Course-1/Week-3/C1_W3_Lab_3_linear_transformations.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "EAt-K2qgcIou"
   },
   "source": [
    "# Linear Transformations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "FZYK-0rin5x7"
   },
   "source": [
    "In this lab you will explore linear transformations, visualize their results and master matrix multiplication to apply various linear transformations."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Table of Contents\n",
    "- [ 1 - Transformations](#1)\n",
    "- [ 2 - Linear Transformations](#2)\n",
    "- [ 3 - Transformations Defined as a Matrix Multiplication](#3)\n",
    "- [ 4 - Standard Transformations in a Plane](#4)\n",
    "  - [ 4.1 - Example 1: Horizontal Scaling (Dilation)](#4.1)\n",
    "  - [ 4.2 - Example 2: Reflection about y-axis (the vertical axis)](#4.2)\n",
    "- [ 5 - Application of Linear Transformations: Computer Graphics](#5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "XI8PBrk_2Z4V"
   },
   "source": [
    "## Packages\n",
    "\n",
    "Run the following cell to load the package you'll need."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "tags": [
     "graded"
    ]
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "# OpenCV library for image transformations.\n",
    "import cv2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name='1'></a>\n",
    "## 1 - Transformations\n",
    "\n",
    "A **transformation** is a function from one vector space to another that respects the underlying (linear) structure of each vector space. Referring to a specific transformation, you can use a symbol, such as $T$. Specifying the spaces containing the input and output vectors, e.g. $\\mathbb{R}^2$ and $\\mathbb{R}^3$, you can write $T: \\mathbb{R}^2 \\rightarrow \\mathbb{R}^3$. Transforming vector $v \\in \\mathbb{R}^2$ into the vector $w\\in\\mathbb{R}^3$ by the transformation $T$, you can use the notation $T(v)=w$ and read it as \"*T of v equals to w*\" or \"*vector w is an **image** of vector v with the transformation T*\".\n",
    "\n",
    "The following Python function corresponds to the transformation $T: \\mathbb{R}^2 \\rightarrow \\mathbb{R}^3$ with the following symbolic formula:\n",
    "\n",
    "$$T\\begin{pmatrix}\n",
    "          \\begin{bmatrix}\n",
    "           v_1 \\\\           \n",
    "           v_2\n",
    "          \\end{bmatrix}\\end{pmatrix}=\n",
    "          \\begin{bmatrix}\n",
    "           3v_1 \\\\\n",
    "           0 \\\\\n",
    "           -2v_2\n",
    "          \\end{bmatrix}\n",
    "          \\tag{1}\n",
    "          $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "id": "Je3yV0Wnn5x8",
    "scrolled": true,
    "tags": [
     "graded"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Original vector:\n",
      " [[3]\n",
      " [5]] \n",
      "\n",
      " Result of the transformation:\n",
      " [[  9.]\n",
      " [  0.]\n",
      " [-10.]]\n"
     ]
    }
   ],
   "source": [
    "def T(v):\n",
    "    w = np.zeros((3,1))\n",
    "    w[0,0] = 3*v[0,0]\n",
    "    w[2,0] = -2*v[1,0]\n",
    "    \n",
    "    return w\n",
    "\n",
    "v = np.array([[3], [5]])\n",
    "w = T(v)\n",
    "\n",
    "print(\"Original vector:\\n\", v, \"\\n\\n Result of the transformation:\\n\", w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name='2'></a>\n",
    "## 2 - Linear Transformations\n",
    "\n",
    "A transformation $T$ is said to be **linear** if the following two properties are true for any scalar $k$, and any input vectors $u$ and $v$:\n",
    "\n",
    "1. $T(kv)=kT(v)$,\n",
    "2. $T(u+v)=T(u)+T(v)$.\n",
    "\n",
    "In the example above $T$ is a linear transformation:\n",
    "\n",
    "$$T (kv) =\n",
    "          T \\begin{pmatrix}\\begin{bmatrix}\n",
    "          kv_1 \\\\\n",
    "          kv_2\n",
    "          \\end{bmatrix}\\end{pmatrix} = \n",
    "          \\begin{bmatrix}\n",
    "           3kv_1 \\\\\n",
    "           0 \\\\\n",
    "           -2kv_2\n",
