CSE559A Lecture 2

The Geometry of Image Formation

Mapping between image and world coordinates.

Today’s focus:

x=K[R\ t]X

Add a barrier to block off most of the rays.

$f$ is the focal length.
$c$ is the center of the aperture.

Focal length: distance between the aperture and the image plane.
Field of View (FOV): the angle between the two rays that pass through the aperture and the image plane.
Zoom: the ratio of the focal length to the image plane.

Beyond the pinhole/perspective camera model, there are other types of projection.

Length and area are not preserved.

Angle is not preserved.

But straight lines are still straight.

Parallel lines in the world intersect at a vanishing point on the image plane.

Vanishing lines: the set of all vanishing points of parallel lines in the world on the same plane in the world.

Vertical vanishing point at infinity.

Linear projection model.

x=K[R\ t]X

converting from homogeneous to inhomogeneous coordinates:

When $w=0$ , the point is at infinity.

Homogeneous coordinates are invariant under scaling (non-zero scalar).

k\begin{bmatrix}x\\y\\w\end{bmatrix}=\begin{bmatrix}kx\\ky\\kw\end{bmatrix}\implies\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}x/k\\y/k\end{bmatrix}

A convenient way to represent a point at infinity is to use a unit vector.

Line equation: $ax+by+c=0$

line_i=\begin{bmatrix}a_i\\b_i\\c_i\end{bmatrix}

Append a 1 to pixel coordinates to get homogeneous coordinates.

pixel_i=\begin{bmatrix}u_i\\v_i\\1\end{bmatrix}

Line given by cross product of two points:

line_i=pixel_1\times pixel_2

Intersection of two lines given by cross product of the lines:

pixel_i=line_1\times line_2

Intrinsic Assumptions:

Extrinsic Assumptions:

x=K[I\ 0]X\implies w\begin{bmatrix}u\\v\\1\end{bmatrix}=\begin{bmatrix}f&0&0&0\\0&f&0&0\\0&0&1&0\end{bmatrix}\begin{bmatrix}x\\y\\z\\1\end{bmatrix}

Removing the assumptions:

Intrinsic assumptions:

Extrinsic assumptions:

x=K[I\ 0]X\implies w\begin{bmatrix}u\\v\\1\end{bmatrix}=\begin{bmatrix}\alpha&0&u_0&0\\0&\beta&v_0&0\\0&0&1&0\end{bmatrix}\begin{bmatrix}x\\y\\z\\1\end{bmatrix}

Adding skew:

x=K[I\ 0]X\implies w\begin{bmatrix}u\\v\\1\end{bmatrix}=\begin{bmatrix}\alpha&s&u_0&0\\0&\beta&v_0&0\\0&0&1&0\end{bmatrix}\begin{bmatrix}x\\y\\z\\1\end{bmatrix}

Finally, adding camera rotation and translation:

x=K[I\ t]X\implies w\begin{bmatrix}u\\v\\1\end{bmatrix}=\begin{bmatrix}\alpha&s&u_0\\0&\beta&v_0\\0&0&1\end{bmatrix}\begin{bmatrix}r_{11}&r_{12}&r_{13}&t_x\\r_{21}&r_{22}&r_{23}&t_y\\r_{31}&r_{32}&r_{33}&t_z\end{bmatrix}\begin{bmatrix}x\\y\\z\\1\end{bmatrix}

What is the degrees of freedom of the camera matrix?

Total: 11