Update notes for Lecture 05

GAMES-202/markdown/Lecture05.md

@@ -4,76 +4,124 @@
 
 ## I. Distance Field Soft Shadows
 
+![image-20230919163306811](../images/Lecture05-img-14.png)
+
+**Pros**: Faster than shadow maps, and looks way better.
+
+**Cons**: Memory cost.
+
+
+
 ### (Signed) Distance Function
 
 **Input**: Coordinate
 
 **Output**: The *minimum distance* from that coordinate to the object being described
 
-**Characteristics**:
+- May use **signs** to represent whether the point is **inside/outside** the object
+
+*Related to **optimal transport***.
 
-- **Preserving Boundary**: Blending two SDF results in a moving boundary (rather than a blurred one)
+
+
+#### Characteristics
+
+- **Preserving Boundary**: Blending two SDF results in a moving boundary (rather than a blurred object)
 
   ![image-20230912183945613](../images/Lecture05-img-1.png)
 
-- **Combing Distance Functions**:
+- **Combining Geometric Shapes**: Combining SDFs results in a blended shape.
 
+  - Therefore, it is easy to compute SDF for moving objects that are interacting with the scene.
+
   ![image-20230912184114916](../images/Lecture05-img-2.png)
+
+- **Hard to apply textures to SDF objects**: Hard to do $uv$ parameterization.
 
 
 
-#### Usages
-
-#### Ray Marching (Sphere Tracing)
+### Ray Marching (Sphere Tracing)
 
 Used in ray marching to perform ray-SDF intersections.
 
+- **Safe distance**: Assume there is a **distance field** for the entire scene, and assume at point $p$, we have $\text{SDF}(p) = d$.
+  - The ray will not intersect any object if it travels $d$ distance (regardless of the direction of traveling).
+
 ![image-20230912184257745](../images/Lecture05-img-3.png)
 
-Each time at a given point $p$, travel $\text{SDF}(p)$ distance.
+Therefore, each time at a given point $p$, just travel $\text{SDF}(p)$ distance.
 
 
 
-#### Computing Occlusions (SDF Soft Shadow)
+### Computing Occlusions (SDF Soft Shadow)
 
 ![image-20230912184425904](../images/Lecture05-img-4.png)
 
 The value of $\text{SDF}$ can be used to determine a "safe" angle seen from the eye:
 
 - The smaller the "safe" angle, the less the visibility
 
-![image-20230912184640509](../images/Lecture05-img-5.png)
 
-To compute the angle, use $\min\left\{\frac{k \cdot \text{SDF} (p)}{p - o}, 1.0\right\} $:
 
-- Larger $k$ leads to earlier cutting off of **penumbra**, resulting in **harder** shadows
+During ray marching
+
+- Calculate the "safe" angle from the eye at every step
+- Keep the minimum $\theta$
+
+![image-20230919165149338](../images/Lecture05-img-15.png)
+
+To compute the visibility (related to the safe angle) during ray marching:
+
+- Use $V = \min\left\{\frac{k \cdot \text{SDF} (p)}{p - o}, 1.0\right\} $:
+  - Larger $k$ <=> smaller safe angle <=> **earlier** cutting off of penumbra <=> **harder** shadows.
+    - If the $k$ is small, then the visibility would be significantly higher for regions that can see *partly* the light source.
 
+- Why not use $\arcsin$?
+  - To avoid expensive computations.
 
+![image-20230912184640509](../images/Lecture05-img-5.png)
 
 ![image-20230912184947922](../images/Lecture05-img-6.png)
 
-#### Anti-Aliased Characters in RTR
+
+
+### Anti-Aliased Characters in RTR
 
 ![image-20230912185138801](../images/Lecture05-img-7.png)
 
 [Troika-Three-Text from GitHub](https://github.com/protectwise/troika/tree/master/packages/troika-three-text)
 
+- Interpolating the SDF to achieve smooth boundary on fonts.
 