    "          \\end{bmatrix} =\n",
    "          k\\begin{bmatrix}\n",
    "           3v_1 \\\\\n",
    "           0 \\\\\n",
    "           -2v_2\n",
    "          \\end{bmatrix} = \n",
    "          kT(v),\\tag{2}$$\n",
    "          \n",
    "$$T (u+v) =\n",
    "          T \\begin{pmatrix}\\begin{bmatrix}\n",
    "          u_1 + v_1 \\\\\n",
    "          u_2 + v_2\n",
    "          \\end{bmatrix}\\end{pmatrix} = \n",
    "          \\begin{bmatrix}\n",
    "           3(u_1+v_1) \\\\\n",
    "           0 \\\\\n",
    "           -2(u_2+v_2)\n",
    "          \\end{bmatrix} = \n",
    "          \\begin{bmatrix}\n",
    "           3u_1 \\\\\n",
    "           0 \\\\\n",
    "           -2u_2\n",
    "          \\end{bmatrix} +\n",
    "          \\begin{bmatrix}\n",
    "           3v_1 \\\\\n",
    "           0 \\\\\n",
    "           -2v_2\n",
    "          \\end{bmatrix} = \n",
    "          T(u)+T(v).\\tag{3}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can change the values of $k$ or vectors $u$ and $v$ in the cell below, to check that this is true for some specific values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "tags": [
     "graded"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "T(k*v):\n",
      " [[ 42.]\n",
      " [  0.]\n",
      " [-56.]] \n",
      " k*T(v):\n",
      " [[ 42.]\n",
      " [  0.]\n",
      " [-56.]] \n",
      "\n",
      "\n",
      "T(u+v):\n",
      " [[ 9.]\n",
      " [ 0.]\n",
      " [-4.]] \n",
      " T(u)+T(v):\n",
      " [[ 9.]\n",
      " [ 0.]\n",
      " [-4.]]\n"
     ]
    }
   ],
   "source": [
    "u = np.array([[1], [-2]])\n",
    "v = np.array([[2], [4]])\n",
    "\n",
    "k = 7\n",
    "\n",
    "print(\"T(k*v):\\n\", T(k*v), \"\\n k*T(v):\\n\", k*T(v), \"\\n\\n\")\n",
    "print(\"T(u+v):\\n\", T(u+v), \"\\n T(u)+T(v):\\n\", T(u)+T(v))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some examples of linear transformations are rotations, reflections, scaling (dilations), etc. In this lab you will explore a few of them."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name='3'></a>\n",
    "## 3 - Transformations Defined as a Matrix Multiplication\n",
    "\n",
    "Let $L: \\mathbb{R}^m \\rightarrow \\mathbb{R}^n$ be defined by a matrix $A$, where $L(v)=Av$, multiplication of the matrix $A$ ($n\\times m$) and vector $v$ ($m\\times 1$) resulting in the vector $w$ ($n\\times 1$).\n",
    "\n",
    "Now try to guess, what should be the elements of matrix $A$, corresponding to the transformation $L: \\mathbb{R}^2 \\rightarrow \\mathbb{R}^3$:\n",
    "\n",
    "$$L\\begin{pmatrix}\n",
    "          \\begin{bmatrix}\n",
    "           v_1 \\\\           \n",
    "           v_2\n",
    "          \\end{bmatrix}\\end{pmatrix}=\n",
    "          \\begin{bmatrix}\n",
    "           3v_1 \\\\\n",
    "           0 \\\\\n",
    "           -2v_2\n",
    "          \\end{bmatrix}=\n",
    "          \\begin{bmatrix}\n",
    "           ? & ? \\\\\n",
    "           ? & ? \\\\\n",
    "           ? & ?\n",
    "          \\end{bmatrix}\n",
    "          \\begin{bmatrix}\n",
    "           v_1 \\\\\n",
    "           v_2\n",
    "          \\end{bmatrix}\n",
    "          \\tag{4}\n",
    "          $$\n",
    "\n",
    "To do that, write the transformation $L$ as $Av$ and then perform matrix multiplication:\n",
    "    $$L\\begin{pmatrix}\n",
    "          \\begin{bmatrix}\n",
    "           v_1 \\\\           \n",
    "           v_2\n",
    "          \\end{bmatrix}\\end{pmatrix}=\n",
    "          A\\begin{bmatrix}\n",
    "           v_1 \\\\           \n",
    "           v_2\n",
    "          \\end{bmatrix}=\n",
    "          \\begin{bmatrix}\n",
    "           a_{1,1} & a_{1,2} \\\\\n",
    "           a_{2,1} & a_{2,2} \\\\\n",
    "           a_{3,1} & a_{3,2}\n",
    "          \\end{bmatrix}\n",
    "          \\begin{bmatrix}\n",
    "           v_1 \\\\           \n",
    "           v_2\n",
    "          \\end{bmatrix}=\n",
