Smoothing Images

Gaussian Blurring

​​The Gaussian filter is the same as the mean filter; however, it involves a weighted average of the surrounding pixels and has sigma as a parameter. The kernel used in Gaussian blurring represents a discrete approximation of a Gaussian distribution. The Gaussian filter helps in preserving the edges of the image. The Open-CV library provides a function called the "GaussianBlur" that helps in implementing a Gaussian filter. The GaussianBlur function allows you to specify the shape of the kernel and standard deviation for the x and y directions separately. If only one sigma value is specified, it is considered the sigma value for both the x and y directions.

The formula and the graph for the Gaussian distribution of a kernel are

Properties of Gaussian Blurring

• The Gaussian blurring is a non-uniform low-pass filter.
• The coefficients of the kernel decrease as the distance from the center increases.
• Pixels at the center have a higher weight than the pixels at the periphery.
• The image's brightness may not be preserved after applying a Gaussian filter.
• The kernel is rotationally symmetric.

Method of calculating Gaussian Blurring

Let us consider the image of size 5 x 5 pixels and a kernel of size 3 x 3 pixels.

Image

12	23	4	32	44
32	65	45	32	42
21	25	67	11	23
45	23	78	32	23
45	34	89	66	21

Kernel

1	2	1
2	4	2
1	2	1

If we want to replace the (3, 3) pixel of the image we have to calculate a weighted average of the pixel.

Weighted average = 

(1/16)(65x1 + 45 x 2 + 32 x 1 + 25 x 2 + 67 x 4 + 11 x 2 + 23 x 1 + 78 x 2 + 32 x 1)

Similarly, update all the pixels of the image. 

Code for Gaussian Blurring

Importing libraries

import numpy as np
import cv2
from matplotlib import pyplot as plt
%matplotlib inline
from PIL import Image
from PIL import ImageFilter

Creating a function that takes the relative path of the image.

def gaussian_blurring(s):
    # reading image
    img = cv2.imread(s)

    # converting from BGR to HSV
    img = cv2.cvtColor(img, cv2.COLOR_BGR2HSV)

    # applying gaussian blurring with kernel dimension of 25 × 25 and standard deviation of 0
    new_image = cv2.GaussianBlur(img,(25, 25), 0)

    # figure size
    plt.figure(figsize=(11,11))

    # plotting original image
    plt.subplot(121), plt.imshow(cv2.cvtColor(img, cv2.COLOR_HSV2RGB)),plt.title('Original')
    plt.xticks([]), plt.yticks([])

    # plotting blur image
    plt.subplot(122), plt.imshow(cv2.cvtColor(new_image, cv2.COLOR_HSV2RGB)),plt.title('Gaussian filter')
    plt.xticks([]), plt.yticks([])

    # displaying plot
    plt.show()

Calling the function

gaussian_blurring('img.jpg')

Median Filtering

The median filter calculates the median of the pixel intensities surrounded by the center pixel in a n x n kernel. The intensity of the center pixel is then replaced with a median. It helps in removing salt and pepper noise from the image. This preserves the edges of an image, but it does not handle speckle noise. The Open-CV library provides a "medianBlur" function that helps in implementing the median blur filter.

Properties of Median Filtering

• Median filtering is a non-linear digital filtering technique.
• It helps to preserve edges while removing noise from the images.
• It is used in removing salt and pepper noise.

Method of calculating Gaussian Blurring

Let us consider the image of size 5 x 5 pixels, and a kernel of size 3 x 3 pixels.

Image

12	23	4	32	44
32	65	45	32	42
21	25	67	11	23
45	23	78	32	23
45	34	89	66	21

If we want to replace the pixel of the image at positions (3, 3) we have to calculate the median of all the nearest pixels.

The median of (65, 45, 32, 25, 67, 11, 23, 78, 32) is 32 so, the pixel at position 3,3 will be replaced with 32

Similarly, update all the pixels of the image. 

