compute_affine_transform.ipynb

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  "metadata": {
    "colab": {
      "name": "compute-affine-transform.ipynb",
      "provenance": [],
      "authorship_tag": "ABX9TyNGZmxopgbjEfz4f/sKP4xC",
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/ajdillhoff/Linear-Algebra-Examples/blob/main/compute_affine_transform.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Computing Affine Transformations\n",
        "\n",
        "One important result of solving linear systems is to compute a transformation between two different sets of points.\n",
        "\n",
        "Given a set of points from the original coordinate system and another set of the same points transformed by $A$, how do we solve for that transformation matrix?\n",
        "\n",
        "We are looking for some $A$ such that\n",
        "\n",
        "$$\n",
        "A \\mathbf{x} = \\hat{\\mathbf{x}}.\n",
        "$$\n",
        "\n",
        "In a perfect world, $\\hat{\\mathbf{x}}$ will match the target points $\\mathbf{y}$. In other words,\n",
        "\n",
        "$$\n",
        "\\|\\hat{\\mathbf{x}} - \\mathbf{y}\\|_2 = 0.\n",
        "$$\n",
        "\n",
        "**For this problem, we will assume this is the case.**\n",
        "\n",
        "For an affine transformation, we would have some matrix of the form\n",
        "\n",
        "$$\n",
        "A = \\begin{bmatrix}\n",
        "a_{11} & a_{12} & a_{13}\\\\\n",
        "a_{21} & a_{22} & a_{23}\n",
        "\\end{bmatrix}.\n",
        "$$\n",
        "\n",
        "Given a homogeneous 2D coordinate $\\mathbf{x}$, we can compute each component $A\\mathbf{x}$ as\n",
        "\n",
        "\\begin{align*}\n",
        "\\hat{x}_1 &= a_{11} * x_1 + a_{12} * x_2 + a_{13} * 1\\\\\n",
        "\\hat{x}_2 &= a_{21} * x_1 + a_{22} * x_2 + a_{23} * 1\\\\\n",
        "\\end{align*}\n",
        "\n",
        "We can fit this using a least squares approach by the following construction.\n",
        "\n",
        "\\begin{equation*}\n",
        "\\begin{bmatrix}\n",
        "x_1^{(1)} & x_2^{(1)} & 1 & 0 & 0 & 0\\\\\n",
        "0 & 0 & 0 & x_1^{(1)} & x_2^{(1)} & 1\\\\\n",
        "&& \\vdots\\\\\n",
        "x_1^{(n)} & x_2^{(n)} & 1 & 0 & 0 & 0\\\\\n",
        "0 & 0 & 0 & x_1^{(n)} & x_2^{(n)} & 1\\\\\n",
        "\\end{bmatrix}\n",
        "\\begin{bmatrix}\n",
        "a_{11}\\\\\n",
        "a_{12}\\\\\n",
        "a_{13}\\\\\n",
        "a_{21}\\\\\n",
        "a_{22}\\\\\n",
        "a_{23}\\\\\n",
        "\\end{bmatrix}=\n",
        "\\begin{bmatrix}\n",
        "\\hat{x}_1^{(1)}\\\\\n",
        "\\hat{x}_2^{(1)}\\\\\n",
        "\\vdots\\\\\n",
        "\\hat{x}_1^{(n)}\\\\\n",
        "\\hat{x}_2^{(n)}\\\\\n",
        "\\end{bmatrix}\n",
        "\\end{equation*}\n",
        "\n",
        "We can solve this using the *normal equations*:\n",
        "\n",
        "$$\n",
        "A^T A \\mathbf{x} = A^T \\mathbf{b}.\n",
        "$$\n"
      ],
      "metadata": {
        "id": "h-gpXkZKGxP_"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ],
      "metadata": {
        "id": "e4maPg_sJ4Yy"
      },
      "execution_count": 3,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "\n",
        "def compute_affine_transform(src, dst):\n",
        "    \"\"\"Computes the affine transformation matrix `A` such that A @ src = dst.\"\"\"\n",
        "    num_samples = src.shape[0]\n",
        "    \n",
        "    # Convert to homogeneous coordinates\n",
        "    src_affine = np.concatenate((src, np.ones((num_samples, 1))), axis=1)\n",
        "    zero_buffer = np.zeros_like(src_affine)\n",
        "    r1 = np.concatenate((src_affine, zero_buffer), axis=1)\n",
        "    r2 = np.concatenate((zero_buffer, src_affine), axis=1)\n",
        "    \n",
        "    # Construct our design matrix X by interleaving the two equations required\n",
        "    # for each point correspondence.\n",
        "    X = np.empty((r1.shape[0] + r2.shape[0], r1.shape[1]), dtype=r1.dtype)\n",
        "    X[0::2] = r1\n",
        "    X[1::2] = r2\n",
        "    \n",
        "    # Flatten the target points into a vector (this is the b vector from above)\n",
        "    y = dst.ravel()\n",
        "    \n",
        "    # TODO: Compute M by using the normal equations.\n",
        "    # Here, M corresponds to the solution vector x from above.\n",
        "    \n",
        "\n",
        "    # Resahpe our solution vector (x in the equations above) into a 3x3\n",
        "    # affine transformation matrix.\n",
        "    M = np.reshape(M, (2, 3))\n",
        "    M = np.vstack((M, np.zeros((1, M.shape[1]))))\n",
        "    M[2, 2] = 1\n",
        "    \n",
        "    return M"
      ],
      "metadata": {
        "id": "_DLoMbxyIylI"
      },
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# Randomly sample a pattern and then transform it by some random rotation.\n",
        "num_points = 100\n",
        "x = np.random.rand(num_points, 2) * 2 - 1\n",
        "\n",
        "# Create a random rotation matrix\n",
        "theta = np.random.uniform(0, np.pi)\n",
        "R = np.array([\n",
        "    [np.cos(theta), -np.sin(theta), 0],\n",
        "    [np.sin(theta), np.cos(theta), 0],\n",
        "    [0, 0, 1]\n",
        "])\n",
        "\n",
        "y = R @ np.concatenate((x, np.ones((100, 1))), axis=1).T\n",
        "y = y.T[:, :2]\n",
        "\n",
        "# Call `compute_affine_transform` to estimate the rotation matrix\n",
        "M = compute_affine_transform(x, y)\n",
        "\n",
        "# Compute the norm between the true matrix and estimated matrix.\n",
        "# This should be a very small number close to 0.\n",
        "estimation_error = np.linalg.norm(R - M)\n",
        "print(f\"Error = {estimation_error}\")\n",
        "\n",
        "plt.scatter(x[:, 0], x[:, 1], c='b')\n",
        "plt.scatter(y[:, 0], y[:, 1], c='r')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 300
        },
        "id": "57x1uU78Jw56",
        "outputId": "2c57ee22-73b7-4339-dc0a-4bfe9d62620c"
      },
      "execution_count": 27,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Error = 8.102184729199735e-16\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.collections.PathCollection at 0x7f656879fe90>"
            ]
          },
          "metadata": {},
          "execution_count": 27
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    }
  ]
}