From 7fec23c54b039baf770224a04c7ef1e7551c4ca9 Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Thu, 20 Mar 2025 23:08:55 -0500 Subject: [PATCH] update --- docker-compose.yaml | 2 +- pages/CSE559A/CSE559A_L15.md | 2 + pages/CSE559A/CSE559A_L16.md | 114 +++++++++++++++++++++++ pages/CSE559A/_meta.js | 1 + pages/Math4121/Math4121_L24.md | 106 ++++++++++++++++++++- pages/Math4121/_meta.js | 8 +- pages/Math416/Math416_L17.md | 154 +++++++++++++++++++++++++++++++ pages/Math416/_meta.js | 1 + pages/Swap/Math401/Math401_N2.md | 9 ++ 9 files changed, 389 insertions(+), 8 deletions(-) create mode 100644 pages/CSE559A/CSE559A_L16.md create mode 100644 pages/Math416/Math416_L17.md create mode 100644 pages/Swap/Math401/Math401_N2.md diff --git a/docker-compose.yaml b/docker-compose.yaml index b5e7fa9..4d9633c 100644 --- a/docker-compose.yaml +++ b/docker-compose.yaml @@ -3,7 +3,7 @@ services: build: context: ./ dockerfile: ./Dockerfile - image: trance0/notenextra:v1.1.7 + image: trance0/notenextra:v1.1.8 restart: on-failure:5 ports: - 13000:3000 diff --git a/pages/CSE559A/CSE559A_L15.md b/pages/CSE559A/CSE559A_L15.md index 13ab90c..edced87 100644 --- a/pages/CSE559A/CSE559A_L15.md +++ b/pages/CSE559A/CSE559A_L15.md @@ -92,6 +92,8 @@ SSD is a multi-resolution object detection RetinaNet combine feature pyramid network with focal loss to reduce the standard cross-entropy loss for well-classified examples. +![RetinaNet](https://notenextra.trance-0.com/CSE559A/RetinaNet.png) + > Cross-entropy loss: > $$CE(p_t) = - \log(p_t)$$ diff --git a/pages/CSE559A/CSE559A_L16.md b/pages/CSE559A/CSE559A_L16.md new file mode 100644 index 0000000..5b9264e --- /dev/null +++ b/pages/CSE559A/CSE559A_L16.md @@ -0,0 +1,114 @@ +# CSE559A Lecture 16 + +## Dense image labelling + +### Semantic segmentation + +Use one-hot encoding to represent the class of each pixel. + +### General Network design + +Design a network with only convolutional layers, make predictions for all pixels at once. + +Can the network operate at full image resolution? + +Practical solution: first downsample, then upsample + +### Outline + +- Upgrading a Classification Network to Segmentation +- Operations for dense prediction + - Transposed convolutions, unpooling +- Architectures for dense prediction + - DeconvNet, U-Net, "U-Net" +- Instance segmentation + - Mask R-CNN +- Other dense prediction problems + +### Fully Convolutional Networks + +"upgrading" a classification network to a dense prediction network + +1. Covert "fully connected" layers to 1x1 convolutions +2. Make the input image larger +3. Upsample the output + +Start with an existing classification CNN ("an encoder") + +Then use bilinear interpolation and transposed convolutions to make full resolution. + +### Operations for dense prediction + +#### Transposed Convolutions + +Use the filter to "paint" in the output: place copies of the filter on the output, multiply by corresponding value in the input, sum where copies of the filter overlap + +We can increase the resolution of the output by using a larger stride in the convolution. + +- For stride 2, dilate the input by inserting rows and columns of zeros between adjacent entries, convolve with flipped filter +- Sometimes called convolution with fractional input stride 1/2 + +#### Unpooling + +Max unpooling: + +- Copy the maximum value in the input region to all locations in the output +- Use the location of the maximum value to know where to put the value in the output + +Nearest neighbor unpooling: + +- Copy the maximum value in the input region to all locations in the output +- Use the location of the maximum value to know where to put the value in the output + +### Architectures for dense prediction + +#### DeconvNet + +![DeconvNet](https://notenextra.trance-0.com/CSE559A/DeconvNet.png) + +_How the information about location is encoded in the network?