Outline
โข Block matrix multiplication
โข 8-point algorithm
โข Factorization
Block matrix multiplication
Block matrix
Just treat them as elements.
Problem 1
โข ๐๐ป = ๐ด, ๐๐ป1, ๐ป2๐ป3, ๐ป4
= ๐ผ3, 0
๐ด๐ป1 + ๐๐ป3 = ๐ผ3
๐ด๐ป2 + ๐๐ป4 = 0
= 3
3 1 3 1
3 1
How to choose H3 and H4?
8-point algorithm
Epipolar geometry
epipolar plane
epipolar line
โข P: object
โข O1, O2: center of camera
โข p1, p2: image point
โข e1,e2: epipole
Fundamental matrix F
โข F is rank 2
โ why? ๐น = ๐พโ๐ ๐ร ๐ ๐พโฒโ1, and ๐ร is rank 2.
โ Use SVD to ensure this property.
โข F has 7 dof
โ 8 independent ratio due to scaling.
โ det ๐น = 0 โ 7 dof
โข Transpose
โ F for cameras (O1, O2) iff FT for cameras (O2, O1)
Fundamental matrix F (cont'd)
โข Epipolar lines: ๐1 = ๐น๐2, ๐1๐ โ ๐1 = 0
โ 2D line:
โข Epipole: โ๐2, ๐1๐ ๐น๐2 = 0
โ e1 is left null vector of F
โ Similarly, โ๐1, ๐1๐๐น ๐2 = 0,
so e2 is right null vector of F
โข Correlation: for epipolar line pair l and l', any point p on l is mapped to l' (no inverse)
Computation of F
For each pair of corresponding points (u', v', 1), (u, v, 1):
8-point algorithm!
Numerical error
Normalized 8-ยญโpoint algorithm
SVD!
SVD!
โขโฏNormalize: qi = Tpi, qiโ = Tโpiโ
โขโฏ8-point algorithm to solve F from qi
T Fq qiโ = 0 โขโฏForce Fq to have rank 2
โขโฏDe-normalize Fq to get F F = TT Fq Tโ
Normalizing data points
โข Goal โ Mean: 0
โ Average distance to the mean: 2
โข Intuitively, we want ๐๐ = ๐๐ โ ๐๐ 2
๐
โ ๐ฅ๐ =1
๐ ๐ฅ๐ , ๐ ๐ฆ๐ =
1
๐ ๐ฆ๐ , ๐
โ ๐ =1
๐ ๐ฅ๐ โ ๐ฅ๐
2 + ๐ฆ๐ โ ๐ฆ๐ 2
๐
โข ๐๐ =2/๐ 0 โ๐ฅ 2/๐
0 2/๐ โ๐ฆ 2/๐0 0 1
๐๐
3x1 3x1
(๐๐ , ๐๐ )
๐
Use SVD on least square problem
โข Solve over-determined ๐ด๐ฅ = 0 min ๐ด๐ฅ 2 s. t. ๐ฅ 2 = 1
From SVD, ๐ด = ๐ฮฃ๐๐, want to minimize ๐ด๐ฅ 2 = ๐ฅ๐๐ด๐๐ด๐ฅ = ๐ฅ๐ ๐ฮฃ๐๐ ๐ ๐ฮฃ๐ ๐ฅ = ๐ฅ๐๐ฮฃ๐๐๐๐ฮฃ๐๐๐ฅ = ๐ฅ๐๐ฮฃ๐ฮฃ๐๐๐ฅ
= ๐๐2 ๐ฃ๐๐๐ฅ2
๐
Choose x to be vk corresponding to smallest ๐๐
Use SVD to reduce rank
โข ๐ด = ๐ฮฃ๐๐ = ๐๐1 โฏ โฏโฎ ๐2 โฎโฆ โฏ โฑ
๐๐ = ๐๐๐ข๐๐ฃ๐๐
๐
โข Intuition: only retain k components
โ Gives best rank k approximation of A
โข For formal proof, see Eckart-Young theorem
Enforcing rank 2 on F
Non-singular F Singular F
Factorization
Structure From Motion
known solve for
Factorization
known solve for
Factorization (1)
(2) SVD
(3) Columns are the 3D points
Factorization
โข DEMO