 
-### Pros and Cons
+
+### Pros and Cons of Distance Field Soft Shadows
 
 **Pros**:
 
-- Fast
+- **Fast***:
+
+  - When compared with hard shadow generating by ray marching: Almost no additional cost (regardless of distance field generation)
+
+  - When compared with traditional shadow mapping: faster precomputation, approximately the same time for querying
 - High quality
 
 **Cons**:
 
-- Needs precomputation
+- Needs precomputation:
+  - When deformation of objects occurred, re-computation is required
+
 - Needs **heavy** storage:
-  - 
+  - 3D texture storage
 - Artifacts:
-  - 
+  - For example, around connections between objects
+
+
+### Acceleration
+
+- Hiearchies:
+  - Trees (KD Trees, ...)
+- Compression by deep learning
 
 
 
@@ -83,36 +131,53 @@ Environment lighting: Simulate lighting coming from the surroundings of a scene.
 
 ![image-20230913185204341](../images/Lecture05-img-8.png)
 
-There are **spherical** maps and **cube** maps.
+There are **spherical** maps and **cube** maps. In the following context we may assume we have already obtained the lighting information regardless of the underlying structure.
+
+The environment lighting is used when solving the rendering equation. In this chapter we will discuss shading only - that is, the visibility term will be ignored.
 
-The environment lighting is used when solving the rendering equation. Environment maps provide **info of incoming radiance** on a specific shading point.
+Environment maps provide **information of incoming radiance** on a specific shading point.
 
 - **General Solution**: Monte Carlo Integration
   - Numerical
   - Large amount of samples required
-  - Can be **slow**
+  - Can be slow:
     - Sampling is not preferred in shaders\* in general: High time complexity.
+      - Recently, many temporal methods that relies on sampling are used.
 
 
 
 ### The Split Sum Approximation
 
 In the industry, the resulting integral is computed using sampling and summing, thus it is called **split sum** rather than split **integral**.
+$$
+\frac{1}{N} \sum_{k=1}^{N} \frac{L_i(\mathbf{l}_k) f(\mathbf{l}_k, \mathbf{v}) \cos \theta_{\mathbf{l}_k}}{p(\mathbf{l}_k, \mathbf{v})}
+\approx
+\left(
+\frac{1}{N} \sum_{k=1}^{N} L_i(\mathbf{l}_k)
+\right)
+\left(
+\frac{1}{N} \sum_{k=1}^{N} \frac{\mathbf{l}_K, \mathbf{v} \cos \theta_{\mathbf{l}_k} \cos \theta_{\mathbf{l}_k}}{p(\mathbf{l}_k, \mathbf{v})}
+\right)
+$$
+
+
+- Used in Unreal Engine.
+- **State of Art**
 
 #### Principle
 
 The split sum approximation **avoids sampling** by approximating the given integral in a way described in Lecture 4:
 $$
 \int_{\Omega} f(x) g(x) \dd{x} \approx 
-\underbrace{\frac{\int_{\Omega_G} f(x) \dd{x}}{\int_{\Omega_G} \dd{x}}}_{\substack{\text{The average value} \\ \text{of $f(x)$ over the} \\ \text{effective interval of }g(x)}}
+\underbrace{\frac{\int_{\Omega_G} f(x) \dd{x}}{\int_{\Omega_G} \dd{x}}}_{\substack{\text{The \textbf{average} value} \\ \text{of $f(x)$ over the} \\ \text{effective interval of }g(x)}}
 \cdot \int_\Omega g(x) \dd{x}
 $$
 
 - **General BRDFs satisfy the requirements for accuracy in all cases**:
   - If the BRDF is **glossy**, then $g(x)$ or $f_r(x)$ in the formula, has small support.
   - If the BRDF is **diffuse**, then $g(x)$ or $f_r(x)$ in the formula, is smooth.
 
-- *Note the slight edit on* $\Omega_G$.
+- *Note the slight edit on* $\Omega_G$: We integrate $f(x)$ only in the support of $g$.
 
