How do we generate our polynomial to solve for the 5-point method?
We generate our coefficient matrix the same way as in the 8 point method, only with 5 points -> 5x9
Use this 5x9 known coefficient matrix to calculate the 4-Dim Basis of the Null Space X,Y,Z,W
Express the unknown essential matrix vector e as a linear combination of the null space basis
where, x,y,z,w are unknown (small letter)
w = 1 (as we only want to solve up to scale)
lastly, introduce a set of constraints to generate a non-linear polynomial system to solve to get fundamental matrix e
What is the goal of introducing constraints to our essential vector e?
introduce constraints to treat e as a function of the coefficients x,y,z,w
we have two constraints that yield 3 equations each
this non-linear system of equations needs to be solved…
How can we decompose the essential matrix?
after solving for x,y,z,w
we can decompose with SVD
Can we directly extract R and t from our SVD result?
no -> as we have no skew-symmetric matrix after the SVD to directly extract it…
How can we transform our SVD resuilt to be able to decompose it? (First step)
rewrite the middle part of the SVD to:
What is a “problematic” thing in the first step?
rewriting it is not unique
-> can change vorzeichen …
What is the second step of rewriting our SVD result to decompose it?
after the first step, write UTU inbetween the two new matrices
=> i.e. we introduce the identitiy matrix
-> now the blue part is skew symmetric (no explanation -> simply take it as it is…)
How do we get from the “box of matrices” in the second step to actually extracting t?
How do we get from the second box of matrices to the rotation matrix?
can be directly extracted and simply matrix multiplied…
How can we express the rotation and translation matrices more concise?
We can replace the “middle part” in both
=> write the middle part as rotation along the z-axis by +/-90 degree (as we have two cases…) above is +90 degree
-> do the same for the t matrix (here we have to include the E part as the rotaiotn only has -1,+1 elements)
Why can we express our rotatoin and translation more concise?
for specific alpha, these two are equal (in our case +pi/2; -pi/2 for negated first part…)
What does the “problem” we encountered when decomposing the middle part of the SVD cause?
we can derive two different translations and to different rotation matrices
-> 4 options we have to choose from
How do we solve the problem that we have 4 candidate combinations of R and t?
there exists only one solution where the points are actually infront of both cameras
What is the idea of 1D correspondance search?
take backprojection of the point we want to find correspondances for
this backprojection ray projected on the camera frame where we search the correspondance is the epipolar plane
=> potential matches have to lie on the epipolar line!
=> so called epipolar constraint!
Why do we want to reduce the correspondance search problem to 1D in the first place?
2D exhaustive search is very expensive!
Where do all epipolar lines intersect?
at the epiploe
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