How to Find a Projection Matrix: The First Step

Grades 11-12
Student math problem: How to Find a Projection Matrix: The First Step
Original question
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Question: explain what is the first step that I should find? for the projection matrix
To find the projection matrix for a camera, the first step is to establish the Intrinsic Matrix, often denoted as K. This matrix describes the internal geometric properties of the camera—essentially, how it converts 3D coordinates in the camera's view into 2D pixel coordinates on the sensor.
To calculate this matrix, you need to synthesize the hardware specifications provided in the text:
  1. Focal Length in Pixels: The focal length is given as 10\text{ mm}. However, the matrix requires this value in "pixel units." You find this by multiplying the physical focal length by the pixel density (pixels per mm). Since a 10\text{ mm} sensor is divided into 400 pixels, the density is 40\text{ px/mm}. Thus, the focal length in pixels ( f_x and f_y) is: f_x = f_y = 10\text{ mm} \times 40\text{ px/mm} = 400\text{ pixels}
  2. Principal Point: This is the pixel coordinate where the optical axis (the center of the lens's view) hits the sensor. The problem states this occurs at (200, 200). These are your u_0 and v_0 values.
  3. The Matrix Structure: The resulting 3 \times 3 Intrinsic Matrix K takes the form: K = \begin{bmatrix} f_x & 0 & u_0 \\ 0 & f_y & v_0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 400 & 0 & 200 \\ 0 & 400 & 200 \\ 0 & 0 & 1 \end{bmatrix}
Once this is established, the next logical step is to determine the Extrinsic Parameters, which involve the Rotation ( R) and Translation ( t) shown in your diagram. This defines where the camera is located in the world and how it is tilted.

Visualizing the Setup

In the 3D model below:
  • The World Coordinate System is shown at the origin on the right. The X-axis is red, Y is green, and Z is blue.
  • The Camera Coordinate System is located 200\text{ mm} away on the left.
  • Note the 45^\circ rotation: The camera's X_c and Y_c axes are rotated relative to the world's X-axis, while the Z_c axis remains parallel to the world's Z-axis.
  • The Translation vector represents the position of the world origin relative to the camera's center.
Drawing these two frames clearly is essential for deriving the Rotation matrix R and the Translation vector t used in the final projection formula P = K[R|t].
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