Multi-Geo_CV.lyx

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\begin_body

\begin_layout Chapter
Intro
\end_layout

\begin_layout Standard
An introduction to the principal ideas.
\end_layout

\begin_layout Section
Projective geometry
\end_layout

\begin_layout Standard
Straightness is preserved, the most general requirement on the mapping.
\end_layout

\begin_layout Standard
In Euclidean space, parallel lines do not meet.
 By adding an infinity point to a Euclidean space, projective space is introduce
d, an extension of Euclidean space in which two lines always meet in a point.
\end_layout

\begin_layout Standard
Points are represented by 
\emph on
equivalence classes
\emph default
 of coordinate triples, where two triples are equivalent when they differ
 by a common multiple, called 
\emph on
homogeneous coordinates.

\emph default
 Given a point 
\begin_inset Formula $\left(x,y,0\right)$
\end_inset

, divide by the last coordinate, 
\begin_inset Formula $\left(x/0,y/0\right)$
\end_inset

 is the point at infinity.
 The Euclidean space 
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset

 can be extended to a project space 
\begin_inset Formula $\mathbb{P}^{n}$
\end_inset

 by representing poitns at homogenous vectors.
\end_layout

\begin_layout Subsection
Homogeneity
\end_layout

\begin_layout Standard
In classiclal Euclidean geometry all points are the same, no distringuished
 point.
 From the point of view of projective geometry, points at infinity are not
 any different from other points.
 By analogy of 
\begin_inset CommandInset href
LatexCommand href
name "Euclidean transformation"
target "https://en.wikipedia.org/wiki/Rigid_transformation"

\end_inset

, a projective transformation of projective space 
\begin_inset Formula $\mathbb{P}^{n}$
\end_inset

 is defined by a linear transformation of homogeneous coordinates
\begin_inset Formula 
\[
X^{'}=H_{\left(n+1\right)\times\left(n+1\right)}X.
\]

\end_inset


\end_layout

\begin_layout Standard
In CV problems, projective space is used as convenient way of repressenting
 the real 3D world.
 In reality, the real world and images of it do not contain points at infinity.
\end_layout

\begin_layout Subsection
Affine geometry
\end_layout

\begin_layout Standard
Parallelism is not a concept of projective geometry, since all lines in
 projective space meet at some point.
 Parallel lines are just lines that intersect at points at infinity.
\end_layout

\begin_layout Standard
The geometry of the projective plane and a distinguished line is known as
 
\emph on
affine geometry
\emph default
 and any projective transformation that maps the distringuished line in
 one space to the distinguished line of the other space is known as an 
\emph on
affine transformation
\emph default
.
\end_layout

\begin_layout Subsection
Euclidean geometry
\end_layout

\begin_layout Standard
Singling out some special feature of the line or plane at infinity affine
 geometry becomes Euclidean geometry.
\end_layout

\begin_layout Section
Camera Projections
\end_layout

\begin_layout Standard
The usual way of modelling the drop from 3D world to a 2D image is by 
\emph on
central projection
\emph default
 in which a ray from a point in space is drawn from a 3D world point through
 a fixed point in space, called the 
\emph on
center of project
\emph default
.
 All rays intersect a specific plane in space chosen as the 
\emph on
image plane
\emph default
.
 Central projection is simply a map from 
\begin_inset Formula $\mathbb{P}^{3}$
\end_inset

 to 
\begin_inset Formula $\mathbb{P}^{2}$
\end_inset

.
\end_layout

\begin_layout Standard
All points in a ray passing through the center of projection project to
 the same point in an image.
 Thus, the set of all image points is the same as the set of rays through
 the camera center.
 The set of rays themselves may be thought of as a representation of the
 image space 
\begin_inset Formula $\mathbb{P}^{2}$
\end_inset

.
 All that is important is the camera center.
\end_layout

\begin_layout Part
Projective Geometry, Transformations and Estimation
\end_layout

\begin_layout Standard
The ideas of notation of projective geometry are central to an analysis
 of multiple view geometry.
\end_layout

\begin_layout Chapter
Projective Geometry and Transformations of 2D
\end_layout

\begin_layout Standard
The main geometric ideas and notation that are required to understand the
 main material.
\end_layout

\begin_layout Standard
A significant advantage of the algebraic approach to geometry is that results
 derived in this way may more easily be used to derive algorithms and practical
 computational methods.
\end_layout

\begin_layout Section
Homogeneous representation of points and lines on a plane
\end_layout

\begin_layout Standard
\begin_inset CommandInset href
LatexCommand href
name "Equivalence class"
target "https://en.wikipedia.org/wiki/Equivalence_class"

\end_inset


\end_layout

\begin_layout Subsection
Lines
\end_layout

\begin_layout Standard
Since 
\begin_inset Formula $ax+by+c=0$
\end_inset

 and 
\begin_inset Formula $kax+kby+kc=0$
\end_inset

 are the same line, any particular 
\begin_inset Formula $(a,b,c)^{T}$
\end_inset

 , except 
\begin_inset Formula $\left(0,0,0\right)^{T}$
\end_inset

, which is not a line, is a representative of the equivalence class of lines.
 The set of equivalence classes of vectors in 
\begin_inset Formula $\mathbb{R}^{3}-\left(0,0,0\right)^{T}$
\end_inset

 forms the projective space 
\begin_inset Formula $\mathbb{P}^{2}$
\end_inset

.
 The study of the geometry of 
\begin_inset Formula $\mathbb{P}^{2}$
\end_inset

 is known as 
\emph on
projective geometry
\emph default
.
\end_layout

\begin_layout Subsection
Points
\end_layout

\begin_layout Standard
It is easy to see that for a point 
\begin_inset Formula $(x,y)$
\end_inset

 and a line 
\begin_inset Formula $\boldsymbol{l}$
\end_inset

, 
\begin_inset Formula $\left(kx,ky,k\right)\boldsymbol{l}=0$
\end_inset

 iff 
\begin_inset Formula $\left(x,y,1\right)\boldsymbol{l}=0$
\end_inset

, thus points as homogeneous vectors are also elements of 
\begin_inset Formula $\mathbb{P}^{2}$
\end_inset

.
\end_layout

\begin_layout Subsection
Degrees of freedeom
\end_layout

\begin_layout Standard
Lines and points need two parameters to specify.
\end_layout

\begin_layout Subsection
Interseciton of lines
\end_layout

\begin_layout Standard
The interseciton of two lines 
\begin_inset Formula $\boldsymbol{l}$
\end_inset

 and 
\begin_inset Formula $\boldsymbol{l}^{\prime}$
\end_inset

 is the point 
\begin_inset Formula $\boldsymbol{x}=\boldsymbol{l}\times\boldsymbol{l}^{\prime}$
\end_inset

.
\end_layout

\begin_layout Subsection
Line joining points
\end_layout

\begin_layout Standard
The line through two points 
\begin_inset Formula $\boldsymbol{x}$
\end_inset

 and 
\begin_inset Formula $\boldsymbol{x}^{\prime}$
\end_inset

 is 
\begin_inset Formula $\boldsymbol{l}=\boldsymbol{x}\times\boldsymbol{x}^{\prime}$
\end_inset

