rotate_matrix_vs_shear.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "fff5227f-1c19-4cf9-a44c-f25dcecbdda2",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "%matplotlib ipympl\n",
    "import ipywidgets as widgets\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "301fe6b9-e37c-4154-9a9c-c86c0c79ef7c",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ipywidgets version:  8.0.4\n",
      "numpy version     :  1.20.3\n"
     ]
    }
   ],
   "source": [
    "print( 'ipywidgets version: ', widgets.__version__ )\n",
    "print( 'numpy version     : ', np.__version__)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "8db88ba2-c70f-4192-9a15-3c1dbd79e0b5",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "# define space invader\n",
    "space_invader = np.array([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],  # row_00\n",
    "                          [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],  # row_01\n",
    "                          [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],  # row_02\n",
    "                          [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],  # row_03\n",
    "                          [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0],  # row_04\n",
    "                          [0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],  # row_05\n",
    "                          [0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0],  # row_06\n",
    "                          [0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0],  # row_07\n",
    "                          [0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0],  # row_08\n",
    "                          [0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0],  # row_09\n",
    "                          [0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0],  # row_10\n",
    "                          [0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0],  # row_11\n",
    "                          [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],  # row_12\n",
    "                          [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],  # row_13\n",
    "                          [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],  # row_14\n",
    "                          [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]) # row_15\n",
    "\n",
    "Nx, Ny = space_invader.shape[0], space_invader.shape[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "f4d4775d-4669-40c9-88b9-c8491db8acd5",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "d177ad049f9742e987eb4921803dde82",
       "version_major": 2,
       "version_minor": 0
      },
      "image/png": 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",
      "text/html": [
       "\n",
       "            <div style=\"display: inline-block;\">\n",
       "                <div class=\"jupyter-widgets widget-label\" style=\"text-align: center;\">\n",
       "                    Figure\n",
       "                </div>\n",
       "                <img 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' width=400.0/>\n",
       "            </div>\n",
       "        "
      ],
      "text/plain": [
       "Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot the original (non-rotated) image\n",
    "fig1 = plt.figure( figsize=(4,4) )   # (width, height) in inches\n",
    "ax1_1= fig1.add_subplot( 1,1,1 )      # rows, cols, plotnumber\n",
    "\n",
    "img  = ax1_1.imshow(space_invader)\n",
    "ax1_1.set_title ('original image')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4e3ac56b-1e13-4737-9df8-5a8d5785b410",
   "metadata": {},
   "source": [
    "# Rotation via Matrices\n",
    "\n",
    "$\n",
    "\\begin{pmatrix} x'\\\\ y' \\end{pmatrix} =\n",
    "\\begin{pmatrix}\n",
    "\\cos\\alpha  & \\sin\\theta\\\\\n",
    "-\\sin\\alpha & \\cos\\theta\\\\\n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix} x\\\\y \\end{pmatrix}\n",
    "$\n",
    "\n",
    "\\begin{align}\n",
    "x' &= x\\cos\\alpha + y\\sin\\alpha \\\\\n",
    "y' &= -x\\sin\\alpha + y\\cos\\alpha\n",
