## Linear Pushbroom Cameras (1994)

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Venue: | IEEE Transactions on Pattern Analysis and Machine Intelligence |

Citations: | 142 - 5 self |

### BibTeX

@ARTICLE{Hartley94linearpushbroom,

author = {Richard I. Hartley and Rajiv Gupta},

title = {Linear Pushbroom Cameras},

journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},

year = {1994},

volume = {19},

pages = {963--975}

}

### Years of Citing Articles

### OpenURL

### Abstract

Modelling th# push broom sensors commonly used in satellite imagery is quite di#cult and computationally intensive due to th# complicated motion ofth# orbiting satellite with respect to th# rotating earth# In addition, th# math#46 tical model is quite complex, involving orbital dynamics, andh#(0k is di#cult to analyze. Inth#A paper, a simplified model of apush broom sensor(th# linear push broom model) is introduced. Ith as th e advantage of computational simplicity wh#A9 atth# same time giving very accurate results compared with th# full orbitingpush broom model. Meth# ds are given for solving th# major standardph# togrammetric problems for th e linear push broom sensor. Simple non-iterative solutions are given for th# following problems : computation of th# model parameters from groundcontrol points; determination of relative model parameters from image correspondences between two images; scene reconstruction given image correspondences and ground-control points. In addition, th# linearpush broom model leads toth#0 retical insigh ts th# t will be approximately valid for th# full model as well.Th# epipolar geometry of linear push broom cameras in investigated and sh own to be totally di#erent from th at of a perspective camera. Neverth eless, a matrix analogous to th e essential matrix of perspective cameras issh own to exist for linear push broom sensors. Fromth#0 it is sh# wn th# t a scene is determined up to an a#ne transformation from two viewswith linearpush broom cameras. Keywords :push broom sensor, satellite image, essential matrixph# togrammetry, camera model The research describ ed in this paper hasb een supportedb y DARPA Contract #MDA97291 -C-0053 1 Real Push broom sensors are commonly used in satellite cameras, notably th# SPOT satellite forth# generatio...

### Citations

1302 |
Three-Dimensional Computer Vision: A Geometric Viewpoint
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- 1993
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Citation Context ... solution is found when more than seven matches are given. The solution of the non-homogeneous set of equations is the singular vector corresponding to the least singular value of the equation matrix(=-=[8]-=-). 8sThe last two rows of M are determined by this method only up to an unknown constant factor. To determine the matrix M that correctly determines which points are in front of the camera, according ... |

651 |
A computer algorithm for reconstructing a scene from two projections
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Citation Context ...tive camera placement of two or more pinhole cameras and consequent determination of pinhole cameras has been extensively considered. Most relevant to the present paper is the work of LonguetHiggins (=-=[8]-=-) who defined the so-called essential matrix Q, which may be determined from eight or more correspondence points between two images by linear techniques. Other non-linear techniques for determining Q,... |

629 |
Mathematica - A System for Doing Mathematics by Computer
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Citation Context ...matrix M is almost uniquely determined by the essential matrix. Under the assumption that M ′ =(I | 0), the essential matrix may be computed explicitly in terms of the entries of M. Using Mathematic=-=a([10]) or by hand it may be computed -=-that. ⎛ ⎞ 0 0 m11m33 − m13m31 m13m21 − m11m23 ⎜ Q =(qij) = ⎜ 0 0 m11m32 − m12m31 m12m21 − m11m22 ⎟ ⎝ ⎠ (14) m22 −m32 m14m32 − m12m34 m12m24 − m14m22 m23 −m33 m14m33 − m... |

509 |
What can be seen in three dimensions with an uncalibrated stereo rig
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Citation Context ...n affine transformation. This has the practical consequence that affine invariants of a scene may be computed from two pushbroom views. A similar result applies to perspective views2 ,as was shown in =-=[1, 6]-=-. It is hoped that the linear pushbroom model may provide the basis for the development of further image understanding algorithms in the same way that the pinhole camera model has given rise to a weal... |

367 | Camera self-calibration: theory and experiments
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Citation Context ...hniques relate especially to so called “calibrated cameras”, for which the internal parameters are known. A paper that deals with the determination of the essential matrix for uncalibrated cameras=-= is [2]-=-. As for the determination of the world coordinates of points see from two pinhole cameras, it has been shown ([1, 7]) that for uncalibrated cameras the position of world points is determined up to an... |

