07-background_removal_with_robust_PCA.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Background Removal with Robust PCA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Parte 1: Execução do tutorial exemplo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Getting Started"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import requests, zipfile, os\n",
    "url = \"https://download1337.mediafire.com/42s21eo7zgjg/rkxwc8t203b1m2m/data.zip\"\n",
    "filename = \"data.zip\"\n",
    "with open(filename, \"wb\") as f:\n",
    "    r = requests.get(url)\n",
    "    f.write(r.content)\n",
    "zip_ref = zipfile.ZipFile(filename)\n",
    "zip_ref.extractall(\"./\")\n",
    "zip_ref.close()\n",
    "os.remove(filename)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Import needed libraries\n",
    "import moviepy.editor as mpe\n",
    "import numpy as np\n",
    "import scipy\n",
    "from PIL import Image\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "video = mpe.VideoFileClip(\"PETS09-S2L2.avi\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "video.subclip(0,50).ipython_display(width=300)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "video.duration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "scale = 25\n",
    "dims = (int(240 * (scale / 100)), int(320 * (scale / 100)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Helper Methods"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def create_data_matrix_from_video(clip, k=5, scale=50):\n",
    "  new_scale = (int(320 * (scale / 100)), int(240 * (scale / 100)))\n",
    "  return np.vstack([np.array(Image.fromarray(rgb2gray(clip.get_frame(i / float(k))).astype(np.uint8)).resize(size=new_scale)).flatten() for i in range(k * int(clip.duration))]).T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def rgb2gray(rgb):\n",
    "  return np.dot(rgb[...,:3], [0.299, 0.587, 0.114])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plt_images(M, A, E, index_array, dims, filename=None):\n",
    "    f = plt.figure(figsize=(15, 10))\n",
    "    r = len(index_array)\n",
    "    # pics = r * 3\n",
    "    for k, i in enumerate(index_array):\n",
    "      for j, mat in enumerate([M, A, E]):\n",
    "        sp = f.add_subplot(r, 3, 3*k + j + 1)\n",
    "        sp.axis('Off')\n",
    "        pixels = mat[:,i]\n",
    "        if isinstance(pixels, scipy.sparse.csr_matrix):\n",
    "            pixels = pixels.todense()\n",
    "        plt.imshow(np.reshape(pixels, dims), cmap='gray')\n",
    "    return f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plots(ims, dims, figsize=(15,20), rows=1, interp=False, titles=None):\n",
    "  if type(ims[0]) is np.ndarray:\n",
    "    ims = np.array(ims)\n",
    "  f = plt.figure(figsize=figsize)\n",
    "  for i in range(len(ims)):\n",
    "    sp = f.add_subplot(rows, len(ims)//rows, i+1)\n",
    "    sp.axis('Off')\n",
    "    plt.imshow(np.reshape(ims[i], dims), cmap=\"gray\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Load and view the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "M = create_data_matrix_from_video(video, 100, scale)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(dims, M.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(M[:, 140], dims), cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.save(\"low_PETS09-S2L2_matrix.npy\", M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(12, 12))\n",
    "plt.imshow(M, cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imsave(fname=\"image1.jpg\", arr=np.reshape(M[:,140], dims), cmap='gray')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### SVD"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn import decomposition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "u, s, v = decomposition.randomized_svd(M, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "u.shape, s.shape, v.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "low_rank = u @ np.diag(s) @ v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "low_rank.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(12, 12))\n",
    "plt.imshow(low_rank, cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(low_rank[:,140], dims), cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(M[:,550] - low_rank[:,550], dims), cmap='gray');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "u, s, v = decomposition.randomized_svd(M, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "u.shape, s.shape, v.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "low_rank = u @ np.diag(s) @ v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "low_rank.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(12, 12))\n",
    "plt.imshow(low_rank, cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(M[:,550] - low_rank[:,550], dims), cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(M[:,140] - low_rank[:,140], dims), cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(M[:,140] - low_rank[:,140], dims)[10:45, 23:70], cmap='gray')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Principal Component Analysis (PCA)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import sparse\n",
    "from sklearn.utils.extmath import randomized_svd\n",
    "import fbpca"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "TOL=1e-9\n",
    "MAX_ITERS=3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def converged(Z, d_norm):\n",
    "    err = np.linalg.norm(Z, 'fro') / d_norm\n",
    "    print('error: ', err)\n",
    "    return err < TOL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def shrink(M, tau):\n",
    "    S = np.abs(M) - tau\n",
    "    return np.sign(M) * np.where(S>0, S, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def _svd(M, rank): return fbpca.pca(M, k=min(rank, np.min(M.shape)), raw=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def norm_op(M): return _svd(M, 1)[1][0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def svd_reconstruct(M, rank, min_sv):\n",
    "    u, s, v = _svd(M, rank)\n",
    "    s -= min_sv\n",
    "    nnz = (s > 0).sum()\n",
    "    return u[:,:nnz] @ np.diag(s[:nnz]) @ v[:nnz], nnz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def pcp(X, maxiter=10, k=10): # refactored\n",
    "    m, n = X.shape\n",
    "    trans = m<n\n",
    "    if trans: X = X.T; m, n = X.shape\n",
    "        \n",
    "    lamda = 1/np.sqrt(m)\n",
    "    op_norm = norm_op(X)\n",
    "    Y = np.copy(X) / max(op_norm, np.linalg.norm( X, np.inf) / lamda)\n",
    "    mu = k*1.25/op_norm; mu_bar = mu * 1e7; rho = k * 1.5\n",
    "    \n",
    "    d_norm = np.linalg.norm(X, 'fro')\n",
    "    L = np.zeros_like(X); sv = 1\n",
    "    \n",
    "    examples = []\n",
    "    \n",
    "    for i in range(maxiter):\n",
    "        print(\"rank sv:\", sv)\n",
    "        X2 = X + Y/mu\n",
    "        \n",
