diff --git a/Baudin.png b/Baudin.png new file mode 100644 index 0000000..1718ee3 Binary files /dev/null and b/Baudin.png differ diff --git a/MRA.png b/MRA.png new file mode 100644 index 0000000..3e97cd7 Binary files /dev/null and b/MRA.png differ diff --git a/coffee_is_essential.jpg b/coffee_is_essential.jpg new file mode 100644 index 0000000..7b9b4bc Binary files /dev/null and b/coffee_is_essential.jpg differ diff --git a/fft_example.R b/fft_example.R index 470e005..11e3164 100644 --- a/fft_example.R +++ b/fft_example.R @@ -1,15 +1,18 @@ -N <- 1024*128 -w1 <- 200 -w2 <- 543 +N <- 1024 +w1 <- 1 +w2 <- 1 a1 <- 1 -a2 <- 2 +a2 <- 0 noise <- 0 -x <- seq(from=0, to=5,length.out = N) +x <- seq(from=0, to=5.5,length.out = N) Dt <- (max(x) - min(x)) / N Dw <- 1 / (N * Dt) y <- a1*sin(w1*x*2*pi) + a2*cos(w2*x*2*pi) + rnorm(N, 0, noise) -plot(x,y,type="l") Y <- fft(y)/N MY <- 2*Mod(Y) freqs <- seq(from=0, to=N-1) * Dw -plot(freqs,MY^2, type="l", xlim=c(0,1000)) + +par(mfrow = c(2,1)) +plot(x,y,type="l") +plot(freqs,MY^2, type="b", xlim=c(0,10)) +par(mfrow = c(1,1)) \ No newline at end of file diff --git a/kprl-example.png b/kprl-example.png new file mode 100644 index 0000000..e56261f Binary files /dev/null and b/kprl-example.png differ diff --git a/main-figure/unnamed-chunk-1-1.png b/main-figure/unnamed-chunk-1-1.png index f3b7069..48d512a 100644 Binary files a/main-figure/unnamed-chunk-1-1.png and b/main-figure/unnamed-chunk-1-1.png differ diff --git a/main-figure/unnamed-chunk-10-1.png b/main-figure/unnamed-chunk-10-1.png index 9aa8a70..260f099 100644 Binary files a/main-figure/unnamed-chunk-10-1.png and b/main-figure/unnamed-chunk-10-1.png differ diff --git a/main-figure/unnamed-chunk-12-1.png b/main-figure/unnamed-chunk-12-1.png index fbf70df..2206f39 100644 Binary files a/main-figure/unnamed-chunk-12-1.png and b/main-figure/unnamed-chunk-12-1.png differ diff --git a/main-figure/unnamed-chunk-2-1.png b/main-figure/unnamed-chunk-2-1.png index d7750bb..1f41ed7 100644 Binary files a/main-figure/unnamed-chunk-2-1.png and b/main-figure/unnamed-chunk-2-1.png differ diff --git a/main-figure/unnamed-chunk-5-1.png b/main-figure/unnamed-chunk-5-1.png index 62ef7c7..d0b828c 100644 Binary files a/main-figure/unnamed-chunk-5-1.png and b/main-figure/unnamed-chunk-5-1.png differ diff --git a/main-figure/unnamed-chunk-6-1.png b/main-figure/unnamed-chunk-6-1.png index 3e36175..02b4ada 100644 Binary files a/main-figure/unnamed-chunk-6-1.png and b/main-figure/unnamed-chunk-6-1.png differ diff --git a/main-figure/unnamed-chunk-7-1.png b/main-figure/unnamed-chunk-7-1.png index f5978de..dc5dec4 100644 Binary files a/main-figure/unnamed-chunk-7-1.png and b/main-figure/unnamed-chunk-7-1.png differ diff --git a/main-figure/unnamed-chunk-8-1.png b/main-figure/unnamed-chunk-8-1.png index 051ac33..142120d 100644 Binary files a/main-figure/unnamed-chunk-8-1.png and b/main-figure/unnamed-chunk-8-1.png differ diff --git a/main-figure/unnamed-chunk-9-1.png b/main-figure/unnamed-chunk-9-1.png index 4a6348c..9758d08 100644 Binary files a/main-figure/unnamed-chunk-9-1.png and b/main-figure/unnamed-chunk-9-1.png differ diff --git a/main.Rpres b/main.Rpres index 8130890..eeb4de0 100644 --- a/main.Rpres +++ b/main.Rpres @@ -29,16 +29,17 @@ $$f(x) = a_0 + a_1 e^{i w_1 x} + a_2 e^{i w_2 x} + \cdots$$ Fourier representations ======================================================== +incremental: true The Fourier representation looks different depending on the domain of the function $f:A\rightarrow\mathbb{C}$, with $A$ being:

$\mathbb{R}$: Real numbers -$\mathbb{Z}$: Integer numbers - $\mathbb{T}_p$: Circle of length $p$ +$\mathbb{Z}$: Integer numbers + $\mathbb{P}_N$: Polygon of $N$ sides Continuous Fourier Transform (CFT) @@ -55,32 +56,32 @@ $$ \end{aligned} $$ -Discrete Time Fourier Transform (DTFT) +Fourier Series ======================================================== -For functions defined on the integers $\mathbb{Z}$. +For functions defined on the integers on an interval $\mathbb{T}_p=[0,p)$. -$\phi:\mathbb{Z}\rightarrow\mathbb{C}$ +$g:\mathbb{T}_p\rightarrow\mathbb{C}$ with $g(p) = g(0)$ $$ \begin{aligned} - \phi(n) &= \int_{0}^p \Phi(s) e^{2\pi isn/p} ds \\ - \Phi(s) &= \frac{1}{p}\sum_{n=-\infty}^\infty \phi(n) e^{-2\pi isn/p} \\ + g(x) &= \sum_{k=-\infty}^\infty G(k) e^{2\pi ikx/p} \\ + G(k) &= \frac{1}{p}\int_{0}^p g(x) e^{-2\pi ikx/p} dx \\ \end{aligned} $$ -Fourier Series +Discrete Time Fourier Transform (DTFT) ======================================================== -For functions defined on the integers on an interval $\mathbb{T}_p=[0,p)$. +For functions defined on the integers $\mathbb{Z}$. -$g:\mathbb{T}_p\rightarrow\mathbb{C}$ with $g(p) = g(0)$ +$\phi:\mathbb{Z}\rightarrow\mathbb{C}$ $$ \begin{aligned} - g(x) &= \sum_{k=-\infty}^\infty G(k) e^{2\pi ikx/p} \\ - G(k) &= \frac{1}{p}\int_{0}^p g(x) e^{-2\pi ikx/p} dx \\ + \phi(n) &= \int_{0}^p \Phi(s) e^{2\pi isn/p} ds \\ + \Phi(s) &= \frac{1}{p}\sum_{n=-\infty}^\infty \phi(n) e^{-2\pi isn/p} \\ \end{aligned} $$ @@ -106,15 +107,16 @@ $$ \begin{aligned} \int_{-\infty}^\infty f(x)\overline{g(x)}dx &= \int_{-\infty}^\infty F(s)\overline{G(x)}ds,\ \ \mathbb{R} \\ - \sum_{n=-\infty}^\infty f(n)\overline{g(n)} - &= p \int_{0}^p F(s)\overline{G(x)}ds,\ \ \mathbb{Z} \\ \int_{0}^p f(x)\overline{g(x)}dx &= p \sum_{k=-\infty}^\infty F(k)\overline{G(k)},\ \ \mathbb{T}_p \\ + \sum_{n=-\infty}^\infty f(n)\overline{g(n)} + &= p \int_{0}^p F(s)\overline{G(x)}ds,\ \ \mathbb{Z} \\ \sum_{n=0}^{N-1} f(n)\overline{g(n)} &= N \sum_{k=0}^{N-1} F(k)\overline{G(k)},\ \ \mathbb{P}_N \end{aligned} $$ + Discretization and periodization ======================================================== @@ -126,6 +128,7 @@ From a function $f$, under some conditions, we can construct a periodic functio $$g(x) = \sum_{m=-\infty}^\infty f(x-mp) $$ + Poisson relations ======================================================== incremental: true @@ -149,6 +152,7 @@ Poisson cube Nyquist-Shannon sampling theorem ======================================================== +incremental: true Assume that $f(t)$ is *$\sigma$-bandlimited* @@ -166,6 +170,7 @@ $$\sigma < \frac{1}{2\Delta t} \equiv \omega_{Nyq}$$ Example: single sinusoid ======================================================== +incremental:true $$ \begin{aligned} @@ -189,12 +194,12 @@ Example: single sinusoid ======================================================== ```{r, echo=FALSE} -N <- 1024*128 -w1 <- 10 +N <- 1024 +w1 <- 1 a1 <- 1 -x <- seq(from=0, to=4/w1,length.out = N) +x <- seq(from=0, to=5.3,length.out = N) y <- a1*sin(w1*x*2*pi) -plot(x*w1,y/a1,type="b", +plot(x*w1,y/a1,type="l", xlab = expression(paste("w"[1],"t")), ylab = expression("f(t)/a"[1])) ``` @@ -247,7 +252,7 @@ Example: Multiple sinusoids with noise ======================================================== ```{r, echo=FALSE} -N <- 1024*128 +N <- 1024*4 w1 <- 10 w2 <- 1 a1 <- 2 @@ -286,7 +291,7 @@ Example: Time-varying signal ======================================================== ```{r, echo=FALSE} -N <- 1024*128 +N <- 1024*4 w1 <- 5 a1 <- 1 mu <- 5 @@ -331,7 +336,7 @@ Example: Time-varying signal 2 ======================================================== ```{r, echo=FALSE} -N <- 1024*256 +N <- 1024*4 w1 <- 5 w2 <- 0.7 w3 <- 1 @@ -427,6 +432,7 @@ text(1,0.8, labels = "a=1.2", col="blue") Definition of wavelet transforms ======================================================== +incremental: true Given $f\in L^2(\mathbb{R})$, we define its *continuous wavelet transform* with respect to the wavelet $\psi$ as @@ -440,6 +446,7 @@ $$ 0 < C_{\psi}\equiv\int_{-\infty}^\infty Example: Time-varying signal CWT ======================================================== +incremental: true ```{r, echo=FALSE} N <- 1024*256 @@ -482,6 +489,7 @@ Mexican hat CWT Example: Time-varying signal CWT 2 ======================================================== +incremental: true Haar CWT @@ -496,6 +504,7 @@ Morlet CWT Example: Time-varying signal CWT 3 ======================================================== +incremental: true Morlet @@ -524,9 +533,16 @@ $$ Inverse of a wavelet transform ======================================================== +incremental: true $$f(t) = \frac{1}{C_\psi}\int_{-\infty}^\infty\int_{-\infty}^\infty \mathcal{W}_\psi[f](a,b) \psi_{a,b}(t) \frac{da\ db}{a^2}$$ +only when + +$$ 0 < C_{\psi}\equiv\int_{-\infty}^\infty +\frac{\left|\hat{\psi}(\omega)\right|^2}{\left|\omega\right|} d\omega +<\infty $$ + Discrete wavelet transform ======================================================== @@ -543,20 +559,59 @@ $$\psi_{m,n}(t)=2^{-m/2} \psi \left( 2^{-m} t - n \right),\ \ \ n,m \in \mathbb{ When can we recover $f(t)$ from $\mathcal{W}_\psi[f](m,n)$ ? -Discrete wavelet transform 2 +Orthonormal basis ======================================================== -When $\psi_{m,n}$ is complete in and orthonormal in $L^2(\mathbb{R})$: +Find $\psi_{m,n}$ such that they form a complete and orthonormal basis in $L^2(\mathbb{R})$: $$f(t) = \sum_{m,n=-\infty}^\infty \langle f,\psi_{m,n}\rangle\ \psi_{m,n}(t)$$ Multiresolution analysis (MRA) ======================================================== +incremental: true + +![](MRA.png) + +> *MRA is really an effective mathematical framework for hierarchical decomposition of an image (or signal) into components of different scales (or frequencies).* + + +Example application +======================================================== + +**Time-frequency analysis of solar $p$-modes** + +*F. Baudin, A. Gabriel, D. Gibert (1994)* + +*** + +![](Baudin.png) + + +Example application 2 +======================================================== + +**Wavelets: a powerful tool for studying rotation, activity, and pulsation in Kepler and CoRoT stellar light curves** + +*J. P. Bravo, S. Roque, R. Estrela, I. C. Leão, and J. R. De Medeiros (2014)* + +*** + +![](kprl-example.png) + + +Thank you! +======================================================== + +![](coffee_is_essential.jpg) + +*** -> MRA is really an effective mathematical