_ + +#### U-Net + +![U-Net](https://notenextra.trance-0.com/CSE559A/U-Net.png) + +- Like FCN, fuse upsampled higher-level feature maps with higher-res, lower-level feature maps (like residual connections) +- Unlike FCN, fuse by concatenation, predict at the end + +#### Extended U-Net Architecture + +Many variants of U-Net would replace the "encoder" of the U-Net with other architectures. + +![Extended U-Net Architecture Example](https://notenextra.trance-0.com/CSE559A/ExU-Net.png) + +##### Encoder/Decoder v.s. U-Net + +![Encoder/Decoder v.s. U-Net](https://notenextra.trance-0.com/CSE559A/EncoderDecoder_vs_U-Net.png) + +### Instance Segmentation + +#### Mask R-CNN + +Mask R-CNN = Faster R-CNN + FCN on Region of Interest + +### Extend to keypoint prediction? + +- Use a similar architecture to Mask R-CNN + +_Continue on Tuesday_ + +### Other tasks + +#### Panoptic feature pyramid network + +![Panoptic Feature Pyramid Network](https://notenextra.trance-0.com/CSE559A/Panoptic_Feature_Pyramid_Network.png) + +#### Depth and normal estimation + +![Depth and Normal Estimation](https://notenextra.trance-0.com/CSE559A/Depth_and_Normal_Estimation.png) + +D. Eigen and R. Fergus, Predicting Depth, Surface Normals and Semantic Labels with a Common Multi-Scale Convolutional Architecture, ICCV 2015 + +#### Colorization + +R. Zhang, P. Isola, and A. Efros, Colorful Image Colorization, ECCV 2016 diff --git a/pages/CSE559A/_meta.js b/pages/CSE559A/_meta.js index 8401059..775b93d 100644 --- a/pages/CSE559A/_meta.js +++ b/pages/CSE559A/_meta.js @@ -18,4 +18,5 @@ export default { CSE559A_L13: "Computer Vision (Lecture 13)", CSE559A_L14: "Computer Vision (Lecture 14)", CSE559A_L15: "Computer Vision (Lecture 15)", + CSE559A_L16: "Computer Vision (Lecture 16)", } diff --git a/pages/Math4121/Math4121_L24.md b/pages/Math4121/Math4121_L24.md index f92a2b5..2520a10 100644 --- a/pages/Math4121/Math4121_L24.md +++ b/pages/Math4121/Math4121_L24.md @@ -1 +1,105 @@ -# Lecture 24 \ No newline at end of file +# Math4121 Lecture 24 + +## Chapter 5: Measure Theory + +### Jordan Measurable + +#### Proposition 5.1 + +A bounded set $S\subseteq \mathbb{R}^n$ is Jordan measurable if + +$$ +c_e(S)=c_i(S)+c_e(\partial S) +$$ + +where $\partial S$ is the boundary of $S$ and $c_e(\partial S)=0$. + +Example: + +1. $S=\mathbb{Q}\cap [0,1]$ is not Jordan measurable. + +Since $c_e(S)=0$ and $\partial S=[0,1]$, $c_i(S)=1$. + +So $c_e(\partial S)=1\neq 0$. + +2. SVC(3) is Jordan measurable. + +Since $c_e(S)=0$ and $\partial S=0$, $c_i(S)=0$. The outer content of the cantor set is $0$. + +> Any set or subset of a set with $c_e(S)=0$ is Jordan measurable. + +3. SVC(4) + +At each step, we remove $2^n$ intervals of length $\frac{1}{4^n}$. + +So $S=\bigcap_{n=1}^{\infty} C_i$ and $c_e(C_k)=c_e(C_{k-1})-\frac{2^{k-1}}{4^k}$. $c_e(C_0)=1$. + +So + +$$ +\begin{aligned} +c_e(S)&\leq \lim_{k\to\infty} c_e(C_k)\\ +&=1-\sum_{k=1}^{\infty} \frac{2^{k-1}}{4^k}\\ +&=1-\frac{1}{4}\sum_{k=0}^{\infty} \left(\frac{2}{4}\right)^k\\ +&=1-\frac{1}{4}\cdot \frac{1}{1-\frac{2}{4}}\\ +&=1-\frac{1}{4}\cdot \frac{1}{\frac{1}{2}}\\ +&=1-\frac{1}{2}\\ +&=\frac{1}{2}. +\end{aligned} +$$ + +And we can also claim that $c_i(S)\geq \frac{1}{2}$. Suppose not, then $\exists \{I_j\}_{j=1}^{\infty}$ such that $S\subseteq \bigcup_{j=1}^{\infty} I_j$ and $\sum_{j=1}^{\infty} \ell(I_j)< \frac{1}{2}$. + +Then $S$ would have gaps with lengths summing to greater than $\frac{1}{2}$. This contradicts with what we just proved. + +So $c_e(SVC(4))=\frac{1}{2}$. + +> General formula for $c_e(SVC(n))=\frac{n-3}{n-2}$, and since $SVC(n)$ is nowhere dense, $c_i(SVC(n))=0$. + +### Additivity of Content + +Recall that outer content is sub-additive. Let $S,T\subseteq \mathbb{R}^n$ be disjoint. + +$$ +c_e(S\cup T)\leq c_e(S)+c_e(T) +$$ + +The inner content is super-additive. Let $S,T\subseteq \mathbb{R}^n$ be disjoint. + +$$ +c_i(S\cup T)\geq c_i(S)+c_i(T) +$$ + +#### Proposition 5.2 + +Finite additivity of Jordan content: + +Let $S_1,\ldots,S_N\subseteq \mathbb{R}^n$ are pairwise disjoint Jordan measurable sets, then + +$$ +c(\bigcup_{i=1}^N S_i)=\sum_{i=1}^N c(S_i) +$$ + +Proof: + +$$ +\begin{aligned} +\sum_{i=1}^N c_i(S_i)&\leq c_i(\bigcup_{i=1}^N S_i)\\ +&\leq c_e(\bigcup_{i=1}^N S_i)\\ +&\leq \sum_{i=1}^N c_e(S_i)\\ +\end{aligned} +$$ + +Since $\sum_{i=1}^N c(S_i)=\sum_{i=1}^N c_e(S_i)=\sum_{i=1}^N c_i(S_i)$, we have + +$$ +c(\bigcup_{i=1}^N S_i)=\sum_{i=1}^N c(S_i) +$$ + +QED + +##### Failure for countable additivity for Jordan content + +Notice that each singleton $\{q\}$ is Jordan measurable and $c(\{q\})=0$. But take $a\in \mathbb{Q}\cap [0,1]$, $Q\cap [0,1]=\bigcup_{q\in Q\cap [0,1]} \{q\}$, but $\mathbb{Q}\cap [0,1]$ is not Jordan measurable. + +Issue is a countable union of Jordan measurable sets is not necessarily Jordan measurable. diff --git a/pages/Math4121/_meta.js b/pages/Math4121/_meta.js index c23a2bb..9e904fa 100644 --- a/pages/Math4121/_meta.js +++ b/pages/Math4121/_meta.js @@ -25,12 +25,8 @@ export default { Math4121_L20: "Introduction to Lebesgue Integration (Lecture 20)", Math4121_L21: "Introduction to Lebesgue Integration (Lecture 21)", Math4121_L22: "Introduction to Lebesgue Integration (Lecture 22)", - Math4121_L23: { - display: 'hidden' - }, - Math4121_L24: { - display: 'hidden' - }, + Math4121_L23: "Introduction to Lebesgue Integration (Lecture 23)", + Math4121_L24: "Introduction to Lebesgue Integration (Lecture 24)", Math4121_L25: { display: 'hidden' }, diff --git a/pages/Math416/Math416_L17.md b/pages/Math416/Math416_L17.md new file mode 100644 index 0000000..8165efa --- /dev/null +++ b/pages/Math416/Math416_L17.md @@ -0,0 +1,154 @@ +# Math416 Lecture 17 + +## Continue on Chapter 7 + +### Harmonic conjugates + +#### Theorem 7.18 + +Existence of harmonic conjugates. + +Let $u$ be a harmonic function on $\Omega$ a convex open subset in $\mathbb{C}$. Then there exists $g\in O(\Omega)$ such that $\text{Re}(g)=u$ on $\Omega$. + +Moreover, $g$ is unique up to an imaginary additive constant. + +Proof: + +Let $f=2\frac{\partial u}{\partial z}=\frac{\partial u}{\partial x}-i\frac{\partial u}{\partial y}$ + +$f$ is holomorphic on $\Omega$ + +Since $\frac{\partial u}{\partial \overline{z}}=0$ on $\Omega$, $f$ is holomorphic on $\Omega$ + +So $f=g'$, fix $z_0\in \Omega$, we can choose $q(z_0)=u(z_0)$ and $g=u_1+iv_1$, $g'=\frac{\partial u_1}{\partial x}+i\frac{\partial v_1}{\partial x}=\frac{\partial v_1}{\partial y}-i\frac{\partial u_1}{\partial y}=\frac{\partial u}{\partial x}-i\frac{\partial u}{\partial y}$, given that $\frac{\partial u_1}{\partial x}=\frac{\partial u}{\partial x}$ and $\frac{\partial u_1}{\partial