.
\end_layout

\begin_layout Section
Ideal points and the line at infinity
\end_layout

\begin_layout Standard
The points with last coordinate 
\begin_inset Formula $x_{3}=0$
\end_inset

 are known as 
\emph on
ideal points
\emph default
, or 
\emph on
points at infinity, written as 
\begin_inset Formula 
\[
\left(x_{1},x_{2},0\right)^{T},
\]

\end_inset

which lies on the line at infinity 
\begin_inset Formula $l_{\infty}=\left(0,0,1\right)^{T}$
\end_inset

.
\end_layout

\begin_layout Standard
Given a line 
\begin_inset Formula $l=\left(a,b,c\right)$
\end_inset

 and a parallel line 
\begin_inset Formula $l^{\prime}=\left(a,b,c^{\prime}\right)$
\end_inset

, both intersects 
\begin_inset Formula $l_{\infty}$
\end_inset

 at 
\begin_inset Formula $\left(b,-a,0\right)$
\end_inset

.
 As the line's direction varies, the ideal point 
\begin_inset Formula $\left(b,-a,0\right)$
\end_inset

 varies over 
\begin_inset Formula $l_{\infty}$
\end_inset

.
 In 
\begin_inset Formula $\mathbb{P}^{2}$
\end_inset

, two distinct lines always meet in a single point and two distincts lie
 on a single line.
\end_layout

\begin_layout Standard
Points and lines of 
\begin_inset Formula $\mathbb{P}^{2}$
\end_inset

 are represented by rays and planes, through the origin in 
\begin_inset Formula $\mathbb{R}^{3}$
\end_inset

.
 Lines lying in the 
\begin_inset Formula $x_{1}x_{2}$
\end_inset

-plane represent ideal points, and the 
\begin_inset Formula $x_{1}x_{2}$
\end_inset

-plane represents 
\begin_inset Formula $l_{\infty}$
\end_inset

.
\end_layout

\begin_layout Proposition
Duality principle
\end_layout

\begin_layout Proposition
To any theorem of 
\begin_inset Formula $2$
\end_inset

-dimensional projective geometry, there corresponds a dual theorem, which
 may be derived by interchanging roles of points and lines in the original
 theorem.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Standard
The proof of hte dual theorem will be the dual of the proof of the original
 theorem.
 Once the original theorem has been proved, it is not necessary to prove
 the dual of a given theorem.
\end_layout

\begin_layout Subsection
Conics and dual conics
\end_layout

\begin_layout Standard
In 2D projective geometry, all non-degenerate conics are equivalent under
 projective transformations.
 The homogeneous form of a conic can be obtained by replacing 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 with 
\begin_inset Formula $x_{1}/x_{3}$
\end_inset

 and 
\begin_inset Formula $x_{2}/x_{3}$
\end_inset

, resulting in 
\begin_inset Formula 
\[
\boldsymbol{x}^{T}C\boldsymbol{x}=0
\]

\end_inset

where the conic coefficient matrix 
\begin_inset Formula $C$
\end_inset

 is given by 
\begin_inset Formula 
\[
C=\begin{bmatrix}a & b/2 & d/2\\
b/2 & c & e/2\\
d/2 & e/2 & f
\end{bmatrix}.
\]

\end_inset


\end_layout

\begin_layout Standard
The conic 
\begin_inset Formula $C$
\end_inset

 is more properly termed a 
\emph on
point 
\emph default
conic, a sit defines an equation on points.
\end_layout

\begin_layout Standard
The conic has five degrees of freedom 
\begin_inset Formula $\left\{ a\colon b\colon c\colon d\colon e\colon f\right\} $
\end_inset

 or equivalently the six elements of a symmetric matrix less one for scale.
\end_layout

\begin_layout Proposition
A conic is determined uniquely (up to scale) by five points in general position.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Standard
The method of fitting a geometric entity or relation by determining a null
 space is frequently used.
\end_layout

\begin_layout Proposition
The line 
\begin_inset Formula $l$
\end_inset

 tangent to a conic 
\begin_inset Formula $C$
\end_inset

 at a point 
\begin_inset Formula $\boldsymbol{x}$
\end_inset

 is given by 
\begin_inset Formula $\boldsymbol{l}=C\boldsymbol{x}$
\end_inset

.
\end_layout

\begin_layout Proof
The line passes through 
\begin_inset Formula $\boldsymbol{x}$
\end_inset

, that is, 
\begin_inset Formula 
\[
l^{T}x=x^{T}Cx=0.
\]

\end_inset


\end_layout

\begin_layout Proof
If 
\begin_inset Formula $l$
\end_inset

 has one point contact with the conic, then we are done with the proof.
 Otherwise it meets the conic at another point 
\begin_inset Formula $y$
\end_inset

.
 Then 
\begin_inset Formula $y^{T}Cy=0$
\end_inset

 and 
\begin_inset Formula $x^{T}Cy=l^{T}y=0.$
\end_inset

 Then 
\begin_inset Formula 
\[
\left(x+\alpha y\right)^{T}C\left(x+\alpha y\right)=0\quad\text{for all }\alpha.
\]

\end_inset

which means the whole line 
\begin_inset Formula $l$
\end_inset

 lies on the conic 
\begin_inset Formula $C$
\end_inset

, which is degenerate.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Definition
If the matrix 
\begin_inset Formula $C$
\end_inset

 is not of full rank, then the conic is termed degenerate.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Standard
Degenerate point conics include two lines (rank 
\begin_inset Formula $2$
\end_inset

), and a repeated line (rank 
\begin_inset Formula $1$
\end_inset

).
\end_layout

\begin_layout Standard
Given the duality result, it is not surprising that there is also a conic
 which defines an equaiton on lines.
\end_layout

\begin_layout Definition
Dual conics, 
\begin_inset Formula $C^{*}$
\end_inset


\end_layout

\begin_layout Definition
A line 
\begin_inset Formula $l$
\end_inset

 tangent to the conic 
\begin_inset Formula $C$
\end_inset

 satisfies 
\begin_inset Formula $l^{T}C^{*}l=0$
\end_inset

.
 
