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MPS-SAGE · Nov 27, 2015 · a8bb1a5 · a8bb1a5
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diff --git a/Baudin.png b/Baudin.png
diff --git a/MRA.png b/MRA.png
diff --git a/coffee_is_essential.jpg b/coffee_is_essential.jpg
diff --git 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))
diff --git a/kprl-example.png b/kprl-example.png
diff --git a/main-figure/unnamed-chunk-1-1.png b/main-figure/unnamed-chunk-1-1.png
diff --git a/main-figure/unnamed-chunk-10-1.png b/main-figure/unnamed-chunk-10-1.png
diff --git a/main-figure/unnamed-chunk-12-1.png b/main-figure/unnamed-chunk-12-1.png
diff --git a/main-figure/unnamed-chunk-2-1.png b/main-figure/unnamed-chunk-2-1.png
diff --git a/main-figure/unnamed-chunk-5-1.png b/main-figure/unnamed-chunk-5-1.png
diff --git a/main-figure/unnamed-chunk-6-1.png b/main-figure/unnamed-chunk-6-1.png
diff --git a/main-figure/unnamed-chunk-7-1.png b/main-figure/unnamed-chunk-7-1.png
diff --git a/main-figure/unnamed-chunk-8-1.png b/main-figure/unnamed-chunk-8-1.png
diff --git a/main-figure/unnamed-chunk-9-1.png b/main-figure/unnamed-chunk-9-1.png
diff --git 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:
 <br/><br/>
 
 $\mathbb{R}$: 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
 ========================================================