Code for Median Filtering

Importing libraries

import numpy as np
import cv2
from matplotlib import pyplot as plt
%matplotlib inline
from PIL import Image
from PIL import ImageFilter

Creating a function that takes the relative path of the image as a parameter.

def median_filtering(s):
    # reading image
    img = cv2.imread(s)

    # converting from BGR to HSV
    img = cv2.cvtColor(img, cv2.COLOR_BGR2HSV)

    # applying median blurring with kernel dimension of 25
    new_image = cv2.medianBlur(img, 25)

    # figure size
    plt.figure(figsize=(11,11))

    # plotting original image
    plt.subplot(121), plt.imshow(cv2.cvtColor(img, cv2.COLOR_HSV2RGB)),plt.title('Original')
    plt.xticks([]), plt.yticks([])

    # plotting blur image
    plt.subplot(122), plt.imshow(cv2.cvtColor(new_image, cv2.COLOR_HSV2RGB)),plt.title('Median filter')
    plt.xticks([]), plt.yticks([])

    # displaying plot
    plt.show()

Calling the function

median_filtering('img.jpg')

4. Bilateral Filtering

A bilateral filter is a non-linear, edge-preserving, noise-reducing filter used for smoothing images. The intensity of each pixel is replaced with a weighted mean of intensity values from nearby pixels. The weights of the filter are based on a Gaussian distribution. Crucially, the weights depend not only on the Euclidean distance of pixels but also on the radiometric differences (e.g., range differences, such as color intensity, depth distance, etc.). This filtering preserves sharp edges.

The following image shows how to preserve edges by spatial and color(intensity) weights.

^source

Properties of Bilateral Filtering

• Two pixels are close to each other in bilateral filtering if they occupy neighboring spatial positions and exhibit some photometric range similarity.
• Bilateral filtering takes the Gaussian filter in space and considers another Gaussian filter that is a function of pixel difference. For this reason, bilateral filters preserve edges as the pixel at the edges have large intensity variation.

The formula for Bilateral filtering

Bilateral filtering is represented by the symbol BF.

Here W_p is the normalization factor. The value of W_p is given below.

In the above equations, Gσs is a spatial Gaussian that decreases the influence of distant pixels. Gσr is a range Gaussian that decreases the influence of pixels q with an intensity value different from Ip. σs is the space parameter it represents the spatial extent of the kernel size of the considered neighborhood and σr is the range parameter that represents the minimum amplitude of an edge. Both the parameters will measure the amount of filtering for image I.

Code for Bilateral Filtering

Importing libraries
import numpy as np
import cv2
from matplotlib import pyplot as plt
%matplotlib inline
from PIL import Image
from PIL import ImageFilter

Creating a function that takes the relative path of an image as an input parameter.

def bilateral_filtering(s):
    # reading image
    img = cv2.imread(s)

    # converting from BGR to HSV
    img = cv2.cvtColor(img, cv2.COLOR_BGR2HSV)

    # applyin bilateral filter with d = 25 and sigmaColor = sigmaSpace = 100
    new_image = cv2.bilateralFilter(img, 25, 100, 100)

    # figure size
    plt.figure(figsize=(11,11))

    # plotting original image
    plt.subplot(121), plt.imshow(cv2.cvtColor(img, cv2.COLOR_HSV2RGB)),plt.title('Original')
    plt.xticks([]), plt.yticks([])

    # plotting blur image
    plt.subplot(122), plt.imshow(cv2.cvtColor(new_image, cv2.COLOR_HSV2RGB)),plt.title('Bilateral filter')
    plt.xticks([]), plt.yticks([])

    # displaying plot
    plt.show()

Calling function

bilateral_filtering("img.jpg")

Also read, Sampling and Quantization

FAQs

1. What are the techniques used for image smoothing?
   Low pass filtering (aka smoothing) removes high spatial frequency noise from a digital image. The low-pass filters usually employ a moving window operator that affects one pixel of the image at a time.
    
2. What is image smoothing and sharping?
   Color image smoothing is part of preprocessing techniques intended for removing possible image perturbations without losing image information. Analogously, sharpening is a preprocessing technique that plays an important role in feature extraction in image processing.
    
3. How does a gaussian blur work?
   In a Gaussian blur, the pixels near the kernel's center are given more weight than those far away from the center. This averaging is done on a channel-by-channel basis, and the value of the pixel is replaced with the average channel value.
    
4. What is meant by Gaussian?
   being or having the shape of a normal curve or a normal distribution.
    
5. What is the difference between mean and median filtering?
   The median is a more robust average than the mean, so a single very unrepresentative pixel in a neighborhood will not significantly affect the median value.

Key Takeaways

In this article, we have studied the following topics:

• Uses of image smoothing
• Mean Blurring
• Gaussian Blurring
• Median Filtering
• Bilateral Filtering

Want to learn more about Machine Learning? Here is an excellent course that can guide you in learning. 

Happy Coding!