Initial Upload for First week

.github/workflows/blank-slate.yml

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+# This is a basic workflow to help you get started with Actions
+
+name: CI
+
+# Controls when the workflow will run
+on:
+  # Triggers the workflow on push or pull request events but only for the "main" branch
+  push:
+    branches: [ "main" ]
+  pull_request:
+    branches: [ "main" ]
+
+  # Allows you to run this workflow manually from the Actions tab
+  workflow_dispatch:
+
+
+jobs:
+  cleanup:
+    runs-on: ubuntu-latest
+    permissions: write-all
+
+    steps:
+      - name: Delete deployment
+        uses: strumwolf/delete-deployment-environment@v2
+        with:
+          token: ${{ secrets.GITHUB_TOKEN }}
+          environment: github-pages
+          onlyRemoveDeployments: true
+

Lecture-Slides/DIP-Slide/ipynb/Mathematical-Fundamentals.ipynb

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+{"cells":[{"cell_type":"markdown","id":"fb204713-3bbd-40e3-a23e-bf628ab08497","metadata":{},"source":"Code for Digital Image Processing\n=================================\n\n"},{"cell_type":"markdown","id":"025d85bb-fe4d-49a0-866e-1bdc325996c5","metadata":{},"source":["These are the code snippets used in Mathematical Fundamentals\npart of Digital Image Processing.\n\n"]},{"cell_type":"markdown","id":"59189747-5928-40e8-a56a-8d44c91f10ca","metadata":{},"source":["### Introduction\n\n"]},{"cell_type":"markdown","id":"15c29ce1-0134-461d-84a1-20582c5fc081","metadata":{},"source":["The following code uses the standard matplotlib along with the\ncustom ChalcedonPy which is detailed in its source code here.\n\nThe code is used primarily in the following code as a means to\nsave figures for use in slides.\n\n"]},{"cell_type":"code","execution_count":1,"id":"45db9e58-7e8e-4f80-ac86-8ca0eb429df5","metadata":{},"outputs":[],"source":["import matplotlib.pyplot as plt\nimport ChalcedonPy as cp\n\n# Initialise ChalcedonPy\ncp.init(save_path=\"Mathematical-Fundamentals\",\n        display_mode=\"slide\")"]},{"cell_type":"markdown","id":"6f895dc3-731f-4c32-91e3-43a122f88d1d","metadata":{},"source":["### Introduction\n\n"]},{"cell_type":"markdown","id":"4ece5898-7146-4d3b-9ab7-e05402f5f650","metadata":{},"source":["The following code covers the topic of mathematical fundamentals, which\nincludes some examples on:\n\n-   convolution,\n-   fourier transform,\n-   spectral leakage,\n-   information theory.\n\n"]},{"cell_type":"markdown","id":"da33ca9c-454d-40d4-81bc-6e592e374308","metadata":{},"source":["### Exercise: Convolution 1D\n\n"]},{"cell_type":"markdown","id":"5cea3c99-c05a-43ff-9fa7-3ebbd040efde","metadata":{},"source":["The following is the solution to the 1D Convolution question given\nin the lecture\n\n"]},{"cell_type":"code","execution_count":1,"id":"4806c080-9900-49fc-af2d-55c376e1c38f","metadata":{},"outputs":[],"source":["import numpy as np\ndef convolve_1d(signal, kernel):\n    kernel = kernel[::-1]\n    k = len(kernel)\n    s = len(signal)\n    signal = [0]*(k-1)+signal+[0]*(k-1)\n    n = s+(k-1)\n    res = []\n    for i in range(s+k-1):\n        res.append(np.dot(signal[i:(i+k)], kernel))\n    return res"]},{"cell_type":"markdown","id":"931edcbf-dfbf-4347-8ecd-362f76641c66","metadata":{},"source":["Now the function is defined, it is a matter of entering the two arrays\nand printing the function.