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content/CSE559A/CSE559A_L2.md

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+# CSE559A Lecture 2
+
+## The Geometry of Image Formation
+
+Mapping between image and world coordinates.
+
+Today's focus:
+
+$$
+x=K[R\ t]X
+$$
+
+### Pinhole Camera Model
+
+Add a barrier to block off most of the rays.
+
+- Reduce blurring
+- The opening known as the **aperture**
+
+$f$ is the focal length.  
+$c$ is the center of the aperture.
+
+#### Focal length/ Field of View (FOV)/ Zoom
+
+- 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.
+
+#### Other types of projection
+
+Beyond the pinhole/perspective camera model, there are other types of projection.
+
+- Radial distortion
+- 360-degree camera
+  - Equirectangular Panoramas
+- Random lens
+- Rotating sensors
+- Photofinishing
+- Tiltshift lens
+
+### Perspective Geometry
+
+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.
+
+### Camera/Projection Matrix
+
+Linear projection model.
+
+$$
+x=K[R\ t]X
+$$
+
+- $x$: image coordinates 2d (homogeneous coordinates)
+- $X$: world coordinates 3d (homogeneous coordinates)
+- $K$: camera matrix (3x3 and invertible)
+- $R$: camera rotation matrix (3x3)
+- $t$: camera translation vector (3x1)
+
+#### Homogeneous coordinates
+
+- 2D: $$(x, y)\to\begin{bmatrix}x\\y\\1\end{bmatrix}$$
+- 3D: $$(x, y, z)\to\begin{bmatrix}x\\y\\z\\1\end{bmatrix}$$
+
+converting from homogeneous to inhomogeneous coordinates:
+
+- 2D: $$\begin{bmatrix}x\\y\\w\end{bmatrix}\to(x/w, y/w)$$
+- 3D: $$\begin{bmatrix}x\\y\\z\\w\end{bmatrix}\to(x/w, y/w, z/w)$$
+
+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:
+
+$$
+pixel_i=line_1\times line_2
+$$
+
+#### Pinhole Camera Projection Matrix
+
+Intrinsic Assumptions:
+
+- Unit aspect ratio
+- No skew
+- Optical center at (0,0)
+
+Extrinsic Assumptions:
+
+- No rotation
+- No translation (camera at world origin)
+
+$$
+x=K[I\ 0]X\implies w\begin{bmatrix}u\\v\\1\end{bmatrix}=\begin{bmatrix}f&0&0&0\\0&f&0&0\\0&0&1&0\end{bmatrix}\begin{bmatrix}x\\y\\z\\1\end{bmatrix}
+$$
+
+Removing the assumptions:
+
+Intrinsic assumptions:
+
+- Unit aspect ratio
+- No skew
+
+Extrinsic assumptions:
+
+- No rotation
+- No translation
+
+$$
+x=K[I\ 0]X\implies w\begin{bmatrix}u\\v\\1\end{bmatrix}=\begin{bmatrix}\alpha&0&u_0&0\\0&\beta&v_0&0\\0&0&1&0\end{bmatrix}\begin{bmatrix}x\\y\\z\\1\end{bmatrix}
+$$
+
+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?
+
+- rotation: 3
+- translation: 3
+- camera matrix: 5
+
+Total: 11