    "\\end{align}\n",
    "\n",
    "Problem: aliasing (sin and cos give real numbers we need to round back to integers, which can lead sometimes to numbers being round to the same coordinate on the grid twice or not at all)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "b9b6a829-e95c-4eb8-a5f9-f57cc26143b5",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "def rotate_via_rotMatrix(coords_in, angle_in_degree, rot_axis='z', rot_center=np.array([0,0])):\n",
    "    alpha = np.radians(angle_in_degree)\n",
    "    \"\"\"\n",
    "    Rotate coordinate vector using the rotation matrix.\n",
    "    \"\"\"\n",
    "    \n",
    "    # transform coordinate system such that rotation center is origin\n",
    "    coords_in[0] -= rot_center[0]\n",
    "    coords_in[1] -= rot_center[1]\n",
    "    \n",
    "    # define rotation matrix\n",
    "    if rot_axis == 'z':\n",
    "        rot_matrix = np.matrix([[np.cos(alpha), -np.sin(alpha)],\n",
    "                                [np.sin(alpha), np.cos(alpha) ]])\n",
    "        \n",
    "    coords_new = rot_matrix * np.array([[coords_in[0]],\n",
    "                                        [coords_in[1]]])\n",
    "    \n",
    "    # transform coordinate system back to original system\n",
    "    coords_new = np.array( [round(coords_new[0,0] + rot_center[0]), \n",
    "                            round(coords_new[1,0] + rot_center[1])] )\n",
    "    \n",
    "    return coords_new"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "31b55ec4-47ac-4684-acc3-fd7c8434344e",
   "metadata": {},
   "source": [
    "# Rotation via Shearing\n",
    "\n",
    "Introduced by Alan W. Paeth in 1986 [1], a rotation can also be performed by 3 consecutive shear operations. Assuming that we have a 2D image, we need 2 quantities to perform the shearing operations: $k_x$ and $k_y$, which represent the shear operation in $x$ and $y$ direction, respectively. Let's assume further that we want to rotate our image by the angle $\\alpha$, then $k_x$ and $k_y$ are calculated as follows\n",
    "\\begin{align}\n",
    "k_x &= -\\tan(\\alpha/2)\\\\\n",
    "k_y &= \\sin(\\alpha).\n",
    "\\end{align}\n",
    "The new coordinates $x'$ and $y'$ are then calculated with the following three steps\n",
    "\\begin{align}\n",
    "x' &= x + k_x\\cdot y\\\\\n",
    "y' &= y + k_y\\cdot x'\\\\\n",
    "x' &= x' + k_x\\cdot y'\n",
    "\\end{align}\n",
    "This corresponds to a rotation around the origin $(0,0)$. To rotate around a different coordinate, we simply perform a coordinate transformation before the shearing operations, and a back-tranformation after the shearing operations.\n",
    "\n",
    "Since there are three separate shearing operations, and each shearing operation is just in one direction and conformal in area, there will be no \"gaps\"/\"holes\" appearing, no aliasing. Also note that this is memory saving as only one plane is treated at once.\n",
    "\n",
    "The shearing operation can also we written in matrix form:\n",
    "\n",
    "$\n",
    "\\begin{pmatrix}x'\\\\y'\\end{pmatrix}\n",
    "= \\begin{pmatrix} 1 & -\\tan\\alpha/2 \\\\ 0 & 1\\end{pmatrix}\n",
    "  \\begin{pmatrix} 1 & 0 \\\\ \\sin\\alpha & 1\\end{pmatrix}\n",
    "  \\begin{pmatrix} 1 & -\\tan\\alpha/2 \\\\ 0 & 1\\end{pmatrix}\n",
    "  \\begin{pmatrix}x\\\\y\\end{pmatrix}\n",
    "$\n",
    "\n",
    "Note: for illustration purposes it might be nice to show how doing the three consecutive shears look like.\n",
    "\n",
    "[1] https://doi.org/10.20380/GI1986.15"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "0eaaaec5-1394-45dc-ad7f-9696506919c6",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "def rotate_via_shearing(coords_in, angle_in_degree, rot_axis='z', rot_center=np.array([0,0])):\n",
    "    \"\"\"\n",
    "    Rotate coordinate vector using the shearing method.\n",
    "    \"\"\"\n",
    "    \n",
    "    alpha = np.radians(angle_in_degree)\n",
    "    \n",
    "    # transform coordinate system such that rotation center is origin\n",
    "    coords_in[0] -= rot_center[0]\n",
