Merge branch 'release/0.2.0'

Author: adamiao

.idea/workspace.xml

@@ -1,2 +1,36 @@
-This is a tutorial about applying Sobel filters to images for edge detection.
-It is currently under construction!
+# Sobel Filter/Operator (Edge Detection)
+
+Let's create a tool to detect edges of an image! To do so we'll use the Sobel operator/filter (there are other techniques out there, e.g. Canny edge detector). The matrices associated with the kernels we'll be using are given by:
+
+<img src="https://latex.codecogs.com/gif.latex?G_x=\begin{bmatrix}&space;1&space;&&space;0&space;&&space;-1&space;\\&space;2&space;&&space;0&space;&&space;-2&space;\\&space;1&space;&&space;0&space;&&space;-1&space;\end{bmatrix}\quad\text{and}\quad&space;G_y=\begin{bmatrix}&space;1&space;&&space;2&space;&&space;1&space;\\&space;0&space;&&space;0&space;&&space;0&space;\\&space;-1&space;&&space;-2&space;&&space;-1&space;\end{bmatrix}" title="G_x=\begin{bmatrix} 1 & 0 & -1 \\ 2 & 0 & -2 \\ 1 & 0 & -1 \end{bmatrix}\quad\text{and}\quad G_y=\begin{bmatrix} 1 & 2 & 1 \\ 0 & 0 & 0 \\ -1 & -2 & -1 \end{bmatrix}" />
+
+Think of Gx and Gy as maps of the form <img src="https://latex.codecogs.com/gif.latex?\mathbb{R}^{3\,\times\,3}\rightarrow\mathbb{R}" />, where we take in a 3 by 3 patch of image (denoted by <img src="https://latex.codecogs.com/gif.latex?$$A\in\mathbb{R}^{3\,\times\,3$$" />), do some operation on it (using the matrices shown above), and return a real number (the value of the pixel for the output image).
+
+So, for example, if we are given the following patch of grayscale image:
+
+![alt-text](Images/grayscale_patch.PNG "Example of a grayscale patch")
+
+which is really the matrix:
+
+<img src="https://latex.codecogs.com/gif.latex?A=\begin{bmatrix}&space;119&space;&&space;80&space;&&space;122&space;\\&space;177&space;&&space;154&space;&&space;212&space;\\&space;89&space;&&space;25&space;&&space;152&space;\end{bmatrix}" />
+
+Then the output from the operations is:
+
+<img src="https://latex.codecogs.com/gif.latex?G_x(A)=1\cdot&space;119+0\cdot&space;80-1\cdot&space;122+2\cdot&space;177+0\cdot&space;154-2\cdot&space;212+1\cdot&space;89+0\cdot&space;25-1\cdot&space;152" />
+<img src="https://latex.codecogs.com/gif.latex?\therefore\quad&space;G_x(A)=-136" />
+
+
+and
+
+<img src="https://latex.codecogs.com/gif.latex?G_x(A)=1\cdot&space;119+2\cdot&space;80+1\cdot&space;122+0\cdot&space;177+0\cdot&space;154+0\cdot&space;212-1\cdot&space;89-2\cdot&space;25-1\cdot&space;152" />
+<img src="https://latex.codecogs.com/gif.latex?\therefore\quad&space;G_y(A)=110" />
+
+To apply the Sobel operation to that patch of image we want to calculate:
+
+<img src="https://latex.codecogs.com/gif.latex?\sqrt{\big[G_x(A)\big]^2+\big[G_y(A)\big]^2}\quad\text{which&space;in&space;the&space;case&space;of&space;this&space;example&space;is&space;equal&space;to&space;}\,174.917" />
+
+This number will be the pixel contained in the output image. For now it's ok to have values which are out of the [0,255] range because at the very end we will normalized the image values. Notice that the output will be always greater than or equal to zero.
+
+We do this operation for the entire input image and create a new image based on these outputs just describe above.
+
+This completes the application of the edge detection technique.

Images/grayscale_patch.PNG

@@ -0,0 +1,19 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+# Example of grayscale patch
+A = np.array([[119, 80, 122], [177, 154, 212], [89, 25, 152]])
+plt.imshow(A.astype('int32'), cmap=plt.get_cmap('gray'))
+
+
+# This chunk of code defines the kernel operations and outputs image
+Gx = np.array([[1.0, 0.0, -1.0], [2.0, 0.0, -2.0], [1.0, 0.0, -1.0]])
+Gy = np.array([[1.0, 2.0, 1.0], [0.0, 0.0, 0.0], [-1.0, -2.0, -1.0]])
+
+# The output of the kernel operation is
+
+GxA = np.sum(np.multiply(Gx, A))
+GyA = np.sum(np.multiply(Gy, A))
+
+output_value = np.sqrt(GxA**2 + GyA**2)
+print(output_value)