diff --git a/pages/CSE559A/CSE559A_L5.md b/pages/CSE559A/CSE559A_L5.md new file mode 100644 index 0000000..f925461 --- /dev/null +++ b/pages/CSE559A/CSE559A_L5.md @@ -0,0 +1,222 @@ +# Lecture 5 + +## Continue on linear interpolation + +- In linear interpolation, extreme values are at the boundary. +- In bicubic interpolation, extreme values may be inside. + +`scipy.interpolate.RegularGridInterpolator` + +### Image transformations + +Image warping is a process of applying transformation $T$ to an image. + +Parametric (global) warping: $T(x,y)=(x',y')$ + +Geometric transformation: $T(x,y)=(x',y')$ This applies to each pixel in the same way. (global) + +#### Translation + +$T(x,y)=(x+a,y+b)$ + +matrix form: + +$$ +\begin{pmatrix} +x'\\y' +\end{pmatrix} += +\begin{pmatrix} +1&0\\0&1 +\end{pmatrix} +\begin{pmatrix} +x\\y +\end{pmatrix} ++ +\begin{pmatrix} +a\\b +\end{pmatrix} +$$ + +#### Scaling + +$T(x,y)=(s_xx,s_yy)$ matrix form: + +$$ +\begin{pmatrix} +x'\\y' +\end{pmatrix} += +\begin{pmatrix} +s_x&0\\0&s_y +\end{pmatrix} +\begin{pmatrix} +x\\y +\end{pmatrix} +$$ + +#### Rotation + +$T(x,y)=(x\cos\theta-y\sin\theta,x\sin\theta+y\cos\theta)$ + +matrix form: + +$$ +\begin{pmatrix} +x'\\y' +\end{pmatrix} += +\begin{pmatrix} +\cos\theta&-\sin\theta\\\sin\theta&\cos\theta +\end{pmatrix} +\begin{pmatrix} +x\\y +\end{pmatrix} +$$ + +To undo the rotation, we need to rotate the image by $-\theta$. This is equivalent to apply $R^T$ to the image. + +#### Affine transformation + +$T(x,y)=(a_1x+a_2y+a_3,b_1x+b_2y+b_3)$ + +matrix form: + +$$ +\begin{pmatrix} +x'\\y' +\end{pmatrix} += +\begin{pmatrix} +a_1&a_2&a_3\\b_1&b_2&b_3 +\end{pmatrix} +\begin{pmatrix} +x\\y\\1 +\end{pmatrix} +$$ + +Taking all the transformations together. + +#### Projective homography + +$T(x,y)=(\frac{ax+by+c}{gx+hy+i},\frac{dx+ey+f}{gx+hy+i})$ + +$$ +\begin{pmatrix} +x'\\y'\\1 +\end{pmatrix} += +\begin{pmatrix} +a&b&c\\d&e&f\\g&h&i +\end{pmatrix} +\begin{pmatrix} +x\\y\\1 +\end{pmatrix} +$$ + +### Image warping + +#### Forward warping + +Send each pixel to its new position and do the matching. + +- May cause gaps where the pixel is not mapped to any pixel. + +#### Inverse warping + +Send each new position to its original position and do the matching. + +- Some mapping may not be invertible. + +#### Which one is better? + +- Inverse warping is better because it usually more efficient, doesn't have a problem with holes. +- However, it may not always be possible to find the inverse mapping. + +## Sampling and Aliasing + +### Naive sampling + +- Remove half of the rows and columns in the image. + +Example: + +When sampling a sine wave, the result may interpret as different wave. + +#### Nyquist-Shannon sampling theorem + +- A bandlimited signal can be uniquely determined by its samples if the sampling rate is greater than twice the maximum frequency of the signal. + +- If the sampling rate is less than twice the maximum frequency of the signal, the signal will be aliased. + +#### Anti-aliasing + +- Sample more frequently. (not always possible) +- Get rid of all frequencies that are greater than half of the new sampling frequency. + - Use a low-pass filter to get rid of all frequencies that are greater than half of the new sampling frequency. (eg, Gaussian filter) + +```python +import scipy.ndimage as ndimage +def down_sample(height, width, image): + # Apply Gaussian blur to the image + im_blur = ndimage.gaussian_filter(image, sigma=1) + # Down sample the image by taking every second pixel + return im_blur[::2, ::2] +``` + +## Nonlinear filtering + +### Median filter + +Replace the value of a pixel with the median value of its neighbors. + +- Good for removing salt and pepper noise. (black and white dot noise) + +### Morphological operations + +Binary image: image with only 0 and 1. + +Let $B$ be a structuring element and $A$ be the original image (binary image). + +- Erosion: $A\ominus B = \{p\mid B_p\subseteq A\}$, this is the set of all points that are completely covered by $B$. +- Dilation: $A\oplus B = \{p\mid B_p\cap A\neq\emptyset\}$, this is the set of all points that are at least partially covered by $B$. +- Opening: $A\circ B = (A\ominus B)\oplus B$, this is the set of all points that are at least partially covered by $B$ after erosion. +- Closing: $A\bullet B = (A\oplus B)\ominus B$, this is the set of all points that are completely covered by $B$ after dilation. + +Boundary extraction: use XOR operation on eroded image and original image. + +Connected component labeling: label the connected components in the image. _use prebuild function in scipy.ndimage_ + +## Light,Camera/Eyes, and Color + +### Principles of grouping and Gestalt Laws + +- Proximity: objects that are close to each other are more likely to be grouped together. +- Similarity: objects that are similar are more likely to be grouped together. +- Closure: objects that form a closed path are more likely to be grouped together. +- Continuity: objects that form a continuous path are more likely to be grouped together. + +### Light and surface interactions + +A photon's life choices: + +- Absorption +- Diffuse reflection (nice to model) (lambertian surface) +- Specular reflection (mirror-like) (perfect mirror) +- Transparency +- Refraction +- Fluorescence (returns different color) +- Subsurface scattering (candles) +- Photosphorescence +- Interreflection + +#### BRDF (Bidirectional Reflectance Distribution Function) + +$$ +\rho(\theta_i,\phi_i,\theta_o,\phi_o) +$$ + +- $\theta_i$ is the angle of incidence. +- $\phi_i$ is the azimuthal angle of incidence. +- $\theta_o$ is the angle of reflection. +- $\phi_o$ is the azimuthal angle of reflection. diff --git a/pages/CSE559A/CSE559A_L6.md b/pages/CSE559A/CSE559A_L6.md new file mode 100644 index 0000000..cb092dd --- /dev/null +++ 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/CSE559A/_meta.js b/pages/CSE559A/_meta.js index e6563d1..649aa52 100644 --- a/pages/CSE559A/_meta.js +++ b/pages/CSE559A/_meta.js @@ -7,4 +7,6 @@ export default { CSE559A_L2: "Computer Vision (Lecture 2)", CSE559A_L3: "Computer Vision (Lecture 3)", CSE559A_L4: "Computer Vision (Lecture 4)", + CSE559A_L5: "Computer Vision (Lecture 5)", + CSE559A_L6: "Computer Vision (Lecture 6)", } diff --git a/pages/Math4111/Math4111_L24.md b/pages/Math4111/Math4111_L24.md index 3f66ca7..a50a313 100644 --- a/pages/Math4111/Math4111_L24.md +++ b/pages/Math4111/Math4111_L24.md @@ -133,6 +133,28 @@ By Theorem 2.28, $\sup f(X)$ and $\inf f(X)$ exist and are in $f(X)$. Let $p_0\i EOP +--- + +Supplemental materials: + +_I found this section is not covered in the lecture but is used in later chapters._ + +#### Definition 4.18 + +Let $f$ be a mapping of a metric space $X$ into a metric space $Y$. $f$ is **uniformly continuous** on $X$ if $\forall \epsilon > 0$, $\exists \delta > 0$ such that $\forall x, y\in X$, $|x-y| < \delta \implies |f(x)-f(y)| < \epsilon$. + +#### Theorem 4.19 + +If $f$ is a continuous mapping of a compact metric space $X$ into a metric space $Y$, then $f$ is uniformly continuous on $X$. + +Proof: + +See the textbook. + +EOP + +--- + ### Continuity and connectedness > **Definition 2.45**: Let $X$ be a metric space. $A,B\subset X$ are **separated** if $\overline{A}\cap B = \phi$ and $\overline{B}\cap A = \phi$. diff --git a/pages/Math4121/Math4121_L7.md b/pages/Math4121/Math4121_L7.md index 9b385d4..f8f848b 100644 --- a/pages/Math4121/Math4121_L7.md +++ b/pages/Math4121/Math4121_L7.md @@ -1 +1,96 @@ -# Lecture 7 \ No newline at end of file +# Lecture 7 + +## Continue on Chapter 6 + +### Riemann integrable + +#### Theorem 6.6 + +A function $f$ is Riemann integrable with respect to $\alpha$ on $[a, b]$ if and only if for every $\epsilon > 0$, there exists a partition $P$ of $[a, b]$ such that $U(f, P, \alpha) - L(f, P, \alpha) < \epsilon$. + +Proof: + +$\impliedby$ + +For every $P$, + +$$ +L(f, P, \alpha) \leq \underline{\int}_a^b f d\alpha \leq \overline{\int}_a^b f d\alpha \leq U(f, P, \alpha) +$$ + +So if $f$ is Riemann integrable with respect to $\alpha$ on $[a, b]$, then for every $\epsilon > 0$, there exists a partition $P$ such that + +$$ +0 \leq \overline{\int}_a^b f d\alpha - \underline{\int}_a^b f d\alpha \leq U(f, P, \alpha) - L(f, P, \alpha) < \epsilon +$$ + +Thus $0 \leq \overline{\int}_a^b f d\alpha - \underline{\int}_a^b f d\alpha < \epsilon,\forall \epsilon > 0$. + +Then, $\overline{\int}_a^b f d\alpha = \underline{\int}_a^b f d\alpha$. + +So, $f$ is Riemann integrable with respect to $\alpha$ on $[a, b]$. + +$\implies$ + +If $f\in \mathscr{R}(\alpha)$ on $[a, b]$, then $f$ is Riemann integrable with respect to $\alpha$ on $[a, b]$. + +Then by the definition of Riemann integrable, $\sup_{P} L(f, P, \alpha) =\int^b_a f d\alpha = \inf_{P} U(f, P, \alpha)$. + +Given any $\epsilon > 0$, by definition of infimum and supremum, there exists a partition $P_1,P_2$ such that + +$$ +\int^b_a f d\alpha - \frac{\epsilon}{2} < L(f, P_1, \alpha) \leq \sup_{P} L(f, P, \alpha) = \inf_{P} U(f, P, \alpha) < \int^b_a f d\alpha + \frac{\epsilon}{2} +$$ + +Taking $P = P_1 \cup P_2$, by [Theorem 6.4](https://notenextra.trance-0.com/Math4121/Math4121_L6#theorem-64) we have + +$$ +U(f, P, \alpha) - L(f, P, \alpha) \leq \left( \int^b_a f d\alpha + \frac{\epsilon}{2} \right) - \left( \int^b_a f d\alpha - \frac{\epsilon}{2} \right) = \epsilon +$$ + +So $f$ is Riemann integrable with respect to $\alpha$ on $[a, b]$. + +EOP + +#### Theorem 6.8 + +If $f$ is continuous on $[a, b]$, then $f$ is Riemann integrable with respect to $\alpha$ on $[a, b]$. + +Proof: + +> Main idea: +> +> $$U(f, P, \alpha) - L(f, P, \alpha) = \sum_{i=1}^n \left( M_i - m_i \right) \Delta \alpha_i$$ +> +> If we can make $M_i - m_i$ small enough, then $U(f, P, \alpha) - L(f, P, \alpha)$ can be made arbitrarily small. +> +> Since $M_i=\sup_{x\in [t_{i-1}, t_i]} f(x)$ and $m_i=\inf_{x\in [t_{i-1}, t_i]} f(x)$, we can make $M_i - m_i$ small enough by making the partition $P$ sufficiently fine. + +Suppose we can find a partition $P$ such that $M_i - m_i < \eta$. Then $U(f, P, \alpha) - L(f, P, \alpha) \leq\eta\sum_{i=1}^n \Delta \alpha_i = \eta (\alpha(b)-\alpha(a))$. + +> Let $\epsilon >0$ and choose $\eta = \frac{\epsilon}{\alpha(b)-\alpha(a)}$. Then there exists a partition $P$ such that $U(f, P, \alpha) - L(f, P, \alpha) < \epsilon$. + +Since $f$ is continuous on $[a, b]$ (a compact set), then $f$ is uniformly continuous on $[a, b]$. [Theorem 4.19](https://notenextra.trance-0.com/Math4111/Math4111_L24#theorem-419) + +> If $f$ is continuous on $x$, then $\forall \epsilon > 0$, $\exists \delta > 0$ such that $|x-y| < \delta \implies |f(x)-f(y)| < \epsilon$. +> +> If $f$ is continuous on $[a, b]$, then $f$ is continuous at $x,\forall x\in [a, b]$. + +So, there exists a $\delta > 0$ such that for all $x, t\in [a, b]$ with $|x-t| < \delta$, we have $|f(x)-f(t)| < \eta$. + +Let $P=\{x_0, x_1, \cdots, x_n\}$ be a partition of $[a, b]$ such that $\Delta x_i < \delta$ for all $i$. + +So, $\sup_{x,t\in [x_{i-1}, x_i]} |f(x)-f(t)| < \eta$ for all $i$. + +$$ +\begin{aligned} +\sup_{x,t\in [x_{i-1}, x_i]} |f(x)-f(t)| &= \sup_{x,t\in [x_{i-1}, x_i]} f(x)-f(t) \\ +&= \sup_{x\in [x_{i-1}, x_i]} f(x)-\sup_{t\in [x_{i-1}, x_i]} -f(t) \\ +&=\sup_{x\in [x_{i-1}, x_i]} f(x)-\inf_{t\in [x_{i-1}, x_i]} f(t) \\ +&= M_i - m_i +\end{aligned} +$$ + +So, $f$ is Riemann integrable with respect to $\alpha$ on $[a, b]$. + +EOP diff --git a/pages/Math416/Math416_L5.md b/pages/Math416/Math416_L5.md new file mode 100644 index 0000000..29c7f00 --- /dev/null +++ b/pages/Math416/Math416_L5.md @@ -0,0 +1,225 @@ +# Lecture 5 + +## Review + +Let $f$ be a complex function. that maps $\mathbb{R}^2$ to $\mathbb{R}^2$. $f(x+iy)=u(x,y)+iv(x,y)$. + +$Df(x+iy)=\begin{pmatrix} +\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\\ +\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} +\end{pmatrix}=\begin{pmatrix} +\alpha & \beta\\ +\sigma & \delta +\end{pmatrix}$ + +So + +$$\begin{aligned} +\frac{\partial f}{\partial \zeta}&=\frac{1}{2}\left(u_x+v_y\right)-i\frac{1}{2}\left(v_x+u_y\right)\\ +&=\frac{1}{2}\left(\alpha+\delta\right)-i\frac{1}{2}\left(\beta-\sigma\right)\\ +\end{aligned}$$ + +$$ +\begin{aligned} +\frac{\partial f}{\partial \overline{\zeta}}&=\frac{1}{2}\left(u_x+v_y\right)+i\frac{1}{2}\left(v_x+u_y\right)\\ +&=\frac{1}{2}\left(\alpha-\delta\right)+i\frac{1}{2}\left(\beta+\sigma\right)\\ +\end{aligned} +$$ + +When $f$ is conformal, $Df(x+iy)=\begin{pmatrix} +\alpha & \beta\\ +-\beta & \alpha +\end{pmatrix}$. + +So $\frac{\partial f}{\partial \zeta}=\frac{1}{2}(\alpha+\alpha)+i\frac{1}{2}(\beta+\beta)=a$ + +$\frac{\partial f}{\partial \overline{\zeta}}=\frac{1}{2}(\alpha-\alpha)+i\frac{1}{2}(\beta-\beta)=0$ + +> Less pain to represent a complex function using four real numbers. + +## Chapter 3: Linear fractional Transformations + +Let $a,b,c,d$ be complex numbers. such that $ad-bc\neq 0$. + +The linear fractional transformation is defined as + +$$ +\phi(\zeta)=\frac{a\zeta+b}{c\zeta+d} +$$ + +If we let $\psi(\zeta)=\frac{e\zeta-f}{-g\zeta+h}$ also be a linear fractional transformation, then $\phi\circ\psi$ is also a linear fractional transformation. + +New coefficients can be solved by + +$$ +\begin{pmatrix} +a & b\\ +c & d +\end{pmatrix} +\begin{pmatrix} +e & f\\ +g & h +\end{pmatrix} += +\begin{pmatrix} +k&l\\ +m&n +\end{pmatrix} +$$ + +So $\phi\circ\psi(\zeta)=\frac{k\zeta+l}{m\zeta+n}$ + +### Complex projective space + +$\mathbb{R}P^1$ is the set of lines through the origin in $\mathbb{R}^2$. + +We defined $(a,b)\sim(c,d),(a,b),(c,d)\in\mathbb{R}^2\setminus\{(0,0)\}$ if $\exists t\neq 0,t\in\mathbb{R}\setminus\{0\}$ such that $(a,b)=t(c,d)$. + +$R\mathbb{P}^1=S^1\setminus\{\pm x\}\cong S^1$ + +Equivalently, + +$\mathbb{C}P^1$ is the set of lines through the origin in $\mathbb{C}$. + +We defined $(a,b)\sim(c,d),(a,b),(c,d)\in\mathbb{C}\setminus\{(0,0)\}$ if $\exists t\neq 0,t\in\mathbb{C}\setminus\{0\}$ such that $(a,b)=(tc,td)$. + +So, $\forall \zeta\in\mathbb{C}\setminus\{0\}$: + +If $a\neq 0$, then $(a,b)\sim(1,\frac{b}{a})$. + +If $a=0$, then $(0,b)\sim(0,-b)$. + +So, $\mathbb{C}P^1$ is the set of lines through the origin in $\mathbb{C}$. + +### Linear fractional transformations + +Let $M=\begin{pmatrix} +a & b\\ +c & d +\end{pmatrix}$ be a $2\times 2$ matrix with complex entries. That maps $\mathbb{C}^2$ to $\mathbb{C}^2$. + +Suppose $M$ is non-singular. Then $ad-bc\neq 0$. + +If $M\begin{pmatrix} +\zeta_1\\ +\zeta_2 +\end{pmatrix}=\begin{pmatrix} +\omega_1\\ +\omega_2 +\end{pmatrix}$, then $M\begin{pmatrix} +t\zeta_1\\ +t\zeta_2 +\end{pmatrix}=\begin{pmatrix} +t\omega_1\\ +t\omega_2 +\end{pmatrix}$. + +So, $M$ induces a map $\phi_M:\mathbb{C}P^1\to\mathbb{C}P^1$ defined by $M\begin{pmatrix} +\zeta\\ +1 +\end{pmatrix}=\begin{pmatrix} +\frac{a\zeta+b}{c\zeta+d}\\ +1 +\end{pmatrix}$. + +$\phi_M(\zeta)=\frac{a\zeta+b}{c\zeta+d}$. + +If we let $M_2=\begin{pmatrix} +e &f\\ +g &h +\end{pmatrix}$, where $ad-bc\neq 0$ and $eh-fg\neq 0$, then $\phi_{M_2}(\zeta)=\frac{e\zeta+f}{g\zeta+h}$. + +So, $M_2M_1=\begin{pmatrix} +a&b\\ +c&d +\end{pmatrix}\begin{pmatrix} +e&f\\ +g&h +\end{pmatrix}=\begin{pmatrix} +\zeta\\ +1 +\end{pmatrix}$. + +This also gives $\begin{pmatrix} +k\zeta+l\\ +m\zeta+n +\end{pmatrix}\sim\begin{pmatrix} +\frac{k\zeta+l}{m\zeta+n}\\ +1 +\end{pmatrix}$. + +So, if $ab-cd\neq 0$, then $\exists M^{-1}$ such that $M_2M_1=I$. + +So non-constant linear fractional transformations form a group under composition. + +When do two matrices gives the $t_0$ same linear fractional transformation? + +$M_2^{-1}M_1=\alpha I$ + +We defined $GL(2,\mathbb{C})$ to be the group of general linear transformations of order 2 over $\mathbb{C}$. + +This is equivalent to the group of invertible $2\times 2$ matrices over $\mathbb{C}$ under matrix multiplication. + +Let $F$ be the function that maps $M$ to $\phi_M$. + +$F:GL(2,\mathbb{C})\to\text{Homeo}(\mathbb{C}P^1)$ + +So the kernel of $F$ is the set of matrices that represent the identity transformation. $\ker F=\left\{\alpha I\right\},\alpha\in\mathbb{C}\setminus\{0\}$. + +#### Corollary of conformality + +If $\phi$ is a non-constant linear fractional transformation, then $\phi$ is conformal. + +Proof: + +Know that $\phi_0\circ\phi(\zeta)=\zeta$, + +Then $\phi(\zeta)=\phi_0^{-1}\circ\phi\circ\phi_0(\zeta)$. + +So $\phi(\zeta)=\frac{a\zeta+b}{c\zeta+d}$. + +$\phi:\mathbb{C}\cup\{\infty\}\to\mathbb{C}\cup\{\infty\}$ which gives $\phi(\infty)=\frac{a}{c}$ and $\phi(-\frac{d}{c})=\infty$. + +So, $\phi$ is conformal. + +EOP + +#### Proposition 3.4 of Fixed points + +Any non-constant linear fractional transformation except the identity transformation has 1 or 2 fixed points. + +Proof: + +Let $\phi(\zeta)=\frac{a\zeta+b}{c\zeta+d}$. + +Case 1: $c=0$ + +Then $\infty$ is a fixed point. + +Case 2: $c\neq 0$ + +Then $\phi(\zeta)=\frac{a\zeta+b}{c\zeta+d}$. + +The solution of $\phi(\zeta)=\zeta$ is $c\zeta^2+(d-a)\zeta-b=0$. + +Such solutions are $\zeta=\frac{-(d-a)\pm\sqrt{(d-a)^2+4bc}}{2c}$. + +So, $\phi$ has 1 or 2 fixed points. + +EOP + +#### Proposition 3.5 of triple transitivity + +If $\zeta_1,\zeta_2,\zeta_3\in\mathbb{C}P^1$ are distinct, then there exists a non-constant linear fractional transformation $\phi$ such that $\phi(\zeta_1)=\zeta_2$ and $\phi(\zeta_3)=\infty$. + +Proof as homework. + +#### Theorem 3.8 Preservation of clircles + +We defined clircle to be a circle or a line. + +If $\phi$ is a non-constant linear fractional transformation, then $\phi$ maps clircles to clircles. + +Proof: + +Continue on next lecture. 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 a1fc1b3..e487ab6 100644 --- a/pages/Math416/_meta.js +++ b/pages/Math416/_meta.js @@ -7,4 +7,6 @@ export default { Math416_L2: "Complex Variables (Lecture 2)", 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 0000000..8a1b4a7 Binary files /dev/null and b/public/CSE559A/DigitalCameraBlockDiagram.png differ