Structure from Motion - Introduction to Computer Version - Lecture Sli, Lecture notes of Computer Science

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Structure from Motion

Computer Vision

This class: structure from motion

• Recap of epipolar geometry
  • Depth from two views
• Affine structure from motion

Recap: Fundamental Matrix

• Fundamental matrix maps from a point in one

image to a line in the other

• If x and x’ correspond to the same 3d point X:

Structure from motion

• Given a set of corresponding points in two or more images, compute the camera parameters and the 3D point coordinates

Camera 1 (^) Camera 2 Camera 3 R 1 ,t (^1) R 2 ,t 2 R^3 ,t^3

? ??^ Noah SnavelySlide credit:

?

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How do we know the scale of image content?

Structure from motion ambiguity

• If we scale the entire scene by some factor k and, at the same time, scale the camera matrices by the factor of 1/ k , the projections of the scene points in the image remain exactly the same
• More generally: if we transform the scene using a transformation Q and apply the inverse transformation to the camera matrices, then the images do not change

x PX PQ QX

-

Projective structure from motion

• Given: m images of n fixed 3D points
  • x ij = P i X j , i = 1 ,… , m, j = 1 , … , n
• Problem: estimate m projection matrices P i and n 3D points X j from the mn corresponding points x ij

x 1 j

x 2 j

x 3 j

Xj

P 1

P 2

P 3

Slides from Lana Lazebnik Docsity.com

Types of ambiguity

vT v

Projective A t 15dof

Affine 12dof

Similarity 7dof

Euclidean 6dof

Preserves intersection and tangency

Preserves parallellism, volume ratios

Preserves angles, ratios of length

0 1

A t T

0 1

R t T

s

0 1

R t T^ Preserves angles, lengths

• With no constraints on the camera calibration matrix or on the scene, we get a projective reconstruction
• Need additional information to upgrade the reconstruction to affine, similarity, or Euclidean Docsity.com

Projective ambiguity

x PX PQ QP X

- P

vT v

A t Q p

Affine ambiguity

x PX PQ QA X

- A

Affine

0 1

A t QA T

Affine ambiguity

Similarity ambiguity

Bundle adjustment

• Non-linear method for refining structure and motion
• Minimizing reprojection error 2

1 1

m

i

n

j

E P X D x ij P i X j

x 1 j x 2 j

x 3 j

X j

P 1

P 2

P 3

P 1 X j

P 2 X j^ P^3 X j