{
 "metadata": {
  "name": "",
  "signature": "sha256:5b44004e97625e733a6588149bed4fc0c9eda5b70ed3c9420b3c41b63e6842a0"
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - compatibility with Python 3\n",
      "from __future__ import print_function  # print('me') instead of print 'me'\n",
      "from __future__ import division  # 1/2 == 0.5, not 0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - show figures inside the notebook\n",
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - import common modules\n",
      "import numpy as np  # the Python array package\n",
      "import matplotlib.pyplot as plt  # the Python plotting package"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - set gray colormap and nearest neighbor interpolation by default\n",
      "plt.rcParams['image.cmap'] = 'gray'\n",
      "plt.rcParams['image.interpolation'] = 'nearest'"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Smoothing and convolution"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Start by smoothing in 1D.  We first load our new fixed functional image, and get the first volume:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - get first volume from fixed functional\n",
      "import nibabel as nib\n",
      "img = nib.load('fds114_sub009_t2r1.nii')\n",
      "data = img.get_data()\n",
      "vol0 = data[..., 0]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now get a slice over the third dimension:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - get slice over third dimension\n",
      "mid_slice = vol0[:, :, 14]\n",
      "plt.imshow(mid_slice)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "<matplotlib.image.AxesImage at 0x109727610>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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RKO6JAhZGZVHfOFiE/x2L9F70Z8++0tjofqy5Ti/es0ktEuH9+HFZFHgkGqMo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\ny0pqeQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x105d0ae50>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Get one line of this slice, from left to right, from the middle of the brain:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - get middle line from middle slice\n",
      "mid_line = mid_slice[:, 25]\n",
      "plt.plot(mid_line)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "[<matplotlib.lines.Line2D at 0x109870290>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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3b4aOHXcNzr59w+ghf52NG8OUsjfe2BnCf/lL+I8eNizcDj4Y9twznFck/9a2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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10974b050>"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We are going to start by smoothing this line (1D smooth)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Copy the `fwhm2sigma` function from the [smoothing as convolution](http://perrin.dynevor.org/smoothing_as_convolution.html) notebook:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# fwhm2sigma\n",
      "def fwhm2sigma(fwhm):\n",
      "    return fwhm / np.sqrt(8 * np.log(2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - check fwhm2sigma\n",
      "assert np.allclose(fwhm2sigma(4), 1.698643600576)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We are going to smooth by 8 pixels FWHM, so we need the corresponding `sigma`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# sigma for FWHM 8\n",
      "sigma = fwhm2sigma(8)\n",
      "sigma"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "3.3972872011520763"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We use a probability density function of a normal Gaussian distribution for our kernel:\n",
      "\n",
      "$$\n",
      "f(x, \\mu, \\sigma) = p e^q\n",
      "$$\n",
      "\n",
      "where:\n",
      "\n",
      "$$\n",
      "p = \\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\\\\n",
      "q = -\\frac{(x-\\mu)^2}{2\\sigma^2}\n",
      "$$\n",
      "\n",
      "See : https://en.wikipedia.org/wiki/Normal_distribution\n",
      "\n",
      "To make our smoothing kernel, we can sample the PDF, for a given mu and sigma.\n",
      "\n",
      "We can use our old friend `scipy.stats` to sample the PDF."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - import the normal distribution function from scipy.stats\n",
      "from scipy.stats import norm\n",
      "norm_pdf = norm.pdf"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we need to make our kernel.\n",
      "\n",
      "* Work out the +/- limit for the kernel x values with `limit = round(sigma * 4)`\n",
      "* Make a vector of integers `x` to sample the PDF from `-limit` to `limit` - *inclusive*;\n",
      "* Sample the PDF at these x values, for $\\mu = 0$ and given $\\sigma$, to get the `kernel` vector;\n",
      "* Plot the x values against the y values;\n",
      "* Work out how many indices from the start of the `kernel` vector, to the value corresponding to x=0; (this is the `kernel_offset`."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Work out the +/- limit for the kernel x values;\n",
      "# Make a vector `x` to sample the PDF;\n",
      "# Get `kernel` vector by sampling the PDF at these x values (mu=0);\n",
      "# Work out `kernel_offset`.\n",
      "\n",
      "limit = round(sigma * 4)\n",
      "# Make an x range between -sigma * 3 and +sigma * 3\n",
      "x_for_kernel = np.arange(-limit, limit+1)\n",
      "# Calculate kernel\n",
      "kernel = norm_pdf(x_for_kernel, 0, sigma)\n",
      "# Plot kernel\n",
      "plt.plot(x_for_kernel, kernel)\n",
      "# Calculate `kernel_offset`\n",
      "kernel_offset = int(limit)\n",
      "kernel_offset"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "14"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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IhQvhnHPUo5fk6Ns37LxUVsKZZ8LEiWHygWSXwr6ArV0bgv2yy+B3vwurV3bv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\nAaZkbVo07NcFAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x109e98f90>"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we can convolve our image line with the `kernel`.  We take the result starting at `kernel_offset`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Convolve image line with kernel\n",
      "# Slice out the result we want using kernel_offset\n",
      "# Plot smoothed with unsmoothed line\n",
      "smoothed = np.convolve(mid_line, kernel)\n",
      "smoothed = smoothed[kernel_offset:len(mid_line)+kernel_offset]\n",
      "plt.plot(mid_line)\n",
      "plt.plot(smoothed)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "[<matplotlib.lines.Line2D at 0x10a0823d0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Xnq1bQfSfUZUfESE+Jp4KzZPJzfV1NcoTNDFsqDD4Z6XM0v6+8or4pvFUaK7t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\nl0p5lCf7/ADdGnbjhf4vMGj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       "text": [
        "<matplotlib.figure.Figure at 0x10a082590>"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we are going to smooth a whole slice.\n",
      "\n",
      "* Make a new array the same shape as `mid_slice` called `smoothed_slice`;\n",
      "* Loop over the first dimension of `mid_slice`, and fill each line of `smoothed_slice` with the smoothed version;\n",
      "* Loop over the second dimension of `smoothed_slice` and replace each line with the smoothed version;\n",
      "* Show."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Make a new array `smoothed_slice` for smoothed output\n",
      "# Loop over first dimension applying kernel as above\n",
      "# Loop over second dimension of smoothed image, applying kernel\n",
      "# Show with `imshow`\n",
      "smoothed_slice = np.zeros(mid_slice.shape)\n",
      "n_x, n_y = mid_slice.shape\n",
      "for x in range(n_x):\n",
      "    line = mid_slice[x, :]\n",
      "    smoothed = np.convolve(line, kernel)\n",
      "    smoothed_slice[x, :] = smoothed[kernel_offset:n_y+kernel_offset]\n",
      "for y in range(n_y):\n",
      "    line = smoothed_slice[:, y]\n",
      "    smoothed = np.convolve(line, kernel)\n",
      "    smoothed_slice[:, y] = smoothed[kernel_offset:n_x+kernel_offset]\n",
      "plt.imshow(smoothed_slice)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "<matplotlib.image.AxesImage at 0x10a0dcc50>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10a04f590>"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now time to do the same thing with `scipy.ndimage.gaussian_filter`."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# - import gaussian_filter\n",
      "from scipy.ndimage import gaussian_filter"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Have a look at the help for `gaussian_filter`.\n",
      "\n",
      "Use `gaussian_filter` to make another smoothed version of `mid_slice`."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Use gaussian filter to smooth `mid_slice`\n",
      "scipy_slice = gaussian_filter(mid_slice, sigma)\n",
      "plt.imshow(scipy_slice)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": [
        "<matplotlib.image.AxesImage at 0x10a320d50>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10a129290>"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Subtract the `gaussian_filter` version from your handcrafted version.\n",
      "\n",
      "Can you explain the difference?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Subtract two versions of the smoothed slice, show\n",
      "diff_slice = smoothed_slice - scipy_slice\n",
      "plt.imshow(diff_slice)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "<matplotlib.image.AxesImage at 0x10a4706d0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10a348a90>"
       ]
      }
     ],
     "prompt_number": 17
    }
   ],
   "metadata": {}
  }
 ]
}