**Zangyueyang Xian Yitong Deng Final Project**
Student name and Netid: Zangyueyang Xian (F0040NB) and Yitong Deng (F003MZT)
(##) Motivational image
Our motivational image is this painting by the Spanish surrealist painter Dalí, featuring his iconic "melting watches" raises
profound questions about the appearance and the reality. The most notable feature in this piece are the molten watches, which
we usually perceive as sturdy and lifeless, are now molten and attracts ants, as if they are deteriorated cheese. These constructs
are obviously abnorml, illogical, and they almost invites denial, as they do not conform with our ingrained understanding
of how the world is and should be. However, at the same time, they can be said to reveal a deeper truth,
as they invoke a kind of confusion, dissonance and irreconcilability that can resonate with the one that we
too feel about the world on some subconscious level. In this project, we also seek to explore this idea, that the truth
of an object can only be revealed when it is contorted from of its typical, accepted form. And only then can we
parse into its intrinsic nature, and see "what's inside". Hence, we design a scene with a statue of lion, made of metal, which
melts under the influence of gravity. As the lion melts, a cat emerges from within the lion and remains upright, as if it is the
last man standing.
(##) Microfacet BRDF --- 4 PTS
We implemented micro-facet brdf basically by following PBRT Ch.8 and Ch.14. We choose Torrance-Sparrow model, with the normal distrubution as Beckmann-Spizzichino distribution. Here is a comparison between our implementation for micro-facet brdf and Bling-Phong brdf:
From left to right, the sigma value in Beckmann-Spizzichino distribution for each sphere is 0.015, 0.03, 0.06,..., 0.48. Exponent value in Bling-Phong brdf is the same as in assignments.
There are noticable noises near the boundary for each sphere. This is because in importance sampling from materials, we only sampled from the normal distrubution but did not account the masking term. When the incident angle is almost parallel to the surface, the normal distribution and the brdf is quite different from each other. We tried to account the masking term, but with lack of details it is hard for us to follow the PBRT codes. However in our real scene these noises do not appear.
(##) Participating Media --- 8 PTS
We implemented participating mediun by reading the PBRT codes. Basically the medium is represented implicitly by its boundary. In our implementation, each surface can define 2 participating medium that attached to it, one outside and one inside. Outside just mean that the medium is lying on the same side as the surface normal, and inside means the other one, so surfaces do not have to be closed.
We implemented homogeneus medium, which could be characterized by absorbtion coefficients, scattering coefficients, phase function and emission. We only implemented isotropic phase function. We implemented integrators both sampling from materials and from both materials and direct illumination. With these we then deliver two examples:
In the first example, there is a noticable artifact on the right-most sphere. This is because when defining medium, we have to introduce some surfaces that are not come from the scene. Thus we implemented a material called "NULL", which is a specular material that can only scatter a ray with the same direction (penetrating through). However, this changes the behaviour of direct illumination sampling. If a direct lighting ray intersects a "NULL" face, it will give 0 contribution, while previously this intersection does not exist so it will works correctly. In this scene part of the right-most sphere is outside the box (outside a "NULL" face), so our MIS integrator degenerated to MATS integrator on that part.
In the second example, It is much noiser than the first one, even with the same SPP. This is due to that there is no emitted surface in the scene, and our integrator does not sample direct illumination from medium. Thus our MIS integrator degenerated to MATS integrator again.
(##) Importance Sampling of Environment --- 4 PTS
The importance sampling of environment maps is implemented according to the lecture note, where the RGB image is first reduced to
a single-channel image by taking the luminance value. Then we compute the marginal probability for each row, and create a CDF for each
row by summing the probabilities up to that row, and use the inverse method to sample a row with a random variable between $[0, 1)$.
Once the row is chosen, we repeat the same process, but this time with the conditional probability for each column being sampled given
that row, and sample the column.
There are some code refactoring that is necessary to enable this feature. In our scene, the environment mapping is applied to a cylinder.
Since the UV mapping scheme from a planar image differs from geometry to geometry, the importance sampling need to be geometry specific.
Hence we will implement a sample_env and pdf_env in the cylinder class, which will be called when the
cylinder is a light textured with an image. How are we supposed to differentiate such cylinders? We implement a Image_Info
struct to store all the pre-computed marginal probabilities and CDFs. We will store a pointer to this struct in the texture base class,
but it will only be initialized in the ImageTexture class. These probabilities will be computed once, and the Cylinder
then needs to generate random variables between $[0, 1)$ and use the std::upper_bound function to find the corresponding column or row.
It is also necessary to take care of the edge case where the generated random float is very close to $1$, which due to numeric issues
may cause the std::upper_bound to return the last of the list, causing out of bound errors.
