Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 1/27
Chapter 5
Reconstruction from Two Views:
Linear Algorithms
Multiple View Geometry
Summer 2021
Prof. Daniel Cremers
Chair for Computer Vision and Artificial Intelligence
Departments of Informatics & Mathematics
Technical University of Munich
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 2/27
Overview
1 The Reconstruction Problem
2 The Epipolar Constraint
3 Eight-Point Algorithm
4 Structure Reconstruction
5 Four-Point Algorithm
6 The Uncalibrated Case
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 3/27
Problem Formulation
In the last sections, we discussed how to identify point
correspondences between two consecutive frames. In this
section, we will tackle the next problem, namely that of
reconstructing the 3D geometry of cameras and points.
To this end, we will make the following assumptions:
• We assume that we are given a set of corresponding
points in two frames taken with the same camera from
different vantage points.
• We assume that the scene is static, i.e. none of the
observed 3D points moved during the camera motion.
• We also assume that the intrinsic camera (calibration)
parameters are known.
We will first estimate the camera motion from the set of
corresponding points. Once we know the relative location and
orientation of the cameras, we can reconstruct the 3D location
of all corresponding points by triangulation.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 4/27
Problem Formulation
Goal: Estimate camera motion and 3D scene structure from
two views.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 5/27
The Reconstruction Problem
In general 3D reconstruction is a challenging problem. If we are
given two views with 100 feature points in each of them, then
we have 200 point coordinates in 2D. The goal is to estimate
• 6 parameters modeling the camera motion R,T and
• 100× 3 coordinates for the 3D points Xj .
This could be done by minimizing the projection error:
E(R,T ,X1, . . . ,X100) =
∑
j
‖x j1 − π(Xj )‖
2 + ‖x j2 − π(R,T ,Xj )‖
2
This amounts to a difficult optimization problem called bundle
adjustment.
Before we look into this problem, we will first study an elegant
solution to entirely get rid of the 3D point coordinates. It leads
to the well-known 8-point algorithm.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 6/27
Epipolar Geometry: Some Notation
The projections of a point X onto the two images are denoted
by x1 and x2. The optical centers of each camera are denoted
by o1 and o2. The intersections of the line (o1,o2) with each
image plane are called the epipoles e1 and e2. The
intersections between the epipolar plane (o1,o2,X ) and the
image planes are called epipolar lines l1 and l2. There is one
epipolar plane for each 3D point X .
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 7/27
The Epipolar Constraint
We know that x1 (in homogeneous coordinates) is the
projection of a 3D point X . Given known camera parameters
(K = 1) and no rotation or translation of the first camera, we
merely have a projection with unknown depth λ1. From the first
to the second frame we additionally have a camera rotation R
and translation T followed by a projection. This gives the
equations:
λ1×1 = X , λ2×2 = RX + T .
Inserting the first equation into the second, we get:
λ2×2 = R(λ1×1) + T .
Now we remove the translation by multiplying with T̂
(T̂ v ≡ T × v ):
λ2T̂ x2 = λ1T̂Rx1
And projection onto x2 gives the epipolar constraint:
x>2 T̂R x1 = 0
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 8/27
The Epipolar Constraint
The epipolar constraint
x>2 T̂R x1 = 0
provides a relation between the 2D point coordinates of a 3D
point in each of the two images and the camera transformation
parameters. The original 3D point coordinates have been
removed. The matrix
E = T̂R ∈ R3×3
is called the essential matrix. The epipolar constraint is also
known as essential constraint or bilinear constraint.
Geometrically, this constraint states that the three vectors
−−→
o1X ,
−−→o2o1 and
−−→
o2X form a plane, i.e. the triple product of these
vectors (measuring the volume of the parallelepiped) is zero: In
coordinates of the second frame Rx1 gives the direction of the
vector
−−→
o1X ; T gives the direction of
−−→o2o1, and x2 is proportional
to the vector
−−→
o2X such that
volume = x>2 (T × Rx1) = x
>
2 T̂R x1 = 0.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 9/27
Properties of the Essential Matrix E
The space of all essential matrices is called the essential
space:
E ≡
{
T̂R | R ∈ SO(3), T ∈ R3
}
⊂ R3×3
Theorem [Huang & Faugeras, 1989] (Characterization of the
essential matrix): A nonzero matrix E ∈ R3×3 is an essential
matrix if and only if E has a singular value decomposition
(SVD) E = UΣV> with
Σ = diag{σ, σ, 0}
for some σ > 0 and U,V ∈ SO(3).