    "    coords_in[1] -= rot_center[1]\n",
    "    \n",
    "    # translations\n",
    "    shear_1 = np.tan(alpha/2.)\n",
    "    shear_2 = -np.sin(alpha)\n",
    "    shear_3 = np.tan(alpha/2.)\n",
    "    \n",
    "    if rot_axis =='z':\n",
    "        y_new = coords_in[1] + round(shear_1*coords_in[0])\n",
    "        x_new = coords_in[0] + round(shear_2*y_new)\n",
    "        y_new = y_new + round(shear_3*x_new)\n",
    "        \n",
    "    # transform coordinate system back to original system\n",
    "    coords_new = np.array( [[x_new + rot_center[0]], [y_new + rot_center[1]]] )\n",
    "        \n",
    "    return coords_new"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "55e6359b-6e2b-4112-80e7-f0455c2c2d9e",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "def check_boundaries(coords_new, coords_min, coords_max):\n",
    "    \"\"\"\n",
    "    Check if the new (rotated) coordinate vector is still within the grid boundaries.\n",
    "    \"\"\"\n",
    "    # check x boundaries\n",
    "    if coords_new[0] < coords_min[0]: return False\n",
    "    elif coords_new[0] >= coords_max[0]: return False\n",
    "    # check y boundaries\n",
    "    if coords_new[1] < coords_min[1]: return False\n",
    "    elif coords_new[1] >= coords_max[1]: return False\n",
    "\n",
    "    return True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "df204df3-0e9b-4e52-9792-27c3c1ccf789",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "def rotate_arr_via_rotMatrix( arr_in, angle_in_degrees,\n",
    "                              rot_axis='z', rot_center=np.array([0,0])):\n",
    "    \"\"\"\n",
    "    Rotate full array using the rotation matrix.\n",
    "    \"\"\"\n",
    "\n",
    "    Ny, Ny = arr_in.shape[0], arr_in.shape[1]\n",
    "    \n",
    "    arr_rotated = np.zeros( (Nx,Ny) )\n",
    "\n",
    "    for xx in range(Nx):\n",
    "        for yy in range(Ny):\n",
    "            if arr_in[xx,yy] > 0:\n",
    "                # rotate via rotation matrix\n",
    "                coords_new = rotate_via_rotMatrix( np.array([xx,yy]), angle_in_degrees, \n",
    "                                                   rot_axis='z', \n",
    "                                                   rot_center=rot_center )\n",
    "                if check_boundaries(coords_new, np.array([0,0]), np.array([Nx, Ny])):\n",
    "                    arr_rotated[ coords_new[0], coords_new[1] ] = 1\n",
    "                \n",
    "    return arr_rotated"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "2e0151c2-b15c-4e8d-a123-8a2b1ba76d5d",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "def rotate_arr_via_shearing( arr_in, angle_in_degrees, \n",
    "                             rot_axis='z', rot_center=np.array([0,0]) ):\n",
    "    \"\"\"\n",
    "    Rotate full array using the shearing method.\n",
    "    \"\"\"\n",
    "    \n",
    "    Ny, Ny = arr_in.shape[0], arr_in.shape[1]\n",
    "        \n",
    "    arr_rotated = np.zeros( (Nx,Ny) )\n",
    "\n",
    "    for xx in range(Nx):\n",
    "        for yy in range(Ny):\n",
    "            if arr_in[xx,yy] > 0:\n",
    "                # rotate via rotation matrix\n",
    "                coords_new = rotate_via_shearing( np.array([xx,yy]), angle_in_degrees, \n",
    "                                                  rot_axis='z', \n",
    "                                                  rot_center=rot_center )\n",
    "                if check_boundaries(coords_new, np.array([0,0]), np.array([Nx, Ny])):\n",
    "                    arr_rotated[ coords_new[0], coords_new[1] ] = 1\n",
    "                \n",
    "    return arr_rotated"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "6528efab-ef29-467c-996c-91d2e0bb6eba",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "2fce546d368e4657abd9baf459397cbf",
       "version_major": 2,
       "version_minor": 0
      },
      "image/png": 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",
      "text/html": [
       "\n",
       "            <div style=\"display: inline-block;\">\n",