    "          \\begin{bmatrix}\n",
    "           a_{1,1}v_1+a_{1,2}v_2 \\\\\n",
    "           a_{2,1}v_1+a_{2,2}v_2 \\\\\n",
    "           a_{3,1}v_1+a_{3,2}v_2 \\\\\n",
    "          \\end{bmatrix}=\n",
    "          \\begin{bmatrix}\n",
    "           3v_1 \\\\\n",
    "           0 \\\\\n",
    "           -2v_2\n",
    "          \\end{bmatrix}\\tag{5}\n",
    "          $$\n",
    "          \n",
    "Can you see now what should be the values of the elements $a_{i,j}$ of matrix $A$ to make the equalities $(5)$ correct? Find out the answer in the following code cell:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "tags": [
     "graded"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Transformation matrix:\n",
      " [[ 3  0]\n",
      " [ 0  0]\n",
      " [ 0 -2]] \n",
      "\n",
      "Original vector:\n",
      " [[3]\n",
      " [5]] \n",
      "\n",
      " Result of the transformation:\n",
      " [[  9]\n",
      " [  0]\n",
      " [-10]]\n"
     ]
    }
   ],
   "source": [
    "def L(v):\n",
    "    A = np.array([[3,0], [0,0], [0,-2]])\n",
    "    print(\"Transformation matrix:\\n\", A, \"\\n\")\n",
    "    w = A @ v\n",
    "    \n",
    "    return w\n",
    "\n",
    "v = np.array([[3], [5]])\n",
    "w = L(v)\n",
    "\n",
    "print(\"Original vector:\\n\", v, \"\\n\\n Result of the transformation:\\n\", w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Every linear transformation can be carried out by matrix multiplication. And vice versa, carrying out matrix multiplication, it is natural to consider the linear transformation that it represents. It means you can associate the matrix with the linear transformation in some way. This is a key connection between linear transformations and matrix algebra."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name='4'></a>\n",
    "## 4 - Standard Transformations in a Plane\n",
    "\n",
    "As discussed above in section [3](#3), a linear transformation $L: \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2$ can be represented as a multiplication of a $2 \\times 2$ matrix and a coordinate vector $v\\in\\mathbb{R}^2.$ Note that so far you have been using some random vector $v\\in\\mathbb{R}^2.$ (e.g. $v=\\begin{bmatrix}3 \\\\ 5\\end{bmatrix}$). To have a better intuition of what the transformation is really doing in the $\\mathbb{R}^2$ space, it is wise to choose vector $v$ in a less random way. \n",
    "\n",
    "A good choice would be vectors of a standard basis $e_1=\\begin{bmatrix}1 \\\\ 0\\end{bmatrix}$ and $e_2=\\begin{bmatrix}0 \\\\ 1\\end{bmatrix}$. Let's apply linear transformation $L$ to each of the vectors $e_1$ and $e_2$: $L(e_1)=Ae_1$ and $L(e_2)=Ae_2$. If you put vectors $\\{e_1, e_2\\}$ into columns of a matrix and perform matrix multiplication\n",
    "\n",
    "$$A\\begin{bmatrix}e_1 & e_2\\end{bmatrix}=\\begin{bmatrix}Ae_1 & Ae_2\\end{bmatrix}=\\begin{bmatrix}L(e_1) & L(e_2)\\end{bmatrix},\\tag{3}$$\n",
    "\n",
    "you can note that $\\begin{bmatrix}e_1 & e_2\\end{bmatrix}=\\begin{bmatrix}1 & 0 \\\\ 0 & 1\\end{bmatrix}$ (identity matrix). Thus, $A\\begin{bmatrix}e_1 & e_2\\end{bmatrix} = AI=A$, and\n",
    "\n",
    "$$A=\\begin{bmatrix}L(e_1) & L(e_2)\\end{bmatrix}.\\tag{4}$$\n",
    "\n",
    "This is a matrix with the columns that are the images of the vectors of the standard basis. \n",
    "\n",
    "This choice of vectors \\{$e_1, e_2$\\} provides opportinuty for the visual representation of the linear transformation $L$ (you will see the examples below)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name='4.1'></a>\n",
    "### 4.1 - Example 1: Horizontal Scaling (Dilation)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Horizontal scaling (factor $2$ in this example) can be defined considering transformation of a vector $e_1=\\begin{bmatrix}1 \\\\ 0\\end{bmatrix}$ into a vector $\\begin{bmatrix}2 \\\\ 0\\end{bmatrix}$ and leaving vector $e_2=\\begin{bmatrix}0 \\\\ 1\\end{bmatrix}$ without any changes. The following function `T_hscaling()` corresponds to the horizontal scaling (factor $2$) of a vector. The second function `transform_vectors()` applies defined transformation to a set of vectors (here two vectors)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "tags": [
     "graded"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Original vectors:\n",
      " e1= \n",
      " [[1]\n",
      " [0]] \n",
      " e2=\n",
      " [[0]\n",
      " [1]] \n",
      "\n",
      " Result of the transformation (matrix form):\n",
      " [[2 0]\n",
      " [0 1]]\n"
     ]
    }
   ],
   "source": [
    "def T_hscaling(v):\n",
    "    A = np.array([[2,0], [0,1]])\n",
    "    w = A @ v\n",
    "    \n",
    "    return w\n",
    "    \n",
    "    \n",
    "def transform_vectors(T, v1, v2):\n",
    "    V = np.hstack((v1.reshape(2,1), v2.reshape(2,1)))\n",
    "    W = T(V)\n",
    "    \n",
    "    return W\n",
    "    \n",
    "e1 = np.array([[1], [0]])\n",
    "e2 = np.array([[0], [1]])\n",
    "\n",
    "transformation_result_hscaling = transform_vectors(T_hscaling, e1, e2)\n",
    "\n",
    "print(\"Original vectors:\\n e1= \\n\", e1, \"\\n e2=\\n\", e2, \n",
    "      \"\\n\\n Result of the transformation (matrix form):\\n\", transformation_result_hscaling)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can get a visual understanding of the transformation, producing a plot which displays input vectors, and their transformations. Do not worry if the code in the following cell will not be clear - at this stage this is not important code to understand."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "tags": [
     "graded"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def plot_transformation(T, e1, e2):\n",
    "    color_original = \"#129cab\"\n",
    "    color_transformed = \"#cc8933\"\n",
    "    \n",
    "    _, ax = plt.subplots(figsize=(7, 7))\n",
    "    ax.tick_params(axis='x', labelsize=14)\n",
    "    ax.tick_params(axis='y', labelsize=14)\n",
    "    ax.set_xticks(np.arange(-5, 5))\n",
    "    ax.set_yticks(np.arange(-5, 5))\n",
    "    \n",
    "    plt.axis([-5, 5, -5, 5])\n",
    "    plt.quiver([0, 0],[0, 0], [e1[0], e2[0]], [e1[1], e2[1]], color=color_original, angles='xy', scale_units='xy', scale=1)\n",
    "    plt.plot([0, e2[0], e1[0], e1[0]], \n",
    "             [0, e2[1], e2[1], e1[1]], \n",
    "             color=color_original)\n",
    "    e1_sgn = 0.4 * np.array([[1] if i==0 else [i] for i in np.sign(e1)])\n",
    "    ax.text(e1[0]-0.2+e1_sgn[0], e1[1]-0.2+e1_sgn[1], f'$e_1$', fontsize=14, color=color_original)\n",
    "    e2_sgn = 0.4 * np.array([[1] if i==0 else [i] for i in np.sign(e2)])\n",
    "    ax.text(e2[0]-0.2+e2_sgn[0], e2[1]-0.2+e2_sgn[1], f'$e_2$', fontsize=14, color=color_original)\n",
    "    \n",
    "    e1_transformed = T(e1)\n",
    "    e2_transformed = T(e2)\n",
    "    \n",
    "    plt.quiver([0, 0],[0, 0], [e1_transformed[0], e2_transformed[0]], [e1_transformed[1], e2_transformed[1]], \n",
    "               color=color_transformed, angles='xy', scale_units='xy', scale=1)\n",
    "    plt.plot([0,e2_transformed[0], e1_transformed[0]+e2_transformed[0], e1_transformed[0]], \n",
    "             [0,e2_transformed[1], e1_transformed[1]+e2_transformed[1], e1_transformed[1]], \n",
    "             color=color_transformed)\n",
    "    e1_transformed_sgn = 0.4 * np.array([[1] if i==0 else [i] for i in np.sign(e1_transformed)])\n",
    "    ax.text(e1_transformed[0]-0.2+e1_transformed_sgn[0], e1_transformed[1]-e1_transformed_sgn[1], \n",
    "            f'$T(e_1)$', fontsize=14, color=color_transformed)\n",
    "    e2_transformed_sgn = 0.4 * np.array([[1] if i==0 else [i] for i in np.sign(e2_transformed)])\n",
    "    ax.text(e2_transformed[0]-0.2+e2_transformed_sgn[0], e2_transformed[1]-e2_transformed_sgn[1], \n",
    "            f'$T(e_2)$', fontsize=14, color=color_transformed)\n",
    "    \n",
    "    plt.gca().set_aspect(\"equal\")\n",
    "    plt.show()\n",
    "    \n",