 
 
@@ -136,20 +201,36 @@ _{\text{Do shading without considering lighting}}
 $$
 
 - **Prefiltering** of the environment lighting: Computing the first term $\frac{
-  	\int_{f_r} L_i (\text{p}, \omega_i) \dd{\omega_i}}{\int_{f_r} \dd{\omega_i}}$
-  - Pre-generating a set of differently filtered environment lighting
+  	\int_{f_r} L_i (\text{p}, \omega_i) \dd{\omega_i}}{\int_{f_r} \dd{\omega_i}}$, the average value of $L_i(p, \omega_i)$ over regions of various size.
+  - Pre-generating a set of differently filtered environment lighting.
   - Filter size in-between can be approximated via (trilinear) interpolation.
+    - Size of the filter should correspond to solid angle (but not a fixed size on the resultant texture)
 
 ![image-20230913191319711](../images/Lecture05-img-9.png)
 
-- And then **querying** the pre-filtered environment lighting at $\omega_i$:
+- And then **querying** the pre-filtered environment lighting at $\omega_i$ (corresponding to $\vec{r}$ in the following figure):
 
+  - For diffusions you may follow the direction of $n$
+
   ![image-20230913191510378](../images/Lecture05-img-10.png)
 
 
 
 #### 2nd Stage
 
+*Note that there is a another solution as of the time writing this note.*
+
+- Linearly Transformed Cosines: [Eric Heitz's Research Page (wordpress.com)](https://eheitzresearch.wordpress.com/415-2/)
+
+  ![image-20230919200658841](../images/Lecture05-img-16.png)
+
+The idea is to:
+
+- **Precompute** the value of the integration left in equation $\ref{eq1}$ for all possible combinations of variables:
+  - Too many dimensions: roughness $\alpha$, color (Fresnel term) (triple terms, incident angle), etc
+
+
+
 Compute the latter part of equation $\ref{eq1}$ by **approximating the surface as a microfacet**, and extract the Fresnel term using the **Schlick's Approximation**:
 $$
 \begin{equation} \tag{The Microfacet Model}
@@ -172,35 +253,40 @@ $$
 
 where
 $$
-R_0 = \left(\frac{n_1 - n_2}{n_1 + n_2}\right)^2.
+\begin{equation} \tag{Basic reflection rate}
+R_0 = \left(\frac{n_1 - n_2}{n_1 + n_2}\right)^2
+\end{equation}
 $$
 
 $$
 \begin{equation} \tag{The NDF, Normal Distribution Function}
-D(h) = \frac{\exp(\tan^2\theta_h)}{\pi \alpha^2 \cos^4 \theta_h}
+D(h) = \frac{\exp(\tan^2\theta_h /\alpha^2)}{\pi \alpha^2 \cos^4 \theta_h}
 \end{equation}
 $$
 
 ![image-20230913192819390](../images/Lecture05-img-11.png)
 
-**The base color $R_0$ is now extracted**:
+**The "base color $R_0$" is now extracted**:
 $$
 \begin{aligned}
 \int_{\Omega+} f_r(\text{p}, \omega_i, \omega_o) \cos \theta_i \dd{\omega_i} \approx{} & R_0 \int_{\Omega_+}
 \frac{f_r}{F} (1 - (1 - \cos \theta_i)^5) \cos \theta_i \dd{\omega_i} + \\
 & \int_{\Omega_+} \frac{f_r}{F} (1 - \cos \theta_i)^5 \cos \theta_i \dd{\omega_i}
 \end{aligned}
 $$
-and both integrals can be **precomputed**:
+and both integrals can be **precomputed** with only 2 parameters, as now there are only roughness $\alpha$ and incident angle of light $\omega_i$ left:
 
 ![image-20230913193058987](../images/Lecture05-img-12.png)
 
-- **Precomputed texture**: Each integral produces one value for each $(roughness, \cos\theta_o)$ angle pair.
+- **Precomputed texture**: Each integral produces one value for each $(\alpha, \cos\theta_o)$ angle pair.
 
 
 
 #### Results
 
+Completely avoided sampling!
+
 Very fast and almost identical:
 
 ![image-20230913193333372](../images/Lecture05-img-13.png)
+