A Tutorial On Affine and Projective Geometries
A Tutorial On Affine and Projective Geometries
A Tutorial On Affine and Projective Geometries
A N I N T R O D U C T I O N TO P O S E E S T I M AT I O N
S N E H A L I . B H AYA N I
201211008 M .T E C H ( I C T ) `
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):14261446, 1989.
2 model
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on camera models can be found in Richard Hartley and Andrew Zisserman. Multiple view geometry in computer vision. Cambridge,
2003.
Figure 3 The euclidean transformation between the world and camera coordinate frames3
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.
Duality
A significant reason for studying projective geometry is its simplicity. dimensional vector space has every 1 dimensional subspace dual to 1 dimensional subspace. Eg in 3 we have a plane to be a dual of a point and lines are their own duals. Hyperplanes,
+1 =1
= 0.
Represented by the point 1 , , +1 in +1 upto a non-zero scalar multiplication. Hence a point in represents a unique hyperplane in +1 and hence a unique hyperplane in .
Figure 4 A case where the centers of image plane and that of camera coordinate system dont coincide.
Extrinsic parameters :
3D Rotation () 3D translation ()
Figure 5 A case where the camera coordinate system is a rotated version of the world coordinate system.
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Figure 6 A homography or projective correspondence between two images of the same object via a third plane .
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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,
= 0 for all correspondences. is of rank 2 with seven degrees of freedom.
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