diff --git a/pages/CSE559A/CSE559A_L22.md b/pages/CSE559A/CSE559A_L22.md index 345474b..5032edc 100644 --- a/pages/CSE559A/CSE559A_L22.md +++ b/pages/CSE559A/CSE559A_L22.md @@ -166,11 +166,11 @@ y \\ \end{pmatrix} $$ -Constraint from a match $(x_i,x_i^')$: $x_i^'≅Hx_i$ +Constraint from a match $(x_i,x_i')$, $x_i'\cong Hx_i$ -How can we get rid of the scale ambiguity? +How can we get rid of the scale ambiguity? -Cross product trick: $x_i^' × Hx_i=0$ +Cross product trick:$x_i' × Hx_i=0$ The cross product is defined as: @@ -181,9 +181,9 @@ $$ Let $h_1^T, h_2^T, h_3^T$ be the rows of $H$. Then $$ -x_i^' × Hx_i=\begin{pmatrix} - x_i^' \\ - y_i^' \\ +x_i' × Hx_i=\begin{pmatrix} + x_i' \\ + y_i' \\ 1 \end{pmatrix} \times \begin{pmatrix} h_1^T x_i \\ @@ -192,18 +192,18 @@ x_i^' × Hx_i=\begin{pmatrix} \end{pmatrix} = \begin{pmatrix} - y_i^' h_3^T x_i−h_2^T x_i \\ - h_1^T x_i−x_i^' h_3^T x_i \\ - x_i^' h_2^T x_i−y_i^' h_1^T x_i + y_i' h_3^T x_i−h_2^T x_i \\ + h_1^T x_i−x_i' h_3^T x_i \\ + x_i' h_2^T x_i−y_i' h_1^T x_i \end{pmatrix} $$ -Constraint from a match $(x_i,x_i^')$: +Constraint from a match $(x_i,x_i')$: $$ -x_i^' × Hx_i=\begin{pmatrix} - x_i^' \\ - y_i^' \\ +x_i' × Hx_i=\begin{pmatrix} + x_i' \\ + y_i' \\ 1 \end{pmatrix} \times \begin{pmatrix} h_1^T x_i \\ @@ -212,9 +212,9 @@ x_i^' × Hx_i=\begin{pmatrix} \end{pmatrix} = \begin{pmatrix} - y_i^' h_3^T x_i−h_2^T x_i \\ - h_1^T x_i−x_i^' h_3^T x_i \\ - x_i^' h_2^T x_i−y_i^' h_1^T x_i + y_i' h_3^T x_i−h_2^T x_i \\ + h_1^T x_i−x_i' h_3^T x_i \\ + x_i' h_2^T x_i−y_i' h_1^T x_i \end{pmatrix} $$ @@ -222,9 +222,9 @@ Rearranging the terms: $$ \begin{bmatrix} - 0^T &-x_i^T &y_i^' x_i^T \\ - x_i^T &0^T &-x_i^' x_i^T \\ - y_i^' x_i^T &x_i^' x_i^T &0^T + 0^T &-x_i^T &y_i' x_i^T \\ + x_i^T &0^T &-x_i' x_i^T \\ + y_i' x_i^T &x_i' x_i^T &0^T \end{bmatrix} \begin{bmatrix} h_1 \\ diff --git a/pages/Math416/Math416_L27.md b/pages/Math416/Math416_L27.md index 8b13789..93c5f9c 100644 --- a/pages/Math416/Math416_L27.md +++ b/pages/Math416/Math416_L27.md @@ -1 +1,195 @@ +# Math416 Lecture 27 +## Continue on Application to evaluate $\int_{-\infty}^\infty \frac{\cos x}{1+x^4}dx$ + +Consider the function$f(z)=\frac{e^{iz}}{1+z^4}=\frac{\cos z+i\sin z}{1+z^4}$. + +Our desired integral can be evaluated by $\int_{-R}^R f(z)dz$ + +To evaluate the singularity, $z^4=-1$ has four roots by the De Moivre's theorem. + +$z^4=-1=e^{i\pi+2k\pi i}$ for $k=0,1,2,3$. + +So $z=e^{i\theta}$ for $\theta=\frac{\pi}{4}+\frac{k\pi}{2}$ for $k=0,1,2,3$. + +So the singularities are $z=e^{i\pi/4},e^{i3\pi/4},e^{i5\pi/4},e^{i7\pi/4}$. + +Only $z=e^{i\pi/4},e^{i3\pi/4}$ are in the upper half plane. + +So we can use the semi-circle contour to evaluate the integral. Name the path as $\gamma$. + +$\int_\gamma f(z)dz=2\pi i\left[\operatorname{Res}_{z=e^{i\pi/4}}(f)+\operatorname{Res}_{z=e^{i3\pi/4}}(f)\right]$. + +The two poles are simple poles. + +$\operatorname{Res}_{z_0}(f)=\lim_{z\to z_0}(z-z_0)f(z)$. + +So + +$$ +\begin{aligned} +\operatorname{Res}_{z=e^{i\pi/4}}(f)&=\lim_{z\to e^{i\pi/4}}(z-e^{i\pi/4})\frac{e^{iz}}{1+z^4}\\ +&=\frac{(z-e^{i\pi/4})e^{iz}}{(z-e^{i\pi/4})(z-e^{i3\pi/4})(z-e^{i5\pi/4})(z-e^{i7\pi/4})}\\ +&=\frac{e^{ie^{i\pi/4}}}{(e^{i\pi/4}-e^{i3\pi/4})(e^{i\pi/4}-e^{i5\pi/4})(e^{i\pi/4}-e^{i7\pi/4})} +\end{aligned} +$$ + +A short cut goes as follows: + +We know $p(z)=1+z^4$ has four roots $z_1,z_2,z_3,z_4$. + +$$ +\lim_{z\to z_0}\frac{(z-z_0)}{p(z)}=\frac{1}{p'(z_0)} +$$ + +So + +$$ +\operatorname{Res}_{z=e^{i\pi/4}}(f)=\frac{e^{ie^{i\pi/4}}}{4e^{i3\pi/4}} +$$ + +Similarly, + +$$ +\operatorname{Res}_{z=e^{i3\pi/4}}(f)=\frac{e^{ie^{i3\pi/4}}}{4e^{i\pi/4}} +$$ + +So the sum of the residues is + +$$ +\begin{aligned} +\operatorname{Res}_{z=e^{i\pi/4}}(f)+\operatorname{Res}_{z=e^{i3\pi/4}}(f)&=\frac{e^{ie^{i\pi/4}}}{4e^{i3\pi/4}}+\frac{e^{ie^{i3\pi/4}}}{4e^{i\pi/4}}\\ +&=\frac{e^{\frac{i}{\sqrt{2}}} e^{-\frac{1}{\sqrt{2}}}}{4[-\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}}]}+\frac{e^{-\frac{i}{\sqrt{2}}}-e^{-\frac{1}{\sqrt{2}}}}{4[\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}}]}\\ +&=\frac{\pi\sqrt{2}}{2}e^{-\frac{1}{\sqrt{2}}}(\cos\frac{1}{\sqrt{2}}+\sin\frac{1}{\sqrt{2}}) +\end{aligned} +$$ + +For the semicircle part, we can bound our estimate by + +$$ +\left|\int_{C_R}f(z)dz\right|\leq\pi R\max_{z\in C_R}|f(z)|\leq \pi \frac{1}{R^4}\to 0 +$$ + +as $R\to\infty$. + +So + +$$ +\int_{-\infty}^\infty\frac{\cos x}{1+x^4}dx=\frac{\pi\sqrt{2}}{2}e^{-\frac{1}{\sqrt{2}}}(\cos\frac{1}{\sqrt{2}}+\sin\frac{1}{\sqrt{2}}) +$$ + +## Big idea of this course + +$f$ is holomorphic $\iff$ $f$ has complex derivative. + +$f$ is holomorphic $\iff$ $f$ satisfies Cauchy-Riemann equations $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$ + +$f$ is holomorphic $\iff$ $f$ is analytic (is locally given by power series). The power series is integrable/differentiable term by term in the radius of convergence. + +### Laurent series + +Similar to power series both with annulus of convergence. + +$f(z)=\sum_{n=-\infty}^\infty a_n(z-z_0)^n$ for $z\in A(z_0,r,R)$. + +Identity theorem: If $f$ is holomorphic on a domain $\Omega$, it is uniquely determined by its values on any sets with a limit point in $\Omega$. + +### Cauchy's Theorem + +$$ +\int_\gamma f(z)dz=0 +$$ + +If $f$ is holomorphic on $\Omega$ and $\gamma$ is a closed path in $\Omega$ and $\gamma\cup \operatorname{int}\gamma\subset \Omega$, then $\int_\gamma f(z)dz=0$. + +### Favorite estimate + +$$ +\left|\int_\gamma f(z)dz\right|\leq \sup_{z\in\gamma}|f(z)|\cdot \operatorname{length}(\gamma) +$$ + +### Cauchy's Integral Formula + +$$ +f(z_0)=\frac{1}{2\pi i}\int_\gamma \frac{f(z)}{z-z_0}dz +$$ + +where $z_0\in \operatorname{int}\gamma$ and $\gamma$ is a closed path. + +Extension: If $f$ is holomorphic on $\Omega$ and $z_0\in \Omega$, then $f$ is infinitely differentiable and + +$$ +f^{(n)}(z_0)=\frac{n!