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What is image + +A 2d array of numbers. + +### Vision is hard + +connection to graphics. + +computer vision need to generate the model from the image. + +#### Are A and B the same color? + +It depends on the context what you mean by "the same". + +todo + +#### Chair detector example. + +double for loops. + +#### Our visual system is not perfect. + +Some optical illusion images. + +todo, embed images here. + +### Ridiculously brief history of computer vision + +1960s: interpretation of synthetic worlds +1970s: some progress on interpreting selected images +1980s: ANNs come and go; shift toward geometry and increased mathematical rigor +1990s: face recognition; statistical analysis in vogue +2000s: becoming useful; significant use of machine learning; large annotated datasets available; video processing starts. +2010s: Deep learning with ConvNets +2020s: String synthesis; continued improvement across tasks, vision-language models. + +## How computer vision is used now + +### OCR, Optical Character Recognition + +Technology to convert scanned docs to text. + + diff --git a/pages/Math401/_meta.js b/pages/CSE559A/_meta.js similarity index 52% rename from pages/Math401/_meta.js rename to pages/CSE559A/_meta.js index 832c974..6558f18 100644 --- a/pages/Math401/_meta.js +++ b/pages/CSE559A/_meta.js @@ -3,13 +3,7 @@ export default { "---":{ type: 'separator' }, - Math401_L1: { - display: 'hidden' - }, - Math401_L2: { - display: 'hidden' - }, - Math401_L3: { + CSE559A_L1: { display: 'hidden' }, } diff --git a/pages/CSE559A/index.md b/pages/CSE559A/index.md new file mode 100644 index 0000000..94943e9 --- /dev/null +++ b/pages/CSE559A/index.md @@ -0,0 +1,4 @@ +# CSE 559A: Computer Vision + +## Course Description + diff --git a/pages/Math401/index.md b/pages/Math401/index.md deleted file mode 100644 index 62a4a05..0000000 --- a/pages/Math401/index.md +++ /dev/null @@ -1,4 +0,0 @@ -# Math 401: Honors Seminar in Mathematics - -## Course Description - diff --git a/pages/Math4351/Math4351_L1.md b/pages/Math4351/Math4351_L1.md new file mode 100644 index 0000000..9fa9945 --- /dev/null +++ b/pages/Math4351/Math4351_L1.md @@ -0,0 +1,160 @@ +# Lecture 1 + +## Toy example (RSA encryption) + +$Enc$ + +1. Choose a letter, count to number of letters in the alphabet. +2. Calculate $s^7$, then divide by 33, take the remainder. +3. Send the remainder. + +$Dec: s = r^3 \mod 33$ + +To build up such system. + +Step 1: Understanding the arithmetic of remainders. + +## Part 1: Divisibility and prime numbers + +### Divisibility and division algorithm + +Let $a, b\in \mathbb{Z}$, with $b>0$. There are unique integers, $q$ (quotient) and $r$ (remainder), such that $a = bq + r$ and $0 \leq r < b$. + +Example: $31=7\times 4 + 3$ + +Proof: + +The quotient and remainder are unique. + +(1) Existence: + +Let $S = \{a - bk \mid k \in \mathbb{Z}, a - bk \geq 0\}$. Choose $r$ to be the smallest non-negative element of $S$. This means $r = a - bq$ for some $q \in \mathbb{Z}$. i.e. $a=bq+r$. + +> New notion: $\triangle FSOC$ means For the sake of contradiction. + +Notice that $r \geq 0$, by contradiction, suppose $r \geq b$, then $r-b \geq 0$ and $r-b \in S$, but $r-b < r$, which contradicts the minimality of $r$. + +Therefore, $r \geq 0$ and $r < b$. + +Example: $a=31, b=7$, $S = \{31-7k \mid k \in \mathbb{Z}, 31-7k \geq 0\}=\{\cdots, -32, -25, -18, -11, -4, 3, 10, 17, 24, 31, \cdots\}$, $r=3$. + +So We choose $q=4$ and $r=3$. + +(2) Uniqueness: + +Suppose we have two pairs $(q, r)$ and $(q', r')$ such that $a = bq + r = bq' + r'$ Suppose $q \neq q'$, without loss of generality, suppose $q > q'$, $q-q' \geq 1$. Then $b(q-q') = r'-r$. + +Since $r'=b(q-q')+r \geq b(q-q') \geq b$, which contradicts that $r' < b$. + +Therefore, $q=q'$ and $r=r'$. + +EOP + +#### Definition: Divisibility + +Let $a, b \in \mathbb{Z}$, we say $b$ divides $a$ and write $b \mid a$ if there exists $k\in \mathbb{Z}$ such that $a = bk$. + +Example: $3 \mid 12$ because $12 = 3 \times 4$. + +#### Properties of divisibility + +Let $a, b, c \in \mathbb{Z}$. + +(1) $b \mid a \iff r=0$ in the division algorithm. + +(2) If $a \mid b$ and $b \mid c$, then $a \mid c$. + +(3) If $a \mid b$ and $b \mid a$, then $a = \pm b$. + +(4) If $a \mid b$ and $a \mid c$, then $a \mid bx + cy$ for all $x, y \in \mathbb{Z}$. (We call such $bx+cy$ a linear combination of $b$ and $c$.) + +(5) If $c\neq 0$ and $a \mid b \iff ac \mid bc$. + +Some proof examples: + +(2) Since $a \mid b$ and $b \mid c$, there exist $k, l \in \mathbb{Z}$ such that $b = ak$ and $c = bl$. Then $c = bl = (ak)l = a(kl)$, so $a \mid c$. + +EOP + +(3) If $a \mid b$ and $b \mid a$, then there exist $k, l \in \mathbb{Z}$ such that $b = ak$ and $a = bl$. Then $a = bl = (ak)l = a(kl)$, so $a(1-kl) = 0$. + +Case 1: $a=0$, then $b=0$, so $a=b$. + +Case 2: $a \neq 0$, then $1-kl=0$, so $kl=1$. Since $k, l \in \mathbb{Z}$, $k=l=\pm 1$, so $a = \pm b$. + +EOP + +#### Definition: Divisor + +Let $a\in \mathbb{Z}$, we define $D(a) = \{d\in \mathbb{Z} \mid d \mid a\}$. + +**Note that $D(0) = \mathbb{Z}$.** + +Example: $D(12) = \{\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\}$. + +#### Definition: Greatest common divisor + +Let $a, b \in \mathbb{Z}$, where $a,b$ not both zero, we define the greatest common divisor of $a$ and $b$ to be the largest element in $D(a) \cap D(b)$. It is denoted by $(a,b)$. + +> Terrible, I really hate this notation. But professor said it's unlikely to be confused with the interval $(a,b)$ since they don't show up in the same context usually. + +Example: + +$(12, 18) = 6$. + +**Note that $(0,0)$ is not defined. (there is no largest element in $D(0) \cap D(0)$.)** + +but it is okay that one of $a, b$ is zero. For example, $(0, 18) = 18$. + +$(n,n) = |n|$ for all $n \in \mathbb{Z}$. + +In general, if $(a,b)=0$ we say $a$ and $b$ are relatively prime, or coprime. + +$\forall a, b \in \mathbb{Z}$, $(a,b) \geq 1$. + +#### Theorem for calculating gcd + +Let $a, b \in \mathbb{Z}$, with $b\neq 0$, then for any $k\in \mathbb{Z}$, $(a,b) = (b,a-bk)$. + +Example: $(12, 18) = (18, 12-18) = (18, -6) = 6$. + +$(938,210)=(210,938-210\times 4)=(210,938-840)=(210,98)$. + +Proof: + +We will prove that $D(a) \cap D(b) = D(b) \cap D(a-bk)$. + +(1) $D(a) \cap D(b) \subseteq D(b) \cap D(a-bk)$: + +Let $d \in D(a) \cap D(b)$, then $d \mid a$ and $d \mid b$. + +By property of divisibility (4), If $a\mid b$ and $b\mid c$, then for all $x,y\in \mathbb{Z}$, $a\mid bx+cy$. + +So $d\mid a+b\cdot (-k) = a-bk$. + +Therefore, $d \in D(b) \cap D(a-bk)$. + +(2) $D(b) \cap D(a-bk) \subseteq D(a) \cap D(b)$: + +Let $d \in D(b) \cap D(a-bk)$, then $d \mid b$ and $d \mid a-bk$. + +By property of divisibility (4), $d \mid bk + (a-bk) = a$. + +Therefore, $d \in D(a) \cap D(b)$. + +EOP + +This theorem gives rise to the Euclidean algorithm which is a efficient way to compute the greatest common divisor of two integers. $2\Theta(\log n)+1=O(\log n)$ ([Proof in CSE 442T Lecture 7](https://notenextra.trance-0.com/CSE442T/CSE442T_L7#euclidean-algorithm)). + +### Euclidean algorithm + +We will skip this part, it's already the third time we see this algorithm in wustl. + +#### Theorem: Euclidean algorithm returns correct gcd + +Let $a>b>0$, be integers. Using the Euclidean algorithm, we can find $b>r_0>r_1>r_2>\cdots>r_n$ such that $a=bq_0+r_0, b=r_0q_1+r_1, \cdots, r_{n-1}=r_nq_{n+1}+r_{n+1}, r_n=0$. Then $(a,b)=r_n$. + +Proof: + +(a) This process terminates. $b>r_0>r_1>r_2>\cdots>r_n$ is a strictly decreasing sequence of positive integers, so it must terminate. + diff --git a/pages/Math4351/_meta.js b/pages/Math4351/_meta.js new file mode 100644 index 0000000..0a1e78a --- /dev/null +++ b/pages/Math4351/_meta.js @@ -0,0 +1,7 @@ +export default { + index: "Course Description", + "---":{ + type: 'separator' + }, + Math4351_L1: "Introduction to Number Theory and Cryptography (Lecture 1)", +} diff --git a/pages/Math4351/index.md b/pages/Math4351/index.md new file mode 100644 index 0000000..4e9da1d --- /dev/null +++ b/pages/Math4351/index.md @@ -0,0 +1,7 @@ +# Math 4351 + +Number theory and cryptography. + +No textbook required. + + diff --git a/pages/_meta.js b/pages/_meta.js index 06af0a6..2ab3b1c 100644 --- a/pages/_meta.js +++ b/pages/_meta.js @@ -29,13 +29,19 @@ export default { Math4121: { type: 'page', }, + Math4351: { + type: 'page', + }, CSE332S: { type: 'page', }, + CSE347: { + type: 'page', + }, CSE442T: { type: 'page', }, - CSE347: { + CSE559A: { type: 'page', }, about: {