       "                <div class=\"jupyter-widgets widget-label\" style=\"text-align: center;\">\n",
       "                    Figure\n",
       "                </div>\n",
       "                <img 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' width=1000.0/>\n",
       "            </div>\n",
       "        "
      ],
      "text/plain": [
       "Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "angle_in_degrees = 20\n",
    "rot_center = np.array( [round(Nx/2), round(Ny/2)] )\n",
    "\n",
    "space_invader_rotated1 = rotate_arr_via_rotMatrix( space_invader, angle_in_degrees, rot_center=rot_center )\n",
    "space_invader_rotated2 = rotate_arr_via_shearing( space_invader, angle_in_degrees, rot_center=rot_center )\n",
    "\n",
    "fig2 = plt.figure( figsize=(10,4) )   # (width, height) in inches\n",
    "\n",
    "# original image\n",
    "ax2_1 = fig2.add_subplot( 1,3,1 )      # rows, cols, plotnumber\n",
    "img_original = ax2_1.imshow(space_invader)\n",
    "ax2_1.set_title ('original image')\n",
    "\n",
    "# rotated via rotation matrix\n",
    "ax2_2  = fig2.add_subplot( 1,3,2 )      # rows, cols, plotnumber\n",
    "img_rotated_shear = ax2_2.imshow(space_invader_rotated1)\n",
    "ax2_2.set_title ('rotated via rotation matrix')\n",
    "\n",
    "# rotated via rotation matrix\n",
    "ax2_3  = fig2.add_subplot( 1,3,3 )      # rows, cols, plotnumber\n",
    "img_rotated_shear = ax2_3.imshow(space_invader_rotated2)\n",
    "ax2_3.set_title ('rotated via shearings')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "adf5ae8c-f31b-44aa-b490-26ed01c01a45",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "def plot_image(ax, arr_in, angle_in_degrees=0, rot='shear', rot_center=np.array([0,0])):\n",
    "    if rot == 'shear':\n",
    "        img_rot = rotate_arr_via_shearing( arr_in, angle_in_degrees, rot_center=rot_center )\n",
    "    elif rot == 'matrix':\n",
    "        img_rot = rotate_arr_via_rotMatrix( arr_in, angle_in_degrees, rot_center=rot_center )\n",
    "    img = ax.imshow(img_rot)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "31701dbd-c904-4228-9ae6-7499b47c2698",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "c260cdbd97354b2291e29b77bd2e4414",
       "version_major": 2,
       "version_minor": 0
      },
      "image/png": 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74YUXstLS0mxrmzO1tbXZwIED8zIry7LsjjvuyCIie+yxx3LLDj300LzH2pRsq8+QIUOyiMgefPDBvOXnn39+FhHZ7373u9yytWvXZsOGDcuGDh2aW68Nrbss+9t6+kczZszISkpK8tbT2Wef3eA6+secbOrfobFjx+b97i644IKstLQ0W716dQNr5G9aKs+3tP42/32bM2dOvZf943N+3rx5WURk11xzTfbqq69mXbt2rfO3FbJM7m7+OcXI3bvvvjuLiOypp55q8DqbtzF79eqVvffee7nl9957bxYR2X333Zdbdvjhh2cjR47MNmzYkPfYDjjggGzEiBG5ZY39vWze1t5pp53qZPi2bIc3dtv12WefzSIiu/POOxtcP5AyHyGERGzatCl++9vfxvjx42OnnXbKLR8wYEB84QtfiMcffzyqq6vzbnPGGWds9RhR8+fPj40bN8ZZZ52Vt/zcc89t9Gxdu3aNL37xi7nzHTt2jFGjRsWrr76aW1ZaWpp796i2tjbee++92LhxY+yzzz6xePHiRv+sxvjyl7+c+3ePHj1il112iS5dusRJJ52UW77LLrtEjx498mYsKyvLHVdg06ZN8e6770bXrl1jl112yZvxwQcfjIEDB8axxx6bW1ZeXh5nnHFG3hxLliyJpUuXxhe+8IV4991345133ol33nkn1q1bF4cffng89thjUVtb26yPHT6pr371q3nnH3jggYiImDp1at7yCy+8MCIi76NsDWnsa6ohjX0Nbdq0KebNmxfjx4+PwYMH527/qU99Kqqqqrb6c0pKSmLChAnxwAMPxAcffJBbfvvtt8fAgQNzH+2pT3Nk27Bhw+rM+cADD8SoUaPyfnbXrl3jK1/5Srz22mvxwgsvbPV+O3XqlPv3unXr4p133okDDjggsiyLZ599tlGzfdwn+Tv0la98Je8jiQcffHBs2rQpXn/99Ub9zELn+daUlZXFlClTGnXdI444Is4888y46qqr4oQTTojy8vJGf5yKtknutnzu9ujRIyIi7r///vjoo4+2eN2TTz45dthhh9z5zXuqbc6a9957Lx5++OE46aSTYu3atbl19e6770ZVVVUsXbo03nzzzYho+u9l0qRJeRm+JY3ZDm/stuvmPazmzZtX78c7IXUKLEjE22+/HevXr49ddtmlzmWf+tSnora2NpYvX563fPNHKbZk838idt5557zlPXv2zPujvSU77rhjnYPC77DDDvH+++/nLZs7d27sscceUV5eHr169Yo+ffrEb37zm2b9TH15eXn06dMnb1lFRUW9M1ZUVOTNWFtbG//rf/2vGDFiRJSVlUXv3r2jT58+8d///d95M77++usxfPjwOvf3j+tw6dKlEfG3jZA+ffrknW644YaoqalxPAGS8Y958frrr0e7du3qPK/79+8fPXr0aFQB0djXVEMa+xp6++2348MPP4wRI0bUuY/6MrM+J598cnz44Ye545Z88MEH8cADD8SECRO2+qUX25pt9WX166+/3mDeb758a954442YPHly9OzZM7p27Rp9+vSJQw89NCLiE2XPJ/k79PH/2EZE7u/KP/59qE9L5PnWDBw4sEkHj/7e974XPXv2jCVLlsSPfvSj6Nu3b6NvS9sjd1s+dw899NA48cQT48orr4zevXvHcccdFzfeeGPeMcY221p+vfzyy5FlWUybNq3Oupo+fXpE/O0jixFN/700Zht+s8Zshzd223XYsGExderUuOGGG6J3795RVVUVs2fPtr3KdsMxsGA71th3brZVQ3t5Zf//eA4