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    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
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      "source": [
        "%matplotlib inline"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "\n# Bilinear Interpolation\nThis example shows how to use the :py:class:`pylops.signalprocessing.Bilinar`\noperator to perform bilinear interpolation to a 2-dimensional input vector.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import numpy as np\nimport matplotlib.pyplot as plt\nimport matplotlib.gridspec as pltgs\nfrom scipy import misc\n\nimport pylops\n\nplt.close('all')\nnp.random.seed(0)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "First of all, we create a 2-dimensional input vector containing an image\nfrom the ``scipy.misc`` family.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "x = misc.face()[::5, ::5, 0]\nnz, nx = x.shape"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "We can now define a set of available samples in the\nfirst and second direction of the array and apply bilinear interpolation.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "nsamples = 2000\niava = np.vstack((np.random.uniform(0, nz-1, nsamples),\n                  np.random.uniform(0, nx-1, nsamples)))\n\nBop = pylops.signalprocessing.Bilinear(iava, (nz, nx))\ny = Bop * x.ravel()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "At this point we try to reconstruct the input signal imposing a smooth\nsolution by means of a regularization term that minimizes the Laplacian of\nthe solution.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "D2op = pylops.Laplacian((nz, nx), weights=(1, 1), dtype='float64')\n\nxadj = Bop.H * y\nxinv = pylops.optimization.leastsquares.NormalEquationsInversion(Bop, [D2op], y,\n                                                                 epsRs=[np.sqrt(0.1)],\n                                                                 returninfo=False,\n                                                                 **dict(maxiter=100))\nxadj = xadj.reshape(nz, nx)\nxinv = xinv.reshape(nz, nx)\n\nfig, axs = plt.subplots(1, 3, figsize=(10, 4))\nfig.suptitle('Bilinear interpolation', fontsize=14,\n             fontweight='bold', y=0.95)\naxs[0].imshow(x, cmap='gray_r', vmin=0, vmax=250)\naxs[0].axis('tight')\naxs[0].set_title('Original')\naxs[1].imshow(xadj, cmap='gray_r', vmin=0, vmax=250)\naxs[1].axis('tight')\naxs[1].set_title('Sampled')\naxs[2].imshow(xinv, cmap='gray_r', vmin=0, vmax=250)\naxs[2].axis('tight')\naxs[2].set_title('2D Regularization')\nplt.tight_layout()\nplt.subplots_adjust(top=0.8)"
      ]
    }
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