\begin_inset Formula $C^{*}$
\end_inset

 is the adjoint matrix of 
\begin_inset Formula $C$
\end_inset

.
\end_layout

\begin_layout Definition
Dual conics are also called conic envelopes because of tangent lines.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Section
Projective transformations
\end_layout

\begin_layout Standard
2D proejctive geometry is the study of properties of the projective palen
 
\begin_inset Formula $\mathbb{P}^{2}$
\end_inset

 that are invariant under a group of transformations known as 
\emph on
projectivities
\emph default
.
\end_layout

\begin_layout Definition
projectivity/collineation/projective transformation/homography
\end_layout

\begin_layout Definition
A projectivity is an invertible (nonsingular) mapping 
\begin_inset Formula $h$
\end_inset

 from 
\begin_inset Formula $\mathbb{P}^{2}$
\end_inset

 to itself s.t.
 three points 
\begin_inset Formula $x_{1}$
\end_inset

, 
\begin_inset Formula $x_{2}$
\end_inset

 and 
\begin_inset Formula $x_{3}$
\end_inset

 lie on the same line iff 
\begin_inset Formula $h\left(x_{1}\right)$
\end_inset

, 
\begin_inset Formula $h\left(x_{2}\right)$
\end_inset

 and 
\begin_inset Formula $h\left(x_{3}\right)$
\end_inset

 do.
\end_layout

\begin_layout Standard
Projectivities form a group since the inverse of a projectivity is also
 a projectivity, and so is the composition of two projectivities.
\end_layout

\begin_layout Theorem
\begin_inset CommandInset label
LatexCommand label
name "thm:non-singular matrix for projectivity"

\end_inset

A mapping 
\begin_inset Formula $h\colon\mathbb{P}^{2}\to\mathbb{P}^{2}$
\end_inset

 is a projectivity iff there exists a non-singular 
\begin_inset Formula $3\times3$
\end_inset

 matrix 
\begin_inset Formula $H$
\end_inset

 s.t.
 for any point in 
\begin_inset Formula $\mathbb{P}^{2}$
\end_inset

represented by a vector 
\begin_inset Formula $\vec{x}$
\end_inset

 it is true that 
\begin_inset Formula $h\left(\vec{x}\right)=H\vec{x}$
\end_inset

.
\end_layout

\begin_layout Standard
Any invertible linear transformation of homogeneous coordinates is a projectivit
y.
\end_layout

\begin_layout Proof
Let 
\begin_inset Formula $x_{1}$
\end_inset

, 
\begin_inset Formula $x_{2}$
\end_inset

 and 
\begin_inset Formula $x_{3}$
\end_inset

 lie on a line 
\begin_inset Formula $l$
\end_inset

.
 Thus 
\begin_inset Formula $l^{T}x_{i}=0$
\end_inset

 for 
\begin_inset Formula $i=1,2,3$
\end_inset

.
 Let 
\begin_inset Formula $H$
\end_inset

 be a non-singular 
\begin_inset Formula $3\times3$
\end_inset

 matrix.
 
\begin_inset Formula $l^{T}H^{-1}Hx_{i}=0$
\end_inset

.
 Thus the points 
\begin_inset Formula $Hx_{i}$
\end_inset

 all lie on the line 
\begin_inset Formula $H^{-1}l$
\end_inset

, and so collinearity is preserved by the transformation.
\end_layout

\begin_layout Proof
The converse is omitted since it is not provided in the book.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Standard
An alternative defition can be provided given the above theorem.
\end_layout

\begin_layout Definition
Projective transformation
\end_layout

\begin_layout Definition
A planar projective transformation is a linear transformation on homogeneous
 
\begin_inset Formula $3$
\end_inset

-vectors represented by a non-singular 
\begin_inset Formula $3\times3$
\end_inset

 matrix
\begin_inset Formula 
\[
\begin{pmatrix}x_{1}^{\prime}\\
x_{2}^{\prime}\\
x_{3}^{\prime}
\end{pmatrix}=\begin{bmatrix}h_{11} & h_{12} & h_{13}\\
h_{21} & h_{22} & h_{23}\\
h_{31} & h_{32} & h_{33}
\end{bmatrix}\begin{pmatrix}x_{1}\\
x_{2}\\
x_{3}
\end{pmatrix}
\]

\end_inset

or more briefly 
\begin_inset Formula $x^{\prime}=Hx$
\end_inset

.
\end_layout

\begin_layout Standard
Note that the matrix 
\begin_inset Formula $H$
\end_inset

 may be changed by multiplication by an arbitrary non-zero scale without
 altering the projective transformation, thus called a 
\emph on
homogenous 
\emph default
matrix.
 Only the ratio of the matrix elements is significant, giving eight degrees
 of freedom.
\end_layout

\begin_layout Subsection
Transformations of lines and conics
\end_layout

\begin_layout Subsubsection
Transformation of lines
\end_layout

\begin_layout Standard
From the proof 
\begin_inset CommandInset ref
LatexCommand vpageref
reference "thm:non-singular matrix for projectivity"

\end_inset

, a line transforms as 
\begin_inset Formula 
\[
l^{\prime}=\left(H^{-1}\right)^{T}l.
\]

\end_inset


\end_layout

\begin_layout Subsubsection
Transformation of conics
\end_layout

\begin_layout Standard
Under a point transformation 
\begin_inset Formula $x^{\prime}=Hx$
\end_inset

, a conic 
\begin_inset Formula $C$
\end_inset

 transforms to 
\begin_inset Formula 
\[
C^{\prime}=\left(H^{-1}\right)^{T}CH^{-1}.
\]

\end_inset

a dual conic 
\begin_inset Formula $C^{*}$
\end_inset

 transforms to 
\begin_inset Formula 
\[
C^{*\prime}=HC^{*}H^{T}.
\]

\end_inset


\end_layout

\begin_layout Section
A hierarchy of transformations
\end_layout

\begin_layout Standard
Projective transformations form a group, called the 
\emph on
projective linear group.
 The following transformations goes from special to general.
\end_layout

\begin_layout Definition
invariant
\end_layout

\begin_layout Definition
A invariant of a geometric configuration is a function of the configuration
 whose value is unchanged by a particular transformation.
\end_layout

\begin_layout Subsection
Class Ⅰ: Isometries
\end_layout

\begin_layout Definition
Isometries are transformations of the plane 
\begin_inset Formula $\mathbb{R}^{2}$
\end_inset

 that preserve Euclidean distance (so called iso-metry), represented as
 
\begin_inset Formula 
\[
\begin{pmatrix}x^{\prime}\\
y^{\prime}\\
1
\end{pmatrix}=\begin{bmatrix}\epsilon\cos\theta & -\sin\theta & t_{x}\\
\epsilon\sin\theta & \cos\theta & t_{y}\\
0 & 0 & 1
\end{bmatrix}\begin{pmatrix}x\\
y\\
1
\end{pmatrix}
\]

\end_inset


\begin_inset space ~
\end_inset

where 
\begin_inset Formula $\epsilon=\pm1$
\end_inset

.
 
\begin_inset Formula $\epsilon=-1$
\end_inset

 reverses orientation.
\end_layout

\begin_layout Example*
Euclidean transformations model the motion of a rigid object, the most important
 isometries in practice, written as
\begin_inset Formula 
\[
\mathbf{x}^{\prime}=H_{E}\mathbf{x}=\begin{bmatrix}R & \mathbf{t}\\
\mathbf{0} & 1
\end{bmatrix}\mathbf{x}
\]

\end_inset

where 
\begin_inset Formula $R$
\end_inset

 is a 
\begin_inset Formula $2\times2$
\end_inset

 rotation matrix (an orthogonal matrix s.t.
 
