166889-HW1

Presented by Dijing Zhang

1. Practicing with Cameras

1.1 360-degree Renders

Create a 360-degree gif video that shows many continuous views of the provided cow mesh

1.2 Re-creating the Dolly Zoom

The Dolly Zoom is a famous camera effect, first used in the Alfred Hitchcock film Vertigo. The core idea is to change the focal length of the camera while moving the camera in a way such that the subject is the same size in the frame, producing a rather unsettling effect.

2. Practicing with Meshes

2.1 Constructing a Tetrahedron

vertices = torch.tensor([[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], 
                         [0.0, 0.0, 0.0], [0.0, 0.0, 1.0]])
faces = torch.tensor([[0, 1, 2], [0, 1, 3], [0, 2, 3], [1, 2, 3]])

Vertices: 4 Faces: 4

2.2 Constructing a Cube

vertices = 0.5 * torch.tensor([[1.0, 1.0, -1.0], [1.0, -1.0, -1.0], [-1.0, -1.0, -1.0], [-1.0, 1.0, -1.0], 
                         [1.0, 1.0, 1.0], [1.0, -1.0, 1.0], [-1.0, -1.0, 1.0], [-1.0, 1.0, 1.0]])
faces = torch.tensor([[0, 1, 2], [0, 2, 3], [3, 6, 7], [2, 3, 6],
                      [1, 2, 6], [1, 5, 6], [4, 5, 7], [4, 6, 7],
                      [0, 4, 7], [0, 3, 7], [1, 4, 0], [1, 5, 4],])

Vertices: 8 Faces: 12

3. Re-texturing a mesh

Retexturing the cow mesh such that the color smoothly changes from the front of the cow to the back of the cow.

color1: [0, 0, 1]

color2: [1, 0, 0]

4. Camera Transformations

When working with 3D, finding a reasonable camera pose is often the first step to producing a useful visualization, and an important first step toward debugging.

Find a set (R_relative, T_relative) such that the new camera extrinsics with R = R_relative @ R_0 and T = R_relative @ T_0 + T_relative produces each of the following images:

For case1, which rotate the cow in xy plane 90 degrees closewise direction, we set R_relative = [[0, 1, 0], [-1, 0, 0], [0, 0, 1]] to rotate the x-axis and y-axis.

For case2, which move camera more far from object, just simply increase the translation by setting T_relative=[0, 0, 3]

For case3, which move the camera up-right, just simply set T_relative=[0.5, -0.5, 0].

For case4, which rotates the cow in x-z plane (along y-axis) 90 degrees closewisely, we can reset x-aixs to [0, 0, 1] and z-aixs to [-1, 0, 0] and remember to remove camera by setting T_relative = [-3, 0, 3]

5. Rendering Generic 3D Representations

5.1 Rendering Point Clouds from RGB-D Images

Practice rendering point clouds constructed from 2 RGB-D images from the Common Objects in 3D Dataset.

Reconstruct pointcloud from two different images (including image, depth, mask) and merged results.

5.2 Parametric Functions

A parametric function generates a 3D point for each point in the source domain. For example, given an elevation theta and azimuth phi, we can parameterize the surface of a unit sphere as (sin(theta) cos(phi), cos(theta), sin(theta) sin(phi)).

Equation of a torus with radius $r$ and $R$:

$$x = (R + r * \cos \theta) * \cos\phi$$$$y = (R + r * \cos \theta) * \sin\phi$$$$z = r * \sin\theta$$

5.3 Implicit Surfaces

Explore representing geometry as a function in the form of an implicit function. In general, given a function F(x, y, z), we can define the surface to be the zero level-set of F i.e. (x,y,z) such that F(x, y, z) = 0. The function F can be a mathematical equation or even a neural network. To visualize such a representation, we can discretize the 3D space and evaluate the implicit function, storing the values in a voxel grid. Finally, to recover the mesh, we can run the marching cubes algorithm to extract the 0-level set.

Discuss some of the tradeoffs between rendering as a mesh vs a point cloud (include rendering speed, rendering quality, ease of use, memory usage, etc.)

Rendering speed: It depends on the quality of mesh or pointcloud (voxel size, number of samples for each respectively). But pc runs faster than mesh.

Quality: It depends on the voxel size and number of samples. But pc looks smoother while mesh looks more complete.

Ease of use: Pc is more easliy to use.

memory usage: Mesh is smaller.

6. Do Something Fun

Generating an iron man!

7. Sampling Points on Meshes

Explore how to obtain point clouds from triangle meshes. One obvious way to do this is to simply discard the face information and treat the vertices as a point cloud. However, this might be unresonable if the faces are not of equal size.

Instead, as we saw in the lectures, a solution to this problem is to use a uniform sampling of the surface using stratified sampling. The procedure is as follows:

1. Sample a face with probability proportional to the area of the face

2. Sample a random barycentric coordinate uniformly'

3. Compute the corresponding point using baricentric coordinates on the selected face.

Takes a triangle mesh and the number of samples and outputs a point cloud. Then, using the cow mesh, randomly sample 10, 100, 1000, and 10000 points.