REf/+7/8ekydPjvHjx8c3vvGN6Nu3b5SWlsaMGTPilVdeKfgsjZnx2muvjWnTpsXpp58eV199dfTs2TPatWsX559//ifaU2rzbb773e82eDyGrl27Nvl+oRAayotP8o2lm23ra6qxr6H6/uPRVPvvv38MHTo07rjjjvjCF74Q9913X3z44Ydx8sknb/F2zZFthcjqTZs2xb/8y7/Ee++9FxdffHHsuuuu0aVLl3jzzTdj8uTJLbb3Z2Oyt6m3bck8b+rv5tlnn839h/X555+PiRMnNun2tC1yt+Vzt6SkJO66665YtGhR3HfffTFv3rw4/fTT4/vf/34sWrQob7tsa1mzeV1ddNFFDe51trkgaurvpSnZsy05W5/vf//7MXny5Lj33nvjt7/9bZx33nkxY8aMWLRoUe5LoSBVCixIRJ8+faJz587x0ksv1bnsxRdfjHbt2sWgQYOafL9DhgyJiL+9i/Txd3vefffdRr1D3lh33XVX7LTTTvHrX/86b8Ns8ztUKbjrrrtizJgx8W//9m95y1evXh29e/fOnR8yZEi88MILkWVZ3mN5+eWX8243fPjwiIjo3r17jB07toCTQ/MbMmRI1NbWxtKlS3N7/URErFq1KlavXp3LjoiG/7PV2NdUQ7dv7GuoT58+0alTp9yeAx9XX2Y25KSTToof/vCHUV1dHbfffnsMHTo09t9//y3eplDZNmTIkAbzfvPlEQ2vu+effz7+9Kc/xdy5c+NLX/pSbvlDDz1U57qN/c9yof4OFcK2Pvc+iXXr1sWUKVNit912iwMOOCBmzpwZxx9/fO7b3mBr5G7L5e7+++8f+++/f3z729+OW265JU499dS47bbb8j62vDWbP0rdoUOHrW7nNfb3UiiN3XbdbOTIkTFy5Mj41re+Fb///e/jwAMPjDlz5sQ111xT8FlhW/gIISSitLQ0jjjiiLj33nvjtddeyy1ftWpV3HLLLXHQQQdF9+7dm3y/hx9+eLRv3z6uu+66vOU/+clPtnXkPJvfHfr4u0FPPPFELFy4sFl/zrYoLS2t827VnXfemTt+wWZVVVXx5ptv5n1F8oYNG+LnP/953vX23nvvGD58eHzve9/LO77DZo35Knkols997nMR8bdvu/q4H/zgBxERcdRRR+WWdenSJVavXl3nPhr7murSpUtERJ37aOxrqLS0NKqqquKee+6JN954I3f5H//4x5g3b94WHmW+k08+OWpqamLu3Lnx4IMP5h1nqSGFyrbPfe5z8eSTT+bdz7p16+JnP/tZDB06NHbbbbeIaHjd1TdXlmXxwx/+sM7Paug+/lGh/g4VwrY+9z6Jiy++ON54442YO3du/OAHP4ihQ4fGpEmTmmVPFdoGuVv43H3//ffrrJ/Ne5o19bXat2/fGD16dFx//fXxl7/8pc7lH9/Oa+zvpVAau+1aXV0dGzduzFs2cuTIaNeunSxju2APLEjINddcEw899FAcdNBBcdZZZ0X79u3j+uuvj5qampg5c+Ynus9+/frF17/+9fj+978fxx57bBx55JHx3HPPxX/+539G7969m+3d6aOPPjp+/etfx/HHHx9HHXVULFu2LObMmRO77bZbvRtIxXD00UfHVVddFVOmTIkDDjggnn/++bj55pvzDlYc8bevnf/JT34SEydOjK9//esxYMCAuPnmm6O8vDwi/v6uZrt27eKGG26IcePGxe677x5TpkyJgQMHxptvvhmPPPJIdO/ePe67774Wf5zQGHvuuWdMmjQpfvazn8Xq1avj0EMPjSeffDLmzp0b48ePjzFjxuSuu/fee8d1110X11xzTey8887Rt2/fOOywwxr9mho+fHj06NEj5syZE926dYsuXbrEfvvtF8OGDWv0a+jKK6+MBx98MA4++OA466yzYuPGjfHjH/84dt999/jv//7vRj3mz3zmM7HzzjvHZZddFjU1NVv9GEtE4bLtkksuiVtvvTXGjRsX5513XvTs2TPmzp0by5Yti1/96le5gwE3tO523XXXGD58eFx00UXx5ptvRvfu3eNXv/pVvXvW7r333hERcd5550VVVVWUlpbGKaecUu9chfg7VAjN8dxriocffjh++tOfxvTp0+Mzn/lMRETceOONMXr06Jg2bVpS64Z0yd3C5+7cuXPjpz/9aRx//PExfPjwWLt2bfz85z+P7t275wrEppg9e3YcdNBBMXLkyDjjjDNip512ilWrVsXChQvjz3/+czz33HO5mRvzeymUxm67Pvzww3HOOefEhAkT4p/+6Z9i48aN8ctf/jJKS0vjxBNPbJFZYZu06HceAlu1ePHirKqqKuvatWvWuXPnbMyYMdnvf//7vOts/vry+r4iePNly5Ytyy3buHFjNm3atKx///5Zp06dssMOOyz74x//mPXq1Sv76le/mrteQ1/fu/vuu9f5OZMmTcqGDBmSO19bW5tde+212ZAhQ7KysrLsn//5n7P777+/zvWyrO7Xttdn81ccf/zrnydNmpR16dKlznUbmnHIkCHZUUcdlTu/YcOG7MILL8wGDBiQderUKTvwwAOzhQsX1vt16a+++mp21FFHZZ06dcr69OmTXXjhhdmvfvWrLCKyRYsW5V332WefzU444YSsV69eWVlZWTZkyJDspJNOyubPn7/FxwgtYfPXub/99tt1Lvvoo4+yK6+8Mhs2bFjWoUOHbNCgQdmll16a93XhWZZlK1euzI466qisW7duWUTkXi9NeU3de++92W677Za1b9++zmu7sa+hRx99NNt7772zjh07ZjvttFM2Z86c3ONrrMsuuyyLiGznnXeu9/J/nL0p2Vaff8yhj3vllVeyz3/+81mPHj2y8vLybNSoUdn9999f53oNrbsXXnghGzt2bNa1a9esd+/e2RlnnJE999xzddbvxo0bs3PPPTfr06dPVlJSkre+6svjbfk7VN/fkfq0ZJ43tP4a+lmbL9t8P9XV1dmQIUOyz3zmM9lHH32Ud70LLrgga9euXbZw4cItPl7aFrmbryVzd/HixdnEiROzwYMHZ2VlZVnfvn2zo48+Onv66adz19m8jfnd7363zu3ry8RXXnkl+9KXvpT1798/69ChQzZw4MDs6KOPzu66667cdRr7e9mckXfeeWedn70t2+FZ1rht11dffTU7/fTTs+HDh2fl5eVZz549szFjxmT/5//8ny2tVkhGSZZ9wqO/Adu11atXxw477BDXXHNNXHbZZcUeZ7swa9asuOCCC+LPf/5zDBw4sNjjAABAg2y70to4Bha0AR9++GGdZZuPvzB69OiWHWY78Y/rbMOGDXH99dfHiBEjbAAAAJAU2660BY6BBW3A7bffHjfddFN87nOfi65du8bjjz8et956axxxxBFx4IEHFnu8JJ1wwgkxePDg2GuvvWLNmjXx7//+7/Hiiy/GzTffXOzRAAAgj21X2gIFFrQBe+yxR7Rv3z5mzpwZ1dXVuQO7+6rchlVVVcUNN9wQN998c2zatCl22223uO222xp18FEAAGhJtl1pCxwDCwAAAICkOQYWAAAAAElTYAEAAACQNMfAotnV1tbGihUrolu3blFSUlLscYDtUJZlsXbt2qisrIx27VrPey3yEdhW8hGgfq01H/k7BRbNbsWKFTFo0KBijwG0AsuXL48dd9yx2GM0G/kINBf5CFC/1paP/J0Ci2bXrVu3iIg4KD4X7aNDkacBtkcb46N4PB7I5UlrIR+BbSUfAerXWvORv1Ng0ew27/bdPjpE+xIbIMAn8P+/H7e1fYxEPgLbTD4C1K+V5iN/54OhAAAAACRNgUW9Zs+eHUOHDo3y8vLYb7/94sknnyz2SADJkJEA9ZOPABSKAos6br/99pg6dWpMnz49Fi9eHHvuuWdUVVXFW2+9VezRAIpORgLUTz4CUEgKLOr4wQ9+EGeccUZMmTIldtttt5gzZ0507tw5/vf//t/FHg2g6GQkQP3kIwCFpMAiz1//+td45plnYuzYsbll7dq1i7Fjx8bChQuLOBlA8clIgPrJRwAKzbcQkuedd96JTZs2Rb9+/fKW9+vXL1588cV6b1NTUxM1NTW589XV1QWdEaBYmpqR8hFoK+QjAIVmDyy22YwZM6KioiJ3GjRoULFHAkiCfASon3wEoKkUWOTp3bt3lJaWxqpVq/KWr1q1Kvr371/vbS699NJYs2ZN7rR8+fKWGBWgxTU1I+Uj0FbIRwAKTYFFno4dO8bee+8d8+fPzy2rra2N+fPnx2c/+9l6b1NWVhbdu3fPOwG0Rk3NSPkItBXyEYBCcwws6pg6dWpMmjQp9tlnnxg1alTMmjUr1q1bF1OmTCn2aABFJyMB6icfASgkBRZ1nHzyyfH222/H5ZdfHitXroy99torHnzwwToH5QRoi2QkQP3kIwCFVJJlWVbsIWhdqquro6KiIkbHcdG+pEOxxwG2Qxuzj2JB3Btr1qxpVR8rkY/AtpKPAPVrrfnI3zkGFgAAAABJU2ABAAAAkDQFFgAAAABJU2ABAAAAkDQFFgAAAABJU2ABAAAAkDQFFgAAAABJU2ABAAAAkDQFFgAAAABJU2ABAAAAkDQFFgAAAABJU2ABAAAAkDQFFgAAAABJU2ABAAAAkDQFFgAAAABJU2ABAAAAkDQFFgAAAABJU2ABAAAAkDQFFgAAAABJU2ABAAAAkDQFFgAAAABJU2ABAAAAkDQFFgAAAABJU2BRx4wZM2LfffeNbt26Rd++fWP8+PHx0ksvFXssgKKTjwD1k48AFJoCizoeffTROPvss2PRokXx0EMPxUcffRRHHHFErFu3rtijARSVfASon3wEoNDaF3sA0vPggw/mnb/pppuib9++8cwzz8QhhxxSpKkAik8+AtRPPgJQaPbAYqvWrFkTERE9e/Ys8iQAaZGPAPWTjwA0N3tgsUW1tbVx/vnnx4EHHhif/vSn671OTU1N1NTU5M5XV1e31HgARSMfAeonHwEoBHtgsUVnn312/N//+3/jtttua/A6M2bMiIqKitxp0KBBLTghQHHIR4D6yUcACqEky7Ks2EOQpnPOOSfuvffeeOyxx2LYsGENXq++d9AGDRoUo+O4aF/SoSVGBVqZjdlHsSDujTVr1kT37t2LPU4d8hEoFvkIUL/U85Ft5yOE1JFlWZx77rlx9913x4IFC7a48RERUVZWFmVlZS00HUDxyEeA+slHAApNgUUdZ599dtxyyy1x7733Rrdu3WLlypUREVFRURGdOnUq8nQAxSMfAeonHwEoNMfAoo7rrrsu1qxZE6NHj44BAwbkTrfffnuxRwMoKvkIUD/5CECh2QOLOhwWDaB+8hGgfvIRgEKzBxYAAAAASVNgAQAAAJA0BRYAAAAASVNgAQAAAJA0BRYAAAAASVNgAQAAAJA0BRYAAAAASVNgAQAAAJA0BRYAAAAASVNgAQAAAJA0BRYAAAAASVNgAQAAAJA0BRYAAAAASVNgAQAAAJA0BRYAAAAASVNgAQAAAJA0BRYAAAAASVNgAQAAAJA0BRYAAAAASWtf7AGgLZi3Ykmz3E9V5V7Ncj/8XXP8bvxegNTINoDCkbFQHPbAAgAAACBpCiwAAAAAkqbAAgAAACBpCiwAAAAAkqbAYov+5//8n1FSUhLnn39+sUcBSIp8BGiYjASguSmwaNBTTz0V119/feyxxx7FHgUgKfIRoGEyEoBCUGBRrw8++CBOPfXU+PnPfx477LBDsccBSIZ8BGiYjASgUBRY1Ovss8+Oo446KsaOHbvV69bU1ER1dXXeCaC1ko8ADWtsRspHAJqqfbEHID233XZbLF68OJ566qlGXX/GjBlx5ZVXFngqgOKTjwANa0pGykcAmsoeWORZvnx5fP3rX4+bb745ysvLG3WbSy+9NNasWZM7LV++vMBTArQ8+QjQsKZmpHwEoKnsgUWeZ555Jt566634zGc+k1u2adOmeOyxx+InP/lJ1NTURGlpad5tysrKoqysrKVHBWhR8hGgYU3NSPkIQFMpsMhz+OGHx/PPP5+3bMqUKbHrrrvGxRdfXOc/ZwBthXwEaJiMBKDQFFjk6datW3z605/OW9alS5fo1atXneUAbYl8BGiYjASg0BwDCwAAAICk2QOLrVqwYEGxRwBIknwEaJiMBKA52QMLAAAAgKQpsAAAAABImo8QQguoqtyrWe5n3ool23wfzTVLCqwPKC6vwcKwXoGI5smCCHnwj2QsbL/sgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACStfbEHgLZg3oolxR4hJ6VZUtAc66Oqcq9tvg/YHqX03Jdt+VrT+kjpeQYtqbme+60pD1KRyjqVj7Q19sACAAAAIGkKLAAAAACSpsACAAAAIGkKLOp4880344tf/GL06tUrOnXqFCNHjoynn3662GMBJEFGAtRPPgJQSA7iTp73338/DjzwwBgzZkz853/+Z/Tp0yeWLl0aO+ywQ7FHAyg6GQlQP/kIQKEpsMjzne98JwYNGhQ33nhjbtmwYcOKOBFAOmQkQP3kIwCF5iOE5PmP//iP2GeffWLChAnRt2/f+Od//uf4+c9/vsXb1NTURHV1dd4JoDVqakbKR6CtkI8AFJoCizyvvvpqXHfddTFixIiYN29efO1rX4vzzjsv5s6d2+BtZsyYERUVFbnToEGDWnBigJbT1IyUj0BbIR8BKLSSLMuyYg9BOjp27Bj77LNP/P73v88tO++88+Kpp56KhQsX1nubmpqaqKmpyZ2vrq6OQYMGxeg4LtqXdCj4zNuDeSuWFHsECqiqcq9ij9DqbMw+igVxb6xZsya6d+9e7HFympqR8rHlyNnWS8bmk480lXxsveRjvlTzkeZjDyzyDBgwIHbbbbe8ZZ/61KfijTfeaPA2ZWVl0b1797wTQGvU1IyUj0BbIR8BKDQFFnkOPPDAeOmll/KW/elPf4ohQ4YUaSKAdMhIgPrJRwAKTYFFngsuuCAWLVoU1157bbz88stxyy23xM9+9rM4++yziz0aQNHJSID6yUcACk2BRZ5999037r777rj11lvj05/+dFx99dUxa9asOPXUU4s9GkDRyUiA+slHAAqtfbEHID1HH310HH300cUeAyBJMhKgfvIRgEKyBxYAAAAASVNgAQAAAJA0HyGErZi3YkmxR8ipqtxrm+8jpcezrVJZH81xH83xWGB71JoyKSWp5GNzaK455Czbm1Reg62NfKxLPrK9sAcWAAAAAElTYAEAAACQNAUWAAAAAElTYAEAAACQNAUWAAAAAElTYAEAAACQNAUWAAAAAElTYAEAAACQNAUWAAAAAElTYAEAAACQNAUWAAAAAElTYAEAAACQNAUWAAAAAElTYAEAAACQNAUWAAAAAElTYAEAAACQtPbFHgAKad6KJcUeoVm1tsezrVrT+miux1JVuVez3A80RkqvweZ47qf0eLZVKo/F74W2KqXnrddhvlQei98LNJ09sAAAAABImgILAAAAgKQpsAAAAABImgKLPJs2bYpp06bFsGHDolOnTjF8+PC4+uqrI8uyYo8GUHQyEqB+8hGAQnMQd/J85zvfieuuuy7mzp0bu+++ezz99NMxZcqUqKioiPPOO6/Y4wEUlYwEqJ98BKDQFFjk+f3vfx/HHXdcHHXUURERMXTo0Lj11lvjySefLPJkAMUnIwHqJx8BKDQfISTPAQccEPPnz48//elPERHx3HPPxeOPPx7jxo0r8mQAxScjAeonHwEoNHtgkeeSSy6J6urq2HXXXaO0tDQ2bdoU3/72t+PUU09t8DY1NTVRU1OTO19dXd0SowK0uKZmpHwE2gr5CECh2QOLPHfccUfcfPPNccstt8TixYtj7ty58b3vfS/mzp3b4G1mzJgRFRUVudOgQYNacGKAltPUjJSPQFshHwEoNAUWeb7xjW/EJZdcEqecckqMHDkyTjvttLjgggtixowZDd7m0ksvjTVr1uROy5cvb8GJAVpOUzNSPgJthXwEoNB8hJA869evj3bt8nvN0tLSqK2tbfA2ZWVlUVZWVujRAIquqRkpH4G2Qj4CUGgKLPIcc8wx8e1vfzsGDx4cu+++ezz77LPxgx/8IE4//fRijwZQdDISoH7yEYBCU2CR58c//nFMmzYtzjrrrHjrrbeisrIyzjzzzLj88suLPRpA0clIgPrJRwAKTYFFnm7dusWsWbNi1qxZxR4FIDkyEqB+8hGAQnMQdwAAAACSpsACAAAAIGk+QghbUVW51