FREE ELECTRONIC LIBRARY - Dissertations, online materials

Pages:     | 1 |   ...   | 5 | 6 || 8 | 9 |   ...   | 16 |

«MULTI-CAMERA SIMULTANEOUS LOCALIZATION AND MAPPING Brian Sanderson Clipp A dissertation submitted to the faculty of the University of North Carolina ...»

-- [ Page 7 ] --

In contrast, with a single camera absolute scale cannot be determined. Omnidirectional cameras based on parabolic mirrors provide wide-angle scene coverage but at the expense of uneven sampling of the visual sphere and also unknowable absolute scale. Some omnidirectional cameras are essentially small clusters of standard perspective cameras (ImmersiveMedia,Imove,Ladybug2) but without a large baseline between the cameras, these cannot be used to measure absolute, scaled ego-motion. This is a problem of signal to noise ratio where the baseline of the cameras is small relative to the scene depth making the scale constraint from the camera geometry weak. Another possible approach to scaled motion estimation from video was presented by Scaramuzza et al. (2009). They use the fact that the camera moves with nonholonomic motion in a plane to calculate the scaled camera motion Figure 3.1: Example of a multi-camera system on a vehicle with one point correspondence.

It can be difficult to avoid losing part of the field-of-view due of a single camera or omnidirectional camera due to occlusion, which may require camera cluster placement high up on a boom. Alternatively, for mounting on a vehicle the system can be split into two clusters so that one can be placed on each side of the vehicle and occlusion problems are minimized while giving a large baseline for scale estimation. In this chapter we will show that by using a system of two camera clusters, consisting of one or more cameras each, separated by a known transformation, the six degrees of freedom (DOF) of camera system motion, including scale, can be recovered.

An example of a multi-camera system for the capture of ground-based video is shown in Figure 3.1. It consists of two camera clusters, one on each side of a vehicle. The cameras are attached tightly to the vehicle and can be considered a rigid object. This system is used for the experimental evaluation of our approach.

Computing the scale, structure and camera motion from video of a general scene is

–  –  –

et al. investigated the properties of visual odometry for single-camera and stereo-camera systems. Their analysis showed that a single camera system is not capable of maintaining a consistent scale over time. Their stereo system is able to maintain absolute scale over

–  –  –

Figure 3.2: (a) Overlapping stereo camera pair, (b) Non-overlapping multi-camera system extended periods of time by using a known baseline and cameras with overlapping fields of view.

Our approach eliminates the requirement for overlapping fields of view and is able to maintain the absolute scale over time.

In the next section 3.2, we introduce our novel solution to finding the 6DOF motion of a two-camera system with non-overlapping views. We derive the mathematical basis for our technique in section 3.3 as well as give a geometrical interpretation of the scale constraint.

The algorithm used to solve for the scaled motion is described in section 3.4. Section 3.5 discusses the evaluation of the technique on synthetic data and on real imagery.

3.2 6DOF Multi-camera Motion The proposed approach addresses the 6DOF motion estimation of multi-camera systems with non-overlapping fields-of-view. Most previous approaches to 6DOF motion estimation have used camera configurations with overlapping fields of view, which allow correspondences to be triangulated simultaneously across multiple views with a known, rigid baseline. Our approach uses a temporal baseline where points are only visible in one camera at a given time. The difference in the two approaches is illustrated in Figure 3.2.

Our technique assumes we can establish at least five temporal correspondences in one of the cameras and one temporal correspondence in any additional camera. In practice, this assumption is not a limitation, as a reliable estimation of camera motion requires multiple correspondences from each camera due to noise.

The essential matrix which defines the epipolar geometry of a single freely moving cal

–  –  –

geometry. The ambiguity can be eliminated with additional points. With oriented geometry the rotation and the translation up to scale of the camera can be extracted from the essential matrix. Consequently, a single camera provides 5DOF of the camera motion. The remaining degree is the scale of the translation. Given these 5DOF of multi-camera system motion (rotation and translation direction), we can compensate for the rotation of the system. Our approach is based on the observation that given the temporal epipolar geometry of one of the cameras, the position of the epipole in each of the other cameras of the multi-camera system is restricted to a line in the image. Hence, the scale as the remaining degree of freedom of the camera motion describes a linear subspace.

