16-889 Assignment 1

Byeongjoo Ahn (bahn@andrew.cmu.edu)

1. Practicing with Cameras

1.1. 360-degree Renders (5 points) `p1_1.ipynb`

The gif of 360-degree video is shown below. The point light used for the illumination is collocated with the camera for better visibility.

1.2 Re-creating the Dolly Zoom (10 points) `p1_2.ipynb`

The gif of dolly zoom effect is shown below.

2. Practicing with Meshes `p2.ipynb`

2.1 Constructing a Tetrahedron (5 points)

The gif of a tetrahedron is shown below.

Number of vertices: 4

Number of faces: 4

2.2 Constructing a Cube (5 points)

The gif of a cube is shown below.

Number of vertices: 8

Number of faces: 12

3. Re-texturing a mesh (10 points) `p3.ipynb`

The gif of the rendered mesh with a new texture is shown below.

Color 1: [1,0,1] (Magenta)

Color 2: [0,1,1] (Cyan)

4. Camera Transformations (20 points) `p4.ipynb`

The renderings and descriptions for target transformations are shown below.

Transformation #1: The rotation matrix should transform $(1,0,0)→(0,1,0)$ , $(0,1,0)→(-1,0,0)$ , and $(0,0,1)→(0,0,1)$ , which results in $R = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ . As the distance from the camera is the same to the original one, $T$ remains the same and $T_\text{relative}$ should be $(0,0,0)$

Transformation #2: The camera is moved away from the object and the rotation is not changed, which leads to $T = (0, 0, 2)$ .

Transformation #3: The center of the object (i.e., origin of the world coordinate system) is moved to bottom-left (i.e., positive $x$ -direction and negative $y$ -direction in the camera coordinate system), which means $T = (0.5, -0.5, 0)$ .

Transformation #4: The rotation matrix should transform $(1,0,0)→(0,0,1)$ , $(0,1,0)→(0,1,0)$ , and $(0,0,1)→(-1,0,0)$ , which makes $R = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \end{pmatrix}$ . As the distance from the camera is the same to the original distance, $T$ should remain the same and thereby $T_\text{relative} = -R_\text{relative} T_0 + T_0$ .

Note that pytorch3d uses the convention of $X’ = XR + T$ , so $R$ matrix is transposed compared to the case of $X’ = RX + T$ .

5. Rendering Generic 3D Representations

5.1 Rendering Point Clouds from RGB-D Images (10 points) `p5_1.ipynb`

The rendering of point clouds from the RGBD images are shown below. For the rendering, we made the initial pose of the camera the same as the camera1 and adjusted a distance to be 6 units as suggested by the instruction, and rotated the camera around the object.

5.2 Parametric Functions (10 points) `p5_2.ipynb`

The gif of the torus point cloud is provided below.

5.3 Implicit Surfaces (15 points) `p5_3.ipynb`

The gif of the torus mesh is provided below.

Rendering as Mesh vs. Point cloud: Compared to a point cloud, a mesh is easy to render with less memory usage (though it needs an additional connectivity information) and high rendering quality as point cloud requires a lot of point samples to achieve a rendering without spaces between the points. Also, the mesh representation includes normal information that allows us more realistic rendering (e.g., specular highlight) based on the rendering equation. Occlusion is also difficult to handle in the point cloud rendering.