Theorem (Pose recovery from the essential matrix): There
exist exactly two relative poses (R,T ) with R ∈ SO(3) and
T ∈ R3 corresponding to an essential matrix E ∈ E . For
E = UΣV> we have:
(T̂1,R1) =
(
URZ (+π2 )ΣU
>, UR>Z (+
π
2 )V
>) , (1)
(T̂2,R2) =
(
URZ (−π2 )ΣU
>, UR>Z (−
π
2 )V
>) , (2)
In general, only one of these gives meaningful (positive) depth
values.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 10/27
A Basic Reconstruction Algorithm
We have seen that the 2D-coordinates of each 3D point are
coupled to the camera parameters R and T through an
epipolar constraint. In the following, we will derive a 3D
reconstruction algorithm which proceeds as follows:
• Recover the essential matrix E from the epipolar
constraints associated with a set of point pairs.
• Extract the relative translation and rotation from the
essential matrix E .
In general, the matrix E recovered from a set of epipolar
constraints will not be an essential matrix. One can resolve this
problem in two ways:
1 Recover some matrix E ∈ R3×3 from the epipolar
constraints and then project it onto the essential space.
2 Optimize the epipolar constraints in the essential space.
While the second approach is in principle more accurate it
involves a nonlinear constrained optimization. We will pursue
the first approach which is simpler and faster.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 11/27
The Eight-Point Linear Algorithm
First we rewrite the epipolar constraint as a scalar product in
the elements of the matrix E and the coordinates of the points
x1 and x2. Let
Es = (e11,e21,e31,e12,e22,e32,e13,e23,e33)
> ∈ R9
be the vector of elements of E and
a ≡ x1 ⊗ x2
the Kronecker product of the vectors x i = (xi , yi , zi ), defined as
a = (x1x2, x1y2, x1z2, y1x2, y1y2, y1z2, z1x2, z1y2, z1z2)
> ∈ R9.
Then the epipolar constraint can be written as:
x>2 E x1 = a
>Es = 0.
For n point pairs, we can combine this into the linear system:
χEs = 0, with χ = (a1,a2, . . . ,an)>.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 12/27
The Eight-Point Linear Algorithm
According to
χEs = 0, with χ = (a1,a2, . . . ,an)>.
we see that the vector of coefficients of the essential matrix E
defines the null space of the matrix χ. In order for the above
system to have a unique solution (up to a scaling factor and
ruling out the trivial solution E = 0), the rank of the matrix χ
needs to be exactly 8. Therefore we need at least 8 point pairs.
In certain degenerate cases, the solution for the essential
matrix is not unique even if we have 8 or more point pairs. One
such example is the case that all points lie on a line or on a
plane.
Clearly, we will not be able to recover the sign of E . Since with
each E , there are two possible assignments of rotation R and
translation T , we therefore end up with four possible solutions
for rotation and translation.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 13/27
Projection onto Essential Space
The numerically estimated coefficients Es will in general not
correspond to an essential matrix. One can resolve this
problem by projecting it back to the essential space.
Theorem (Projection onto essential space): Let F ∈ R3×3 be
an arbitrary matrix with SVD
F = U diag{λ1, λ2, λ3}V>, λ1 ≥ λ2 ≥ λ3. Then the essential
matrix E which minimizes the Frobenius norm ‖F − E‖2f is
given by
E = U diag{σ, σ, 0}V>, with σ =
λ1 + λ2
2
.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 14/27
Eight Point Algorithm (Longuet-Higgins ’81)
Given a set of n = 8 or more point pairs x i1,x
i
2:
• Compute an approximation of the essential matrix.
Construct the matrix χ = (a1,a2, . . . ,an)>, where
ai = x i1 ⊗ x
i
2. Find the vector E
s ∈ R9 which minimizes
‖χEs‖ as the ninth column of Vχ in the SVD χ = UχΣχV>χ .
Unstack Es into 3× 3-matrix E .
• Project onto essential space. Compute the SVD
E = U diag{σ1, σ2, σ3}V>. Since in the reconstruction, E
is only defined up to a scalar, we project E onto the
normalized essential space by replacing the singular
values σ1, σ2, σ3 with 1,1,0.
• Recover the displacement from the essential matrix. The
four possible solutions for rotation and translation are:
R = UR>Z (±
π
2 )V
>, T̂ = URZ (±π2 )ΣU
>,
with a rotation by ±π2 around z:
R>Z (±
π
2 ) =
0 ±1 0∓1 0 0
0 0 1
.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 15/27
Do We Need Eight Points?
The above reasoning showed that we need at least eight points
in order for the matrix χ to have rank 8 and therefore guarantee
a unique solution for E . Yet, one can take into account the
special structure of E . The space of essential matrices is
actually a five-dimensional space, i.e. E only has 5 (and not 9)
degrees of freedom.