    "plot_transformation(T_hscaling, e1, e2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can observe that the polygon has been stretched in the horizontal direction as a result of the transformation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name='4.2'></a>\n",
    "### 4.2 - Example 2: Reflection about y-axis (the vertical axis)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Function `T_reflection_yaxis()` defined below corresponds to the reflection about y-axis:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "tags": [
     "graded"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Original vectors:\n",
      " e1= \n",
      " [[1]\n",
      " [0]] \n",
      " e2=\n",
      " [[0]\n",
      " [1]] \n",
      "\n",
      " Result of the transformation (matrix form):\n",
      " [[-1  0]\n",
      " [ 0  1]]\n"
     ]
    }
   ],
   "source": [
    "def T_reflection_yaxis(v):\n",
    "    A = np.array([[-1,0], [0,1]])\n",
    "    w = A @ v\n",
    "    \n",
    "    return w\n",
    "    \n",
    "e1 = np.array([[1], [0]])\n",
    "e2 = np.array([[0], [1]])\n",
    "\n",
    "transformation_result_reflection_yaxis = transform_vectors(T_reflection_yaxis, e1, e2)\n",
    "\n",
    "print(\"Original vectors:\\n e1= \\n\", e1,\"\\n e2=\\n\", e2, \n",
    "      \"\\n\\n Result of the transformation (matrix form):\\n\", transformation_result_reflection_yaxis)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can visualize this transformation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "tags": [
     "graded"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_transformation(T_reflection_yaxis, e1, e2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are many more standard linear transformations to explore. But now you have the required tools to apply them and visualize the results."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a name='5'></a>\n",
    "## 5 - Application of Linear Transformations: Computer Graphics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A large number of basic geometric shapes is used in computer graphics. Such shapes (e.g. triangles, quadrilaterals) are defined by their vertexes (corners). Linear transformations are often used to generate complex shapes from the basic ones, though scaling, reflection, rotation, shearing etc. It provides opportunity to manipulate those shapes efficiently. \n",
    "\n",
    "The software responsible for rendering of a graphics, has to process the coordinates of millions of vertexes. The use of matrix multiplication to manipulate coordinates helps to merge multiple transformations together, just applying matrix multiplication one by one in a sequence. And another advantage is that the dedicated hardware, such as Graphics Processing Units (GPUs), is designed specifically to handle these calculations in large numbers with high speed.\n",
    "\n",
    "So, matrix multiplication and linear transformations give you a super power, especially on scale!\n",
    "\n",
    "Here is an example where linear transformations could have helped to reduce the amount of work preparing the image:\n",
    "\n",
    "\n",
    "<img src = \"images/barnsley_fern.png\" width=\"200\" align=\"center\"/>\n",
    "\n",
    "\n",
    "All of the subleafs are similar and can be prepared as just linear transformations of one original leaf.\n",
    "\n",
    "Let's see a simple example of two transformations applied to a leaf image. For the image transformations you can use an `OpenCV` library. First, upload and show the image:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7f1e05309b80>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "img = cv2.imread('images/leaf_original.png', 0)\n",
    "plt.imshow(img)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course, this is just a very simple leaf image (not a real example in preparation of the proper art work), but it will help you to get the idea how a few transformations can be applied in a row. Try to rotate the image 90 degrees clockwise and then apply a shear transformation, which can be visualized as:\n",