\begin_inset Formula $R^{T}R=RR^{T}=I$
\end_inset

), 
\begin_inset Formula $\mathbf{t}$
\end_inset

 a translation 
\begin_inset Formula $2$
\end_inset

-vector.
 A Euclidenan transformation is also known as a 
\emph on
displacement
\emph default
.
\end_layout

\begin_layout Example*
A planar Euclidean transformation has three degrees of freedom.
 The transformation can be computed from two point correspondences.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Standard
Length, angle and area are invariants under isometries.
\end_layout

\begin_layout Subsection
Class Ⅱ: Similarity transformations
\end_layout

\begin_layout Definition
A similarity transformation (similarity) is an isometry composed with an
 
\begin_inset CommandInset href
LatexCommand href
name "isotropic"
target "https://en.wikipedia.org/wiki/Isotropy"

\end_inset

 scaling
\begin_inset Formula 
\[
\begin{pmatrix}x^{\prime}\\
y^{\prime}\\
1
\end{pmatrix}=\begin{bmatrix}s\cos\theta & -s\sin\theta & t_{x}\\
s\sin\theta & s\cos\theta & t_{y}\\
0 & 0 & 1
\end{bmatrix}\begin{pmatrix}x\\
y\\
1
\end{pmatrix}
\]

\end_inset

or 
\begin_inset Formula 
\[
\mathbf{x}^{\prime}=H_{S}\mathbf{x}=\begin{bmatrix}sR & \mathbf{t}\\
\mathbf{0} & 1
\end{bmatrix}\mathbf{x}
\]

\end_inset


\end_layout

\begin_layout Standard
where the scalar 
\begin_inset Formula $s$
\end_inset

 is the isotropic scaling.
 A similary transformation is also known as an equi-form
\emph on
 transformation
\emph default
 because it preserves 
\begin_inset Quotes eld
\end_inset

shape
\begin_inset Quotes erd
\end_inset

.
\end_layout

\begin_layout Standard
A planar similarity has four degrees of freedom, making it determinable
 from two point correspondences.
\end_layout

\begin_layout Standard
Parallelism, ratio of line lengths, ratio of areas, angles.
\end_layout

\begin_layout Subsection
Class Ⅲ: Affine transformations
\end_layout

\begin_layout Definition
An affine transformation (affinity) is a general non-singular linear transformat
ion of inhomogeneous coordinates followed by a translation.
\begin_inset Formula 
\[
\begin{pmatrix}x^{\prime}\\
y^{\prime}\\
1
\end{pmatrix}=\begin{bmatrix}a_{11} & a_{12} & t_{x}\\
a_{21} & a_{22} & t_{y}\\
0 & 0 & 1
\end{bmatrix}\begin{pmatrix}x\\
y\\
1
\end{pmatrix}
\]

\end_inset

or
\begin_inset Formula 
\[
\mathbf{x}^{\prime}=H_{A}\mathbf{x}=\begin{bmatrix}A & \mathbf{t}\\
\mathbf{0} & 1
\end{bmatrix}\mathbf{x}
\]

\end_inset

with 
\begin_inset Formula $A$
\end_inset

 a 
\begin_inset Formula $2\times2$
\end_inset

 non-singular matrix.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Standard
From the SVD, the linear component 
\begin_inset Formula $A$
\end_inset

 can be decomposed into 
\begin_inset Formula 
\[
A=UDV^{T}=\left(UV^{T}\right)\left(VDV^{T}\right)=R\left(\theta\right)\left(R\left(-\phi\right)DR\left(\phi\right)\right)
\]

\end_inset

where
\begin_inset Formula 
\[
D=\begin{bmatrix}\lambda_{1} & 0\\
0 & \lambda_{2}
\end{bmatrix}
\]

\end_inset

since 
\begin_inset Formula $U$
\end_inset

 and 
\begin_inset Formula $V$
\end_inset

 are orthogonal matrices, namely 
\emph on
rotations
\emph default
 and 
\emph on
non-isotropic
\emph default
 scalings.
 The essence of an affinity is the scaling in orthogonal directions, oriented
 at a particular angle.
\end_layout

\begin_layout Standard
A planar affine transformation has six degrees of freedom, determinable
 from three points correspondence.
\end_layout

\begin_layout Standard
Length ratios and angels between lines are not preserved because of affine
 transformation.
 Three important invariants, line parallelism, ratio of lengths of parallel
 line segments (the scaling is common to all lines with the same direction),
 ratios of areas (the area is scaled by 
\begin_inset Formula $\det A=\lambda_{1}\lambda_{2}$
\end_inset

).
 An affinity is orientation-preserving or -reversing according to whether
 
\begin_inset Formula $\det A$
\end_inset

 is positive or negative respectively.
\end_layout

\begin_layout Subsection
Class Ⅵ: Projective transformations
\end_layout

\begin_layout Definition
Non singular linear transformation of homogeneous coordinates.
\begin_inset Formula 
\[
\mathbf{x}^{\prime}=H_{p}\mathbf{x}=\begin{bmatrix}A & \mathbf{t}\\
\mathbf{v}^{T} & v
\end{bmatrix}\mathbf{x}
\]

\end_inset

where the vector 
\begin_inset Formula $\mathbf{v}=\left(v_{1},v_{2}\right)^{T}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Standard
A projective transformation has eight degrees of freedom, and can be computed
 from four point correspondences between two planes with no three collinear
 on either plane.
\end_layout

\begin_layout Standard
The most fundamental projective invariant is the corss ratio of four collinear
 points.
 For a given projective transformation, area scaling varies with position
 and the orientation of a transformed line depends on both the orientation
 and position of the source line.
\end_layout

\begin_layout Standard
The vector 
\begin_inset Formula $\mathbf{v}$
\end_inset

 is responsible for the non-linear effects of the projectivity, mapping
 a point at infinity to a finite point.
\begin_inset Formula 
\[
\begin{bmatrix}A & \mathbf{t}\\
\mathbf{v}^{T} & v
\end{bmatrix}\begin{pmatrix}x_{1}\\
x_{2}\\
0
\end{pmatrix}=\begin{pmatrix}A\begin{pmatrix}x_{1}\\
x_{2}
\end{pmatrix}\\
v_{1}x_{1}+v_{2}x_{2}
\end{pmatrix}.
\]

\end_inset


\end_layout

\begin_layout Subsection
Decomposition of a projective transformation
\end_layout

\begin_layout Standard
A projective transformation can be decomposed into a chain of transformations.
\begin_inset Formula 
\[
H=H_{S}H_{A}H_{P}=\begin{bmatrix}sR & \mathbf{t}\\
\mathbf{0}^{T} & 1
\end{bmatrix}\begin{bmatrix}K & 0\\
\mathbf{0}^{T} & 1
\end{bmatrix}\begin{bmatrix}I & \mathbf{0}\\
\mathbf{v^{T}} & v
\end{bmatrix}=\begin{bmatrix}A & \mathbf{t}\\
\mathbf{v}^{T} & v
\end{bmatrix}.
\]

