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Affine Structure from Motion: Understanding the Affine Fundamental Matrix and Epipoles, Study notes of Computer Science

Georgia Institute of Technology - Main CampusComputer Science

These notes provide clarification on the affine fundamental matrix, a concept used in computer vision for modeling cameras with the assumption of a scene far away and small depth variation. The calculation of the affine epipolar matrix and the use of ransac for finding correspondences. With only 4 degrees of freedom, 4 correspondences are needed to recover the matrix.

Typology: Study notes

Pre 2010

Uploaded on 08/05/2009

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koofers-user-6nr 🇺🇸

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Affine Structure from Motion
(These notes are meant to clarify the slides which are the primary reference)
Affine fundamental matrix
Affine cameras model the case where
camera centers at infinity,
scene far away and depth variation is small
all lines to camera are parallel
all parallel lines in the scene map to parallel lines in the image
i.e., 3D points at infinity map to 2D points at infinity
The general case:
1000
24232221
14131211
MMMM
MMMM
The calibrated case:
1000
2232221
1131211
tRRR
tRRR
This assumes that we know the skew and stretching so you end up with
rotation matrices. Note the t’s are the projection of the world origin.
To find the epipoles, we need to ask where C' is projected onto C.
The affine epipolar matrix has a special form:
0''
0
1
00
00
1''
edycxbyax
y
x
edc
b
a
yx
There are only 4 degrees of freedom. Hence, to recover F only 4 correspondences are
needed. This is not surprising: if you have 2 affine pictures of a building, one can take the
building itself as a calibration object, and we can easily figure out the transformation to
the 2nd image.
pf3

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Affine Structure from Motion

(These notes are meant to clarify the slides which are the primary reference)

Affine fundamental matrix

Affine cameras model the case where  camera centers at infinity,  scene far away and depth variation is small  all lines to camera are parallel  all parallel lines in the scene map to parallel lines in the image  i.e., 3D points at infinity map to 2D points at infinity The general case:           0 0 0 1 21 22 23 24 11 12 13 14 M M M M M M M M The calibrated case:           0 0 0 1 21 22 23 2 11 12 13 1 R R R t R R R t This assumes that we know the skew and stretching so you end up with rotation matrices. Note the t’s are the projection of the world origin. To find the epipoles, we need to ask where C' is projected onto C. The affine epipolar matrix has a special form:

' ' 0 0 1 0 0 0 0 ' ' 1                            ax by cx dy e y x c d e b a x y There are only 4 degrees of freedom. Hence, to recover F only 4 correspondences are needed. This is not surprising: if you have 2 affine pictures of a building, one can take the building itself as a calibration object, and we can easily figure out the transformation to the 2nd image.

RANSAC

Suppose we have 2 images, as above, and want to estimate a transformation from one image to the other. The biggest problem when extracting features and matching them based on appearance is the existence of outliers, which make the estimation process impossible. RANSAC is a way to get rid of these outliers. RANSAC is best explained with a simple line fitting example:

  1. randomly sample minimal needed to estimate a line, i.e., 2 points
  2. threshold on error and count inliers
  3. pick model with most inliers The number of data-points or correspondences needed depends on the degrees of freedom of the transformation, and on the algorithm that you have available to estimate it. Using SVD, we need:  For 8DOF homography -> 4 corresponding points  For 4DOF affine fundamental matrix -> -> 4 corresponding points  For 7DOF fundamental matrix -> 8 corresponding points The number of times N we are required to sample to obtain a minimal set with no outliers, with p percent probability, is

N>log(1-p)/log(1-w^s)

Where w = 1- etha , the fraction of outliers. If etha is increased, there is an increase in number of times we are required to sample. If we increase the size s of the minimal set, then there is a more significant nonlinear increase. The COOL thing about RANSAC: N only depends on the percentage of mismatches etha, not the number of data points.