## Direction of Heading from Image Deformations (1993)

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Citations: | 25 - 3 self |

### BibTeX

@INPROCEEDINGS{Tomasi93directionof,

author = {Carlo Tomasi and Jianbo Shi},

title = {Direction of Heading from Image Deformations},

booktitle = {},

year = {1993},

pages = {422--427}

}

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### Abstract

We propose a method to compute the direction of heading from the differential changes in the angles between the projection rays of pairs of point features. These angles, the image deformations, do not depend on viewer rotation, so the key problem of separating the effects of rotation from those of translation is solved at the input. Experiments show both the feasibility of the method on real images and the advantages of using deformations rather than optical flow. 1 Introduction As we walk down a hallway, the moving images on our retinas convey enough information to determine our direction of heading. Several researchers have investigated how this direction could be computed, either in the human visual system or by a computer processing images from a moving camera. The main difficulty of this computation is to separate the effects of viewer rotation from those of viewer translation. In fact, with only translation the task would be quite simple: features in the image move toward or awa...

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Citation Context ...re continuous velocity fields or second derivatives of image motion [9] [20] [19], since we use as input the instantaneous image velocities of a set of discrete image points. The methods presented in =-=[10]-=- [18] [21] are based on the following observation: if a point p in the first image moves to q in the second, then the vectors q; Rp;T, where R is rotation and T is translation, are coplanar: q \Delta ... |

480 | Computational Geometry
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Citation Context ...arrow triangles (p 1 ; p 2 ; p i ) yielding a numerically poor system of equations. Instead, we measure angles ff for pairs of image points that are connected by the edges of a Delaunay triangulation =-=[13]-=- of all the points (see figure 5). This triangulation is a planar graph, and thus has O(n) edges. Every vertex is connected to at least two others, leading to a sufficient set of measurement equations... |

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Citation Context ...ntinuous velocity fields or second derivatives of image motion [9] [20] [19], since we use as input the instantaneous image velocities of a set of discrete image points. The methods presented in [10] =-=[18]-=- [21] are based on the following observation: if a point p in the first image moves to q in the second, then the vectors q; Rp;T, where R is rotation and T is translation, are coplanar: q \Delta (Rp \... |

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Citation Context ...as image deformations become smaller and smaller relative to noise, the uncertainty in the direction of heading grows, but no outright failure occurs. An observation by Helmholtz [7] has been used in =-=[11]-=- [14] [3] [8]: the vector difference in velocity between two points that are nearby in the image but at different depths is nearly independent of rotation (and exactly so when the points are at the sa... |

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Citation Context ...f determining pairs of image features along depth boundaries and measuring their image velocities (given the interference of the boundary) are thereby avoided. Our approach is similar to those in [2] =-=[1]-=- [12] [6] in that we minimize some residual over the measurements in the least squares sense. Like Heeger and Jepson, we reduce minimization to that of a function of two variables (the parameters for ... |

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Citation Context ... about thirty degrees and the objects in the scene are between about 50 and 100 cm away. About 230 features were automatically selected in the first frame and tracked using the algorithm described in =-=[17]-=-. Of those, 44 were handpicked to provide a roughly uniform distribution over the image. The camera was moved by a Puma arm proceeding in small steps, first along a constant reference direction (rough... |

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Citation Context ...ous velocity fields or second derivatives of image motion [9] [20] [19], since we use as input the instantaneous image velocities of a set of discrete image points. The methods presented in [10] [18] =-=[21]-=- are based on the following observation: if a point p in the first image moves to q in the second, then the vectors q; Rp;T, where R is rotation and T is translation, are coplanar: q \Delta (Rp \Theta... |

178 | Direct Passive Navigation
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Citation Context ...ning pairs of image features along depth boundaries and measuring their image velocities (given the interference of the boundary) are thereby avoided. Our approach is similar to those in [2] [1] [12] =-=[6]-=- in that we minimize some residual over the measurements in the least squares sense. Like Heeger and Jepson, we reduce minimization to that of a function of two variables (the parameters for the direc... |