Test sampling images
Simple scenes with NEE with max_bounce == 1
Without importance sampling environment:
With importance sampling environment:
Without importance sampling environment:
With importance sampling environment:
The environment mapping with importance sampling will be used in our final image to render the cylindrical wall
which is a diffuse light textured with an image of a cat.
(##) Simulation of Lion Deformation --- 4 PTS
We opt to simulate the melting effect of the lion statue as weakly compressible viscous fluid, using the Smoothed Particle Hydrodynamics (SPH) method (which we will implemented ourselves for this project).
To simulate the lion statue as fluid, we need to first discretize the entire fluid domain. We are given the triangle mesh of the surface, and we need to
use particles to sample the entire volume enclosed by the mesh. To do so, we adapted from this project
which use the Generalized Winding Number approach.
This plot depicts the surface vertices:
And this plot depicts the surface vertices plus the sampled volume (since SPH requires well-distribution, we sample with a grid pattern):
We create a custom heat function which corresponds to the extend that the material is subject to gravitational force:
The key idea in SPH is to approximate spatially-varying quantities with neighboring particles, using some neighbor searcher and some smoothing kerenl.
In this project we use the KD-Tree of SciPy, and we implemented the 3 dimensional Cubic Spline kernel ourselves. This plot shows that the behavior in two dimensions
is ideal:
With this functionality, we will be able to compute the essential quantity in SPH, which is the number density. This plot shows the initial number density, which is
what we expect, as the head of the lion has more detail and thus is sampled with more particles.
We further test this with simulating (without adding pressure force or viscosity) for 20 steps:
Here, more particles are gathered around the bottom-left of the statue, hence there will be a larger number density as expected.
Below depicts 10 frames of the melting process:
(##) Generation of Fire --- 0 PTS
Physically fire should be considered as heterogeneus medium for rendering purpose. We do not implemented fire by that. This is due to first it is hard to control the shape of fire with a density field, and second there is no implementation in PBRT that support heterogeneus medium with coefficients different in channels. We are not sure that whether integrate colors channel by channel is viable.
We adopt the method explained here to
generate fire procedurally with Perlin noise. Like in the tutorial, we use 4-dimensional (with time) noise values to generate the RGB
values for each grid point. Then, we compute the luminance of each grid point, and filter out the grid points with luminance lower than
6, which is depicted below:
Then, we will use PyVista to convert the particles into a mesh.
We will use homogeneus medium but with multiple layers with varying emission values to mimic fires. We tune some values and finally get the feeling of fire:
Simple scenes with NEE with 
With importance sampling environment:
Without importance sampling environment:
With importance sampling environment:
The environment mapping with importance sampling will be used in our final image to render the cylindrical wall
which is a diffuse light textured with an image of a cat.
(##) Simulation of Lion Deformation --- 4 PTS
We opt to simulate the melting effect of the lion statue as weakly compressible viscous fluid, using the Smoothed Particle Hydrodynamics (SPH) method (which we will implemented ourselves for this project).
To simulate the lion statue as fluid, we need to first discretize the entire fluid domain. We are given the triangle mesh of the surface, and we need to
use particles to sample the entire volume enclosed by the mesh. To do so, we adapted from this project
which use the Generalized Winding Number approach.
This plot depicts the surface vertices:
And this plot depicts the surface vertices plus the sampled volume (since SPH requires well-distribution, we sample with a grid pattern):
We create a custom heat function which corresponds to the extend that the material is subject to gravitational force:
The key idea in SPH is to approximate spatially-varying quantities with neighboring particles, using some neighbor searcher and some smoothing kerenl.
In this project we use the KD-Tree of SciPy, and we implemented the 3 dimensional Cubic Spline kernel ourselves. This plot shows that the behavior in two dimensions
is ideal:
With this functionality, we will be able to compute the essential quantity in SPH, which is the number density. This plot shows the initial number density, which is
what we expect, as the head of the lion has more detail and thus is sampled with more particles.
We further test this with simulating (without adding pressure force or viscosity) for 20 steps:
Here, more particles are gathered around the bottom-left of the statue, hence there will be a larger number density as expected.
Below depicts 10 frames of the melting process:
(##) Generation of Fire --- 0 PTS
Physically fire should be considered as heterogeneus medium for rendering purpose. We do not implemented fire by that. This is due to first it is hard to control the shape of fire with a density field, and second there is no implementation in PBRT that support heterogeneus medium with coefficients different in channels. We are not sure that whether integrate colors channel by channel is viable.
Then, we will use PyVista to convert the particles into a mesh.
We will use homogeneus medium but with multiple layers with varying emission values to mimic fires. We tune some values and finally get the feeling of fire:
