{
  "cells": [
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "%matplotlib inline"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "\n# 1D Smoothing\n\nThis example shows how to use the :py:class:`pylops.Smoothing1D` operator\nto smooth an input signal along a given axis.\n\nDerivative (or roughening) operators are generally used *regularization*\nin inverse problems. Smoothing has the opposite effect of roughening and\nit can be employed as *preconditioning* in inverse problems.\n\nA smoothing operator is a simple compact filter on lenght $n_{smooth}$\nand each elements is equal to $1/n_{smooth}$.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import matplotlib.pyplot as plt\nimport numpy as np\n\nimport pylops\n\nplt.close(\"all\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Define the input parameters: number of samples of input signal (``N``) and\nlenght of the smoothing filter regression coefficients ($n_{smooth}$).\nIn this first case the input signal is one at the center and zero elsewhere.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "N = 31\nnsmooth = 7\nx = np.zeros(N)\nx[int(N / 2)] = 1\n\nSop = pylops.Smoothing1D(nsmooth=nsmooth, dims=[N], dtype=\"float32\")\n\ny = Sop * x\nxadj = Sop.H * y\n\nfig, ax = plt.subplots(1, 1, figsize=(10, 3))\nax.plot(x, \"k\", lw=2, label=r\"$x$\")\nax.plot(y, \"r\", lw=2, label=r\"$y=Ax$\")\nax.set_title(\"Smoothing in 1st direction\", fontsize=14, fontweight=\"bold\")\nax.legend()\nplt.tight_layout()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Let's repeat the same exercise with a random signal as input. After applying smoothing,\nwe will also try to invert it.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "N = 120\nnsmooth = 13\nx = np.random.normal(0, 1, N)\nSop = pylops.Smoothing1D(nsmooth=13, dims=(N), dtype=\"float32\")\n\ny = Sop * x\nxest = Sop / y\n\nfig, ax = plt.subplots(1, 1, figsize=(10, 3))\nax.plot(x, \"k\", lw=2, label=r\"$x$\")\nax.plot(y, \"r\", lw=2, label=r\"$y=Ax$\")\nax.plot(xest, \"--g\", lw=2, label=r\"$x_{ext}$\")\nax.set_title(\"Smoothing in 1st direction\", fontsize=14, fontweight=\"bold\")\nax.legend()\nplt.tight_layout()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Finally we show that the same operator can be applied to multi-dimensional\ndata along a chosen axis.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "A = np.zeros((11, 21))\nA[5, 10] = 1\n\nSop = pylops.Smoothing1D(nsmooth=5, dims=(11, 21), axis=0, dtype=\"float64\")\nB = Sop * A\n\nfig, axs = plt.subplots(1, 2, figsize=(10, 3))\nfig.suptitle(\n    \"Smoothing in 1st direction for 2d data\", fontsize=14, fontweight=\"bold\", y=0.95\n)\nim = axs[0].imshow(A, interpolation=\"nearest\", vmin=0, vmax=1)\naxs[0].axis(\"tight\")\naxs[0].set_title(\"Model\")\nplt.colorbar(im, ax=axs[0])\nim = axs[1].imshow(B, interpolation=\"nearest\", vmin=0, vmax=1)\naxs[1].axis(\"tight\")\naxs[1].set_title(\"Data\")\nplt.colorbar(im, ax=axs[1])\nplt.tight_layout()\nplt.subplots_adjust(top=0.8)"
      ]
    }
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      "file_extension": ".py",
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      "pygments_lexer": "ipython3",
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