    "\n",
    "<img src = \"images/shear_transformation.png\" width=\"400\" align=\"center\"/>\n",
    "\n",
    "Rotate the image:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7f1e052d7460>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "image_rotated = cv2.rotate(img, cv2.ROTATE_90_CLOCKWISE)\n",
    "\n",
    "plt.imshow(image_rotated)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Applying the shear you will get the following output:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7f1e0522ed30>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "rows,cols = image_rotated.shape\n",
    "# 3 by 3 matrix as it is required for the OpenCV library, don't worry about the details of it for now.\n",
    "M = np.float32([[1, 0.5, 0], [0, 1, 0], [0, 0, 1]])\n",
    "image_rotated_sheared = cv2.warpPerspective(image_rotated, M, (int(cols), int(rows)))\n",
    "plt.imshow(image_rotated_sheared)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What if you will apply those two transformations in the opposite order? Do you think the result will be the same? Run the following code to check that:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7f1e052137c0>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "image_sheared = cv2.warpPerspective(img, M, (int(cols), int(rows)))\n",
    "image_sheared_rotated = cv2.rotate(image_sheared, cv2.ROTATE_90_CLOCKWISE)\n",
    "plt.imshow(image_sheared_rotated)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comparing last two images, you can clearly see that the outputs are different. This is because linear transformation can be defined as a matrix multiplication. Then, applying two transformations in a row, e.g. with matrices $A$ and $B$, you perform multiplications $B(Av)=(BA)v$, where $v$ is a vector. And remember, that generally you cannot change the order in the matrix multiplication (most of the time $BA\\neq AB$). Let's check that! Define two matrices, corresponding to the rotation and shear transformations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "90 degrees clockwise rotation matrix:\n",
      " [[ 0  1]\n",
      " [-1  0]]\n",
      "Matrix for the shear along x-axis:\n",
      " [[1.  0.5]\n",
      " [0.  1. ]]\n"
     ]
    }
   ],
   "source": [
    "M_rotation_90_clockwise = np.array([[0, 1], [-1, 0]])\n",
    "M_shear_x = np.array([[1, 0.5], [0, 1]])\n",
    "\n",
    "print(\"90 degrees clockwise rotation matrix:\\n\", M_rotation_90_clockwise)\n",
    "print(\"Matrix for the shear along x-axis:\\n\", M_shear_x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now check that the results of their multiplications `M_rotation_90_clockwise @ M_shear_x` and `M_shear_x @ M_rotation_90_clockwise` are different:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "M_rotation_90_clockwise by M_shear_x:\n",
      " [[ 0.   1. ]\n",
      " [-1.  -0.5]]\n",
      "M_shear_x by M_rotation_90_clockwise:\n",
      " [[-0.5  1. ]\n",
      " [-1.   0. ]]\n"
     ]
    }
   ],
   "source": [
    "print(\"M_rotation_90_clockwise by M_shear_x:\\n\", M_rotation_90_clockwise @ M_shear_x)\n",
    "print(\"M_shear_x by M_rotation_90_clockwise:\\n\", M_shear_x @ M_rotation_90_clockwise)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This simple example shows that you need to be aware of the mathematical objects and their properties in the applications.\n",
    "\n",
    "Congratulations on completing this lab!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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  "accelerator": "GPU",
  "colab": {
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   "name": "C1_W1_Assignment_Solution.ipynb",
   "provenance": []
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  "coursera": {
   "schema_names": [
    "AI4MC1-1"
   ]
  },
  "grader_version": "1",
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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