zbfx7wVS7b5PkiX5wjbo9b2nGttjwcontaWJ63t8QBtlz2wAAAAAEiaAgsAAACApCmwAAAAAEiaAgsAAACApCmwAAAAAEiaAgsAAACApCmwAAAAAEiaAgsAAACApCmwAAAAAEiaAgsAAACApCmwAAAAAEiaAgsAAACApCmwAAAAAEiaAgsAAACApCmwAAAAAEiaAgsAAACApLUv9gBA41VV7lXsEVqdeSuWFHsE2G41Ryal9BqUsflS+d34vbA9ko+tWyq/G78X2hp7YAEAAACQNAUWAAAAAElTYAEAAACQNAVWG/PYY4/FMcccE5WVlVFSUhL33HNP3uVZlsXll18eAwYMiE6dOsXYsWNj6dKlxRkWoAXJR4CGyUgAik2B1casW7cu9txzz5g9e3a9l8+cOTN+9KMfxZw5c+KJJ56ILl26RFVVVWzYsKGFJwVoWfIRoGEyEoBi8y2Ebcy4ceNi3Lhx9V6WZVnMmjUrvvWtb8Vxxx0XERG/+MUvol+/fnHPPffEKaec0pKjArQo+QjQMBkJQLHZA4ucZcuWxcqVK2Ps2LG5ZRUVFbHffvvFwoULizgZQHHJR4CGyUgAWoI9sMhZuXJlRET069cvb3m/fv1yl9WnpqYmampqcuerq6sLMyBAkchHgIZ9koyUjwA0lT2w2GYzZsyIioqK3GnQoEHFHgkgCfIRoH7yEYCmUmCR079//4iIWLVqVd7yVatW5S6rz6WXXhpr1qzJnZYvX17QOQFamnwEaNgnyUj5CEBTKbDIGTZsWPTv3z/mz5+fW1ZdXR1PPPFEfPazn23wdmVlZdG9e/e8E0BrIh8BGvZJMlI+AtBUjoHVxnzwwQfx8ssv584vW7YslixZEj179ozBgwfH+eefH9dcc02MGDEihg0bFtOmTYvKysoYP3588YYGaAHyEaBhMhKAYlNgtTFPP/10jBkzJnd+6tSpERExadKkuOmmm+Kb3/xmrFu3Lr7yla/E6tWr46CDDooHH3wwysvLizUyQIuQjwANk5EAFJsCq40ZPXp0ZFnW4OUlJSVx1VVXxVVXXdWCUwEUn3wEaJiMBKDYHAMLAAAAgKQpsAAAAABImo8Q0qpVVe5V7BEiovnmmLdiSbPcz7ZqjsfTmh5Lc0hlDtheyaV8qTwWoPhSyZRU5mgOttugOOyBBQAAAEDSFFgAAAAAJE2BBQAAAEDSFFgAAAAAJE2BBQAAAEDSFFgAAAAAJE2BBQAAAEDSFFgAAAAAJE2BBQAAAEDSFFgAAAAAJE2BBQAAAEDSFFgAAAAAJE2BBQAAAEDSFFgAAAAAJE2BBQAAAEDSFFgAAAAAJK19sQcAGq+qcq9ij9BsWtNjge1RKq/BVOaISGuWbdWaHgu0tFReP801x7wVS7b5PlrbOgG2T/bAAgAAACBpCiwAAAAAkqbAAgAAACBpCqw25rHHHotjjjkmKisro6SkJO65557cZR999FFcfPHFMXLkyOjSpUtUVlbGl770pVixYkXxBgZoIfIRoGEyEoBiU2C1MevWrYs999wzZs+eXeey9evXx+LFi2PatGmxePHi+PWvfx0vvfRSHHvssUWYFKBlyUeAhslIAIrNtxC2MePGjYtx48bVe1lFRUU89NBDect+8pOfxKhRo+KNN96IwYMHt8SIAEUhHwEaJiMBKDZ7YLFFa9asiZKSkujRo0exRwFIinwEaJiMBKC52QOLBm3YsCEuvvjimDhxYnTv3r3B69XU1ERNTU3ufHV1dUuMB1A08hGgYY3JSPkIQFPZA4t6ffTRR3HSSSdFlmVx3XXXbfG6M2bMiIqKitxp0KBBLTQlQMuTjwANa2xGykcAmkqBRR2bNzxef/31eOihh7a4d0FExKWXXhpr1qzJnZYvX95CkwK0LPkI0LCmZKR8BKCpfISQPJs3PJYuXRqPPPJI9OrVa6u3KSsri7KyshaYDqB45CNAw5qakfIRgKZSYLUxH3zwQbz88su588uWLYslS5ZEz549Y8CAAfH5z38+Fi9eHPfff39s2rQpVq5cGRERPXv2jI4dOxZrbICCk48ADZORABRbSZZlWbGHoOUsWLAgxowZU2f5pEmT4oorrohhw4bVe7tHHnkkRo8e3aifUV1dHRUVFTE6jov2JR22ZVygjdqYfRQL4t5Ys2bNVj+m11zkI7A9KEY+RhQ+I+Vj4cxbsWSb76Oqcq9tvg8otGLlIy3HHlhtzOjRo2NLnaU+E2ir5CNAw2QkAMXmIO4AAAAAJE2BBQAAAEDSFFgAAAAAJM0xsAAAAFopB2AHWgt7YAEAAACQNAUWAAAAAElTYAEAAACQNAUWAAAAAElTYAEAAACQNAUWAAAAAElTYAEAAACQNAUWAAAAAElTYAEAAACQNAUWAAAAAElTYAEAAACQNAUWAAAAAElTYAEAAACQNAUWAAAAAElTYAEAAACQNAUWAAAAAElTYAEAAACQNAUWAAAAAElTYAEAAACQNAUWAAAAAElTYLUxjz32WBxzzDFRWVkZJSUlcc899zR43a9+9atRUlISs2bNarH5AIpFPgI0TEYCUGwKrDZm3bp1seeee8bs2bO3eL277747Fi1aFJWVlS00GUBxyUeAhslIAIqtfbEHoGWNGzcuxo0bt8XrvPnmm3HuuefGvHnz4qijjmqhyQCKSz4CNExGAlBsCizy1NbWxmmnnRbf+MY3Yvfdd2/UbWpqaqKmpiZ3vrq6ulDjARSNfARoWFMzUj4C0FQ+Qkie73znO9G+ffs477zzGn2bGTNmREVFRe40aNCgAk4IUBzyEaBhTc1I+QhAUymwyHnmmWfihz/8Ydx0001RUlLS6NtdeumlsWbNmtxp+fLlBZwSoOXJR4CGfZKMlI8ANJUCi5zf/e538dZbb8XgwYOjffv20b59+3j99dfjwgsvjKFDhzZ4u7KysujevXveCaA1kY8ADfskGSkfAWgqx8Ai57TTTouxY8fmLauqqorTTjstpkyZUqSpAIpPPgI0TEYC0BIUWG3MBx98EC+//HLu/LJly2LJkiXRs2fPGDx4cPTq1Svv+h06dIj+/fvHLrvs0tKjArQo+QjQMBkJQLEpsNqYp59+OsaMGZM7P3Xq1IiImDRpUtx0001Fmgqg+OQjQMNkJADFpsBqY0aPHh1ZljX6+q+99lrhhgFIiHwEaJiMBKDYHMQdAAAAgKQpsAAAAABImgILAAAAgKQpsAAAAABImgILAAAAgKQpsAAAAABImgILAAAAgKQpsAAAAABImgILAAAAgKQpsAAAAABImgILAAAAgKQpsAAAAABImgILAAAAgKQpsAAAAABImgILAAAAgKQpsAAAAABImgILAAAAgKQpsAAAAABIWvtiD0Drk2VZRERsjI8isiIPA2yXNsZHEfH3PGkt5COwreQjQP1aaz7ydwosmt3atWsjIuLxeKDIkwDbu7Vr10ZFRUWxx2g28hFoLvIRoH6tLR/5u5JMPUkzq62tjRUrVkS3bt2ipKSk3utUV1fHoEGDYvny5dG9e/cWnrB1sk4Lw3ptfo1Zp1mWxdq1a6OysjLatWs9n3aXj8VhnRaG9dr85KN8bGnWaWFYr82vLecjf2cPLJpdu3btYscdd2zUdbt37y7Um5l1WhjWa/Pb2jptje+cycfisk4Lw3ptfvJxyzznmp91WhjWa/Nri/nI36klAQAAAEiaAgsAAACApCmwKIqysrKYPn16lJWVFXuUVsM6LQzrtflZp1tm/TQ/67QwrNfmZ51umfXT/KzTwrBem591SoSDuAMAAACQOHtgAQAAAJA0BRYAAAAASVNgAQAAAJA0BRYtbvbs2TF06NAoLy+P/fbbL5588slij7Rdu+KKK6KkpCTvtOuuuxZ7rO3KY489Fsccc0xUVlZGSUlJ3HPPPXmXZ1kWl19+eQwYMCA6deoUY8eOjaVLlxZn2O3I1tbr5MmT6zx3jzzyyOIMmwj52Lzk47aTj4UhHz8ZGdl85GPzkJHNTz6yJQosWtTtt98eU6dOjenTp8fixYtjzz33jKqqqnjrrbeKPdp2bffdd4+//OUvudPjjz9e7JG2K+vWrYs999wzZs+eXe/lM2fOjB/96EcxZ86ceOKJJ6JLly5RVVUVGzZsaOFJty