\n\n"]},{"cell_type":"code","execution_count":1,"id":"634fde00-ba71-4607-b631-194620958165","metadata":{},"outputs":[],"source":["A = [1,1,2,2,1]\nB = [1,1,1,3]\n\nprint(convolve_1d(A, B))"]},{"cell_type":"markdown","id":"a71d0184-79c8-4315-bb06-b62c717680f9","metadata":{},"source":["### Shannon Nyquist Sampling Theorem\n\n"]},{"cell_type":"markdown","id":"7c72dfc9-5045-4fd0-9264-1c1b2dc29636","metadata":{},"source":["Below is an unused code which showcases the practical\nlimitation of the Shannon-Nyquist theorem.\n\n"]},{"cell_type":"code","execution_count":1,"id":"930f5784-b6f1-4dd2-9299-e12c6e028ca9","metadata":{},"outputs":[],"source":["import numpy as np\nfrom matplotlib import pyplot as plt\n\nwave_freq = 5\ndomain = 1\n\nsample_rates = (\n    (wave_freq * 2 - 1, 'below Nyquist rate'),\n    (wave_freq * 2, '(exclusive) lower bound according to Nyquist?'),\n    (wave_freq * 2 + 1, 'bit more than Nyquist'),\n    (wave_freq * 4, 'double Nyquist'),\n)\n\nn_plots = len(sample_rates) + 1\n\nfig = plt.figure(figsize=(8, 10))\nax = fig.add_subplot(n_plots, 1,  1)\ntime = np.linspace(0, domain, 1000, endpoint=False) # take enough samples to make a seemingly perfect sine wave\nhi_res = np.sin(2 * np.pi * wave_freq * time)\nax.axhline(color=\"0.7\")\nax.plot(time, hi_res, label=f'{wave_freq}Hz wave')\nax.legend(loc=1)\nylim = ax.get_ylim()\n\nfor i, (sample_rate, title) in enumerate(sample_rates):\n    ax = fig.add_subplot(n_plots, 1, i+2)\n\n    t = np.linspace(0, domain, sample_rate, endpoint=False)\n    samples = np.sin((2 * np.pi * wave_freq) * t)\n    ax.axhline(color=\"0.7\")\n    ax.plot(time, hi_res, label=f'{wave_freq}Hz wave', color='0.8') # show what we're aiming for\n    ax.plot(t, samples, marker='o', label=f'{sample_rate}Hz sample rate ({sample_rate/wave_freq}x)')\n    ax.set_ylim(*ylim)\n    ax.legend(loc=1)\n    ax.set_title(title)\n    \nplt.show()"]},{"cell_type":"markdown","id":"c6ee94b6-7e93-4595-98d7-5f138199c0a3","metadata":{},"source":["### Spectral Leakage\n\n"]},{"cell_type":"markdown","id":"e82448b9-f165-46c6-a3e8-0040ce0adc2c","metadata":{},"source":["Let’s take a look at an example using a simple sine wave.\nFirst import the necessary modules.\n\n"]},{"cell_type":"code","execution_count":1,"id":"d0e371cc-53f8-4e60-af5f-00d7d15a8436","metadata":{},"outputs":[],"source":["import numpy as np\nimport matplotlib.pyplot as plt\nimport scipy.stats as st\nimport scipy.signal as sig"]},{"cell_type":"markdown","id":"a7c74e57-661a-4c50-8f1b-67814d7b8f3d","metadata":{},"source":["Let’s define a sine function on a time domain, such that\nthe sine function completes $N$, integer cycles, i.e.,\nno cycles are cut-off within the domain.\n\n"]},{"cell_type":"code","execution_count":1,"id":"8f9332cf-4a87-4092-9f5d-ded6fc63c47e","metadata":{},"outputs":[],"source":["import numpy as np\n\nT = 8 * np.pi  # length of time domain\nt = np.arange(0, T + 0.05 * np.pi, 0.05 * np.pi)\nfreq = np.arange(0, int(len(t) / 2.0) + 1) / len(t)\ny_sin = np.sin(t) # define the sin function"]},{"cell_type":"markdown","id":"722179f4-c67c-4f4e-a8be-3e4b1637b0d8","metadata":{},"source":["Now, we can define and plot our sine function.\n\n"]},{"cell_type":"code","execution_count":1,"id":"8cd15673-c5df-40d3-a42b-7d2828c4e7f0","metadata":{},"outputs":[],"source":["plt.plot(t,y_sin)\nplt.xlabel(\"Data Points\")\nplt.ylabel(\"Amplitude\")\nplt.title(\"Sine Wave\")\nplt.show()"]},{"cell_type":"markdown","id":"30c8ad06-152d-4f97-9715-ba6f23308bab","metadata":{},"source":["Here, we have 4 complete cycles of our sine wave.