Title: CS4277_3CV / Transformation and Transformation Groups (ECE 661)
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Title: CS4277_3CV::Transformation and Transformation Groups (ECE 661)
Title: CS4277_3CV::Transformation and Transformation Groups (ECE 661)

Front: What is a Planar Projective Transform?
Back: - linear transformation on homogenenous vectors
- represented by non-singular matrix H
[$]x' = Hx[/$]
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Front: What is the fundamental property of a homography?
Back: It always maps a straight line to a straight line.
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Front: What is the group of all 3x3 homographies?
Back: PL(3) (projective linear)
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Front: How is PL(3) constructed?
Back: We partition GL(3) into equivalence classes where all the 3x3 matrices in the same class are related by scalar multiples.
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Front: What subgroups of PL(3) are we interested in?
Back: Projective <- Affine <- Similarity <- Euclidean
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Front: What is the **general planar projective transform?
Back: [$$]\begin{bmatrix} A & \vec{t} \\ \vec{v}^T & v\end{bmatrix}[/$$]
with [$]A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}[/$], [$]\vec{t} = \begin{pmatrix} t_x \\ t_y \end{pmatrix}[/$], [$]\vec{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}[/$]
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Front: What implication does it have for homography H to be homogeneous?
Back: Only the ratios of elements of H are important.
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Front: How are affine transformations commonly expressed as?
Back: [$]\begin{pmatrix} x_1' \\ x_2' \\ 1\end{pmatrix} = \begin{bmatrix} a_{11} & a_{12} & t_x \\ a_{21} & a_{22} & t_y \\ 0 & 0 & 1\end{bmatrix} \begin{pmatrix} x_1 \\ x_2 \\ 1\end{pmatrix}[/$]
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Front: Is the affine transformations Matrix A singular?
Back: No. It is non-singular
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Q: A not only takes straight lines into straight lines, it also keeps [...] lines [...].
A: parallel; parallel

Q: A homography H is *affine* if and only if the [...] is mapped to the [...]
A: line at infinity; line at infinity

Front: How is the similarity group obtained from the affine group?
Back: By requiring that A be orthogonal: [$]A^TA = \lambda^2I[/$]
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Q: Similarity transform preserves [...]
A: angles

Front: How is the euclidean group obtained from the similarity group?
Back: By further requiring that A be orthonormal: [$]A^TA = I[/$].
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Q: The group of euclidean transformations corresponds to [...]
A: rigid body motions

Q: Euclidean transformations preserve the [...] between
any two points in the domain plane.
A: euclidean distance