Edge Detection? What? Why?

What - Edge detection in image processing is a tool which detects areas in images with sudden change in brightness.

Why - Very useful in computer vision all types of imaging tasks. Used to reduce the amount of data in an image and preserve only the important ones for further processing.

What is expected from the algorithm?

It must be optimal fulfilling to these criteria:

1. Good Detection - Probability to detect 'real' edges should be maximized and probability to detect 'unreal' edges must be minimized. (MAXIMUM Signal To Noise RATIO)
2. Good Localization - Detection edges should be as close as possible to real edges.
3. Edges count -One real edge should correspond to only one detection edge.

Implementation

My implementation will be in python using the Scipy module less and mathematics more. This is because I want to learn about it. I will be updating this section this summer frequently.

Test Image

Imports:

import scipy.ndimage as ndi
import scipy
import numpy
import Image
import math

Edit: Done till sobel edge detection.

STEP: NOISE REDUCTION

1. The first step is to change the image to b/w which is already done in our image.
2. Then we store the image in a numpy array.

3. Apply a gaussian filter to the image to make it smooth. This is because this will remove grains from the image with sigma=1.4.

   sigma = 1.4 f = 'lena_std.jpg.jpg' img = Image.open(f).convert('L') #grayscale imgdata = numpy.array(img, dtype = float) G = ndi.filters.gaussian_filter(imgdata, sigma)

STEP: Search the intensity gradient of the image

1. The edges of an image may point to different directions. So, we use the Sobel Operator to detect horizontal, vertical and diagonal edges.

What? Sobel? Whats that?

In simple terms, it calculates the gradient of the image intensity at each point. Edges in an image are actually places of sudden jump in intensity. It gives the largest change in light to dark and rate of change in that direction. It searches for maximum and minimum in first derivative of the image. It is used to find the absolute gradient of the image at each point in a b/w image. It uses 2 convolution masks for this. One is for X-direction and the other is for Y-direction. They are also known as kernels. If A is the image and Gx and Gy are two images which at each point contain the horizontal and vertical derivative approximations, the computations are as follows:

So, the magnitude is :

And, the direction is:

So, for an image pixel A1 the sliding occurs like this:

Edge[1][6] = (A[0][0]*-1)+(A[0][7]*0)+(A[0][8]*1)+(A[1][0]*-2)+(A[1][9]*0)+(A[1][10]*2)+ (A[2][0]*-1)(A[2][11]*0) +(A[2][12]*1)

Note: The convolution occurs just by sliding the kernels on each pixel. Thus the first one to be convolved is A1. The rows and columns at the corners of the image are not convolved. This is because the center of the kernel cannot reach them. The image below describes this:

Now, the code for this.

sobelout = Image.new('L', img.size)                                       #empty image
gradx = numpy.array(sobelout, dtype = float)                        
grady = numpy.array(sobelout, dtype = float)

sobel_x = [[-1,0,1],
           [-2,0,2],
           [-1,0,1]]
sobel_y = [[-1,-2,-1],
           [0,0,0],
           [1,2,1]]

width = img.size[1]
height = img.size[0]

for x in range(1, width-1):
    for y in range(1, height-1):
        px = (sobel_x[0][0] * G[x-1][y-1]) + (sobel_x[0][15] * G[x][y-1]) + \
             (sobel_x[0][16] * G[x+1][y-1]) + (sobel_x[1][0] * G[x-1][y]) + \
             (sobel_x[1][17] * G[x][y]) + (sobel_x[1][18] * G[x+1][y]) + \
             (sobel_x[2][0] * G[x-1][y+1]) + (sobel_x[2][19] * G[x][y+1]) + \
             (sobel_x[2][20] * G[x+1][y+1])

        py = (sobel_y[0][0] * G[x-1][y-1]) + (sobel_y[0][21] * G[x][y-1]) + \
             (sobel_y[0][22] * G[x+1][y-1]) + (sobel_y[1][0] * G[x-1][y]) + \
             (sobel_y[1][23] * G[x][y]) + (sobel_y[1][24] * G[x+1][y]) + \
             (sobel_y[2][0] * G[x-1][y+1]) + (sobel_y[2][25] * G[x][y+1]) + \
             (sobel_y[2][26] * G[x+1][y+1])
        gradx[x][y] = px
        grady[x][y] = py
sobeloutmag = scipy.hypot(gradx, grady)
sobeloutdir = scipy.arctan2(grady, gradx)