A simple way to take into account the algebraic properties of E
is to make use of the fact that det E = 0. If now we have only 7
point pairs, the null space of χ will have (at least) Dimension 2,
spanned by two vectors Es1 and E
s
2 . Then we can solve for E by
determining α such that:
det E = det(E1 + αE2) = 0.
Along similar lines, Kruppa proved in 1913 that one needs only
five point pairs to recover (R,T ). In the case of degenerate
motion (for example planar or circular motion), one can resolve
the problem with even fewer point pairs.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 16/27
Limitations and Further Extensions
Among the four possible solutions for R and T , there is
generally only one meaningful one (which assigns positive
depth to all points).
The algorithm fails if the translation is exactly 0, since then
E = 0 and nothing can be recovered. Due to noise this
typically does not happen.
In the case of infinitesimal view point change, one can adapt
the eight point algorithm to the continuous motion case, where
the epipolar constraint is replaced by the continuous epipolar
constraint. Rather than recovering (R,T ) one recovers the
linear and angular velocity of the camera.
In the case of independently moving objects, one can
generalize the epipolar constraint. For two motions for example
we have:
(x>2 E1x1)(x
>
2 E2x1) = 0
with two essential matrices E1 and E2. Given a sufficiently large
number of point pairs, one can solve the respective equations
for multiple essential matrices using polynomial factorization.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 17/27
Structure Reconstruction
The linear eight-point algorithm allowed us to estimate the
camera transformation parameters R and T from a set of
corresponding point pairs. Yet, the essential matrix E and
hence the translation T are only defined up to an arbitrary
scale γ ∈ R+, with ‖E‖ = ‖T‖ = 1. After recovering R and T ,
we therefore have for point X j :
λ
j
2x
j
2 = λ
j
1Rx
j
1 + γT , j = 1, . . . ,n,
with unknown scale parameters λji . We can eliminate one of
these scales by applying x̂ j2:
λ
j
1x̂
j
2Rx
j
1 + γx̂
j
2T = 0, j = 1, . . . ,n.
This corresponds to n linear systems of the form(
x̂ j2Rx
j
1, x̂
j
2T
)(
λ
j
1
γ
)
= 0. j = 1, . . . ,n.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 18/27
Structure Reconstruction
Combining the parameters ~λ = (λ11, λ
2
1, . . . , λ
n
1, γ)
> ∈ Rn+1, we
get the linear equation system
M~λ = 0
with
M ≡
x̂12Rx
1
1 0 0 0 0 x̂
1
2T
0 x̂22Rx
2
1 0 0 0 x̂
2
2T
0 0
. . . 0 0
…
0 0 0 x̂n−12 Rx
n−1
1 0 x̂
n−1
2 T
0 0 0 0 x̂n2Rx
n
1 x̂
n
2T
.
The linear least squares estimate for ~λ is given by the
eigenvector corresponding to the smallest eigenvalue of M>M.
It is only defined up to a global scale. It reflects the ambiguity
that the camera has moved twice the distance, the scene is
twice larger and twice as far away.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 19/27
Example
Left image right image
Reconstruction (Author: Jana Košecká)
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 20/27
Degenerate Configurations
The eight-point algorithm only provides unique solutions (up to
a scalar factor) if all 3D points are in a “general position”. This
is no longer the case for certain degenerate configurations, for
which all points lie on certain 2D surfaces which are called
critical surfaces.
Typically these critical surfaces are described by a quadratic
equation in the three point coordinates, such that they are
referred to as quadratic surfaces.
While most critical configurations do not actually arise in
practice, a specific degenerate configuration which does arise
often is the case that all points lie on a 2D plane (such as
floors, table, walls,…).
For the structure-from-motion problem in the context of points
on a plane, one can exploit additional constraints which leads
to the so-called four-point algorithm.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 21/27
Planar Homographies
Let us assume that all points lie on a plane. If X 1 ∈ R3 denotes
the point coordinates in the first frame, and these lie on a plane
with normal N ∈ S2, then we have:
N>X 1 = d ⇔
1
d
N>X 1 = 1.
In frame two, we therefore have the coordinates:
X 2 = RX 1 +T = RX 1 +T
1
d
N>X 1 =
(
R +
1
d
TN>
)
X 1 ≡ HX 1,
where
H = R +
1
d
TN> ∈ R3×3
is called a homography matrix. Inserting the 2D coordinates,
we get:
λ2×2 = Hλ1×1 ⇔ x2 ∼ Hx1,
where ∼ means equality up to scaling. This expression is
called a planar homography. H depends on camera and plane
parameters.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 22/27
From Point Pairs to Homography
For a pair of corresponding 2D points we therefore have
λ2×2 = Hλ1×1.