    "        # update estimate of Sparse Matrix by \"shrinking/truncating\": original - low-rank\n",
    "        S = shrink(X2 - L, lamda/mu)\n",
    "        \n",
    "        # update estimate of Low-rank Matrix by doing truncated SVD of rank sv & reconstructing.\n",
    "        # count of singular values > 1/mu is returned as svp\n",
    "        L, svp = svd_reconstruct(X2 - S, sv, 1/mu)\n",
    "        \n",
    "        # If svp < sv, you are already calculating enough singular values.\n",
    "        # If not, add 20% (in this case 240) to sv\n",
    "        sv = svp + (1 if svp < sv else round(0.05*n))\n",
    "        \n",
    "        # residual\n",
    "        Z = X - L - S\n",
    "        Y += mu*Z; mu *= rho\n",
    "        \n",
    "        examples.extend([S[140,:], L[140,:]])\n",
    "        \n",
    "        if m > mu_bar: m = mu_bar\n",
    "        if converged(Z, d_norm): break\n",
    "    \n",
    "    if trans: L=L.T; S=S.T\n",
    "    return L, S, examples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m, n = M.shape\n",
    "round(m * .05)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "L, S, examples = pcp(M, maxiter=5, k=10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plots(examples, dims, rows=5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = plt_images(M, S, L, [140], dims)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.save(\"high_res_L.npy\", L)\n",
    "np.save(\"high_res_S.npy\", S)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = plt_images(M, S, L, [0, 100, 1000], dims)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Parte 2: Aplicação em Problema Real"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exemplo 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "real_video_1 = mpe.VideoFileClip(\"TUD-Crossing.avi\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "M_1 = create_data_matrix_from_video(real_video_1, 100, scale)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(dims, M_1.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(M_1[:, 140], dims), cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.save(\"low_TUD-Crossing_matrix.npy\", M_1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(12, 12))\n",
    "plt.imshow(M_1, cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imsave(fname=\"image2.jpg\", arr=np.reshape(M_1[:,140], dims), cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "u_1, s_1, v_1 = decomposition.randomized_svd(M_1, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "u_1.shape, s_1.shape, v_1.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "low_rank_1 = u_1 @ np.diag(s_1) @ v_1\n",
    "low_rank_1.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(12, 12))\n",
    "plt.imshow(low_rank_1, cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(low_rank_1[:,140], dims), cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(M_1[:,550] - low_rank_1[:,550], dims), cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "u_1, s_1, v_1 = decomposition.randomized_svd(M_1, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "u_1.shape, s_1.shape, v_1.shape\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "low_rank_1 = u_1 @ np.diag(s_1) @ v_1\n",
    "low_rank_1.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(12, 12))\n",
    "plt.imshow(low_rank_1, cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(M_1[:,550] - low_rank_1[:,550], dims), cmap='gray')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(M_1[:,140] - low_rank_1[:,140], dims), cmap='gray')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(M_1[:,140] - low_rank_1[:,140], dims)[10:45, 23:70], cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m, n = M_1.shape\n",
    "round(m * .05)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "L, S, examples = pcp(M_1, maxiter=5, k=10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = plt_images(M_1, S, L, [140], dims)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = plt_images(M_1, S, L, [140], dims)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.save(\"high_res_L1.npy\", L)\n",
    "np.save(\"high_res_S1.npy\", S)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = plt_images(M_1, S, L, [0, 100, 1000], dims)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exemplo 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "real_video_2 = mpe.VideoFileClip(\"AVG-TownCentre.avi\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "M_2 = create_data_matrix_from_video(real_video_2, 100, scale)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(dims, M_2.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(M_2[:, 140], dims), cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.save(\"low_TUD-Crossing_matrix.npy\", M_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(12, 12))\n",
    "plt.imshow(M_2, cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imsave(fname=\"image3.jpg\", arr=np.reshape(M_2[:,140], dims), cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "u, s, v = decomposition.randomized_svd(M_2, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "u.shape, s.shape, v.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "low_rank = u @ np.diag(s) @ v\n",
    "low_rank.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(12, 12))\n",
    "plt.imshow(low_rank, cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(low_rank[:,140], dims), cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(M_2[:,550] - low_rank[:,550], dims), cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "u, s, v = decomposition.randomized_svd(M_2, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "u.shape, s.shape, v.shape\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "low_rank = u @ np.diag(s) @ v\n",
    "low_rank.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(12, 12))\n",
    "plt.imshow(low_rank, cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(M_2[:,550] - low_rank[:,550], dims), cmap='gray')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(M_2[:,140] - low_rank[:,140], dims), cmap='gray')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.imshow(np.reshape(M_2[:,140] - low_rank[:,140], dims)[10:45, 23:70], cmap='gray')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m, n = M_2.shape\n",
    "round(m * .05)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "L, S, examples = pcp(M_2, maxiter=5, k=10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = plt_images(M_2, S, L, [140], dims)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = plt_images(M_2, S, L, [140], dims)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.save(\"high_res_L2.npy\", L)\n",
    "np.save(\"high_res_S2.npy\", S)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = plt_images(M_2, S, L, [0, 100, 1000], dims)"
   ]
  }
 ],
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