An Introduction to Camera Self-Calibrati

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1 Research Seminar3D Vision Team – Le Creusot David FofiLe2i UMR CNRS 5158 – IUT Le CreusotDavid.Fofi@u-bourgogne.fr An Introduction to Camera Self-Calibration 9 avril 2009

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2 1. Introduction 2. Basic Principles 2.1 Camera modeling 2.2 Preliminary results 2.3 A counting argument 2.4 Notations 3. Camera Intrinsic Constraints 3.1 Absolute conic/quadric 3.2 Self-calibration 3.3 Results 4. Scene Constraints 4.1 Principle 4.2 Metric constraints 4.3 Results 5. How to Deal with ProCam Systems? 6. References CONTENTS

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3 INTRODUCTION CalibrationCamera calibration is the process of finding the « true » parameters of the camera that took your photographs. Some of these parameters are focal length, format size, principal point and lens distortion. Self-Calibration Self-calibration is calibration with no user intervention nor calibration pattern, performed on-line during the acquisition process. - Accuracy - Metric reconstruction at scale - Varying parameters - Automatic process

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4 BASIC PRINCIPLES 2.1 CAMERA MODELING ? ? ?

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5 BASIC PRINCIPLES 2.1 CAMERA MODELING If Ks are known (calibrated cameras):

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6 BASIC PRINCIPLES 2.2 PRELIMINARY RESULTS Rough approach: If P and M are solutions, then PW and W-1M are also solutions W is a 4x4 invertible matrix with no particular structure: it represents a projective transformation!

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7 BASIC PRINCIPLES 2.2 PRELIMINARY RESULTS Distance, orthogonality and parallelism are not preserved under projective transformation. Colinearities, coplanarities and the cross-ratio are preserved under projective transformation. With self-calibration, you can only do this…

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8 BASIC PRINCIPLES 2.3 A COUNTING ARGUMENT From a projective space to a metric space… 8 constraints are required!! Knowing an intrinsic camera parameter for n views gives n constraints, fixing one yields only n - 1 constraints.

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9 BASIC PRINCIPLES 2.3 A COUNTING ARGUMENT

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10 BASIC PRINCIPLES 2.4 NOTATIONS

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11 INTRINSIC CONSTRAINTS Absolute Conic The absolute conic ? is invariant under Euclidean transformation (e.g. displacement). Thus, its image depends only upon the intrinsic parameters… It is more convenient to use the absolute dual quadric which encodes both the plane at infinity and the absolute conic… 3.1 ABSOLUTE CONIC/QUADRIC Similarity transformation

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12 INTRINSIC CONSTRAINTS 3.2 SELF-CALIBRATION When only two views are available the solution is only determined up to a one parameter family : this results in up to 4 possible solutions. A possible parameterization for the absolute dual quadric… which simplifies the transformation from projective to metric… i.e. to transform O* from a generic position to its canonical position

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13 INTRINSIC CONSTRAINTS 3.2 SELF-CALIBRATION Summarizing… ? Computing a projective reconstruction ? Pi ? Computing the absolute dual quadric ? O* ? Retrieving the absolute dual conic ? ?*i ? Extracting the intrinsic parameters and the plane at infinity ? K, a ? Upgrading the projective reconstruction to metric ? Performing bundle adjustment

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14 INTRINSIC CONSTRAINTS 3.3 RESULTS

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15 SCENE CONSTRAINTS 4.1 PRINCIPLE Metric transformations are a sub-group of projective transformations… A projective transformation W exists that upgrades a projective reconstruction into a metric reconstruction! W is a 4x4 invertible matrix with 15 degrees of freedom… 15 independent metric constraints are necessary to compute W! (Problem: how to extract them from a priorily unknown scene?)

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16 SCENE CONSTRAINTS 4.2 METRIC CONSTRAINTS Alignments, parallelisms, distances, orthogonalities… Fixing the origin Point lying on the horizontal plane Fixing a distance

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17 SCENE CONSTRAINTS 4.3 RESULTS

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18 PROJECTOR-CAMERA How to deal with a Projector-Camera System ?... A displacement of a projector causes a sliding (thus a loss) of the 3D points: self-calibration must be restricted to 2 views (one projector view and one camera view). ? Two views and varying intrinsic parameters: 4 solutions!

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19 REFERENCES M. Pollefeys, R. Koch, L. Van Gool, Self-Calibration and Metric Reconstruction Inspite of Varying and Unknown Intrinsic Camera Parameters, International Journal of Computer Vision; 32(1), pp. 7–25, 1999. A. Fusiello, Uncalibrated Euclidean Reconstruction: a Review, Image and Vision Computing, 18, pp. 555–563, 2000. D. Fofi, J. Salvi, E. Mouaddib, Uncalibrated Reconstruction: an Adaptation to Structured Light Vision, Pattern Recognition, 36, pp. 1631 – 1644, 2003. E.E. Hemayed, A Survey of Camera Self-Calibration, IEEE Conference on Advanced Video and Signal based Surveillance, pp. 351-357, 2003. M. Pollefeys, Visual 3D Modeling from Images, Tutorial Notes, University of North Carolina – Chapel Hill, 2004.

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20 QUESTIONS QUESTIONS QUESTIONS QUESTIONS QUESTIONS