In the next section, we derive the mathematical basis of our approach to motion recovery.


–  –  –

We consider a system involving two cameras, rigidly coupled with respect to each other.

The cameras are assumed to be calibrated. Figure 3.3 shows the configuration of the twocamera system. The cameras are denoted by C1 and C2, at the starting position and C1 and C2 after a rigid motion.

We will consider the motion of the camera-pair to a new position. Our purpose is to determine the motion using image measurements. It is possible through standard techniques to compute the motion of the cameras up to scale, by determining the motion of just one of the cameras using point correspondences from that camera. However, from one camera, motion can be determined only up to scale. The direction of the camera translation may be determined, but not the magnitude of the translation. It will be demonstrated in this chapter that a single correspondence from the second camera is sufficient to determine the scale of the motion, that is, the magnitude of the translation. This result is summarized in the following theorem.

Theorem 1. Let a two camera system have initial configuration determined by camera matrices P1 = [I | 0] and P2 = [R2 | − R2 C2 ].

Suppose it moves rigidly to a new position for which the first camera is specified by P1 = [R1 | − λR1 C1 ]. Then the scale of the translation, λ, is determined by a single point correspondence x ↔ x seen in the second camera according to the formula

–  –  –

where A = R2 R1 [(R1 − I)C2 ]× R2 and B = R2 R1 [C1 ]× R2. In this chapter [a]× b denotes the skew-symmetric matrix inducing the cross product a × b.

–  –  –

Figure 3.3: Motion of a multi-camera system consisting of two rigidly coupled conventional cameras.

In order to simplify the derivation we assume the coordinate system is centered on the initial position of the first camera, so that P1 = [I | 0]. Any other coordinate system is easily transformed to this one by a Euclidean change of coordinates.

Observe also that after the motion, the first camera has moved to a new position with camera center at λC1. The scale is unknown at this point because in our method we propose as a first step determining the motion of the cameras by computing the essential matrix of the first camera over time. This allows us to compute the motion up to scale only. Thus, the scale λ remains unknown. We now proceed to derive Theorem 1. Our immediate goal is to determine the camera matrix for the second camera after the motion. First note that the camera P1 may be written as

–  –  –

where the matrix T, is the Euclidean transformation induced by the motion of the camera pair. Since the second camera undergoes the same Euclidean motion, we can compute the camera P2 to be

–  –  –

Now, given a single point correspondence x ↔ x as seen in the second camera, we may determine the value of λ, the scale of the camera translation. The essential matrix

equation x E2 x = 0 yields x Ax + λx Bx = 0, and hence:

–  –  –


So each correspondence in the second camera provides a measure for the scale. In the next section, we give a geometric interpretation for this constraint.

3.3.1 Geometric Interpretation The situation may be understood via a different geometric interpretation, shown in Figure 3.4. We note from (3.2) that the second camera moves to a new position C2 (λ) = R1 C2 + λC1. The locus of this point for varying values of λ is a straight line with its direction vector C1, passing through the point R1 C2. From its new position, the camera observes a point at position x in its image plane. This image point corresponds to a ray v along which the 3D point X must lie. If we think of the camera as moving along the line C2 (λ) (the locus of possible final positions of the second camera center), then this ray traces out a plane Π; the 3D point X must lie on this plane.

On the other hand, the point X is also seen (as image point x) from the initial position of the second camera, and hence lies along a ray v through C2. The point where this ray meets the plane Π must be the position of the point X. In turn, this determines the scale factor λ.

3.3.2 Critical configurations This geometric interpretation allows us to identify critical configurations in which the scale factor λ cannot be determined. As shown in Figure 3.4, the 3D point X is the intersection of

–  –  –

Figure 3.4: The 3D point X must lie on the plane traced out by the ray corresponding to x for different values of the scale λ.

It also lies on the ray corresponding to x through the initial camera center C2.

the plane Π with a ray v through the camera center C2. If the plane does not pass through C2, then the point X can be located as the intersection of plane and ray. Thus, a critical configuration can only occur when the plane Π passes through the second camera center, C2.

According to the construction, the line C2 (λ) lies on the plane Π. For different 3D points X, and corresponding image measurement x, the plane will vary, but always contain the line C2 (λ). Thus, the planes Π corresponding to different points X form a pencil of planes hinged around the axis line C2 (λ). Unless this line actually passes through C2, there will be at least one point X for which C2 does not lie on the plane Π, and this point can be used to determine the point X, and hence the scale.

