Simulink 3D Animation: how to compute the projection matrix

In Simulink 3D Animation, the projection matrix of a camera determines how a 3D scene is projected onto a 2D plane. To compute the projection matrix for a camera view in a virtual reality world, you can use the following steps in MATLAB.

Steps to Compute the Projection Matrix:

1. Access the Camera Node:

   • Load the virtual reality world using vrworld.
   • Access the specific camera node using getfield or directly accessing the camera properties.

2. Extract Camera Parameters:

   • Retrieve the camera's intrinsic parameters (field of view, aspect ratio, etc.).
   • Get the extrinsic parameters (position and orientation of the camera).

3. Build the Projection Matrix:

   • Combine the intrinsic and extrinsic parameters to construct the projection matrix.
   • The projection matrix $P$ maps a 3D point $\mathbf{X}$ in world coordinates to a 2D point $\mathbf{x}$ on the image plane: $$\mathbf{x}=P\mathbf{X}$$

Example Code to Compute the Projection Matrix:

% Load the virtual reality world
vr_world = vrworld('my_vr_world.wrl');
open(vr_world);

% Access the camera node
camera_node = vrnode(vr_world, 'CameraNodeName'); % Replace with the actual camera node name

% Retrieve camera parameters
field_of_view = camera_node.fieldOfView; % Horizontal field of view in radians
aspect_ratio = camera_node.aspectRatio;  % Aspect ratio (width/height)
position = camera_node.translation;     % Camera position [x, y, z]
orientation = camera_node.rotation;     % Camera orientation [axis_x, axis_y, axis_z, angle]

% Compute intrinsic matrix (K)
f = 1 / tan(field_of_view / 2); % Focal length (assumes normalized coordinates)
K = [f / aspect_ratio, 0, 0;
     0, f, 0;
     0, 0, 1];

% Compute extrinsic matrix (R and t)
R = vrrotvec2mat(orientation); % Rotation matrix from orientation
t = -R * position';            % Translation vector
Rt = [R, t];                   % Combine rotation and translation

% Combine intrinsic and extrinsic to get the projection matrix
P = K * [R, t];

% Display the result
disp('Projection Matrix:');
disp(P);

Explanation of Parameters:

1. Intrinsic Parameters:

   • field_of_view: Determines the horizontal field of view in radians.
   • aspect_ratio: Used to adjust the focal length for the height of the image.

2. Extrinsic Parameters:

   • position: Camera position in the world coordinates.
   • orientation: A rotation vector (axis-angle format) representing the camera's orientation.

3. Projection Matrix:

   • The projection matrix $P$ combines intrinsic (camera-specific) and extrinsic (position and orientation in the world) transformations.