Finding the linear mapping between homogeneous coordinates

In summary, the conversation discusses the formation of a matrix representing the linear mapping between world and image points in an affine camera with a projection relationship governed by a 2x3 matrix and a 2x1 vector. The individual is trying to understand how an affine camera differs from a camera with perspective projection and how to fit a linear mapping based on the given equation. They mention using a 3x4 projection matrix for a camera with perspective projection, but are unsure of how to apply it to an affine camera.
  • #1
stephchia
1
0

Homework Statement


If I have an affine camera with a projection relationship governed by:

\begin{equation}
\begin{bmatrix}
x & y
\end{bmatrix}^T = A
\begin{bmatrix}
X & Y & Z
\end{bmatrix}^T + b
\end{equation}
where A is a 2x3 matrix and b is a 2x1 vector. How can I form a matrix representing the linear mapping between the world point (X,Y,Z) and image point (x,y) if they are represented by homogeneous vectors?

Homework Equations


NIL

The Attempt at a Solution


I understand that for a camera perspective projection, the linear mapping between homogeneous coordinates where the equation is only up to a scale factor can be written as a 3x4 projection matrix that represents a map from 3D to 2D.
\begin{equation}
\begin{bmatrix}
x \\
y \\
w
\end{bmatrix}=
\begin{bmatrix}
f & 0 & 0 & 0 \\
0 & f & 0 & 0 \\
0 & 0 & 1 & 0
\end{bmatrix}
\begin{bmatrix}
X_c \\
Y_c \\
Z_c \\
1
\end{bmatrix}
\end{equation}
However, I am unclear as to how an affine camera differs and how to fit a linear mapping based on the above given governing equation.
 
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  • #2
Remember you want your original vector plus the displacement. Start with the identity matrix components in the first three columns and then ask what effect a non-zero value in the fourth column will have.
 

1. What are homogeneous coordinates and why are they used in linear mapping?

Homogeneous coordinates are a mathematical representation of points in space that allow for translation, rotation, and scaling operations to be represented as linear transformations. They are used in linear mapping because they simplify the calculations and make it easier to perform transformations on points.

2. How do you find the linear mapping between two sets of homogeneous coordinates?

To find the linear mapping between two sets of homogeneous coordinates, you can use a matrix equation. The matrix will have the coordinates of the points you want to map from as the first row, and the coordinates of the points you want to map to as the second row. By solving this matrix equation, you can find the linear mapping between the two sets of coordinates.

3. Can homogeneous coordinates be used for non-linear transformations?

No, homogeneous coordinates are only useful for representing linear transformations. Non-linear transformations, such as bending or twisting, cannot be represented using homogeneous coordinates.

4. How do you apply a linear mapping to a set of homogeneous coordinates?

To apply a linear mapping to a set of homogeneous coordinates, you can multiply the coordinates by the matrix representing the linear transformation. This will result in a new set of coordinates that have been transformed by the linear mapping.

5. What is the significance of the last coordinate in a set of homogeneous coordinates?

The last coordinate in a set of homogeneous coordinates is known as the "homogeneous coordinate" and it is always equal to 1. This coordinate allows for the representation of translation operations in addition to rotation and scaling, making it a crucial component in linear mapping.

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