\end_inset

where 
\begin_inset Formula $A=sRK+\mathbf{tv}^{T}$
\end_inset

 and 
\begin_inset Formula $K$
\end_inset

 is an upper-triangular matrix normalized as 
\begin_inset Formula $\det K=1$
\end_inset

.
 This decomposition is valid provided 
\begin_inset Formula $v\neq0$
\end_inset

 and is unique if 
\begin_inset Formula $s$
\end_inset

 is chosen positive.
 The transformation 
\begin_inset Formula $H_{p}$
\end_inset

 is an 
\emph on
elation.
 The inverse of 
\begin_inset Formula $H$
\end_inset

 is still projective, giving rise to the alternative decomposed form
\begin_inset Formula 
\[
H=H_{p}H_{A}H_{S}.
\]

\end_inset


\end_layout

\begin_layout Subsection
Invariants
\end_layout

\begin_layout Standard
The number of functionally independent invariants is equal to or greater
 than the number of degrees of freedom of the configuration less the number
 of degrees of freedom of the transformation.
\end_layout

\begin_layout Section
The projective geometry of 1D
\end_layout

\begin_layout Standard
The cross ratio is the basic projective invariant of 
\begin_inset Formula $\mathbb{P}^{2}$
\end_inset

.
\end_layout

\begin_layout Definition
Cross ratio
\end_layout

\begin_layout Definition
Given 4 points 
\begin_inset Formula $\bar{\mathbf{x}}_{i}$
\end_inset

 (1D points in homogeneous coordinates), the cross ratio is defined as 
\begin_inset Formula 
\[
\text{Cross}\left(\bar{\mathbf{x}}_{1},\bar{\mathbf{x}}_{2},\bar{\mathbf{x}}_{3},\bar{\mathbf{x}}_{4}\right)=\frac{\left|\bar{\mathbf{x}}_{1}\bar{\mathbf{x}}_{2}\right|\left|\bar{\mathbf{x}}_{3}\bar{\mathbf{x}}_{4}\right|}{\left|\bar{\mathbf{x}}_{1}\bar{\mathbf{x}}_{3}\right|\left|\bar{\mathbf{x}}_{2}\bar{\mathbf{x}}_{4}\right|}
\]

\end_inset

where 
\begin_inset Formula 
\[
\left|\bar{\mathbf{x}}_{i}\bar{\mathbf{x}}_{j}\right|=\det\begin{bmatrix}x_{i1} & x_{j1}\\
x_{i2} & x_{j2}
\end{bmatrix}.
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Definition
concurrent lines
\end_layout

\begin_layout Definition
three or more lines in a plane or higher-dimensional space are said to be
 concurrent if they intersect at a single point.
\end_layout

\begin_layout Standard
The cross ratio of concurrent lines is an invariant to projective transformation
 of the plane.
\end_layout

\begin_layout Section
Recovery of affine and metric properties from images
\end_layout

\begin_layout Standard
It is only necessary to specify 
\begin_inset Formula $4$
\end_inset

 degrees of freedom in order to determine metric properties.
 These 
\begin_inset Formula $4$
\end_inset

 dof are associated with geometric objects: the line at infinity 
\begin_inset Formula $l_{\infty}$
\end_inset

 and the two circular points on 
\begin_inset Formula $l_{\infty}$
\end_inset

.
\end_layout

\begin_layout Subsection
Line at infinity
\end_layout

\begin_layout Proposition
The line at infinity 
\begin_inset Formula $l_{\infty}$
\end_inset

 is a fixed line under the projective transformation 
\begin_inset Formula $H$
\end_inset

 iff 
\begin_inset Formula $H$
\end_inset

 is an affinity.
\end_layout

\begin_layout Standard
However, 
\begin_inset Formula $l_{\infty}$
\end_inset

 is not fixed pointwise under an affine transformation.
\end_layout

\begin_layout Standard
Once the limaged line at infinity is identified in an image of a plane,
 it is then possible to make affine measurements on the original plane.
 Lines may be identified as parallel on the original plane if the imaged
 lines intersect on the imaged 
\begin_inset Formula $l_{\infty}$
\end_inset

.
 A length ratio on a line may be computer from the cross ratio of the three
 points specifying the lengths together with the intersection of the line
 with 
\begin_inset Formula $l_{\infty}$
\end_inset

, which provides the fourth point.
\end_layout

\begin_layout Standard
Given line at infinity imaged at 
\begin_inset Formula $\left(l_{1},l_{2},l_{3}\right)^{T}$
\end_inset

 where 
\begin_inset Formula $l_{3}\neq0$
\end_inset

, a suitable projective point transformation which map 
\begin_inset Formula $l$
\end_inset

 back to 
\begin_inset Formula $l_{\infty}=\left(0,0,1\right)^{T}$
\end_inset

 is 
\begin_inset Formula 
\[
H=H_{A}\begin{bmatrix}1 & 0 & 0\\
0 & 1 & 0\\
l_{1} & l_{2} & l_{3}
\end{bmatrix}.
\]

\end_inset

Appy this matrix to every imaged point to affinely rectify the image.
\end_layout

\begin_layout Subsection
The circular points
\end_layout

\begin_layout Definition
Under any similarity transformation there are two points on 
\begin_inset Formula $l_{\infty}$
\end_inset

 which are fixed, called the 
\emph on
circular points 
\emph default
or 
\emph on
absolute points.
\begin_inset Formula 
\[
I=\begin{pmatrix}1\\
i\\
0
\end{pmatrix}\quad J=\begin{pmatrix}1\\
-i\\
0
\end{pmatrix}.
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Proposition
The circuluar points 
\begin_inset Formula $I,J$
\end_inset

 are fixed points under the projective transformation 
\begin_inset Formula $H$
\end_inset

 iff 
\begin_inset Formula $H$
\end_inset

 is a similarity.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Standard
Every circle intersects 
\begin_inset Formula $l_{\infty}$
\end_inset

 at the circular points.
 A circle can be computed using the general formula for a conic defined
 by five points, where the five points are the three points that specify
 the circle in Euclidean geometry augmented with the two circular points.
\end_layout

\begin_layout Standard
Also, since 
\begin_inset Formula $I=\left(1,0,0\right)+i\left(0,1,0\right)$
\end_inset

, decomposed int othe orthogonal directions of Euclidean gemotry, it is
 not surprising that once the circular points are identified, orthogonality,
 and other metric properties are then determined.
\end_layout

\begin_layout Definition
the conic 
\begin_inset Formula 
\[
C_{\infty}^{*}=IJ^{T}+JI^{T}
\]

\end_inset

is dual to the circular points.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Proposition
The dual conic 
\begin_inset Formula $C_{\infty}^{*}$
\end_inset

 is fixed under the projective transformation 
\begin_inset Formula $H$
\end_inset

 iff 
\begin_inset Formula $H$
\end_inset

 is a similarity.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Standard
TODO
\end_layout