}{2\pi i}\int_\gamma \frac{f(z)}{(z-z_0)^{n+1}}dz +$$ + +### Residue theorem + +If $f$ is holomorphic on $\Omega$ except for a finite number of isolated singularities $z_1,z_2,\dots,z_p$, and $\Gamma$ is a curve inside $\Omega$ that don't pass through any of the singularities ($\Gamma\subset \Omega\setminus \{z_1,z_2,\dots,z_p\}$), then + +$$ +\int_\Gamma f(z)dz=2\pi i\sum_{z_i}\operatorname{ind}_{\Gamma}(z_i) \operatorname{res}_{z_i}(f) +$$ + +### Harmonic conjugate + +Locally, always have harmonic conjugates. + +Globally can do this iff domain is simply connected. + +### Schwarz-pick's Lemma: + +If $f$ maps $D$ to $D$ and $f(0)=0$, then $|f(z)|\leq |z|$ for all $z\in D$. and $|f'(0)|\leq 1$. + +For mobius map, $f:D\to D$ holds, $\varphi(f(z),f(w))=\varphi(z,w)$ for all $z,w\in D$. + +$$ +\varphi(z,w)=\frac{z-w}{1-\overline{w}z} +$$ + +### Convergence + +#### Types of convergence + +**Converge pointwise** (Not very strong): + +$\forall x\in X, \lim_{n\to\infty}f_n(x)=f(x)$. + +Or, $\forall x\in X, \forall \epsilon>0, \exists N>0, \forall n\geq N \implies |f_n(x)-f(x)|<\epsilon$. + +**Converge uniformly** (Much better): + +$\forall \epsilon>0, \exists N>0, \forall n\geq N \implies \forall x\in X, |f_n(x)-f(x)|<\epsilon$. + +**Converge locally uniformly** (Strong): + +$\forall x\in X$, $\exists$ open $x\in U$, such that $f_n\to f$ uniformly on $U$. + +**Converge uniformly on compact subsets** (Good enough for local properties): + +$\forall$ compact $K\subset X$, $f_n\to f$ uniformly on $K$. + +#### Weierstrass' Theorem + +If $f_n\in O(\Omega)$ and $f_n\to f$ locally uniformly, then $f\in O(\Omega)$. + +#### Cauchy-Hadamard's Theorem + +For a power series, $\sum_{n=0}^\infty a_n(z-z_0)^n$, the radius of convergence is + +$$ +R=\frac{1}{\limsup_{n\to\infty}|a_n|^{1/n}} +$$ + +On $B(z_0,R)$, the series converges locally uniformly and absolutely. + +## Argument and Logarithm + +$\arg z$ is any $\theta$ such that $z=re^{i\theta}$. + +$\operatorname{Arg} z$ is the principal value of the argument, $-\pi<\operatorname{Arg} z\leq \pi$. + +$\log z$ is the principal value of the logarithm, $\log z=\ln |z|+i\arg z$. + +$\operatorname{Log} z$ is the set of all logarithms of $z$, $\operatorname{Log} z=\{\log z+2k\pi i: k\in\mathbb{Z}\}$. + +END diff --git a/public/CSE559A/Epipolar_geometry_setup.png b/public/CSE559A/Epipolar_geometry_setup.png new file mode 100644 index 0000000..03061d5 Binary files /dev/null and b/public/CSE559A/Epipolar_geometry_setup.png differ diff --git a/public/CSE559A/Epipolar_line_for_converging_cameras.png b/public/CSE559A/Epipolar_line_for_converging_cameras.png new file mode 100644 index 0000000..fbd1437 Binary files /dev/null and b/public/CSE559A/Epipolar_line_for_converging_cameras.png differ diff --git a/public/CSE559A/Epipolar_line_for_parallel_cameras.png b/public/CSE559A/Epipolar_line_for_parallel_cameras.png new file mode 100644 index 0000000..4241a17 Binary files 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