A Tutorial on Affine and Projective Geometries

A tutorial on affine and projective geometriesAN IN TRODUC TI ON TO POSE EST I MATION

S N E H A L I . B H AYA N I

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M .T EC H ( I C T )

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What is pose estimation?The problem

Methods, a broad classification

The analytical and geometric approach

The four main types of pose estimation problems1

2D-2D pose estimation problem

3D-3D pose estimation problem Relates to rigid motions in 3D coordinate system.

2D perspective-3D pose estimation problem

2D perspective-2D perspective pose estimation problem Related to two-view geometry.

1R.M. Haralick, Hyonam joo, D. Lee, S. Zhuang, V.G.Vaidya, and M.B. Kim. Pose estimation from corresponding point data. Systems, Man and cybernetics,ieee transactions on, 19(6):1426–1446, 1989.

Pin hole camera model2

Figure 1 A simple pin-hole camera Figure 2 A coordinate system for a pin-hole camera model

2More on camera models can be found in Richard Hartley and Andrew Zisserman. Multiple view geometry in computer vision. Cambridge, 2003.

Camera rotation and translation

Figure 3 The euclidean transformation between the world and camera coordinate frames3

Why projective geometry?Study of Invariants

Which properties of Euclidean geometry are invariant?3

Pros of a projective geometry: Camera as a projective model is naturally a homography or a projective transform. Lesser invariants Other geometries are special cases of projective geometry. Duality of lines, points, planes

Homogeneous coordinates help in linearizing a certain mathematical relationship. And hence we represent the relationship as a matrix multiplication.

. Where is the desired matrix mapping in 3D camera coordinate system to in 2D image plane coordinate system

3A systematic study of invariants of transformations for geometrical objects can be found in James R smart. Modern geometries. Cengage learning, 1997 and Rey casse. Projective geometry: an introduction. Oxford university press, 2006.

From affine to projectiveAffine completion for projective space.

The points at infinity, a change in representation.

The invariants.

Parallelism invariant to affine transform but not to a projective transform.

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DualityA significant reason for studying projective geometry is it’s simplicity.

dimensional vector space has every dimensional subspace dual to dimensional subspace. Eg in we have a plane to be a dual of a point and lines are their own duals.

Hyperplanes,

Represented by the point in upto a non-zero scalar multiplication. Hence a point in represents a unique hyperplane in and hence a unique hyperplane in .

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The need for homographyCamera calibration.

Intrinsic parameters : Focal length () Scaling along image axes () Skew () Translation ()

Extrinsic parameters : 3D Rotation () 3D translation ()

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Figure 4 A case where the centers of image plane and that of camera coordinate system

don’t coincide.

Figure 5 A case where the camera coordinate system is a rotated version of the world

coordinate system.

The need for homography Epipolar geometry.#

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Figure 6 A homography or projective correspondence between two images of the same object via a third plane .

Fundamental matrix Point – point correspondence

Point – line correspondence

Given a set of point correspondences for two images of the same scene but with different poses,

for all correspondences.

is of rank 2 with seven degrees of freedom.

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Properties and Computation of

Epipolar line homography.

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Figure 7a Figure 7b Line-line correspondence between two projective planes.

Where from hereVarious approaches to pose estimation and computer vision.

Calibrated and non-calibrated approached.

Degeneracy and noise elimination.

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Thank you

Any questions?

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