\begin_layout Chapter
Projective Geometry and Transformations of 3D
\end_layout

\begin_layout Standard
In 
\begin_inset Formula $\mathbb{P}^{3}$
\end_inset

, there is a 
\emph on
plane 
\emph default
at infinity 
\begin_inset Formula $\pi_{\infty}$
\end_inset

, analog of 
\begin_inset Formula $l_{\infty}$
\end_inset

 of 
\begin_inset Formula $\mathbb{P}^{2}$
\end_inset

.
 Parallel lines and parallel palens intersect on 
\begin_inset Formula $\pi_{\infty}$
\end_inset

.
\end_layout

\begin_layout Section
Points and projective transformation
\end_layout

\begin_layout Standard
Points are represented by a 
\begin_inset Formula $4$
\end_inset

-vector.
 A projective transformation acting on 
\begin_inset Formula $\mathbb{P}^{3}$
\end_inset

 is a linear transformation on homogeneous 
\begin_inset Formula $4$
\end_inset

-vectors represented by a non-singular 
\begin_inset Formula $4\times4$
\end_inset

 matrix
\begin_inset Formula 
\[
\mathbf{x}^{\prime}=H\mathbf{x}
\]

\end_inset

where 
\begin_inset Formula $H$
\end_inset

 has 
\begin_inset Formula $15$
\end_inset

 degrees of freedom.
 The map is a collineation (lines are mapped to lines), which preserves
 incidence relations such as the intersection point of a line with a plane,
 and order of contact.
\end_layout

\begin_layout Standard
In 
\begin_inset Formula $\mathbb{P}^{3}$
\end_inset

 points and planes are dual.
 Lines are self-dual in 
\begin_inset Formula $\mathbb{P}^{3}$
\end_inset

.
\end_layout

\begin_layout Subsection
Planes
\end_layout

\begin_layout Standard
A plane in 3D space may be written as 
\begin_inset Formula 
\[
\pi_{1}x+\pi_{2}y+\pi_{3}z+\pi_{4}=0,
\]

\end_inset

represented in homogenous coordinates 
\begin_inset Formula $\boldsymbol{\mathbf{\pi}}=\left(\pi_{1},\pi_{2},\pi_{3},\pi_{4}\right)^{T}$
\end_inset

.
\end_layout

\begin_layout Proposition
Three points define a plane.
\end_layout

\begin_layout Remark*
Given three point 
\begin_inset Formula $\mathbf{\mathbf{x}_{1}}$
\end_inset

, 
\begin_inset Formula $\mathbf{x}_{2}$
\end_inset

 and 
\begin_inset Formula $\mathbf{x_{3}}$
\end_inset

 that are linearly independent, and a plane vector 
\begin_inset Formula $\boldsymbol{\pi}$
\end_inset

, the system of linear equations
\begin_inset Formula 
\[
\begin{bmatrix}\mathbf{x}_{1}\\
\mathbf{x}_{2}\\
\mathbf{x}_{3}
\end{bmatrix}\boldsymbol{\pi}=0
\]

\end_inset

is full rank matrix, giving a unique solution/kernel to 
\begin_inset Formula $\boldsymbol{\pi}$
\end_inset

.
\end_layout

\begin_layout Remark*
To determine this plane more directly, consider a general point 
\begin_inset Formula $\mathbf{x}$
\end_inset

 on 
\begin_inset Formula $\boldsymbol{\pi}$
\end_inset

 and three points 
\begin_inset Formula $\mathbf{x}_{1}$
\end_inset

, 
\begin_inset Formula $\mathbf{x}_{2}$
\end_inset

 and 
\begin_inset Formula $\mathbf{x}_{3}$
\end_inset

.
 Since this general point is a linear combination of the latter three, giving
 a linear dependent list, the determinant of the matrix 
\begin_inset Formula 
\[
\det\begin{bmatrix}\mathbf{x} & \mathbf{x}_{1} & \mathbf{x}_{2} & \mathbf{x}_{3}\end{bmatrix}=0
\]

\end_inset

expanding it along 
\begin_inset Formula $\mathbf{x}$
\end_inset

 gives 
\begin_inset Formula 
\[
x_{1}D_{234}-x_{2}D_{134}+x_{3}D_{124}-x_{4}D_{123}=0.
\]

\end_inset

Then the plane coeffcients 
\begin_inset Formula 
\[
\boldsymbol{\pi}=\left(D_{234},-D_{134},D_{123},-D_{123}\right)^{T}.
\]

\end_inset

In vector algebra 
\begin_inset Formula 
\[
\boldsymbol{\pi}=\begin{pmatrix}\left(\tilde{\mathbf{x}}_{1}-\tilde{\mathbf{x}}_{3}\right)\times\left(\tilde{\mathbf{x}}_{2}-\tilde{\mathbf{x}}_{3}\right)\\
-\tilde{\mathbf{x}}_{3}^{T}\left(\tilde{\mathbf{x}}_{1}\times\tilde{\mathbf{x}}_{2}\right)
\end{pmatrix}
\]

\end_inset

where 
\begin_inset Formula $\tilde{\mathbf{x}}_{i}$
\end_inset

 is the Euclidean coordinates of point 
\begin_inset Formula $\mathbf{x}_{i}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Proposition
Three planes define a point.
\end_layout

\begin_layout Remark*
This is the dual case of three points defining a plane.
 Consider the following the 
\begin_inset Formula 
\[
\begin{bmatrix}\boldsymbol{\pi}_{1}^{T}\\
\boldsymbol{\pi}_{2}^{T}\\
\boldsymbol{\pi}_{3}^{T}
\end{bmatrix}\mathbf{x}=0.
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Proposition
Under the point transformation 
\begin_inset Formula $\mathbf{x}^{\prime}=H\mathbf{x}$
\end_inset

, a plane transformation is 
\begin_inset Formula 
\[
\boldsymbol{\pi}^{\prime}=\left(H^{-1}\right)^{T}\boldsymbol{\pi}.
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Proposition
Parameterized points on a plane
\end_layout

\begin_layout Proposition
???
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Subsection
Lines
\end_layout

\begin_layout Definition
A line is defined by the join of two points or the intersection of two planes.
\end_layout

\begin_layout Standard
Think of a line as defined by its intersection with two orthogonal planes.
 Lines have 
\begin_inset Formula $4$
\end_inset

 degrees of freedom in 3D space.
 Representing lines in homogeneous coordinates requires 
\begin_inset Formula $5$
\end_inset

-vectors, inconvenient to use in mathematical expressions.
\end_layout

\begin_layout Subsubsection
Null-space and representation
\end_layout

\begin_layout Standard
TODO
\end_layout

\begin_layout Section
The hierarchy of transformation
\end_layout

\begin_layout Subsection
The screw decomposition
\end_layout

\begin_layout Standard
The screw decomposition enables any Euclidean action (a rotation composed
 with a translation) to be reduced to a situation almost as simple as the
 2D case.
\end_layout

\begin_layout Definition
\begin_inset CommandInset href
LatexCommand href
name "Screw axis"
target "https://en.wikipedia.org/wiki/Screw_axis"

\end_inset


\end_layout

\begin_layout Proposition
Any particular translation and rotation is equivalent to a rotation about
 a screw axis together with a translation along the screw axis.
 The screw axis is parallel to the rotation axis.
\end_layout

\begin_layout Standard
TODO
\end_layout

\begin_layout Chapter
Estimation - 2D Projective Transformations
\end_layout

\begin_layout Standard
Consider a set of point correspondences 
\begin_inset Formula $\mathbf{x_{i}\leftrightarrow\mathbf{x}_{i}^{\prime}}$
\end_inset

 between two images.
 The goal is to compute a 
\begin_inset Formula $3\times3$
\end_inset

 matrix 
\begin_inset Formula $H$
\end_inset

 s.t.
 
