Title: CS4277_3CV / 3D Projective Geometry
On every note of this deck, unless a note says otherwise:
Title: CS4277_3CV::3D Projective Geometry
Logo: tum_logo.png

Front: How are 3-space planes written in euclidean coordinates?
Back: [$$]\pi_1X + \pi_2Y + \pi_3Z + \pi_4 = 0[/$$]

Front: How are 3-space planes written in HC?
Back: [$](\pi_1, \pi_2, \pi_3, \pi_4)^T[/$]

Front: How does a plane transform under a point transformation [$]X' = HX[/$]?
Back: [$$]\pi' = H^{-T}\pi[/$$]

Front: What is the point-plane duality?
Back: Three points define a plane, and three planes define a point.

Front: How does a line transform under a point transformation [$]X' = HX[/$]?
Back: [$$]l' = H^{-T}l[/$$] or [$$]l^{T'} = l^TH^{-1}[/$$]

Front: How are quadrics defined in [$]P^3[/$]?
Back: A quadric is a surface defined by [$$]X^TQX = 0[/$$], where Q is a symmetric 4x4 matrix.

Front: How many DOF does a quadric have?
Back: 9

Front: How many points in general position define a quadric?
Back: 9

Front: What happens if quadric Q is singular?
Back: The quadric is degenerate

Front: How does a quadric transform under a point transformation [$]X' = HX[/$]
Back: [$$]Q' = H^{-T}QH^{-1}[/$$]

Front: Is the dual of a quadric also a quadric?
Back: Yes.

Front: What are dual quadrics?
Back: Equations on planes: the tangent planes [$]\pi[/$] to the point quadric Q satisfy [$]\pi^TQ^*\pi = 0[/$].

Front: How does a dual quadric transform under a point transformation [$]X' = HX[/$]?
Back: [$$]Q^{*'} = HQ^*H^T[/$$]

Front: What are invariant properties of the projective transformation group?
Back: Intersection and tangency of surfaces in contact

Front: What are invariant properties of the affine transformation group?
Back: Parallelism of planes, volume ratios, centroids, plane at infinity

Front: What are invariant properties of the similarity transformation group?
Back: Absolute conic

Front: What are invariant properties of the euclidean transformation group?
Back: Volume

Q: The **projective distortion** may be **removed** once [...] is specified.
A: the vanishing line
Title: CS4277_3CV::3D Projective Geometry

Q: The **affine distortion** may be **removed** once the projective distortion
is removed, and the [...] is specified.
A: image of the circular points
Title: CS4277_3CV::3D Projective Geometry

Front: What remains when both projective and affine distortion were removed from the image?
Back: Similarity distortion

Front: True or False? The line at infinity is guaranteed to stay at infinity after a projective transformation?
Back: False

Front: What is the decomposition of a projective transformation in pure-projective, affine, and similarity matrices?
Back: [$$]H = H_S H_A H_P = \begin{bmatrix}sR & t \\ \bold{0}^T & 1\end{bmatrix} \begin{bmatrix} K & \bold{0} \\ \bold{0}^T & 1\end{bmatrix} \begin{bmatrix} I & \bold{0} \\ \bold{v}^T & v \end{bmatrix} = \begin{bmatrix} A & t \\ v^T & v \end{bmatrix}[/$$]

Front: How is the imaged vanishing line of the plane l computed?
Back: From the intersection of two sets of imaged parallel lines.

Front: How does the recovery of affine properties of an image work?
Back: 1. The imaged vanishing line of the plane is computed from the intersection of two sets of imaged parallel lines.
2. Compute [$]H_p' = H_A H_P^{-1}[/$], by choosing an aribitrary affinity [$]H_A[/$]

Front: How to compute a vanishing point from a length ratio?
Back: 1. Measure the distance ratio in the image [$]d(\bold{a}', \bold{b}') : d(\bold{b}', \bold{c}') = a' : b'[/$]
2. Represent [$]\bold{a}, \bold{b}, \bold{c}[/$] are represented as coordinates [$](0, 1)^T, (a, 1)^T, (a + b, 1)^T[/$] in a coordinate frame on the line [$]\langle \bold{a}, \bold{b}, \bold{c} \rangle[/$]. Similarly for [$]\langle \bold{a}', \bold{b}', \bold{c}' \rangle[/$]
3. Relative to these coordinate frames, compute the 1D projective transformation [$]H_{2 \times 2}[/$] mapping [$]\bold{a} \mapsto \bold{a}', \ \bold{b} \mapsto \bold{b}', \ \bold{c} \mapsto \bold{c}'[/$]
4. The image of the point at infinity under [$]H_{2 \times 2}[/$] is the vanishing point on the line [$]\langle \bold{a}', \bold{b}', \bold{c}' \rangle[/$]