Chuan 川 Charles

Ray Tracing 4 Path TracingMonte Carlo Integration Definite intergral: $\int_{a}^{b}{f(x)dx}$ Random variable: $X_i\sim pdf(x)$ , pdf is the Probability Distribution Function Monte Carlo estimator: $F_N=\frac{1}{N}\sum_{i=1}^{N}{\frac{f(x_i)}{pdf(x_i)}}$ 我自己证明一下：中心极限定理，当 $ N\rightarrow \infty $ $\frac{1}{N}\sum_{i=1}^{N}{\frac{f(x_i)}{pdf(x_i)}}\rightarrow E(\frac{f(x_i)}{pdf(x_i)})$ $= \int_{a}^{b}{pdf(x_i)\times \frac{f(x_i)}{pdf(x_i)}}dx$ $=\int_{a}^{b}{f(x_i)}dx$ $= F_N|_{a}^{b}$ $F_N|_{a}^ ...

15 Ray Tracing 3 BRDFBasic Radiometry 辐射度量学基础Irradiance Definition: Power pur unit area. $ E(x) \equiv \frac{d\Phi_{\perp}(x)}{dA}=\frac{\Phi}{A}cos\theta $ 单位面积上接收到的能量 Radiance Definition: Power pur unit solid angle pur unit area $L(p,\omega)\equiv \frac{d^2\Phi(p,\omega)}{d\omega dAcos\theta}$ 单位立体角单位面积上接收到的能量 Radiance 对单位立体角积分，就是Irradiance: $E_i(\omega_i)=L_i(\omega_i)cos\theta_i d\omega_i$ Bidirectional Reflectance Distribution Function (BRDF)Reflection at a Point 光在一个点的反射 可以理解成，从外界来的Irrad ...

14 Ray Tracing 2 AccelerateAcceleration: Bounding Volumes Ray-Intersection With a Box Understanding : box is the intersection of 3 pairs of slabs. Axis-Aligned Bounding Box: AABB 轴对齐包围盒 光线进入所有包围面，进入盒子。出去其中一个包围面，出去包围盒。 $t_{enter}=max{t_{min}}, t_{exit}=min{t_{max}}$ If $t_{enter}<t_{exit}$, ray stay a while in the box. $t_{enter}<t_{exit} and t_{exit}>=0$, ray and AABB intersection. Grid Solution: 在bounding box 里面再弄一堆格子 Spatial Partitioning Examples Oct-Tree KD-Tree BSP-Tree Object P ...

13 Ray Tracing 1 Whitted-StyleShadow Mapping before we move onTBD Whitted-Style Ray TracingTBD Why Ray Tracing? Rasterization couldn’t handle global effects well Soft shadow Light bounces more than once Ray Tracing is accurate, but is very slow Rasterization: real-time, ray tracing: offline ~10k CPU core hours to render one frame in production. Definition: Light Rays: Light travels in straight lines 直线传播 Light rays do not collide with each other if they cross 不相互影响 Light rays travel from th ...

Geometry 3 Mesh Operationby @SankHyan24 Mesh Subdivision: Loop Subdivision:(used only on triangle) Add the number of triangle Assign new vertex positions according to weight New/old vertices updated differently For new vertices: Updated to $\frac{3}{8}(A+B)+\frac{1}{8}(C+D)$ For old vertices: Updated to $(1-nu)\times original position + u neighbor position sum$ $n: vertex degree$ $u:\frac{3}{16} if n=3, \frac{3}{8n} otherwise $ Catmull-Clark Subdivision (General Mesh) Definition: Quad fa ...

11 Geometry 2 Curves and Surface 曲线和表面 Explicit Representations Curves Bezier Curves De Casteljau’s Algorithm B-splines, etc. Surfaces Bezier surfaces Triangles & quads Subdivision, simplification, regulation Explicit Representations Point Cloud (Explicit) Easiest representation: list of points(x,y,z) 只要点足够密，表示任何面 经常被转化成多边形面 Polygon Mesh (Explicit) Store vertices & polygons(一般是三角形/四边形) 更复杂的数据结构，更普遍的表达形式 The Wavefront Object File (.obj) Format Curves Bezier Curves 贝塞尔曲线 Defini ...

Lecture 10 Geometry 1 Introduction 几何 About how to describe a stuff by geometry. A string -> fiber->ply->thread very complex geometry: a city a dog .etc Not every geometry can be described as triangles. Many Ways to Represent Geometry Implicit: 隐式 points satisfy some specified relationship, E.g. sphere:all points in 3D, where $x^2+y^2+z^2=1$ more generally, $f(x,y,z)=0$ Implict Surface: Sampling can be hard,Inside/Outside tests is easy. Explicit: 显式 All points are given directly or ...

Texture Mapping 纹理映射Interpolation Across Triangles: Barycentric Coordinates 重心坐标 A coordinate system for triangles ( $\alpha ,\beta ,\gamma$ ) $(x,y)=\alpha A+\beta B+\gamma C, where \alpha ,\beta ,\gamma>0 and \alpha ,\beta ,\gamma+1=0$ How to Calculate them? Use the triangle area:$$ \alpha = \frac{A_A}{A_A+A_B+A_C}, \ \beta = \frac{A_B}{A_A+A_B+A_C}, \ \gamma = \frac{A_C}{A_A+A_B+A_C}$$ 重心坐标不具有投影变换不变性，所以先三维插值，再投影到二维面上去 Texture Magnification: What if the texture is too small?) A pixel o ...

08 Shading 着色Blinn-Phong Reflection Model Specular highlights Term 高光 Intensity depends on view direction Bright near mirror reflection direction V close to mirror direction <–> half vector near normal(h) Measure “near” by dot productof unit vectors Formula: $L_s = k_s(I/r^2)max(0,<n,h>)^p$ What is the p in $max(0,<n,h>)^p$ for? Make it real: If $\alpha $ is too big, $<n,h>^p$ is small. Ambient Term 环境光照 Assumption: Shading that does not depend on anything Add ...

07 Shading 着色解决遮挡(Visibility)与着色的问题———Z-bufferingBlinn-Phong Reflection Model现在只针对于一个点的着色1.Painter’s Algorithm 遮挡和覆盖overwrite 但是对于物体，填充顺序是个问题 nlog(n) 2.Z-Buffer 深度缓存,n triangle idea ：存储min.z-value, for every pixel Frame buffer Depth buffer Algorithm 维护当前看到的最浅深度的信息 initialize depth buffer to infinity zbuffer[x,y] denotes the screen’s pixel buffer framebuffer[x,y] denotes the information(r,g,b) of the pixel for(every triangle) for(every sample(x,y,z)in Triangle) if(z<zbuffer[x,y]) framebuffe ...