Tensor - Theory If not specifically indicated, the linear spaces discussed below are all finite-dimensional. Definition of Tensor ProductUniversal PropertyLet $V, W, Z$ be linear spaces over the field $k$. $\mathcal{ 2021-07-22 Mathematics #Mathematics #Algebra #Linear Algebra
Precomputed Radiance Transfer Diffuse ObjectFor objects with only diffuse reflection on the surface, the reflection equation can be written as \begin{aligned} L_o(\boldsymbol{\omega}_o)=\frac{\mathrm{base\_color}}{\pi} \int_{ 2021-04-30 Computer Graphics #Computer Graphics #PBR #Real-time Rendering
Image-Based Lighting Image Based Real-time Environment LightingImage-based lighting refers to storing the radiance of environment light from each direction with a texture, which is then used for shading calculation. This 2021-04-27 Computer Graphics #Computer Graphics #PBR
Yoneda Lemma Yoneda lemma is a philosophical theorem in category theory. Dan piponi called it “the most difficult ordinary thing in mathematics”. In this article, we try to describe and prove Yoneda lemma in a cle 2021-01-29 Mathematics #Mathematics #Algebra #Category Theory
Volume Rendering Volume rendering should consider not only the interaction between lights and object surfaces, but also the interaction with a large number of particles in space. In this article, we first introduce th 2021-01-17 #Computer Graphics #PBR #Volume Rendering
Differentiable Manifold Basic Concepts of ManifoldsTopological ManifoldDefinition Definition. An $n$-dimensional topological manifold is a Hausdorff topological space which is locally homeomorphic to an $n$-dimen 2020-12-04 Mathematics #Mathematics #Geometry #Differentiable Manifold
Position Based Dynamics AlgorithmThe dynamics of an object can be described by $n$ vertices and $M$ constraints on vertex position. The mass, position and velocity of the $i$-th vertex are denoted as m_i,\mathbf{x}_i,\mathbf 2020-11-22 Computer Graphics #Computer Graphics #Physical Simulation #PBD
Path Tracing Rendering Algorithm Light Transport EquationBasic FormPreviously, we have seen the reflection equation: L_o(\mathbf{p}, \boldsymbol{\omega}_o) = \int_{H^2} f_{\mathrm{r}} \left(\mathbf{p}, \boldsymbol{\omega}_o, \boldsy 2020-10-13 Computer Graphics #Computer Graphics #PBR
PBR - Surface Reflection Bidirectional Reflectance Distribution FunctionGiven a reflection plane, the irradiance from the incident light at the point $\mathbf{p}$ can be written as E_i(\mathbf{p})=\int_{H^2} L_i(\mathbf{p},\ 2020-10-13 Computer Graphics #Computer Graphics #PBR
Importance Sampling Monte Carlo Integration MethodThe Monte Carlo integration method is a numerical method for solving definite integrals. Suppose we need to compute the following integral: I = \int_a^b f(x)dx.First, se 2020-10-13 Computer Graphics #Computer Graphics #Sampling Theory