y}=\frac{\partial u}{\partial y}$ + +So $u_1=u$ on $\Omega$ + +$\text{Re}(g)=u_1=u$ on $\Omega$ + +If $u+iv$ is holomorphic, $v$ is harmonic conjugate of $u$ + +QED + +### Corollary For Harmonic functions + +#### Theorem 7.19 + +Harmonic functions are $C^\infty$ + +$C^\infty$ is a local property. + +#### Theorem 7.20 + +Mean value property for harmonic functions. + +Let $u$ be harmonic on an open set of $\Omega$ + +Then $u(z_0)=\frac{1}{2\pi}\int_0^{2\pi}u(z_0+re^{i\theta})d\theta$ + +Proof: + +$\text{Re}g(z_0)=\frac{1}{2\pi}\int_0^{2\pi}\text{Re}g(z_0+re^{i\theta})d\theta$ + +QED + +#### Theorem 7.21 + +Identity theorem for harmonic functions. + +Let $u$ be harmonic on a domain $\Omega$. If $u=0$ on some open set $G\subset \Omega$, then $u\equiv 0$ on $\Omega$. + +_If $u=v$ on $G\subset \Omega$, then $u=v$ on $\Omega$._ + +Proof: + +We proceed by contradiction. + +Let $H=\{z\in \Omega:u(z)=0\}$ be the interior of $G$ + +$H$ is open and nonempty. If $H\neq \Omega$, then $\exists z_0\in \partial H\cap \Omega$. Then $\exists r>0$ such that $B_r(z_0)\subset \Omega$ such that $\exists g\in O(B_r(z_0))$ such that $\text{Re}g=u$ on $B_r(z_0)$ + +Since $H\cap B_r(z_0)$ is nonempty open set, then $g$ is constant on $H\cap B_r(z_0)$ + +So $g$ is constant on $B_r(z_0)$ + +So $u$ is constant on $B_r(z_0)$ + +So $D(z_0,r)\subset H$. This is a contradiction that $z_0\in \partial H$ + +QED + +#### Theorem 7.22 + +Maximum principle for harmonic functions. + +A non-constant harmonic function on a domain cannot attain a maximum or minimum on the interior of the domain. + +Proof: + +We proceed by contradiction. + +Suppose $u$ attains a maximum at $z_0\in \Omega$. + +For all $z$ in the neighborhood of $z_0$, $u(z)0$ such that $B_r(z_0)\subset \Omega$. + +By the mean value property, $u(z_0)=\frac{1}{2\pi}\int_0^{2\pi}u(z_0+re^{i\theta})d\theta$ + +So $0= \frac{1}{2\pi}\int_0^{2\pi}u[z_0+re^{i\theta}-u(z_0)]d\theta$ + +We can prove the minimum is similar. + +QED + +> Maximum/minimum (modulus) principle for holomorphic functions. +> +> If $f$ is holomorphic on a domain $\Omega$ and attains a maximum on the boundary of $\Omega$, then $f$ is constant on $\Omega$. +> +> Except at $z_0\in \Omega$ where $f'(z_0)=0$, if $f$ attains a minimum on the boundary of $\Omega$, then $f$ is constant on $\Omega$. + +### Dirichlet problem for domain $D$ + +Let $h: \partial D\to \mathbb{R}$ be a continuous function. Is there a harmonic function $u$ on $D$ such that $u$ is continuous on $\overline{D}$ and $u|_{\partial D}=h$? + +We can always solve the problem for the unit disk. + +$$ +u(z)=\frac{1}{2\pi}\int_0^{2\pi}h(e^{i t})\text{Re}\left(\frac{e^{it}+z}{e^{it}-z}\right)dt +$$ + +Let $z=re^{i\theta}$ + +$$ +\text{Re}\left(\frac{e^{it}+re^{i\theta}}{e^{it}-re^{i\theta}}\right)=\frac {1-r^2}{1-2r\cos(\theta-t)+r^2} +$$ + +_This is called Poisson kernel._ + +$Pr(\theta, t)>0$ and $\int_0^{2\pi}Pr(\theta, t)dt=1$, $\forall r,t$ + +## Chapter 8 Laurent series + +when $\sum_{n=-\infty}^{\infty}a_n(z-z_0)^n$ converges? + +Claim $\exists R>0$ such that $\sum_{n=-\infty}^{\infty}a_n(z-z_0)^n$ converges if $|z-z_0|R$ + +Proof: + +Let $u=\frac{1}{z-z_0}$ + +$\sum_{n=0}^{\infty}a_n(z-z_0)^n$ has radius of convergence $\frac{1}{R}$ + +So the series converges if $|u|<\frac{1}{R}$ + +So $|z-z_0|=\frac{1}{|u|}>\frac{1}{\frac{1}{R}}=R$ + +QED + +### Laurent series + +A Laurent series is a series of the form $\sum_{n=-\infty}^{\infty}a_n(z-z_0)^n$ + +The series converges in some annulus shape $A=\{z:r_1<|z-z_0|