\begin_inset Formula $H\mathbf{x}_{i}=\mathbf{x}_{i}^{\prime}$
\end_inset

 for each 
\begin_inset Formula $i$
\end_inset

.
 First, the number of corresponding points should be determined.
 The given corresponding points are always the ideal choice and there may
 be more.
 There are algorithms that find the best solution.
\end_layout

\begin_layout Section
The Direct Linear Transformation (DLT) algorithm
\end_layout

\begin_layout Standard
Given a set of four point correspondences 
\begin_inset Formula $\mathbf{x}_{i}\leftrightarrow\mathbf{x}_{i}^{\prime}$
\end_inset

, and the transformation 
\begin_inset Formula $\mathbf{x}_{i}^{\prime}=H\mathbf{x}_{i}$
\end_inset

, the equation may be written as 
\begin_inset Formula 
\[
\mathbf{x}_{i}^{\prime}\times H\mathbf{x}_{i}=\mathbf{0}.
\]

\end_inset


\end_layout

\begin_layout Part
Camera Geometry and Single View Geometry
\end_layout

\begin_layout Chapter
Camera models
\end_layout

\begin_layout Standard
The camera mapping is represented by a matrix, and in case of mapping points
 a 
\begin_inset Formula $3\times4$
\end_inset

 matrix 
\begin_inset Formula $P$
\end_inset

 which maps from homogeneous coordinates of a world point in 
\begin_inset Formula $3$
\end_inset

D space to homogeneous coordinates of the imaged point on the image plane.
 The matrix has in general 
\begin_inset Formula $11$
\end_inset

 degrees of freedom and the properties of the camera, such as its center
 and focal length may be extracted from it.
\end_layout

\begin_layout Standard
A camera is a mapping between the 3D world (object space) and a 2D image.
 The principal camera of interest is 
\emph on
central projection.
 
\emph default
However all cameras modeling central projection are specializations of the
 
\emph on
general projective camera
\emph default
.
 The specialized models fall into two major classes: those with a finite
 center and those with center at infinity.
 Of the cameras at infinity, the 
\emph on
affine camera
\emph default
 is of particular importance because it is the natural generalization of
 parallel projection.
\end_layout

\begin_layout Section
Finite cameras
\end_layout

\begin_layout Standard
These models are designed for CCD-like sensors, but also applicable to other
 cameras like X-ray images, scanned photographic negatives, scanned photographs
 from enlarged negatives.
\end_layout

\begin_layout Subsection
The basic pinhole model
\end_layout

\begin_layout Standard
\begin_inset Graphics
	filename pinhole_camera.png
	scale 70
	rotateOrigin center

\end_inset


\end_layout

\begin_layout Definition
basic definitions
\end_layout

\begin_layout Definition
\begin_inset Formula $C$
\end_inset

: camera center;
\end_layout

\begin_layout Definition
\begin_inset Formula $Z$
\end_inset

 axis: principal ray/axis;
\end_layout

\begin_layout Definition
\begin_inset Formula $p$
\end_inset

: principal point, where the principal axis meets the image plane
\end_layout

\begin_layout Definition
principal plane: the plane through the camera center parallel to the image
 plane
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Subsubsection
Homogeneous coordinates representation
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\begin{pmatrix}X\\
Y\\
Z\\
1
\end{pmatrix}\mapsto\begin{pmatrix}fX\\
fY\\
1
\end{pmatrix}=\begin{bmatrix}f &  &  & 0\\
 & f &  & 0\\
 &  & 1 & 0
\end{bmatrix}\begin{pmatrix}X\\
Y\\
Z\\
1
\end{pmatrix}.
\]

\end_inset

or 
\begin_inset Formula 
\[
\mathbf{x}=P\mathbf{X}
\]

\end_inset

where 
\begin_inset Formula $P$
\end_inset

 is the 
\begin_inset Formula $3\times4$
\end_inset

 camera projection matrix, also written as 
\begin_inset Formula 
\[
P=\text{diag}\left(f,f,1\right)\begin{bmatrix}I & \mathbf{0}\end{bmatrix}.
\]

\end_inset


\end_layout

\begin_layout Standard
The above representatio assumes that the origin of coordinates in the image
 plane is at the principal point, which in practice is often not the case.
 The transformation matrix may be rewritten as 
\begin_inset Formula 
\[
P=\begin{bmatrix}f &  & p_{x} & 0\\
 & f & p_{y} & 0\\
 &  & 1 & 0
\end{bmatrix}.
\]

\end_inset

where 
\begin_inset Formula $\left(p_{x},p_{y}\right)$
\end_inset

 is the coordinates of the principal point.
 The transformation is now 
\begin_inset Formula 
\[
\mathbf{x}=K\begin{bmatrix}I & \mathbf{0}\end{bmatrix}\mathbf{X_{cam}}.
\]

\end_inset

where 
\begin_inset Formula $K$
\end_inset

 is called the camera calibration matrix.
 The camera here is assumed to be located at the origin of a Euclidean coordinat
e system with the principal axis of the camera pointing straight down the
 
\begin_inset Formula $Z$
\end_inset

-axis and 
\begin_inset Formula $\mathbf{X_{cam}}$
\end_inset

 is expresses in this coordinate system, called the 
\emph on
camera coordinate system.
\end_layout

\begin_layout Subsubsection
Camera rotation and translation
\end_layout

\begin_layout Standard
In general, points in space will be expressed in terms of a different Euclidean
 coordinate frame other than camera coordinate system, known as the 
\emph on
world coordinate frame
\emph default
.
 Suppose 
\begin_inset Formula $\mathbf{X}$
\end_inset

 is the homogeneous coordinates of a point, then 
\begin_inset Formula 
\[
\mathbf{X_{cam}}=\begin{bmatrix}R & -R\widetilde{C}\\
0 & 1
\end{bmatrix}\mathbf{X}
\]

\end_inset

where 
\begin_inset Formula $\widetilde{C}$
\end_inset

 is the Euclidean coordinates of the camera center in the world coordinate
 frame and 
\begin_inset Formula $R$
\end_inset

 is a 
\begin_inset Formula $3\times3$
\end_inset

 rotation matrix representing the orientation of the camera coordinate frame.
\end_layout

\begin_layout Standard
The transformation is then rewritten as 
\begin_inset Formula 
\[
\mathbf{x}=KR\begin{bmatrix}I & -\widetilde{C}\end{bmatrix}X.
\]

\end_inset

Here 
\begin_inset Formula $K$
\end_inset

 is the 
\emph on
internal camera parameters
\emph default
, or the 
\emph on
internal orientation.
 