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Citation Context ...s residual does not depend on d, and can be minimized with respect to t. Efficient algorithms for computing (7) are given in [5]. The minimization problem can be solved by variable projection methods =-=[4]-=- [15] on the unit sphere jtj = 1. Local minima do exist, but the more points are available in the field of view, the smoother the residual (7) turns out to be. In our experiments, we usually find one ... |

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Citation Context ...re our method with others regarding how the effects of rotation are eliminated from the images. We do not discuss methods that require continuous velocity fields or second derivatives of image motion =-=[9]-=- [20] [19], since we use as input the instantaneous image velocities of a set of discrete image points. The methods presented in [10] [18] [21] are based on the following observation: if a point p in ... |

75 |
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Citation Context ...des more gracefully: as image deformations become smaller and smaller relative to noise, the uncertainty in the direction of heading grows, but no outright failure occurs. An observation by Helmholtz =-=[7]-=- has been used in [11] [14] [3] [8]: the vector difference in velocity between two points that are nearby in the image but at different depths is nearly independent of rotation (and exactly so when th... |

57 |
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Citation Context ...sidual does not depend on d, and can be minimized with respect to t. Efficient algorithms for computing (7) are given in [5]. The minimization problem can be solved by variable projection methods [4] =-=[15]-=- on the unit sphere jtj = 1. Local minima do exist, but the more points are available in the field of view, the smoother the residual (7) turns out to be. In our experiments, we usually find one or tw... |

55 |
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Citation Context ...age deformations become smaller and smaller relative to noise, the uncertainty in the direction of heading grows, but no outright failure occurs. An observation by Helmholtz [7] has been used in [11] =-=[14]-=- [3] [8]: the vector difference in velocity between two points that are nearby in the image but at different depths is nearly independent of rotation (and exactly so when the points are at the same im... |

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Citation Context ...following bilinear system: b = A(t)d : (6) If t; d is a solution to equation (6), so is ct; d=c for any nonzero c, consistently with the fact that absolute scale cannot be recovered from images alone =-=[16]-=-. With the additional constraint that the translation t have unit norm, t is the viewer's direction of heading. 5 Solving for the direction of heading In the presence of noise, equation (6) will only ... |

25 |
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Citation Context ...ur method with others regarding how the effects of rotation are eliminated from the images. We do not discuss methods that require continuous velocity fields or second derivatives of image motion [9] =-=[20]-=- [19], since we use as input the instantaneous image velocities of a set of discrete image points. The methods presented in [10] [18] [21] are based on the following observation: if a point p in the f... |

23 | Recovering heading for visually-guided navigation
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Citation Context ...mations become smaller and smaller relative to noise, the uncertainty in the direction of heading grows, but no outright failure occurs. An observation by Helmholtz [7] has been used in [11] [14] [3] =-=[8]-=-: the vector difference in velocity between two points that are nearby in the image but at different depths is nearly independent of rotation (and exactly so when the points are at the same image loca... |

23 |
A theoretical study of optical flow
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Citation Context ...termining pairs of image features along depth boundaries and measuring their image velocities (given the interference of the boundary) are thereby avoided. Our approach is similar to those in [2] [1] =-=[12]-=- [6] in that we minimize some residual over the measurements in the least squares sense. Like Heeger and Jepson, we reduce minimization to that of a function of two variables (the parameters for the d... |

13 |
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Citation Context ...thod with others regarding how the effects of rotation are eliminated from the images. We do not discuss methods that require continuous velocity fields or second derivatives of image motion [9] [20] =-=[19]-=-, since we use as input the instantaneous image velocities of a set of discrete image points. The methods presented in [10] [18] [21] are based on the following observation: if a point p in the first ... |

5 |
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Citation Context ...eformations become smaller and smaller relative to noise, the uncertainty in the direction of heading grows, but no outright failure occurs. An observation by Helmholtz [7] has been used in [11] [14] =-=[3]-=- [8]: the vector difference in velocity between two points that are nearby in the image but at different depths is nearly independent of rotation (and exactly so when the points are at the same image ... |