Question

Let Pᵢbe points on a 3D object that are projected onto an image plane using a projection M. Mark all the statements that are correct.

A) The fundamental relationship between world and image points is derived using triangle sim- ilarity.
B) In the projection matrix expression M = K [I|0]| the 0 indicates a 3 * 3 matrix of zeros
C) The projection of a 3D point P is non-linear be- cause it is not possible to express the coordi- nates of the projected point as a weighted sum of the coordinates of P.
D) The projection of a 3D object gets bigger when increasing its distance from the camera.
E) The 3D camera projection matrix M is a 3 * 4 matrix

Answer 1

To answer the student's question, the correct statements are: A) Triangle similarity is used in projection, C) The projection is non-linear, and E) The 3D camera projection matrix is indeed a 3 * 4 matrix. Statement B) is incorrect because '0' represents a zero vector and not a 3 * 3 matrix, and D) is also incorrect as the projection of a 3D object gets smaller, not bigger, as it moves away from the camera.

Step-by-step explanation:

The student has asked about the correctness of various statements regarding points on a 3D object projected onto an image plane using a projection matrix. Here are the correct statements:

1. Triangle similarity is used to derive the fundamental relationship between world and image points in projection.
2. The projection matrix expression M = K [I|0] does not mean that the 0 indicates a 3 * 3 matrix of zeros. In this context, '0' represents a zero vector, making M a 3 * 4 matrix that combines the camera's intrinsic parameters (K) and its position and orientation in the world (I for an identity matrix, and a zero vector for no translation).
3. The projection of a 3D point is indeed non-linear, primarily because the projection involves division by the z-coordinate, which cannot be expressed as a weighted sum of the coordinates.
4. The projection of a 3D object on an image plane actually gets smaller as the object's distance from the camera increases, contrary to statement D. This follows from the principles of perspective projection.
5. The 3D camera projection matrix M is indeed a 3 * 4 matrix, which aligns with projective geometry conventions.