Finally, if the line C2 (λ) passes through the point C2, then the method will fail. In this case, the ray corresponding to any point X will lie within the plane Π, and a unique point of intersection cannot be found.

In summary, if the line C2 (λ) does not pass through the initial camera center C2, almost any point correspondence x ↔ x may be used to determine the point X and the translation scale λ. The exceptions are point correspondences given by points X that lie in the plane defined by the camera center C2 and the line C2 (λ) as well as far away points for which Π and v are almost parallel. The interested reader may wish to read another analysis of the critical configurations for scale estimation in non-overlapping multi-camera systems given in (Kim and Chung, 2006).

If on the other hand, the line C2 (λ) passes through the center C2, then the method will always fail. It may be seen that this occurs most importantly if there is no camera rotation, namely R1 = I. In this case, we see that C2 (λ) = C2 + λC1, which passes through C2. It is easy to give an algebraic condition for this critical condition. Since C1 is the direction vector of the line, the point C2 will lie on the line precisely when the vector R1 C2 − C2 is in the direction C1. This gives a condition for singularity (R1 C2 − C2 ) × C1 = 0, or rearranging this expression, and observing that the vector C2 × C1 is perpendicular to the plane of the three camera centers C2, C1 and C1 (the last of these being the coordinate

origin), we may state:

Theorem 2. The critical condition for singularity for scale determination is

–  –  –

In particular, the motion is not critical unless the axis of rotation is perpendicular to the plane determined by the three camera centers C2, C1 and C1.

Intuitively, critical motions occur when the rotation induced translation R1 C2 − C2 is aligned with the translation C1. The most common motion that causes a critical condition is when the camera system translates but has no rotation. Another common, but less obvious, critical motion occurs when both camera paths move along concentric circles. This configuration is illustrated in Figure 3.5. A vehicle borne multi-camera system turning at a constant rate undergoes critical motion, but not when it enters and exits a turn.

Detecting critical motions is important to determining when the scale estimates are

–  –  –

reliable. One method to determine the criticality of a given motion is to use the approach of (Frahm and Pollefeys, 2006). We need to determine the dimension of the space that includes our estimate of the scale. To do this we double the scale λ and measure the difference in the fraction of inliers to the essential matrix of our initial estimate and the doubled scale essential matrix. If a large proportion of inliers are not lost when the scale is doubled then the scale is not observable from the data. If the scale is observable, the deviation from the estimated scale value would cause the correspondences to violate the epipolar constraint, which means they are outliers to the constraint for the doubled scale.

When the scale is ambiguous, doubling the scale does not cause correspondences to be classified as outliers. This method proved to work on real data sets in practice.

3.4 Algorithm Figure 3.6 shows an algorithm to solve relative motion of two generalized cameras from 6 rays with two centers where 5 rays meet one center and a sixth ray meets the other center. First, we use 5 correspondences in one ordinary camera to estimate an essential Figure 3.6: Algorithm for estimating 6DOF motion of a multi-camera system with nonoverlapping fields of view.

matrix between two frames in time. The algorithm used to estimate the essential matrix

–  –  –

Pages:     | 1 |   ...   | 5 | 6 || 8 | 9 |   ...   | 16 |

Similar works:

«Controlled, Encouraged or Adrift? Sources of Variation in Adolescent Substance Use by Tara Leah Fidler A thesis submitted in conformity with the requirements for the degree of Doctorate of Philosophy Department of Sociology University of Toronto © Copyright by Tara Leah Fidler 2012 Controlled, Encouraged or Adrift? Sources of Variation in Adolescent Substance Use Tara Leah Fidler Doctorate of Philosophy Department of Sociology University of Toronto Abstract The frequent consumption of alcohol...»

«Truthfulness, Lies, and Moral Philosophers: What Can We Learn from Mill and Kant? ALASDAIR MACINTYRE THE TANNER LECTURES ON HUMAN VALUES Delivered at Princeton University April 6 and 7, 1994 ALASDAIR MACINTYRE is Arts and Sciences Professor of Philosophy at Duke University. He was educated at Queen Mary College, University of London, and at the University of Manchester. He taught at various British universities, including Oxford and Essex, until 1970. Since then he has taught at a number of...»

