16-889 Homework One

Question 1

Question 1 focuses on practicing with cameras.

Q1.1

Using pytorch3d.renderer.look_at_view_transform we can create a continuously rotating cow. In order to make this we had to render 360 images (one for each degree) and then create the gif with 30 frames per second.

Q1.2

Part 2 of this question asks us to implement the dolly zoom. This camera effect works by changing the focal length of the camera and moving the camera towards the object in question. We calculate the distance to the object with \[ d = \frac{w}{2\tan(\frac{1}{2}FOV)}, \] where $d$ is the distance to place the camera from the object, $w$ is the width of the frame in the camera coordinates, and $FOV$ is the current field of view of the camera.

Question 2

For question 2 we practice generating meshes.

Q2.1

Part 1 of this question asks us to construct a tetrahedron. A tetrahedron is a pyramid, with 4 vertices, and four triangular faces that define it. In some sense this can be thought of as a simple watertight mesh primitive.

Q2.2

A bit more challenging is defining a cube by hand, because now we have eight vertices, and 12 triangular faces that we need to specify in order to define it properly.

Question 3

Question 3 asks us to re-texture the cow mesh from before, using a nice "gradient" color texture.

For my colors I chose color1 = [255, 102, 0], color2 = [0, 0, 1].

Question 4

Question 4 asks us to determine the rotations and translations used to create the provided example images. Image one looks straight on to the cow so a simple swapping of the $x$ and $y$ axes produces the desired transformation:

\[ R = \begin{bmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \qquad t = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \]

And for the second image:

\[ R = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \qquad t = \begin{bmatrix} 0 \\ 0 \\ 5 \end{bmatrix} \]

The third:

\[ R = \begin{bmatrix} 0.995 & 0 & 0.99 \\ 0 & 1 & 0 \\ -0.99 & 0 & 0.995 \end{bmatrix}, \qquad t = \begin{bmatrix} 0.2 \\ 0 \\ 0 \end{bmatrix} \]

And lastly:

\[ R = \begin{bmatrix} 0 & 0 & -1 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}, \qquad t = \begin{bmatrix} -3 \\ 0 \\ 3 \end{bmatrix} \]

Question 5

Question 5 asks us to get familiar with PyTorch3D's point cloud interface.

Q5.1

We are given two sets of plant RGB-D data and asked to generate a point cloud of each set.

The first gif is image one's pointcloud, the second is image two's pointcloud and the last gif is the union of the two.

Q5.2

Here's a parametric torus that I generated using the following parametric equations:

\[ \begin{align} x &= (R + r\cos\theta)\cos\phi \\ y &= (R + r\cos\theta)\sin\phi \\ z &= r\sin\theta \end{align}, \]

where $R$ is the distance from the center of the tube to the center of the torus, $r$ is the tube's radius, and $\phi, \theta = [0, 2\pi)$.

Q5.3

Next we use an implicit definition of a torus and then perform the marching cubes algorithm to reproduce a mesh. The implicit equation I used for the torus is:

\[ f(x, y, z) = \left(\sqrt{x^2 + y^2} - R\right)^2 + z^2 -r^2. \]

Some considerations to keep in mind when rendering a point cloud versus a mesh are that since a mesh has vertices and faces to process, it will take longer to process. This is in contrast to the pointcloud which is just a collection of unordered points. Theoretically this makes sense, because a pointcloud doesn't need to reason about connectivity.

Question 6

For question 6 we were tasked with something fun and creative. What I chose to do was import a mesh of a pedestal, and then create a cool figure eight knot on top of it. Since the pedestal was a mesh, I had to sample points from it in order to put both the knot function and the pedestal in the same space. Ideally I'd have made the knot a torus, but I wasn't sure how to do that; it would probably require re writing my parametric equation as using two variables. The equation of the knot is:

\[ \begin{align} x &= (R + r\cos(2t))\cos(3t) \\ y &= (R + r\cos(2t))\sin(3t) \\ z &= r\sin(4t) \end{align}, \]