Assignment 1: Rendering Basics with PyTorch3D

Name: Edward Li
Andrew ID: edwardli
Late Days Used:

1. Practicing with Cameras

1.1 360-degree Renders (5 points)

A basic render of a cow on a turntable. See code in starter/render_mesh_turntable.py.

1.2 Re-creating the Dolly Zoom (10 points)

With an FOV of $\theta$, we move the camera a distance along the $z$-axis proportional to $\cot(\frac{\theta}{2})$. See code in starter/dolly_zoom.py.

2. Practicing with Meshes

2.1 Constructing a Tetrahedron (5 points)

This tetrahedron has 4 vertices and 4 triangle faces. See code in starter/render_tetrahedron.py.

2.2 Constructing a Cube (5 points)

This cube has 8 vertices and 12 triangle faces. See code in starter/render_cube.py.

3. Re-texturing a mesh (10 points)

I chose to make a gradient from brown ([0.26, 0.14, 0.09]) to white ([1, 1, 1]), as cows come in brown and white. Perhaps this can be some sort of hybrid unity cow thing. See code in starter/render_retexture.py.

4. Camera Transformations (20 points)

• We want R_relative to rotate the points in the XY-plane 90 degrees clockwise. This corresponds to $R_{relative} = \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$.

• Here, we just want T_relative to move the points away from us another 3 units. This is just $T_{relative} = \begin{bmatrix} 0 \ 0 \ 3 \end{bmatrix}$.

• Here, we want T_relative to translate the points along -Y and +X. We do this with $T_{relative} = \begin{bmatrix} 0.5 \ -0.5 \ 0 \end{bmatrix}$.

• Here, we want R_relative to rotate the points 90 degrees clockwize in the XZ-plane. After, we need T_relative to translate the points back from 3-away in the X axis to 3-away in the Z axis. This is done with $R_{relative} = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \end{bmatrix}$ and $T_{relative} = \begin{bmatrix} -3 \ 0 \ 3 \end{bmatrix}$.

See code in starter/camera_transforms.py.

5. Rendering Generic 3D Representations

5.1 Rendering Points Clouds from RGB-D Images (10 points)

Below are the first image, second image, and combined images in order. See code in starter/render_plant.py.

Image 1	Image 2	Combined

5.2 Parametric Functions (10 points)

A torus! With 1000 samples. We texture the same way that was originally given. See samples/render_torus.py.

5.3 Implicit Surfaces (15 points)

We use marching squares to render a torus implicit surface. See code in starter/render_implicit.py.

In terms of mesh and point cloud tradeoffs, we have:

• Rendering Speed - Meshes are generally easier to render than point clouds, especially if the number of vertices/faces are small. This is because it is very easy to find ray intersections with planar faces, making rendering very efficient if there is a small number faces. On the other hand, point clouds generally require dense sets of points for a rendering to make sense, decreasing speed. However, if we use marching cubes to generate a mesh, this will likely be slower or comparable to point cloud, as the mesh will be highly complicated.
• Rendering Quality - Generally, meshes will have higher rendering quality than point clouds due to possible holes found in point cloud renders, as long as the mesh is correct. However, meshes that are derived from level sets (marching cubes) might look worse than a point cloud representation due to the possibility of creating badly shaped and globally suboptimal triangles.
• Ease of Use - Point clouds are generally easier to use than meshes, as continuous deformations are very flexible and allow for significant connectivity changes, whereas meshes are unable to change connectivity easily. Additionally, a point cloud is more easily collected using sensors and easy to convert into.
• Memory Usage - Meshes are generally more memory efficient than point clouds, as a high quality point cloud render requires a high number of samples, whereas meshes only require a sparse set of vertices.

6. Do Something Fun (10 points)

We all know that approximating cows as spheres is overly simplistic. Finding the appropriate approximation for a cow in physics problems is still an open research question. However, a common first-order attempt at this is of course to approximate the cow as a torus.

For this part, I retexture the point cloud torus as a cow. We observe that the cow texture works as a torus, as its edges are all just the same base cow color. Then, we load the texture and rescale it across $\phi$ and $\theta$ to align with points in a sensible way. See code in starter/render_cow_torus.py.

(Extra Credit) 7. Sampling Points on Meshes (10 points)

Code for this part can be found in samples/extra_credit.py.

Mesh	10 points	100 points	1000 points	10000 points