---ECCV'98, Fifth European Conference on Computer Vision, H. Burkhardt and B. Neumann, eds., vol. 1407 of Lecture Notes in Computer Science, pp. 305--321, Springer, Berlin, (Freiburg, Germany, June 2--6, 1998), 1998.

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M. Muhlich and R. Mester, "The role of total least squares in motion analysis, " in Computer Vision---ECCV'98, Fifth European Conference on Computer Vision, H. Burkhardt and B. Neumann, Eds., Freiburg, Germany, June 2--6, 1998, 1998, vol. 1407 of Lecture Notes in Computer Science, pp. 305--321, Springer, Berlin.

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