Perspective Projection with Intrinsics

Implement a perspective projection that maps 3D world points to 2D image coordinates using a camera's intrinsic matrix. This process is crucial in computer vision for understanding how 3D scenes are projected onto 2D images.

The intrinsic matrix, denoted as $K$, encapsulates the camera's internal parameters, including focal lengths ($f_x$, $f_y$) in pixels and the principal point ($c_x$, $c_y$), which represents the optical center of the camera. The intrinsic matrix $K$ is given by $K = \begin{pmatrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{pmatrix}$ To project a 3D point ($X$, $Y$, $Z$) to a 2D image point, the following steps are involved:

1. Perform perspective division by dividing the $X$ and $Y$ coordinates by $Z$.
2. Apply the intrinsic matrix $K$ to the normalized coordinates. The projection equation can be represented as $\mathbf{p} = K \cdot [X/Z, Y/Z, 1]^T$.

This technique is widely used in 3D reconstruction and image processing applications.