319 | Uniqueness and Estimation of Three-Dimensional Motion Parameters of Rigid Objects with Curved Surfaces
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Citation Context ... be determined from eight or more correspondence points between two images by linear techniques. Other non-linear techniques for determining Q, morestableinthe presence of noise, have been published (=-=[19, 18, 9, 16]). T-=-hose techniques relate especially to so called “calibrated cameras”, for which the internal parameters are known. A paper that deals with the determination of the essential matrix for uncalibrated... |

286 | Estimation of relative camera positions for uncalibrated cameras
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- 1992
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Citation Context ...age point (ui,vi) ⊤ in the first image with its corresponding point (u ′ i ,v′ i )⊤ in the second image (Section 5). We show that a matrix analogous to the fundamental matrix for perspective cameras (=-=[2, 3, 4]-=-) exists for linear pushbroom cameras as well. In particular, we prove that there exists a 4 × 4 matrix F ,whichwecalltheLP (linear pushbroom) fundamental matrix, such that (u ′ i ,u′ i v′ i ,v′ i , 1... |

243 |
Stereo from uncalibrated cameras
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- 1992
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Citation Context ...that deals with the determination of the essential matrix for uncalibrated cameras is [2]. As for the determination of the world coordinates of points see from two pinhole cameras, it has been shown (=-=[1, 7]-=-) that for uncalibrated cameras the position of world points is determined up to an unknown projective transform by their images in two separate views. A similar result for linear pushbroom cameras wi... |

181 |
Motion and structure from two perspective views: algorithms, error analysis and error estimation
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- 1989
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Citation Context ... be determined from eight or more correspondence points between two images by linear techniques. Other non-linear techniques for determining Q, morestableinthe presence of noise, have been published (=-=[19, 18, 9, 16]). T-=-hose techniques relate especially to so called “calibrated cameras”, for which the internal parameters are known. A paper that deals with the determination of the essential matrix for uncalibrated... |

149 | On the geometry and algebra of the point and line correspondences between images
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Citation Context ...be satisfied by any pair of corresponding image points (u, v) and (u ′ ,v ′ ). The same basic proof method used above may be used to prove the existence of the fundamental matrix for pinhole cameras (=-=[19]-=-). It is seen that if either M or M ′ is replaced by an equivalent matrix by multiplying the last two rows by a constant c, then the effect is to multiply det A(M,M ′ ), and hence the 12sfundamental p... |

136 | In Defence of the 8-point Algorithm
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- 1995
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Citation Context ...s relate especially to so called “calibrated cameras”, for which the internal parameters are known. Some papers that deal with the determination of the fundamental matrix for uncalibrated cameras are =-=[17, 18]-=-. As for the determination of the world coordinates of points see from two pinhole cameras, it has been shown ([5, 4]) that for uncalibrated cameras the position of world points is determined up to an... |

134 | Relative Orientation
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- 1990
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Citation Context ...rix satisfying these conditions using explicit assumptions about the source of error to formulate a cost function to be minimized. This has been shown to be the best approach for perspective cameras (=-=[21, 16]-=-). The question of numerical stability is important in implementing algorithms using the linear pushbroom model. In particular, it is easy to encounter situations in which the determination of the lin... |

81 |
Decomposition of Transformation Matrices for Robot Vision
- Ganapathy
- 1984
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Citation Context ...om parameters may be computed given the matrix M. This comes down to finding a factorization of M of the kind given in (7). The corresponding problem for pinhole cameras has been solved by Ganapathy (=-=[3]-=-) and Strat ([14]). First of all we determine the position of the camera at time t = 0, referred to subsequently as the initial position of the camera. Multiplying out the product (7) it may be seen t... |

60 | Canonical frames for planar object recognition - Rothwell, Zisserman, et al. - 1992 |

41 | Computing matched-epipolar projections
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- 1993
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Citation Context ...r image taken by different linear pushbroom camera. Many resampling operation — e.g. resampling images in a stereo pair so that the match point disparities are only along one of the image coordinates =-=[6]-=- — cannot be performed on linear pushbroom imagery without breaking the mapping encoded in (7). Points in Front of the Camera. Recall that the camera coordinate frame was set up in such a way that the... |

35 | Relative orientation revisited
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- 1991
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Citation Context ... be determined from eight or more correspondence points between two images by linear techniques. Other non-linear techniques for determining Q, morestableinthe presence of noise, have been published (=-=[19, 18, 9, 16]). T-=-hose techniques relate especially to so called “calibrated cameras”, for which the internal parameters are known. A paper that deals with the determination of the essential matrix for uncalibrated... |