framework for hierarchical decomposition of an image (or signal) into components of different scales (or frequencies). +[https://github.molgen.mpg.de/saravia/wavelets-SAGE](https://github.molgen.mpg.de/saravia/wavelets-SAGE) +saravia@mps.mpg.de +ags3006@gmail.com Orthogonality relations ======================================================== diff --git a/main.md b/main.md index 27eb132..0c42fb6 100644 --- a/main.md +++ b/main.md @@ -29,16 +29,17 @@ $$f(x) = a_0 + a_1 e^{i w_1 x} + a_2 e^{i w_2 x} + \cdots$$ Fourier representations ======================================================== +incremental: true The Fourier representation looks different depending on the domain of the function $f:A\rightarrow\mathbb{C}$, with $A$ being:

$\mathbb{R}$: Real numbers -$\mathbb{Z}$: Integer numbers - $\mathbb{T}_p$: Circle of length $p$ +$\mathbb{Z}$: Integer numbers + $\mathbb{P}_N$: Polygon of $N$ sides Continuous Fourier Transform (CFT) @@ -55,32 +56,32 @@ $$ \end{aligned} $$ -Discrete Time Fourier Transform (DTFT) +Fourier Series ======================================================== -For functions defined on the integers $\mathbb{Z}$. +For functions defined on the integers on an interval $\mathbb{T}_p=[0,p)$. -$\phi:\mathbb{Z}\rightarrow\mathbb{C}$ +$g:\mathbb{T}_p\rightarrow\mathbb{C}$ with $g(p) = g(0)$ $$ \begin{aligned} - \phi(n) &= \int_{0}^p \Phi(s) e^{2\pi isn/p} ds \\ - \Phi(s) &= \frac{1}{p}\sum_{n=-\infty}^\infty \phi(n) e^{-2\pi isn/p} \\ + g(x) &= \sum_{k=-\infty}^\infty G(k) e^{2\pi ikx/p} \\ + G(k) &= \frac{1}{p}\int_{0}^p g(x) e^{-2\pi ikx/p} dx \\ \end{aligned} $$ -Fourier Series +Discrete Time Fourier Transform (DTFT) ======================================================== -For functions defined on the integers on an interval $\mathbb{T}_p=[0,p)$. +For functions defined on the integers $\mathbb{Z}$. -$g:\mathbb{T}_p\rightarrow\mathbb{C}$ with $g(p) = g(0)$ +$\phi:\mathbb{Z}\rightarrow\mathbb{C}$ $$ \begin{aligned} - g(x) &= \sum_{k=-\infty}^\infty G(k) e^{2\pi ikx/p} \\ - G(k) &= \frac{1}{p}\int_{0}^p g(x) e^{-2\pi ikx/p} dx \\ + \phi(n) &= \int_{0}^p \Phi(s) e^{2\pi isn/p} ds \\ + \Phi(s) &= \frac{1}{p}\sum_{n=-\infty}^\infty \phi(n) e^{-2\pi isn/p} \\ \end{aligned} $$ @@ -106,15 +107,16 @@ $$ \begin{aligned} \int_{-\infty}^\infty f(x)\overline{g(x)}dx &= \int_{-\infty}^\infty F(s)\overline{G(x)}ds,\ \ \mathbb{R} \\ - \sum_{n=-\infty}^\infty f(n)\overline{g(n)} - &= p \int_{0}^p F(s)\overline{G(x)}ds,\ \ \mathbb{Z} \\ \int_{0}^p f(x)\overline{g(x)}dx &= p \sum_{k=-\infty}^\infty F(k)\overline{G(k)},\ \ \mathbb{T}_p \\ + \sum_{n=-\infty}^\infty f(n)\overline{g(n)} + &= p \int_{0}^p F(s)\overline{G(x)}ds,\ \ \mathbb{Z} \\ \sum_{n=0}^{N-1} f(n)\overline{g(n)} &= N \sum_{k=0}^{N-1} F(k)\overline{G(k)},\ \ \mathbb{P}_N \end{aligned} $$ + Discretization and periodization ======================================================== @@ -126,6 +128,7 @@ From a function $f$, under some conditions, we can construct a periodic functio $$g(x) = \sum_{m=-\infty}^\infty f(x-mp) $$ + Poisson relations ======================================================== incremental: true @@ -149,6 +152,7 @@ Poisson cube Nyquist-Shannon sampling theorem ======================================================== +incremental: true Assume that $f(t)$ is *$\sigma$-bandlimited* @@ -166,6 +170,7 @@ $$\sigma < \frac{1}{2\Delta t} \equiv \omega_{Nyq}$$ Example: single sinusoid ======================================================== +incremental:true $$ \begin{aligned} @@ -275,6 +280,7 @@ $$\psi_{a,b}(t)=\frac{1}{\sqrt{a}}\psi\left(\frac{t-b}{a}\right)$$ Definition of wavelet transforms ======================================================== +incremental: true Given $f\in L^2(\mathbb{R})$, we define its *continuous wavelet transform* with respect to the wavelet $\psi$ as @@ -288,6 +294,7 @@ $$ 0 < C_{\psi}\equiv\int_{-\infty}^\infty Example: Time-varying signal CWT ======================================================== +incremental: true ![plot of chunk unnamed-chunk-12](main-figure/unnamed-chunk-12-1.png) @@ -299,6 +306,7 @@ Mexican hat CWT Example: Time-varying signal CWT 2 ======================================================== +incremental: true Haar CWT @@ -313,6 +321,7 @@ Morlet CWT Example: Time-varying signal CWT 3 ======================================================== +incremental: true Morlet @@ -341,9 +350,16 @@ $$ Inverse of a wavelet transform ======================================================== +incremental: true $$f(t) = \frac{1}{C_\psi}\int_{-\infty}^\infty\int_{-\infty}^\infty \mathcal{W}_\psi[f](a,b) \psi_{a,b}(t) \frac{da\ db}{a^2}$$ +only when + +$$ 0 < C_{\psi}\equiv\int_{-\infty}^\infty +\frac{\left|\hat{\psi}(\omega)\right|^2}{\left|\omega\right|} d\omega +<\infty $$ + Discrete wavelet transform ======================================================== @@ -360,20 +376,59 @@ $$\psi_{m,n}(t)=2^{-m/2} \psi \left( 2^{-m} t - n \right),\ \ \ n,m \in \mathbb{ When can we recover $f(t)$ from $\mathcal{W}_\psi[f](m,n)$ ? -Discrete wavelet transform 2 +Orthonormal basis ======================================================== -When $\psi_{m,n}$ is complete in and orthonormal in $L^2(\mathbb{R})$: +Find $\psi_{m,n}$ such that they form a complete and orthonormal basis in $L^2(\mathbb{R})$: $$f(t) = \sum_{m,n=-\infty}^\infty \langle f,\psi_{m,n}\rangle\ \psi_{m,n}(t)$$ Multiresolution analysis (MRA) ======================================================== +incremental: true + +![](MRA.png) + +> *MRA is really an effective mathematical framework for hierarchical decomposition of an image (or signal) into components of different scales (or frequencies).* + + +Example application +======================================================== + +**Time-frequency analysis of solar $p$-modes** + +*F. Baudin, A. Gabriel, D. Gibert (1994)* + +*** + +![](Baudin.png) + + +Example application 2 +======================================================== + +**Wavelets: a powerful tool for studying rotation, activity, and pulsation in Kepler and CoRoT stellar light curves** + +*J. P. Bravo, S. Roque, R. Estrela, I. C. Leão, and J. R. De Medeiros (2014)* + +*** + +![](kprl-example.png) + + +Thank you! +======================================================== + +![](coffee_is_essential.jpg) + +*** -> MRA is really an effective mathematical framework for hierarchical decomposition of an image (or signal) into components of different scales (or frequencies). +[https://github.molgen.mpg.de/saravia/wavelets-SAGE](https://github.molgen.mpg.de/saravia/wavelets-SAGE) +saravia@mps.mpg.de +ags3006@gmail.com Orthogonality relations ========================================================