Spatial Resolution (down sampling and up sampling) in image processing

A digital image is a two-dimensional array of size M x N where M is the number of rows and N is the number of columns in the array.  A digital image is made up of a finite number of discrete picture elements called a pixel. The location of each pixel is given by coordinates (x, y) and the value of each pixel is given by intensity value f. Hence, the elements in a digital image can be represented by f(x, y). 

Spatial Resolution

The term spatial resolution corresponds to the total number of pixels in the given image. If the number of pixels is more, then the resolution of the image is more.

Down-sampling

In the down-sampling technique, the number of pixels in the given image is reduced depending on the sampling frequency. Due to this, the resolution and size of the image decrease.

Up-sampling

The number of pixels in the down-sampled image can be increased by using up-sampling interpolation techniques. The up-sampling technique increases the resolution as well as the size of the image.

Some commonly used up-sampling techniques are

• Nearest neighbor interpolation
• Bilinear interpolation
• Cubic interpolation

In this article, we have implemented upsampling using Nearest neighbor interpolation by replicating rows and columns. As the sampling rate increases, artifacts will be seen in the reconstructed image. 

However, by using bilinear interpolation or cubic interpolation, a better quality of the reconstructed image can be attained. Both Down sampling and Up Sampling can be illustrated in grayscale for better understandings because while reading images using OpenCV, some color values are manipulated, So we are going to convert the original input image into a black and white image.

The below program depicts the down sampled and up sampled representation of a given image:

# Import cv2, matplotlib, numpy
import cv2
import matplotlib.pyplot as plt
import numpy as np


# Read the original image and know its type
img1 = cv2.imread('g4g.png', 0)

# Obtain the size of the original image
[m, n] = img1.shape
print('Image Shape:', m, n)

# Show original image
print('Original Image:')
plt.imshow(img1, cmap="gray")


# Down sampling

# Assign a down sampling rate
# Here we are down sampling the
# image by 4
f = 4

# Create a matrix of all zeros for
# downsampled values
img2 = np.zeros((m//f, n//f), dtype=np.int)

# Assign the down sampled values from the original
# image according to the down sampling frequency.
# For example, if the down sampling rate f=2, take
# pixel values from alternate rows and columns
# and assign them in the matrix created above
for i in range(0, m, f):
    for j in range(0, n, f):
        try:

            img2[i//f][j//f] = img1[i][j]
        except IndexError:
            pass


# Show down sampled image
print('Down Sampled Image:')
plt.imshow(img2, cmap="gray")


# Up sampling

# Create matrix of zeros to store the upsampled image
img3 = np.zeros((m, n), dtype=np.int)
# new size
for i in range(0, m-1, f):
    for j in range(0, n-1, f):
        img3[i, j] = img2[i//f][j//f]

# Nearest neighbour interpolation-Replication
# Replicating rows

for i in range(1, m-(f-1), f):
    for j in range(0, n-(f-1)):
        img3[i:i+(f-1), j] = img3[i-1, j]

# Replicating columns
for i in range(0, m-1):
    for j in range(1, n-1, f):
        img3[i, j:j+(f-1)] = img3[i, j-1]

# Plot the up sampled image
print('Up Sampled Image:')
plt.imshow(img3, cmap="gray")

Input:

Output: