From 19886c5210a40dd4afa700e5a0cc1eff2ba25658 Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Thu, 30 Jan 2025 13:38:23 -0600 Subject: [PATCH] update --- pages/CSE559A/CSE559A_L6.md | 128 +++++++++++ pages/Math416/Math416_L6.md | 219 +++++++++++++++++++ pages/Math416/_meta.js | 1 + public/CSE559A/DigitalCameraBlockDiagram.png | Bin 0 -> 60375 bytes 4 files changed, 348 insertions(+) create mode 100644 pages/Math416/Math416_L6.md create mode 100644 public/CSE559A/DigitalCameraBlockDiagram.png diff --git a/pages/CSE559A/CSE559A_L6.md b/pages/CSE559A/CSE559A_L6.md index e69de29..cb092dd 100644 --- a/pages/CSE559A/CSE559A_L6.md +++ b/pages/CSE559A/CSE559A_L6.md @@ -0,0 +1,128 @@ +# Lecture 6 + +## Continue on Light, eye/camera, and color + +### BRDF (Bidirectional Reflectance Distribution Function) + +$$ +\rho(\theta_i,\phi_i,\theta_o,\phi_o) +$$ + +#### Diffuse Reflection + +- Dull, matte surface like chalk or latex paint + +- Most often used in computer vision +- Brightness _does_ depend on direction of illumination + +Diffuse reflection governed by Lambert's law: $I_d = k_d N\cdot L I_i$ + +- $N$: surface normal +- $L$: light direction +- $I_i$: incident light intensity +- $k_d$: albedo + +$$ +\rho(\theta_i,\phi_i,\theta_o,\phi_o)=k_d \cos\theta_i +$$ + +#### Photometric Stereo + +Suppose there are three light sources, $L_1, L_2, L_3$, and we have the following measurements: + +$$ +I_1 = k_d N\cdot L_1 +$$ + +$$ +I_2 = k_d N\cdot L_2 +$$ + +$$ +I_3 = k_d N\cdot L_3 +$$ + +We can solve for $N$ by taking the dot product of $N$ and each light direction and then solving the system of equations. + +Will not do this in the lecture. + +#### Specular Reflection + +- Mirror-like surface + +$$ +I_e=\begin{cases} +I_i & \text{if } V=R \\ +0 & \text{if } V\neq R +\end{cases} +$$ + +- $V$: view direction +- $R$: reflection direction +- $\theta_i$: angle between the incident light and the surface normal + +Near-perfect mirror have a high light around $R$. + +common model: + +$$ +I_e=k_s (V\cdot R)^{n_s}I_i +$$ + +- $k_s$: specular reflection coefficient +- $n_s$: shininess (imperfection of the surface) +- $I_i$: incident light intensity + +#### Phong illumination model + +- Phong approximation of surface reflectance + - Assume reflectance is modeled by three compoents + - Diffuse reflection + - Specular reflection + - Ambient reflection + +$$ +I_e=k_a I_a + I_i \left[k_d (N\cdot L) + k_s (V\cdot R)^{n_s}\right] +$$ + +- $k_a$: ambient reflection coefficient +- $I_a$: ambient light intensity +- $k_d$: diffuse reflection coefficient +- $k_s$: specular reflection coefficient +- $n_s$: shininess +- $I_i$: incident light intensity + +Many other models. + +#### Measuring BRDF + +Use Gonioreflectometer. + +- Device for measuring the reflectance of a surface as a function of the incident and reflected angles. +- Can be used to measure the BRDF of a surface. + +BRDF dataset: + +- MERL dataset +- CURET dataset + +### Camera/Eye + +#### DSLR Camera + +- Pinhole camera model +- Lens +- Aperture (the pinhole) +- Sensor +- ... + +#### Digital Camera block diagram + +![Digital Camera block diagram](https://static.notenextra.trance-0.com/images/CSE559A/DigitalCameraBlockDiagram.png) + +Scanning protocols: + +- Global shutter: all pixels are exposed at the same time +- Interlaced: odd and even lines are exposed at different times +- Rolling shutter: each line is exposed as it is read out + diff --git a/pages/Math416/Math416_L6.md b/pages/Math416/Math416_L6.md new file mode 100644 index 0000000..3c3ed28 --- /dev/null +++ b/pages/Math416/Math416_L6.md @@ -0,0 +1,219 @@ +# Lecture 6 + +## Review + +### Linear Fractional Transformations + +Transformations of the form $f(z)=\frac{az+b}{cz+d}$,$a,b,c,d\in\mathbb{C}$ and $ad-bc\neq 0$ are called linear fractional transformations. + +#### Theorem 3.8 Preservation of clircles + +We defined clircle to be a circle or a line. + +The circle equation is: + +Let $\zeta=u+iv$ be the center of the circle, $r$ be the radius of the circle. + +$$ +circle=\{z\in\mathbb{C}:|\zeta-c|=r\} +$$ + +This is: + +$$ +|\zeta|^2-c\overline{\zeta}-\overline{c}\zeta+|c|^2-r^2=0 +$$ + +If $\phi$ is a non-constant linear fractional transformation, then $\phi$ maps clircles to clircles. + +We claim that a map is circle preserving if and only if for some $\alpha,\beta,\gamma,\delta\in\mathbb{R}$. + +$$ +\alpha|\zeta|^2+\beta Re(\zeta)+\gamma Im(\zeta)+\delta=0 +$$ + +when $\alpha=0$, it is a line. + +when $\alpha\neq 0$, it is a circle. + +Proof: + +Let $w=u+iv=\frac{1}{\zeta}$, so $\frac{1}{w}=\frac{u}{u^2+v^2}-i\frac{v}{u^2+v^2}$. + +Then the original equation becomes: + +$$ +\alpha\left(\frac{u}{u^2+v^2}\right)^2+\beta\left(\frac{u}{u^2+v^2}\right)+\gamma\left(-\frac{v}{u^2+v^2}\right)+\delta=0 +$$ + +Which is in the form of circle equation. + +EOP + +## Chapter 4 Elements of functions + +> $e^t=\sum_{n=0}^{\infty}\frac{t^n}{n!}$ + +So, following the definition of $e^\zeta$, we have: + +$$ +\begin{aligned} +e^{x+iy}&=e^xe^{iy} \\ +&=e^x\left(\sum_{n=0}^{\infty}\frac{(iy)^n}{n!}\right) \\ +&=e^x\left(\sum_{n=0}^{\infty}\frac{(-1)^ny^n}{n!}\right) \\ +&=e^x(\cos y+i\sin y) +\end{aligned} +$$ + +### $e^\zeta$ + +The exponential of $e^\zeta=x+iy$ is defined as: + +$$ +e^\zeta=exp(\zeta)=e^x(\cos y+i\sin y) +$$ + +So, + +$$ +|e^\zeta|=|e^x||\cos y+i\sin y|=e^x +$$ + +#### Theorem 4.3 $e^\zeta$ is holomorphic + +$e^\zeta$ is holomorphic on $\mathbb{C}$. + +Proof: + +$$ +\begin{aligned} +\frac{\partial}{\partial\zeta}e^\zeta&=\frac{1}{2}\left(\frac{\partial}{\partial x}+\frac{i}{\partial y}\right)e^x(\cos y+i\sin y) \\ +&=\frac{1}{2}e^x(\cos y+i\sin y)+ie^x(-\sin y+i\cos y) \\ +&=0 +\end{aligned} +$$ + +EOP + +#### Theorem 4.4 $e^\zeta$ is periodic + +$e^\zeta$ is periodic with period $2\pi i$. + +Proof: + +$$ +e^{\zeta+2\pi i}=e^\zeta e^{2\pi i}=e^\zeta\cdot 1=e^\zeta +$$ + +EOP + +#### Theorem 4.5 $e^\zeta$ as a map + +$e^\zeta$ is a map from $\mathbb{C}$ to $\mathbb{C}$ with period $2\pi i$. + +$$ +e^{\pi i}+1=0 +$$ + +This is a map from cartesian coordinates to polar coordinates, where $e^x$ is the radius and $y$ is the angle. + +This map attains every value in $\mathbb{C}\setminus\{0\}$. + +#### Definition 4.6-8 $\cos\zeta$ and $\sin\zeta$ + +$$ +\cos\zeta=\frac{1}{2}(e^{i\zeta}+e^{-i\zeta}) +$$ + +$$ +\sin\zeta=\frac{1}{2i}(e^{i\zeta}-e^{-i\zeta}) +$$ + +$$ +\cosh\zeta=\frac{1}{2}(e^\zeta+e^{-\zeta}) +$$ + +$$ +\sinh\zeta=\frac{1}{2}(e^\zeta-e^{-\zeta}) +$$ + +From this definition, we can see that $\cos\zeta$ and $\sin\zeta$ are no longer bounded. + +And this definition is still compatible with the previous definition of $\cos$ and $\sin$ when $\zeta$ is real. + +Moreover, + +$$ +\cosh(i\zeta)=\cos\zeta +$$ + +$$ +\sinh(i\zeta)=i\sin\zeta +$$ + +### Logarithm + +#### Definition 4.9 Logarithm + +A logarithm of $a$ is any $b$ such that $e^b=a$. + +If $a=0$, then no logarithm exists. + +If $a\neq 0$, then there exists infinitely many logarithms of $a$. + +Let $a=re^{i\theta}$, $b=x+iy$ be a logarithm of $a$. + +Then, + +$$ +e^{x+iy}=re^{i\theta} +$$ + +Since logarithm is not unique, we can always add $2k\pi i$ to the angle. + +If $y\in(-\pi,\pi]$, then $\log a=b$ means $e^b=a$ and $Im(b)\in(-\pi,\pi]$. + +If $a=re^{i\theta}$, then $\log a=\log r+i(\theta_0+2k\pi)$. + +#### Definition 4.10 + +Let $G$ be an open connected subset of $\mathbb{C}\setminus\{0\}$. + +A branch of $\arg(\zeta)$ in $G$ is a continuous function $\alpha$, such that $\alpha(\zeta)$ is a value of $\arg(\zeta)$. + +A branch of $\log(\zeta)$ in $G$ is a continuous function $\beta$, such that $e^{\beta(\zeta)}=\zeta$. + +Note: $G$ has a branch of $\arg(\zeta)$ if and only if it has a branch of $\log(\zeta)$. + +If $G=\mathbb{C}\setminus\{0\}$, then not branch of $\arg(\zeta)$ exists. + +Suppose $\alpha_1$ and $\alpha_2$ are two branches of $\arg(\zeta)$ in $G$. + +Then, + +$$ +\alpha_1(\zeta)-\alpha_2(\zeta)=2k\pi i +$$ + +for some $k\in\mathbb{Z}$. + +#### Theorem 4.11 + +$\log(\zeta)$ is holomorphic on $\mathbb{C}\setminus\{0\}$. + +Proof: + +Method 1: Use polar coordinates. (See in homework) + +Method 2: Use the fact that $\log(\zeta)$ is the inverse of $e^\zeta$. + +Suppose $h=s+it$, $e^h=e^s(\cos t+i\sin t)$, $e^h-1=e^s(\cos t-1)+i\sin t$. So + +$$ +\begin{aligned} +\frac{e^h-1}{h}&=\frac{(s+it)e^s(\cos t-1)+i\sin t}{s^2+t^2} \\ +&=\frac{e^s(\cos t-1)}{s^2+t^2}+i\frac{\sin t}{s^2+t^2} +\end{aligned} +$$ + +Continue next time. diff --git a/pages/Math416/_meta.js b/pages/Math416/_meta.js index 1fd32f5..e487ab6 100644 --- a/pages/Math416/_meta.js +++ b/pages/Math416/_meta.js @@ -8,4 +8,5 @@ export default { Math416_L3: "Complex Variables (Lecture 3)", Math416_L4: "Complex Variables (Lecture 4)", Math416_L5: "Complex Variables (Lecture 5)", + Math416_L6: "Complex Variables (Lecture 6)", } diff --git a/public/CSE559A/DigitalCameraBlockDiagram.png b/public/CSE559A/DigitalCameraBlockDiagram.png new file mode 100644 index 0000000000000000000000000000000000000000..8a1b4a7c96b652cedac6a2193a6705308360bfd6 GIT binary patch literal 60375 zcmb5Vby!sIxBoq~Qqo8%-7pe@G}4WXG%_eKAX3sWv;qQ3cQ;7G&@q65ba#W&-CfVd z@A;hbIoI#}^ZdcZ#GbwHz4p4-eXq6Nub@|I@_5*k*dP!HPfGr*zbjMB 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