Multilinear Systems and Invariant Theory

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Multi-linear Systems and Invariant Theory in
the Context of Computer Vision and
Graphics Class 5 Self Calibration CS329 Stanfo
rd University
Amnon Shashua
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Material We Will Cover Today

• The basic equations and counting arguments

• The absolute conic and its image.

• Kruppas equations

• Recovering internal parameters.

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The Basic Equations and Counting Arguments
Recall,
3D-gt2D from Euclidean world frame to image
world frame to first camera frame
Let K,K be the internal parameters of camera 1,2
and choose canonical
frame in which RI and T0 for first camera.
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The Basic Equations and Counting Arguments
where
(8 unknown parameters)
maps from the projective frame to Euclidean
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The Basic Equations and Counting Arguments
are the points on the plane at infinity (in Euc
frame)
is the plane at infinity
is the plane at infinity in Proj frame
(recall if W maps points to points (Euc -gt
Proj), then the dual
maps planes to planes)
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The Basic Equations and Counting Arguments
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The Basic Equations and Counting Arguments
Projective frame
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The Basic Equations and Counting Arguments
since
then,
but
provides 5 (non-linear) constraints!
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The Basic Equations and Counting Arguments
Since the right-hand side is symmetric and up to
scale, we have 5 constraints.
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The Basic Equations and Counting Arguments
Lets do some counting
Let
be the number of internal parameters
be the number of views
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The Basic Equations and Counting Arguments
not enough measurements (!)
(fixed internal params)
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The remainder of this lecture is about a
geometric insight of
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The Absolute Conic
where
represents a conic in 2D
are the points on the plane at infinity (in Euc
frame)
is the plane at infinity
is conic on the plane at infinity
when
is the absolute conic (imaginary circle)
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The Absolute Conic
Plane at infinity is preserved under affine
transformations
because
is preserved under similarity transformation (R,t
up to scale)
and
if
then
but
so in order that
we must have
is orthogonal
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The Image of the Absolute Conic
Image of points at infinity
let
if
is a conic on the plane at infinity
then
is the projected conic onto the image
then
since
the image of
is
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The Image of the Dual Absolute Conic
is tangent to the conic at p
is the image of the dual absolute conic
The basic equation
Becomes
Why 8 parameters? 5 for the conic, 3 for the plane
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Geometric Interpretation of
p
direction of optical ray
The angle between two optical rays
given
one can measure angles
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Kruppas Equations
General idea eliminate n from the basic equation.
are degenerate (rank 2) conics
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Kruppas Equations
Note
is a degenerate conic
iff
or
Let
be the homography induced by the plane of the
conic
(slide 14)
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Kruppas Equations
Recall
and the conic is
In our case
Likewise
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Determining K given
the location of the plane at inifinity in the
projective coordinate frame.
Recall
We wish to represent the homography
induced by
be a point on the plane at infinity.
Let
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Determining K given
Recall
(slide 16)
Note
this could be derived from first principles as
well
tangents lines to the image of the absolute conic
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Determining K given
Assume fixed internal parameters
Note
Provides 4 independent linear constraints on
Why 4 and not 5?
we need 3 views (since
has 5 unknowns)
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Why 4 Constraints?
and
are similar matrices, i.e., have the same
eigenvalues
be the axis of rotation, i.e.,
Let
has an eigenvalue 1, with eigenvector
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Why 4 Constraints?
if
is a solution to
then
is also a solution
We need one more camera motion (with a different
axis of rotation).
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Kruppas Equations (revisited)
Kruppas equations
Start with the basic equation
Multiply the terms by
on both sides