Representation of three-dimensional object structure as cross-ratios of determinants of stereo image points

Barrett, EC; Gheen, Gregory; Payton, Paul Max

doi:10.1007/3-540-58240-1_3

Cited by 12 publications

(7 citation statements)

References 4 publications

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“…It is an effective tool for solving problems in several major topics of Computer Vision [1,14,17,6]. The major contribution of this paper is the introduction of an affine trifocal tensor.…”

Section: Discussionmentioning

confidence: 99%

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Analysis and Computation of an Affine Trifocal Tensor

Mendonça¹,

Cipolla²

1998

Procedings of the British Machine Vision Conference 1998

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This paper investigates the trifocal tensor for an affine trinocular rig and defines an affine trifocal tensor. The question of the degrees of freedom of the tensor entries will be addressed, and a novel algorithm to compute the tensor by a linear technique is presented. It will be shown that 4 point or 8 line correspondences along the three images are enough for a reliable computation of the trifocal tensor in the affine case (uncalibrated, weak perspective camera model), in contrast to the projective case, where at least 7 point or 13 line correspondences are needed for a linear computation (and usually with poor results). An analysis of the error in the approximation of a generic trifocal tensor by the affine trifocal tensor is carried out and preliminary experiments with synthetic and real data show the reliability and robustness of the approximation under a wide range of conditions.

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“…It is an effective tool for solving problems in several major topics of Computer Vision [1,14,17,6]. The major contribution of this paper is the introduction of an affine trifocal tensor.…”

Section: Discussionmentioning

confidence: 99%

“…The trifocal tensor is an important tool for several tasks in Computer Vision, like reconstruction [1,14], self-calibration [6] and motion segmentation [17], and its computation is still a research topic [5,18]. The linear algorithms [10,9] have simple conception and execution, but they do not take the appropriate constraints into account.…”

Section: Introductionmentioning

confidence: 99%

Analysis and Computation of an Affine Trifocal Tensor

Mendonça¹,

Cipolla²

1998

Procedings of the British Machine Vision Conference 1998

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“…That I (ι G ), defined in Eq. (20), is invariant under translations of the contour C is trivial, since, as defined in Section 4.2, ι G is calculated with the origin at the centroid of the contour C. Rotation and dilation invariance can be proved by construction. Since the transformations for rotation and dilation in the plane commute, we can consider a twoparameter Abelian group corresponding to a rotation by an angle φ and a dilation by a positive factor β.…”

Section: Theorem 1 the Invariance Measure Density I (ι G ) Is Invamentioning

confidence: 97%

“…Since projective transformations are many-to-one, matching cross-ratios are a necessary, but not sufficient, condition for feature-matching. Cross-ratios are used in many recognition problems involving projective geometry [19][20][21] and can be defined on the basis of four collinear points or four concurrent lines [22]. In practice, this means that such labeled reference points or lines must be extracted from the image before the technique can be applied.…”

Section: Cross-ratiosmentioning

confidence: 99%

Invariance Signatures: Characterizing Contours by Their Departures from Invariance

Squire

Caelli

2000

Computer Vision and Image Understanding

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In this paper, a new invariant feature of two-dimensional contours is reported: the invariance signature. The invariance signature is a measure of the degree to which a contour is invariant under a variety of transformations, derived from the theory of Lie transformation groups. It is shown that the invariance signature is itself invariant under shift, rotation, and scaling of the contour. Since it is derived from local properties of the contour, it is well-suited to a neural network implementation. It is shown that a model-based neural network (MBNN) can be constructed which computes the invariance signature of a contour and classifies patterns on this basis. Experiments demonstrate that invariance signature networks can be employed successfully for shift-, rotation-, and scale-invariant optical character recognition.

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“…where w (2) is introduced in (18) and denotes the template around x (2) . Applying the Bayes' rule to this likelihood term results in:…”

Section: Quality Of Tracked Featurementioning

confidence: 99%

Good features to track: A view geometric approach

Jiang

Yilmaz

2011

2011 IEEE International Conference on Computer Vision Workshops (ICCV Workshops)

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Despite many alternatives to point tracking problem, iterative least squares solution solving the optical flow constraint has been the most popular approach used by many in the field. This paper attempts to leverage the former efforts to enhance such tracking methods by introducing geometric constraint to the tracking problem. In contrast to alternative geometry based methods, the proposed approach provides an online solution to optical flow estimation by exploiting Horn and Schunck flow estimation regularized by view geometric constraints. We particularly use the geometric invariants in the projective coordinates and conjecture that the traditional appearance solution needs to be geometrically regularized. Geometric regularization is achieved by estimating the posterior distribution of possible locations given the samples drawn from other tracked points which specify the scene geometry. Our experiments demonstrate that tracking can still be performed with great accuracy even when the features undergo appearance changes due to projective deformation of the template. Hence the resulting algorithm judges the quality of the tracked points based on not only appearance similarity but also geometric consistency.

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Representation of three-dimensional object structure as cross-ratios of determinants of stereo image points

Cited by 12 publications

References 4 publications

Analysis and Computation of an Affine Trifocal Tensor

Analysis and Computation of an Affine Trifocal Tensor

Invariance Signatures: Characterizing Contours by Their Departures from Invariance

Good features to track: A view geometric approach

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