\nNow, let’s compute the discrete power spectrum of this sine wave.\n\n"]},{"cell_type":"code","execution_count":1,"id":"a7561cab-6d6d-4a75-aeb0-323dddaad111","metadata":{},"outputs":[],"source":["# Do simple fft of the signal\nyfft1_raw = np.fft.fft(y_sin)\nyfft1 = yfft1_raw/len(y_sin)\n\n# compute variance as a function of frequency (spectral power)\nPSD = 2*np.abs(yfft1[0:int(len(t)/2.0+1)])**2"]},{"cell_type":"markdown","id":"826ea80c-df29-4356-aa39-84816f477d42","metadata":{},"source":["And time for plotting out FFT results\n\n"]},{"cell_type":"code","execution_count":1,"id":"91a68382-99b1-4361-bd11-35332c411c73","metadata":{},"outputs":[],"source":["plt.bar(freq, PSD, width=0.0025)\n\nplt.xlabel('Frequency (cycles per time step)')\nplt.ylabel('Normalized Power')\nplt.title(\"D-PSD for Un-windowed Sine Wave\")\nplt.xlim(0,0.1)\n\nplt.show()"]},{"cell_type":"markdown","id":"0a803b6f-9bf1-4e00-91f4-a32af10b24a1","metadata":{},"source":["We get a single spectral peak corresponding to the frequency of our\nsine wave.\n\nNow, let’s see what happens if we apply a window to our sine wave\nthat cuts off the sine wave such that the sine function does not\ncomplete an integer number of cycles within the time domain.\n\n"]},{"cell_type":"code","execution_count":1,"id":"f387abb0-7f6d-4691-a197-deae7aa07739","metadata":{},"outputs":[],"source":["# Now let's window our data, such that we are not cleanly sampling our sine function\ny2 = []\nbxcr = []\nfor i in t:\n    if (i < 1.5*np.pi) or (i > 6.5*np.pi):\n        y2.append(0)\n        bxcr.append(0)\n    else:\n        y2.append(np.sin(i))\n        bxcr.append(1)\ny2 = np.asarray(y2)\nbxcr = np.asarray(bxcr)"]},{"cell_type":"markdown","id":"13627d17-21e8-4360-bf24-31400d7bbb3b","metadata":{},"source":["Results calculated, lets do some plotting.\n\n"]},{"cell_type":"code","execution_count":1,"id":"414c6f3d-8cd2-4d1e-814a-d560114e8d28","metadata":{},"outputs":[],"source":["# Plot windowed sine wave, window\nplt.subplot(2,1,1)\nplt.plot(t,y2, label=\"windowed sine wave\")\nplt.plot(t,bxcr, label=\"box wave\")\nplt.title('Windowed Sine Wave')\nplt.legend(loc=\"lower right\")\n\nplt.subplot(2,1,2)\nplt.plot(t,y2*bxcr)\nplt.title('Also a Windowed Sine Wave')\n\nplt.show()"]},{"cell_type":"markdown","id":"364e2787-f301-495c-9741-27172e45dc89","metadata":{},"source":["To demonstrate what spectral leakage is, we will now compute\nthe discrete power spectrum of the windowed sine wave to see what happens.\n\n"]},{"cell_type":"code","execution_count":1,"id":"63921acc-b477-4e55-b747-711f4592d70b","metadata":{},"outputs":[],"source":["# compute FFT\nyfft2_raw = np.fft.fft(y2)\nyfft2 = yfft2_raw/len(y2)\n\n# compute variance as a function of frequency (spectral power)\nck2y2 = 2*np.abs(yfft2[0:int(len(t)/2.0+1)])**2"]},{"cell_type":"code","execution_count":1,"id":"b9df5d3f-f598-43ec-b693-15ceeee3761a","metadata":{},"outputs":[],"source":["# Plot Power Spectrum of windowed data\nplt.bar(freq,ck2y2,width=0.0025)\nplt.xlabel('Frequency [Hz]')\nplt.ylabel('PSD [W]')\nplt.title(\"Discrete Power Spectrum for Windowed Sine Wave\")\nplt.xlim(0,0.1)\n\nplt.show()"]}],"metadata":{"org":null,"kernelspec":{"display_name":"Python 3","language":"python","name":"python3"},"language_info":{"codemirror_mode":{"name":"ipython","version":3},"file_extension":".py","mimetype":"text/x-python","name":"python","nbconvert_exporter":"python","pygments_lexer":"ipython3","version":"3.5.2"}},"nbformat":4,"nbformat_minor":5}