Update 04_INT/tutorial_4_part01_lagrange.ipynb,...

Update 04_INT/tutorial_4_part01_lagrange.ipynb, 04_INT/tutorial_4_part02_lokale_interpolation.ipynb, 04_INT/tutorial_4_part05_interpolation_anwendungen.ipynb, 04_INT/tutorial_4_part01_lagrange_example.ipynb, 04_INT/tutorial_4_part03_4_hermite_splines.ipynb

04_INT/tutorial_4_part01_lagrange.ipynb

+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Lagrange Interpolation"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Globale Interpolation Wiederholung\n",
+    "\n",
+    "Gegeben: $n$ Datenpunkte \n",
+    "\n",
+    "\\begin{equation}\n",
+    "(x_1, f_1), (x_2, f_2), \\dots , (x_n, f_n)\n",
+    "\\end{equation}\n",
+    "\n",
+    "Gesucht: Interpolierendes Polynom von Grad $n-1$.\n",
+    "\n",
+    "\\begin{equation}\n",
+    "    p(x) \n",
+    "    = \n",
+    "    a_0 + a_1 x + a_2 x^2 + \\cdots + a_{n-1} x^{n-1}\n",
+    "\\end{equation}\n",
+    "\n",
+    "\n",
+    "Ansatz: Löse   $V \\vec{a} = \\vec{f}$\n",
+    "\n",
+    "\n",
+    "\\begin{equation}\n",
+    "    \\label{eq:int:matf}\n",
+    "    \\begin{pmatrix}\n",
+    "        1 & x_1 & x_1^2 & \\cdots & x_1^{n-1}\\\\\n",
+    "        1 & x_2 & x_2^2 & \\cdots & x_2^{n-1}\\\\\n",
+    "        \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\  \n",
+    "        1 & x_n & x_n^2 & \\cdots & x_n^{n-1}\n",
+    "    \\end{pmatrix}\n",
+    "    \\begin{pmatrix}\n",
+    "        a_0\\\\\n",
+    "        a_1\\\\\n",
+    "        \\vdots\\\\\n",
+    "        a_k    \n",
+    "    \\end{pmatrix}\n",
+    "    =\n",
+    "    \\begin{pmatrix}\n",
+    "        f_1\\\\\n",
+    "        f_2\\\\\n",
+    "        \\vdots\\\\\n",
+    "        f_n    \n",
+    "    \\end{pmatrix}\n",
+    "\\end{equation}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Ein anderer Ansatz\n",
+    "\n",
+    "Ein Polynom für jeden Punkt bilden und alle Polynome summieren. \n",
+    "\n",
+    "\\begin{equation}\n",
+    "    p(x) = \\sum_{i=1}^n l_i(x)\\, f_i \\,\n",
+    "    \\label{eq:lagrange_interpoly}\n",
+    "\\end{equation}\n",
+    "\n",
+    "**Lagrange Basispolynome**:\n",
+    "\n",
+    "\\begin{equation}\n",
+    "    l_i(x) \n",
+    "    = \n",
+    "    \\frac{x-x_1}{x_i-x_1}\n",
+    "    \\cdot\\dots\\cdot\n",
+    "    \\frac{x-x_{i-1}}{x_i-x_{i-1}}\n",
+    "    \\cdot\n",
+    "    \\frac{x-x_{i+1}}{x_i-x_{i+1}}\n",
+    "    \\cdot\\dots\\cdot\n",
+    "    \\frac{x-x_n}{x_i-x_n}\n",
+    "    =\n",
+    "    \\prod_{k=1 \\atop k \\neq i}^n \n",
+    "    \\frac{x-x_k}{x_i - x_k}\\,\n",
+    "\\end{equation}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "### Beispiele für jeweils 2, 3, 4 Basispolynome\n",
+    "\n",
+    "<img src=\"imgs/Basispolynome.png\" />"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Beispiel\n",
+    "\n",
+    "Gegeben Messpunkte (x, y): (1, 1), (2, 2), (3, 2), bestimme Lagrange Interpolationspolynom."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "A. Lagrange Basispolynome berechnen:\n",
+    "\n",
+    "$l_i(x) = \\prod_{k=1 \\atop k \\neq i}^n \\frac{x-x_k}{x_i - x_k}$\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "$l_1(x) = \\frac{x-x_2}{x_1-x_2} \\cdot \\frac{x-x_3}{x_1-x_3}$\n",
+    "$= \\frac{x-2}{1-2} \\cdot \\frac{x-3}{1-3}$\n",
+    "$= \\frac{x^2 - 5x + 6}{2}$ "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "$l_2(x) = \\frac{x-x_1}{x_2-x_1} \\cdot \\frac{x-x_3}{x_2-x_3}$\n",
+    "$= \\frac{x-1}{2-1} \\cdot \\frac{x-3}{2-3}$\n",
+    "$= -x^2 + 4x -3$ "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "$l_3(x) = \\frac{x-x_1}{x_3-x_1} \\cdot \\frac{x-x_2}{x_3-x_2}$\n",
+    "$= \\frac{x-1}{3-1} \\cdot \\frac{x-2}{3-2}$\n",
+    "$= \\frac{x^2 - 3x + 2}{2}$ "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "Datenpunkte: (1, 1), (2, 2), (3, 2)\n",
+    "\n",
+    "B. Produkte von Basispolynomen und f-Werten bilden, dann summieren:\n",
+    "\n",
+    "$p(x) = \\sum_{i=1}^3 l_i(x)\\, f_i$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "&emsp;&emsp; $= l_1(x) \\cdot f_1$ &emsp; $+$ &emsp; $l_2(x) \\cdot f_2$ &emsp; $+$ &emsp; $l_3(x) \\cdot f_3$\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "&emsp;&emsp; $= \\frac{x^2 - 5x + 6}{2} \\cdot 1$ &emsp; $+$ &emsp; $(-x^2 + 4x -3) \\cdot 2$ &emsp; $+$ &emsp; $\\frac{x^2 - 3x + 2}{2} \\cdot 2$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "&emsp;&emsp; $ = \\frac{x^2 - 5x + 6}{2}$ &emsp; $+$ &emsp; $(-2x^2 + 8x -6)$ &emsp; $+$ &emsp; $x^2 - 3x + 2$\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "&emsp;&emsp; $ = -\\frac{x^2}{2} + \\frac{5}{2}x -1$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Lagrange und Vandermonde\n",
+    "\n",
+    "Lagrange Basispolynome $l_i(x)$ als Spaltenvektoren einer Matrix $L$:\n",
+    "\n",
+    "&emsp; $\\vec{a} = L \\vec{f}$\n",
+    "\n",
+    "Mit der Vandermonde Matrix:\n",
+    "\n",
+    "&emsp; $V \\vec{a} = \\vec{f}$ &emsp; $\\Rightarrow$ &emsp; $\\vec{a} = V^{-1}\\vec{f}$\n",
+    "\n",
+    "&emsp; $\\Rightarrow$ &emsp; $L = V^{-1}$\n"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9.1"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}