35 |
Three-Dimensional Data Input by Tablet
- Sutherland
- 1974
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Citation Context ... Matrix In this section it will be shown how a linear pushbroom camera matrix may be computed given a set of ground control points. The method is an adaptation of the method of Roberts or Sutherland (=-=[15]) used f-=-or the pinhole cameras. In particular, denoting by m1 ⊤ , m2 ⊤ and m3 ⊤ the three rows of the matrix M and x =(x, y, z, 1) ⊤ a ground control point, (7) may be written in the form of three equ... |

25 | Depth computations from polyhedral images - Sparr - 1992 |

23 |
Bootstrap Stereo
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- 1980
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Citation Context ...s for these two images using a set of 25 ground control points, visible in both images, picked form USGS maps and several automatically generated image to image correspondences found using STEREOSYS (=-=[5]-=-) Two performance metrics were computed. The accuracy with which the camera model maps the ground points to their corresponding image points is important. The RMS difference between the known image co... |

19 |
Recovering the Camera Parameters from a Transformation Matrix
- Strat
- 1984
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Citation Context ...y be computed given the matrix M. This comes down to finding a factorization of M of the kind given in (7). The corresponding problem for pinhole cameras has been solved by Ganapathy ([3]) and Strat (=-=[14]-=-). First of all we determine the position of the camera at time t = 0, referred to subsequently as the initial position of the camera. Multiplying out the product (7) it may be seen that M is of the f... |

11 | X-Ray metrology for quality assurance
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Citation Context ...age which is orthographic in the direction of motion and perspective in the orthogonal direction. Very good results have been obtained in modeling this imaging setup as a linear pushbroom camera (see =-=[1]-=- for details). 1.1 Overview In this paper, a linear approximation to the pushbroom model is introduced. This new model very greatly simplifies the computations involved in working with pushbroom image... |

11 |
Optimal visual motion estimation: a note
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- 1992
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Citation Context ...determined from eight or more correspondence points between two images by linear techniques. Other non-linear techniques for determining F , more stable in the presence of noise, have been published (=-=[13, 14, 15, 16]-=-). Those techniques relate especially to so called “calibrated cameras”, for which the internal parameters are known. Some papers that deal with the determination of the fundamental matrix for uncalib... |

5 | An object-oriented approach to scene reconstruction
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- 1996
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Citation Context ...on, fix the known parameters to known values, and finally carry out an iterative parameter fitting algorithm to get a more exact estimate of the 9scamera mapping. Our camera modelling program Carmen (=-=[12]-=-) uses this approach, allowing any of the parameters to be fixed absolutely, or with a specified standard deviation. It is possible to parametrize the camera in different ways to allow for different t... |

4 |
Terrain elevation extraction from digital SPOT satellite imagery
- Tam
- 1990
(Show Context)
Citation Context ...ven in [4]. The full model takes account of orbital dynamics, provided ephemeris data and attitude drift data to model the imaging process as accurately as possible. A different model is discussed in =-=[17]-=-. Using the full pushbroom model parametrized to an actual orbit and ephemeris data, and an artificial terrain model, a set of ground to image correspondences were computed, one such ground control po... |

4 | Camera estimation for orbiting pushbrooms
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- 1995
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Citation Context ... the linear pushbroom model are described. 19sIn the first experiment, the accuracy of the linear pushbroom model was compared with a full model of SPOT’s HRV camera. This model, which is detailed in =-=[22]-=-, takes into account the orbital dynamics, earth rotation, attitude drift as measured by on-board systems, ephemeris data, and several other phenomena to emulate the imaging process as accurately as p... |

2 |
Invariants of 3D point sets
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(Show Context)
Citation Context ...n affine transformation. This has the practical consequence that affine invariants of a scene may be computed from two pushbroom views. A similar result applies to perspective views2 ,as was shown in =-=[1, 6]-=-. It is hoped that the linear pushbroom model may provide the basis for the development of further image understanding algorithms in the same way that the pinhole camera model has given rise to a weal... |

2 |
Spetsakis and Yiannis Aloimonos, “Optimal Visual Motion Estimation
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- 1992
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Citation Context |

1 |
A camera model for space-borne, pushbroom imaging systems
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- 1230
(Show Context)
Citation Context ...the first experiment, the accuracy of the linear pushbroom model was compared with the full model. In order to make this comparison, a full SPOT model was used. The details of this model are given in =-=[4]-=-. The full model takes account of orbital dynamics, provided ephemeris data and attitude drift data to model the imaging process as accurately as possible. A different model is discussed in [17]. Usin... |