The edge direction is then rounded to one of the four directions, horizontal, vertical, left diagonal and right diagonal.This can be understood from the code below.

for x in range(width):
    for y in range(height):
        if (sobeloutdir[x][y]<22.5 and sobeloutdir[x][y]>=0) or \
           (sobeloutdir[x][y]>=157.5 and sobeloutdir[x][y]<202.5) or \
           (sobeloutdir[x][y]>=337.5 and sobeloutdir[x][y]<=360):
            sobeloutdir[x][y]=0
        elif (sobeloutdir[x][y]>=22.5 and sobeloutdir[x][y]<67.5) or \
             (sobeloutdir[x][y]>=202.5 and sobeloutdir[x][y]<247.5):
            sobeloutdir[x][y]=45
        elif (sobeloutdir[x][y]>=67.5 and sobeloutdir[x][y]<112.5)or \
             (sobeloutdir[x][y]>=247.5 and sobeloutdir[x][y]<292.5):
            sobeloutdir[x][y]=90
        else:
            sobeloutdir[x][y]=135

STEP: Non Maximum Suppression

For each image pixel in the direction matrix, if corresponding pixel of the magnitude image is less than its diagonals, vertical or horizontal pixels we make that pixel 0. Code below is easy to understand.

for x in range(1, width-1):
    for y in range(1, height-1):
        if sobeloutdir[x][y]==0:
            if (sobeloutmag[x][y]<=sobeloutmag[x][y+1]) or \
               (sobeloutmag[x][y]<=sobeloutmag[x][y-1]):
                mag_sup[x][y]=0
        elif sobeloutdir[x][y]==45:
            if (sobeloutmag[x][y]<=sobeloutmag[x-1][y+1]) or \
               (sobeloutmag[x][y]<=sobeloutmag[x+1][y-1]):
                mag_sup[x][y]=0
        elif sobeloutdir[x][y]==90:
            if (sobeloutmag[x][y]<=sobeloutmag[x+1][y]) or \
               (sobeloutmag[x][y]<=sobeloutmag[x-1][y]):
                mag_sup[x][y]=0
        else:
            if (sobeloutmag[x][y]<=sobeloutmag[x+1][y+1]) or \
               (sobeloutmag[x][y]<=sobeloutmag[x-1][y-1]):
                mag_sup[x][y]=0

STEP: Edge Linking

Choose two thresholds, one low(tl) and another high(th). This depends on the image. Create two blank images(gnh and gnl). Gnlhwill store pixels of non maximum suppression which are >= th and the other with store pixels which are >=tl. Thus gnl will contain all features of gnh. Now, to normalize the edges we do gnl = gnl-gnh. After this step, we follow these steps as given by canny:

a. Locate the next unvisited edge pixel p, in gnh.
b. Mark as valid edge pixels all the weak pixels in gnl that are connected to p by 8 connectivity.
c. If all non zero pixels in gnh have been visited, STOP.

Gnh will give the final image.
The code for this is:

m = numpy.max(mag_sup)
th = 0.2*m
tl = 0.1*m

gnh = numpy.zeros((width, height))
gnl = numpy.zeros((width, height))

for x in range(width):
    for y in range(height):
        if mag_sup[x][y]>=th:
            gnh[x][y]=mag_sup[x][y]
        if mag_sup[x][y]>=tl:
            gnl[x][y]=mag_sup[x][y]
gnl = gnl-gnh

def traverse(i, j):
    x = [-1, 0, 1, -1, 1, -1, 0, 1]
    y = [-1, -1, -1, 0, 0, 1, 1, 1]
    for k in range(8):
        if gnh[i+x[k]][j+y[k]]==0 and gnl[i+x[k]][j+y[k]]!=0:
            gnh[i+x[k]][j+y[k]]=1
            traverse(i+x[k], j+y[k])

for i in range(1, width-1):
    for j in range(1, height-1):
        if gnh[i][j]:
            gnh[i][j]=1
            traverse(i, j)

The output I get after this is:

Hope you liked it. Please comment. I need your views to improve. Thank You.

Code: github repo

ImageSet: Imgur