04_INT/tutorial_4_part02_lokale_interpolation.ipynb

+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Lokale Interpolation: Einführung"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "**Runges Phänomen:** Polynome mit hohem Grad neigen zum Schwingen.\n",
+    "\n",
+    "Stückweise interpolieren -> Polynome von kleinem Grad aneinandersetzen."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "#### Stückweise quadratische, kubische, lineare Polynominterpolation\n",
+    "\n",
+    "<img src='imgs/Piecewise_Interpolation.png' />"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "#### Kubische Interpolationspolynome für zwei Stützstellen mit variierenden Steigungen\n",
+    "<img src='imgs/Kubisch.png' />\n",
+    "\n",
+    "Steigungen an den Stützstellen explizit vorgeben..."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9.1"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}

04_INT/tutorial_4_part05_interpolation_anwendungen.ipynb

+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Aufgabe 4 und Anwendungsbeispiele"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Strichfigur Animation\n",
+    "\n",
+    "**Idee:** Für jedes Körperglied Positionen zu festen Zeitpunkten angeben.\n",
+    "\n",
+    "Wie viele solche Zustände (Frames) brauchen wir für eine Animation?\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "**Ziel**: Gegeben wenige Keyframes, Animation mit kontinuierlich aussehender Bewegung herstellen. \n",
+    "\n",
+    "**Wie**: Kubische Spline Interpolation mit unterschiedlichen Randbedingungen. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## \"Tweening\"\n",
+    "\n",
+    "<img src='imgs/tween1.jpeg' />"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "<img src='imgs/tween2.jpeg' />\n",
+    "\n",
+    "\n",
+    "Taken from *tips.clip-studio.com*"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Bayerfilter\n",
+    "\n",
+    "<img src='imgs/Bayer_pattern.png' />"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "<img src='imgs/bayer_profile.png' />"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "### Demosaicing: Interpolation mit benachbarten Pixeln\n",
+    "\n",
+    "<img src='imgs/bayer_array.png' />"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9.1"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}