\emph default
The parameters of 
\begin_inset Formula $R$
\end_inset

 and 
\begin_inset Formula $\widetilde{C}$
\end_inset

 which relate the camera orientation and position to a world coordinate
 system are called 
\emph on
external parameter
\emph default
 or the 
\emph on
exterior orientation.
\end_layout

\begin_layout Standard
The transformation is often also written as 
\begin_inset Formula 
\[
P=K\begin{bmatrix}R & \mathbf{t}\end{bmatrix}
\]

\end_inset

where 
\begin_inset Formula $\mathbf{t}=-R\widetilde{C}$
\end_inset

, to represent the world-to-image transformation as 
\begin_inset Formula $\widetilde{\mathbf{X}}_{\text{cam}}=R\widetilde{\mathbf{X}}+\mathbf{t}$
\end_inset

.
\end_layout

\begin_layout Subsubsection
CCD cameras
\end_layout

\begin_layout Standard
In case of CCD cameras, there is the additional possibility of having non-square
 pixels.
 This has the extra effect of introducing unequal scale factors in each
 direction.
\end_layout

\begin_layout Standard
Suppose the number of pixels per unit distance in image coordinates are
 
\begin_inset Formula $m_{x}$
\end_inset

 and 
\begin_inset Formula $m_{y}$
\end_inset

 in the 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 directions.
 The pixel coordinates is obtained by multiplying 
\begin_inset Formula $K$
\end_inset

 on the left by an extra factor 
\begin_inset Formula $\text{diag}\left(m_{x},m_{y},1\right)$
\end_inset

.
 Thus, the general form of the calibration matrix of a CCD camera is 
\begin_inset Formula 
\[
K=\begin{bmatrix}\alpha_{x} &  & x_{0}\\
 & \alpha_{y} & y_{0}\\
 &  & 1
\end{bmatrix}
\]

\end_inset

where 
\begin_inset Formula $\alpha_{x}=fm_{x}$
\end_inset

 and 
\begin_inset Formula $\alpha_{y}=fm_{y}$
\end_inset

 and 
\begin_inset Formula $\left(x_{0},y_{0}\right)=\left(m_{x}p_{x},m_{y}p_{y}\right)$
\end_inset

 is the principal point in terms of pixel dimensions.
\end_layout

\begin_layout Standard
Moreover, a 
\emph on
skew 
\emph default
parameter 
\begin_inset Formula $s$
\end_inset

 is introduced.
\begin_inset Formula 
\[
K=\begin{bmatrix}\alpha_{x} & s & x_{0}\\
 & \alpha_{y} & y_{0}\\
 &  & 1
\end{bmatrix}.
\]

\end_inset


\end_layout

\begin_layout Standard
Combining all those model together, we have a 
\emph on
finite projective camera, 
\begin_inset Formula 
\[
P=KR\begin{bmatrix}I & -\widetilde{C}\end{bmatrix},
\]

\end_inset


\emph default
It has a 
\begin_inset Formula $11$
\end_inset

 degrees of freedom, exactly the number of dof of a 
\begin_inset Formula $3\times4$
\end_inset

 matrix.
 
\begin_inset Formula $KR$
\end_inset

 is nonsingular, and can be decomposed using 
\begin_inset Formula $RQ$
\end_inset

 matrix decomposition into 
\begin_inset Formula $K$
\end_inset

 and 
\begin_inset Formula $R$
\end_inset

.
\end_layout

\begin_layout Proposition
The set of camera matrices of finite projective cameras is identical with
 the set of homogenous 
\begin_inset Formula $3\times4$
\end_inset

 matrices for which the left hand 
\begin_inset Formula $3\times3$
\end_inset

 submatrix is non-singular.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Section
General projective cameras
\end_layout

\begin_layout Standard
A 
\emph on
general projective camera 
\emph default
is one represented by an arbitrary homogeneous 
\begin_inset Formula $3\times4$
\end_inset

 matrix of rank 
\begin_inset Formula $3$
\end_inset

 (so that the range of the transformation is a full plane) and has 
\begin_inset Formula $11$
\end_inset

 dof.
\end_layout

\begin_layout Subsection
Camera anatomy
\end_layout

\begin_layout Standard
A general projective camera may be decomposed into blocks according to 
\begin_inset Formula 
\[
P=\begin{bmatrix}M & p_{4}\end{bmatrix}.
\]

\end_inset


\end_layout

\begin_layout Standard
The camera center is the null space of 
\begin_inset Formula $P$
\end_inset

.
 A finite camera has the center at 
\begin_inset Formula $\left(\widetilde{C},1\right)$
\end_inset

.
\end_layout

\begin_layout Proposition
All points on a ray from the camera center are mapped to the same image.
\end_layout

\begin_layout Proof
Given the camera center 
\begin_inset Formula $C$
\end_inset

 and any point 
\begin_inset Formula $A$
\end_inset

, any point on 
\begin_inset Formula $CA$
\end_inset

 is represented by 
\begin_inset Formula 
\[
\mathbf{X}=\lambda A+\left(1-\lambda\right)C.
\]

\end_inset

Apply 
\begin_inset Formula $P$
\end_inset

 to both sides, 
\begin_inset Formula 
\[
\mathbf{x}=P\mathbf{X}=\lambda PA.
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Proposition
With the columns of 
\begin_inset Formula $P$
\end_inset

 denoted by 
\begin_inset Formula $p_{1},p_{2},p_{3},p_{4}$
\end_inset

, then 
\begin_inset Formula $p_{1},p_{2},p_{3}$
\end_inset

 are the vanishing point of the world coordinate 
\begin_inset Formula $X,Y,Z$
\end_inset

 axes respectively, 
\begin_inset Formula $p_{4}$
\end_inset

 is the image of the world origin.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Proposition
The principal plane consists of the set of points 
\begin_inset Formula $\mathbf{x}$
\end_inset

 which are imaged on the line at infinity of the image.
 A point lies on the principal plane of the camera iff 
\begin_inset Formula $p^{3T}\mathbf{x}=0$
\end_inset

, where 
\begin_inset Formula $p^{3T}$
\end_inset

 is the third row vector of 
\begin_inset Formula $P$
\end_inset

.
 In other words, 
\begin_inset Formula $p^{3}$
\end_inset

 is the vector representating the principal plane of the camera.
 The other two axis planes of the camera coordinate frame have similar results.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Proposition
Points 
\begin_inset Formula $D=\left(d^{T},0\right)$
\end_inset

 on the plane at infinity represent vanishing points, mapping to 
\begin_inset Formula 
\[
\mathbf{x}=PD=\begin{bmatrix}M & p_{4}\end{bmatrix}D=Md.
\]

\end_inset


\end_layout

\end_body
\end_document