By multiplying with x̂2 we can eliminate λ2 and obtain:
x̂2Hx1 = 0
This equation is called the planar epipolar constraint or planar
homography constraint.
Again, we can cast this equation into the form
a>Hs = 0,
where we have stacked the elements of H into a vector
Hs = (H11,H21, . . . ,H33)
> ∈ R9,
and introduced the matrix
a ≡ x1 ⊗ x̂2 ∈ R9×3.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 23/27
The Four Point Algorithm
Let us now assume we have n ≥ 4 pairs of corresponding 2D
points {x j1,x
j
2}, j = 1, . . . ,n in the two images. Each point pair
induces a matrix aj , we integrate these into a larger matrix
χ ≡ (a1, . . . ,an)> ∈ R3n×9,
and obtain the system
χHs = 0.
As in the case of the essential matrix, the homography matrix
can be estimated up to a scale factor.
This gives rise to the four point algorithm:
• For the point pairs, compute the matrix χ.
• Compute a solution Hs for the above equation by singular
value decomposition of χ.
• Extract the motion parameters from the homography
matrix H = R + 1d TN
>.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 24/27
General Comments
Clearly, the derivation of the four-point algorithm is in close
analogy to that of the eight-point algorithm.
Rather than estimating the essential matrix E one estimates
the homography matrix H to derive R and T . In the four-point
algorithm, the homography matrix is decomposed into R, N
and T/d . In other words, one can reconstruct the normal of the
plane, but the translation is only obtained in units of the offset d
of the plane and the origin.
The 3D structure of the points can then be computed in the
same manner as before.
Since one uses the strong constraint that all points lie in a
plane, the four-point algorithm only requires four
correspondences.
There exist numerous relations between the essential matrix
E = T̂R and the corresponding homography matrix
H = R + Tu> with some u ∈ R3, in particular:
E = T̂H, H>E + E>H = 0.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 25/27
The Case of an Uncalibrated Camera
The reconstruction algorithms introduced above all assume
that the camera is calibrated (K = 1). The general
transformation from a 3D point to the image is given by:
λx ′ = K Π0 g X = (KR,KT )X ,
with the intrinsic parameter matrix or calibration matrix:
K =
fsx fsθ ox0 fsy oy
0 0 1
∈ R3×3.
The calibration matrix maps metric coordinates into image
(pixel) coordinates, using the focal length f , the optical center
ox ,oy , the pixel size sx , sy and a skew factor sθ. If these
parameters are known then one can simply transform the pixel
coordinates x ′ to normalized coordinates x = K−1x ′ to obtain
the representation used in the previous sections. This amounts
to centering the coordinates with respect to the optical center
etc.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 26/27
The Fundamental Matrix
If the camera parameters K cannot be estimated in a
calibration procedure beforehand, then one has to deal with
reconstruction from uncalibrated views.
By transforming all image coordinates x ′ with the inverse
calibration matrix K−1 into metric coordinates x , we obtain the
epipolar constraint for uncalibrated cameras:
x>2 T̂Rx1 = 0 ⇔ x
′>
2 K
−>T̂RK−1x ′1 = 0,
which can be written as
x ′>2 Fx
′
1 = 0,
with the fundamental matrix defined as:
F ≡ K−>T̂RK−1 = K−>EK−1.
Since the invertible matrix K does not affect the rank of this
matrix, we know that F has an SVD F = UΣV> with
Σ = diag(σ1, σ2,0). In fact, any matrix of rank 2 can be a
fundamental matrix.
Reconstruction from
Two Views: Linear
Algorithms
Prof. Daniel Cremers
The Reconstruction
Problem
The Epipolar
Constraint
Eight-Point Algorithm
Structure
Reconstruction
Four-Point Algorithm
The Uncalibrated Case
updated April 12, 2021 27/27
Limitations
While it is straight-forward to extend the eight-point algorithm,
such that one can extract a fundamental matrix from a set of
corresponding image points, it is less straight forward how to
proceed from there.
Firstly, one cannot impose a strong constraint on the specific
structure of the fundamental matrix (apart from the fact that the
last singular value is zero).
Secondly, for a given fundamental matrix F , there does not
exist a finite number of decompositions into extrinsic
parameters R,T and intrinsic parameters K (even apart from
the global scale factor).
As a consequence, one can only determine so-called
projective reconstructions, i.e. reconstructions of geometry and
camera position which are defined up to a so-called projective
transformation.
As a solution, one typically choses a canonical reconstruction
from the family of possible reconstructions.
The Reconstruction Problem
The Epipolar Constraint
Eight-Point Algorithm
Structure Reconstruction
Four-Point Algorithm
The Uncalibrated Case