Motion of points and lines in the uncalibrated case

Abstract

In the present paper we address the problem of computing structure and motion, given a set point and/or line correspondences, in a monocular image sequence, when the camera is not calibrated.Considering point correspondences first, we analyse how to parameterize the retinal correspondences, in function of the chosen geometry: Euclidean, affine or projective geometry. The simplest of these parameterizations is called the FQs-representation and is a composite projective representation. The main result is that considering N + 1 views in such a monocular image sequence, the retinal correspondences are parameterized by 11 N -4 parameters in the general projective case. Moreover, 3 other parameters are required to work in the affine case and 5 additional parameters in the Euclidean case. These 8 parameters are "calibration" parameters and must be calculated considering at least 8 external informations or constraints. The method being constructive, all these representations are made explicit.Then, considering line correspondences, we show how the the same parameterizations can be used when we analyse the motion of lines, in the uncalibrated case. The case of three views is extensively studied and a geometrical interpretation is proposed, introducing the notion of trifocal geometry which generalizes the well known epipolar geometry. It is also discussed how to introduce line correspondences, in a framework based on point correspondences, using the same equations.Finally, considering the FQs-representation, one implementation is proposed as a "motion module", taking retinal correspondences as input, and providing and estimation of the 11 N -4 retinal motion parameters. As discussed in this paper, this module can also estimate the 3D depth of the points up to an affine and projective transformation, defined by the 8 parameters identified in the first section. Experimental results are provided.

A general method for the point of regard estimation in 3D space

“…The pose of the left and right eye cameras relative to the left scene camera ( Figure 6) can be estimated by means of the correspondent image features x, l and l . The geometry of the system is described by the Trifocal Tensor [14,18,7], which relates the involved features according to the incidence relation [8]:…”

Experimental self-calibration from four views

“…We compute the output intrinsic and extrinsic parameters replacing in the next criterion each F by the equation 3, and 9 9 , , minimizing: Using the Model 2, we do not a priori set any parameter as being constant. If N is the number of views [7], we have 11N-15 independent parameters, which is in our case 11'4-15 = 29. In the table 1 we have represented the different models.…”

“…Most authors [2,5,8] studied the case of point correspondences, but have restricted their approach to the case where the intrinsic parameters of the camera are constant~ while only 2 or 3 views have been taken into account, or studied the monocular case for the long sequences as in [7].…”

Uncalibrated 1D projective camera and 3D affine reconstruction of lines

“…Using line segments instead of points as features has attracted the attention of many researchers [11,2,29,28,27,1] for various tasks such as pose estimation, stereo and structure from motion. In this paper, we are interested in structure from motion using line correspondences across mutiple images.…”

“…This provides a very heavy overparametrization of the problem that definitely leads to the instability of the algorithm reported in [11]. The thirteenline algorithm was extended to uncalibrated camera case in [7,27]. The situation for uncalibrated camera case might be expected to be better, as more free parameters are needed.…”