9bW68REUceeWTec/fWW29twQnTIh8LQz5uG/lYGPKx6WRk85OP205GNj/5yBZl0IJGjRqVnX322bnzmzZtyiorK7MZM2YUcart2/Tp07M999yz2GO0GhGR3X333bnztbW1Wf/+/bPvfve7uWWrV6/OysrKsltvvbUIE26f/nG9ZlmWTZo0KTvuuOOKMk+K5GPzk4/NSz4WhnxsHBnZvORj85ORzU8+8o/sgUWL+etf/xrPPPNMjB07NresXbt2MXbs2Fi4cGERJ9v+LV26NCorK2OnnXaKU089Nd54441ij9RqLFu2LFauXJn3vK2oqIj99tvP87YZLFiwIPr27Ru77LJLfO1rX4t333232CMVhXwsHPlYOPKxsOTj38nIwpCPhSUjC0c+tl0KLFrMO++8E5s2bYp+/frlLe/Xr1+sXLmySFNt//bbb7+46aab4sEHH4zrrrsuli1bFgcffHCsXbu22KO1Cpufm563ze/II4+MX/ziFzF//vz4zne+E48++miMGzcuNm3aVOzRWpx8LAz5WFjysXDkYz4Z2fzkY+HJyMKQj21b+2IPAGybcePG5f69xx57xH777RdDhgyJO+64I/7H//gfRZwMtuyUU07J/XvkyJGxxx57xPDhw2PBggVx+OGHF3EyWgv5yPZKPlJo8pHtlXxs2+yBRYvp3bt3lJaWxqpVq/KWr1q1Kvr371+kqVqfHj16xD/90z/Fyy+/XOxRWoXNz03P28Lbaaedonfv3m3yuSsfW4Z8bF7yseW05XyMkJEtQT42PxnZMtp6PrY1CixaTMeOHWPvvfeO+fPn55bV1tbG/Pnz47Of/WwRJ2tdPvjgg3jllVdiwIABxR6lVRg2bFj0798/73lbXV0dTzzxhOdtM/vzn/8c7777bpt87srHliEfm5d8bDltOR8jZGRLkI/NT0a2jLaej22NjxDSoqZOnRqTJk2KffbZJ0aNGhWzZs2KdevWxZQpU4o92nbroosuimOOOSaGDBkSK1asiOnTp0dpaWlMnDix2KNtNz744IO8d22WLVsWS5YsiZ49e8bgwYPj/PPPj2uuuSZGjBgRw4YNi2nTpkVlZWWMHz++eENvB7a0Xnv27BlXXnllnHjiidG/f/945ZVX4pvf/GbsvPPOUVVVVcSpi0c+Nj/5uO3kY2HIx6aTkc1LPjYPGdn85CNbVOyvQaTt+fGPf5wNHjw469ixYzZq1Khs0aJFxR5pu3byySdnAwYMyDp27JgNHDgwO/nkk7OXX3652GNtVx555JEsIuqcJk2alGXZ374Gedq0aVm/fv2ysrKy7PDDD89eeuml4g69HdjSel2/fn12xBFHZH369Mk6dOiQDRkyJDvjjDOylStXFnvsopKPzUs+bjv5WBjy8ZORkc1HPjYPGdn85CNbUpJlWdYSRRkAAAAAfBKOgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0hRYAAAAACRNgQUAAABA0v4fcMBPvMU3Ff0AAAAASUVORK5CYII=",
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       "\n",
       "            <div style=\"display: inline-block;\">\n",
       "                <div class=\"jupyter-widgets widget-label\" style=\"text-align: center;\">\n",
       "                    Figure\n",
       "                </div>\n",
       "                <img src='data:image/png;base64,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' 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       "            </div>\n",
       "        "
      ],
      "text/plain": [
       "Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "69d92d256cc64e58a831c610e98e198a",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "IntSlider(value=0, max=90)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "angle_in_degrees = 10\n",
    "rot_center = np.array( [round(Nx/2), round(Ny/2)] )\n",
    "\n",
    "fig3 = plt.figure( figsize=(12,6) )   # (width, height) in inches\n",
    "\n",
    "# original image\n",
    "ax3_1 = fig3.add_subplot( 1,3,1 )      # rows, cols, plotnumber\n",
    "# rotated via rotation matrix\n",
    "ax3_2 = fig3.add_subplot( 1,3,2 )      # rows, cols, plotnumber\n",
    "# rotated via rotation matrix\n",
    "ax3_3 = fig3.add_subplot( 1,3,3 )      # rows, cols, plotnumber\n",
    "\n",
    "img_original = ax3_1.imshow(space_invader)\n",
    "ax3_1.set_title ('original image')\n",
    "\n",
    "plot_image(ax3_2, space_invader, angle_in_degrees=angle_in_degrees, rot='matrix', rot_center=rot_center)\n",
    "ax3_2.set_title ('rotated via rotation matrix')\n",
    "\n",
    "plot_image(ax3_3, space_invader, angle_in_degrees=angle_in_degrees, rot='shear', rot_center=rot_center)\n",
    "ax3_3.set_title ('rotated via shearings')\n",
    "\n",
    "plt.show()\n",
    "\n",
    "slider_angle = widgets.IntSlider(value=0, min=0, max=90, step=1)\n",
    "\n",
    "# callback function for the slider widget\n",
    "def handle_slider_change(change):\n",
    "    # update rotation via matrix \n",
    "    plot_image(ax3_2, space_invader, angle_in_degrees=change.new, rot='matrix', rot_center=rot_center)\n",
    "    # update rotation via shear\n",
    "    plot_image(ax3_3, space_invader, angle_in_degrees=change.new, rot='shear', rot_center=rot_center)\n",
    "    fig3.canvas.draw()\n",
    "\n",
    "# connect handle_slider_change function to slider widget using the .observe() method, observing changes in value attribute\n",
    "slider_angle.observe(handle_slider_change, 'value')\n",
    "slider_angle"
   ]
  }
 ],
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