From bc18850a8fe42d909d26c90090ad91f724f55609 Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Mon, 22 Sep 2025 11:52:39 -0500 Subject: [PATCH] updates --- content/CSE5519/CSE5519_A2.md | 5 + content/CSE5519/CSE5519_E2.md | 3 + content/Math4201/Math4201_L12.md | 180 +++++++++++++++++++++++++++++++ 3 files changed, 188 insertions(+) create mode 100644 content/Math4201/Math4201_L12.md diff --git a/content/CSE5519/CSE5519_A2.md b/content/CSE5519/CSE5519_A2.md index 9ab8dfe..0ecb884 100644 --- a/content/CSE5519/CSE5519_A2.md +++ b/content/CSE5519/CSE5519_A2.md @@ -1,2 +1,7 @@ # CSE5519 Advances in Computer Vision (Topic A: 2022: Semantic Segmentation) +## Masked-Attention Mask Transformer for Universal Image Segmentation + +[link to the paper](https://openaccess.thecvf.com/content/CVPR2022/papers/Cheng_Masked-Attention_Mask_Transformer_for_Universal_Image_Segmentation_CVPR_2022_paper.pdf) + +### Novelty in Masked-Attention Mask Transformer diff --git a/content/CSE5519/CSE5519_E2.md b/content/CSE5519/CSE5519_E2.md index be745e6..b6c97c7 100644 --- a/content/CSE5519/CSE5519_E2.md +++ b/content/CSE5519/CSE5519_E2.md @@ -1,2 +1,5 @@ # CSE5519 Advances in Computer Vision (Topic E: 2022: Deep Learning for Geometric Computer Vision) +## MeshLoc: Mesh-Based Visual Localization + +[link to the paper](https://arxiv.org/pdf/2210.05494) \ No newline at end of file diff --git a/content/Math4201/Math4201_L12.md b/content/Math4201/Math4201_L12.md new file mode 100644 index 0000000..b87c60a --- /dev/null +++ b/content/Math4201/Math4201_L12.md @@ -0,0 +1,180 @@ +# Math4201 Topology I (Lecture 12) + +## Metric spaces + +### Basic properties and definitions + +#### Definition of metric space + +A metric space is a set $X$ with a function $d:X\times X\to \mathbb{R}$ that satisfies the following properties: + +1. $\forall x,y\in X, d(x,y)\geq 0$ and $d(x,y)=0$ if and only if $x=y$. (positivity) +2. $\forall x,y\in X, d(x,y)=d(y,x)$. (symmetry) +3. $\forall x,y,z\in X, d(x,z)\leq d(x,y)+d(y,z)$. (triangle inequality) + +
+Example of metric space + +Let $X=\mathbb{R}$ and $d(x,y)=|x-y|$. + +Check definition of metric space: + +1. Positivity: $d(x,y)=|x-y|\geq 0$ and $d(x,y)=0$ if and only if $x=y$. +2. Symmetry: $d(x,y)=|x-y|=|y-x|=d(y,x)$. +3. Triangle inequality: $d(x,z)=|x-z|\leq |x-y|+|y-z|=d(x,y)+d(y,z)$ since $|a+b|\leq |a|+|b|$ for all $a,b\in \mathbb{R}$. + +--- + +Let $X$ be arbitrary. The trivial metric is $d(x,y)=\begin{cases} +0 & \text{if } x=y \\ +1 & \text{if } x\neq y +\end{cases}$ + +Check definition of metric space: + +1. Positivity: $d(x,y)=\begin{cases} +0 & \text{if } x=y \\ +1 & \text{if } x\neq y +\end{cases}\geq 0$ and $d(x,y)=0$ if and only if $x=y$. +1. Symmetry: $d(x,y)=\begin{cases} +0 & \text{if } x=y \\ +1 & \text{if } x\neq y +\end{cases}=d(y,x)$. +1. Triangle inequality use case by case analysis. + +
+ +#### Balls of a metric space forms a basis for a topology + +Let $(X,d)$ be a metric space. $x\in X$ and $r>0, r\in \mathbb{R}$. We define the ball of radius $r$ centered at $x$ as $B_r(x)=\{y\in X:d(x,y)0,r\in \mathbb{R}\}\text{ is a basis for a topology on }X +$$ + +