«Learning a Terrain Model for Autonomous Navigation in Rough Terrain Carl Wellington CMU-RI-TR-05-59 Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Robotics The Robotics Institute Carnegie Mellon University Pittsburgh, Pennsylvania 15213 December 2005 Thesis Committee: Anthony Stentz, Chair Alonzo Kelly Jeff Schneider John Reid, John Deere c Carl Wellington MMV This research was sponsored in part by John Deere under contract 476169 Abstract...»

«MANUFACTURING OF POROUS SURFACES WITH MICRO-SCALE FEATURES FOR ADVANCED HEAT TRANSFER by Peng Chen A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mechanical Engineering) in The University of Michigan 2008 Doctoral Committee: Professor Jun Ni, Chair Professor Elijah Kannatey-Asibu Jr. Professor John Halloran Assistant Professor Gap-Yong Kim © Peng Chen All rights reserved 2008 DEDICATION. to the memory of my dear uncle Mr. Ximing...»

«Essays on Collaboration, Innovation, and Network Change in Organizations by Russell James Funk A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Sociology) in The University of Michigan Doctoral Committee: Professor Jason D. Owen-Smith, Chair Professor Gautam Ahuja Professor Mark S. Mizruchi Assistant Professor Maxim Vitalyevich Sytch c Russell James Funk 2014 All Rights Reserved For Kylee, whose love and encouragement made this...»


«AVOIDING THE RECENT PAST: WHICH STIMULUS DIMENSIONS INFLUENCE PROACTIVE INTERFERENCE? by Kimberly S. Craig A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Psychology) in the University of Michigan 2013 Doctoral Committee: Associate Professor Cindy A. Lustig, Chair Professor John Jonides Professor John E. Laird Professor Patricia A. Reuter-Lorenz © Kimberly S. Craig All Rights Reserved 2013 Dedication For my family. I would not have...»

«Handbook CKE55 MA Sociology CKE56 Sociology of Development and Globalization CKG 55 PhD Track Sociology CKH 57 PhD Sociology Discipline of Sociology School of Sociology and Philosophy University College Cork Ireland 2016-2017 CONTACT DETAILS Department Address: Askive, UCC, Donovan’s Road, Cork, Ireland Tel: +353-214902318/2894 Fax: +35321-4272004 General Enquiries, Department of Sociology: Eleanor O’Connor e.oconnor@ucc.ie; Jerry O’Sullivan, jerry.osullivan@ucc.ie Tel: +353-21-4902318...»

«Teaching and Learning Critical Reading with Transnational Texts at a Mexican University: An Emergentist Case Study by Moisés Damián Perales Escudero A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (English and Education) in the University of Michigan Doctoral Committee Professor Diane Larsen-Freeman, Co-Chair Professor Mary Schleppegrell, Co-Chair Professor Anne R. Gere Professor Annemarie Palincsar Associate Professor Désirée...»

«Justifications and Excuses Marcia Baron∗ The distinction between justifications and excuses is a familiar one to most of us who work either in moral philosophy or legal philosophy. But exactly how it should be understood is a matter of considerable disagreement. My aim in this paper is, first, to sort out the differences and try to figure out what underlying disagreements account for them. I give particular attention to the following question: Does a person who acts on a reasonable but...»

«On Ethical Thoughtfulness Item type text; Electronic Dissertation Authors Matteson, Jason Kent Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Downloaded 23-Nov-2016 06:46:30 Link to item...»

«SURFACES, SCALES, AND SYNTHESIS SCIENTIFIC REASONING AT THE NANOSCALE by Julia R. Bursten B.A., Philosophy, Rice University, 2008 M.A., Philosophy, University of Pittsburgh, 2010 Submitted to the Graduate Faculty of the Kenneth P. Dietrich School of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh UNIVERSITY OF PITTSBURGH KENNETH P. DIETRICH SCHOOL OF ARTS AND SCIENCES This dissertation was presented by Julia R....»

<<  HOME   |    CONTACTS
2016 www.dissertation.xlibx.info - Dissertations, online materials

Materials of this site are available for review, all rights belong to their respective owners.
If you do not agree with the fact that your material is placed on this site, please, email us, we will within 1-2 business days delete him.