+Example of balls of a metric space + +Let $X=\mathbb{R}$ and $d(x,y)=\begin{cases} +0 & \text{if } x=y \\ +1 & \text{if } x\neq y +\end{cases}$ + +The balls of this metric space are: + +$$ +B_r(x)=\begin{cases} +\{x\} & \text{if } r<1 \\ +X & \text{if } r\geq 1 +\end{cases} +$$ + +> [!NOTE] +> +> This basis generate the discrete topology of $X$. + +--- + +Let $X=\mathbb{R}$ and $d(x,y)=|x-y|$. + +The balls of this metric space are: + +$$ +B_r(x)=\{(x-r,x+r)\} +$$ + +This basis is the set of all open sets in $\mathbb{R}$, which generates the standard topology of $\mathbb{R}$. + +
+ +
+Proof + +Let's check the two properties of basis: + +1. $\forall x\in X$, $\exists B_r(x)\in \{B_r(x)|x\in X,r>0,r\in \mathbb{R}\}$ such that $x\in B_r(x)$. (Trivial by definition of non-zero radius ball) +2. $\forall B_r(x),B_r(y)\in \{B_r(x)|x\in X,r>0,r\in \mathbb{R}\}$, $\forall z\in B_r(x)\cap B_r(y)$, $\exists B_r(z)\in \{B_r(x)|x\in X,r>0,r\in \mathbb{R}\}$ such that $z\in B_r(z)\subseteq B_r(x)\cap B_r(y)$. + +Observe that for any $z\in B_r(x)$, then there exists $\delta>0$ such that $B_\delta(z)\subseteq B_r(x)$. + +Let $\delta=r-d(x,z)$, then $B_\delta(z)\subseteq B_r(x)$ (by triangle inequality) + +Similarly, there exists $\delta'>0$ such that $B_\delta'(z)\subseteq B_r(y)$. + +Take $\lambda=min\{\delta,\delta'\}$, then $B_\lambda(z)\subseteq B_r(x)\cap B_r(y)$. + +
+ +#### Definition of Metric topology + +For any metric space $(X,d)$, the topology generated by the balls of the metric space is called metric topology. + +#### Definition of metrizable + +A topological space $(X,\mathcal{T})$ is metrizable if it is the metric topology for some metric $d$ on $X$. + +> Q: When is a topological space metrizable? + +#### Lemma: Every metric topology is Hausdorff + +If a topology isn't Hausdorff, then it isn't metrizable. + +
+Example of non-metrizable space + +Trivial topology **with at least two points** is not Hausdorff, so it isn't metrizable. + +--- + +Finite complement topology on infinite set is not Hausdorff. + +Suppose there exists $x,y\in X$ such that $x\neq y$ and $x\in U\subseteq X$ and $y\in V\subseteq X$ such that $X-U$ and $X-V$ are finite. + +Since $U\cap V=\emptyset$, we have $V\subseteq X-U$, which is finite. So $X-V$ is infinite. (contradiction that $X-V$ is finite) + +So $X$ with finite complement topology is not Hausdorff, so it isn't metrizable. + +
+ +
+Proof + +Let $x,y\in (X,d)$ and $x\neq y$. To show that $X$ is Hausdorff, it is suffices to show that there exists $r,r'>0$ such that $B_r(x)\cap B_r'(y)=\emptyset$. + +Take $r=r'=\frac{1}{2}d(x,y)$, then $B_r(x)\cap B_r'(y)=\emptyset$. (by triangle inequality) + +We prove this by contradiction. + +Suppose $\exists z\in B_r(x)\cap B_r'(y)$, then $d(x,z) + +### Other metrics on $\mathbb{R}^n$ + +Let $\mathbb{R}^n$ be the set of all $n$-tuples of real numbers with standard topology. + +Let $d: \mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}$ be defined by (the Euclidean distance) + +$$ +d(u,v)=\sqrt{\sum_{i=1}^n (u_i-v_i)^2} +$$ + +In $\mathbb{R}^2$ the ball is a circle. + +Let $\rho(u,v)=\max_{i=1}^n |u_i-v_i|$. (Square metric) + +In $\mathbb{R}^2$ the ball is a square. + +Let $m(u,v)=\sum_{i=1}^n |u_i-v_i|$. (Manhattan metric) + +In $\mathbb{R}^2$ the ball is a diamond. + +#### Lemma: Square metric, Manhattan metric, and Euclidean metric are well defined metrics on $\mathbb{R}^n$ + +Proof ignored. Hard part is to show the triangle inequality. May use Cauchy-Schwarz inequality.