date: 2024-09-20
title: CV-HW-3
status: DONE
author:
- AllenYGY
tags:
- CV
- HOMEWORK
publish: TrueCV-HW-3
Using a Gaussian filter to smooth the image and then downsample the image by a factor of 2.
We can apply twice the 1D Gaussian filter to the image in the x direction and then in the y direction. then we can downsample the image by a factor of 2 in the x direction and then in the y direction.
Suppose the image is a 2D matrix.
The Gaussian filter is a 2D matrix
The Downsample matrix is a 2D matrix.
The downsampled image is
import random
import numpy as np
def generate_matrix():
matrix = []
random.seed(0)
for i in range(8):
row = []
for j in range(8):
row.append(random.randint(0, 9))
matrix.append(row)
return matrix
def downsample(matrix, G_x, G_y):
print("Apply G_x to matrix")
new_matrix = np.matmul(G_x, matrix)*0.0625
print(new_matrix)
print("Apply G_y to new_matrix")
new_matrix = np.matmul(new_matrix, G_y)*0.0625
print("Downsampled Matrix:")
print(new_matrix)
return new_matrix
G_x = [[6, 4, 1, 0, 0, 0, 0, 0], [1, 4, 6, 4, 1, 0, 0, 0], [0, 0, 1, 4, 6, 4, 1, 0], [0, 0, 0, 0, 1, 4, 6, 4]]
G_y = [[6,1,0,0],
[4,4,0,0],
[1,6,1,0],
[0,4,4,0],
[0,1,6,1],
[0,0,4,4],
[0,0,1,6],
[0,0,0,4]]
matrix = generate_matrix()
matrix = np.array(matrix)
G_x = np.array(G_x)
G_y = np.array(G_y)
print("Original Matrix:")
print(matrix)
print("G_x:")
print(G_x)
print("G_y:")
print(G_y)
downsampled_matrix = downsample(matrix, G_x, G_y)
Original Matrix (8x8):
[[6 6 0 4 8 7 6 4]
[7 5 9 3 8 2 4 2]
[1 9 4 8 9 2 4 1]
[1 5 7 8 1 5 6 5]
[9 3 8 7 7 8 4 0]
[8 0 1 6 0 9 7 5]
[3 5 1 3 9 3 3 2]
[8 7 1 1 5 8 7 1]]
G_x Filter:
[[6 4 1 0 0 0 0 0]
[1 4 6 4 1 0 0 0]
[0 0 1 4 6 4 1 0]
[0 0 0 0 1 4 6 4]]
G_y Filter:
[[6 1 0 0]
[4 4 0 0]
[1 6 1 0]
[0 4 4 0]
[0 1 6 1]
[0 0 4 4]
[0 0 1 6]
[0 0 0 4]]
Apply G_x to matrix
[[4.0625 4.0625 2.5 2.75 5.5625 3.25 3.5 2.0625]
[3.3125 6.4375 6. 6.4375 6.5625 3.4375 4.625 2.375 ]
[5.875 3.25 5.3125 6.8125 4. 6.8125 5.1875 2.6875]
[5.6875 3.8125 1.375 3.3125 5.0625 5.875 4.875 2.25 ]]
Apply G_y to new_matrix
Downsampled Matrix:
[[2.6953125 3.2421875 3.9609375 2.98828125]
[3.2265625 6.0859375 5.59375 3.59765625]
[3.34765625 5.125 5.5625 4.5703125 ]
[3.171875 2.96875 4.5859375 4.17578125]]
Firstly, fill the zeros in the image by the nearest neighbor interpolation.
Then apply the 2D Gaussian filter to the image.
Suppose the image is a 2D matrix
Insert zeros in the image to get a new image
The Gaussian filter is a 2D matrix
import numpy as np
from scipy.ndimage import convolve
def upsample(matrix, G):
print("Original Matrix (4x4):")
print(matrix)
new_matrix = np.zeros((8, 8))
new_matrix[::2, ::2] = matrix
print("\nUpsampled Matrix (Insert Zeros):")
print(new_matrix)
new_matrix = convolve(new_matrix, G, mode='reflect')
print("\nUpsampled Matrix (After Applying Gaussian Filter):")
print(new_matrix)
return new_matrix
matrix = generate_matrix(4)
matrix = np.array(matrix)
G = [[1, 2, 1], [2, 4, 2], [1, 2, 1]]
G = np.array(G)
upsampled_matrix = upsample(matrix, G)
Original Matrix (4x4):
[[6 6 0 4]
[8 7 6 4]
[7 5 9 3]
[8 2 4 2]]
Upsampled Matrix (Insert Zeros):
[[6. 0. 6. 0. 0. 0. 4. 0.]
[0. 0. 0. 0. 0. 0. 0. 0.]
[8. 0. 7. 0. 6. 0. 4. 0.]
[0. 0. 0. 0. 0. 0. 0. 0.]
[7. 0. 5. 0. 9. 0. 3. 0.]
[0. 0. 0. 0. 0. 0. 0. 0.]
[8. 0. 2. 0. 4. 0. 2. 0.]
[0. 0. 0. 0. 0. 0. 0. 0.]]
Upsampled Matrix (After Applying Gaussian Filter):
[[54. 36. 36. 18. 0. 12. 24. 12.]
[42. 27. 26. 19. 12. 14. 16. 8.]
[48. 30. 28. 26. 24. 20. 16. 8.]
[45. 27. 24. 27. 30. 22. 14. 7.]
[42. 24. 20. 28. 36. 24. 12. 6.]
[45. 22. 14. 20. 26. 18. 10. 5.]
[48. 20. 8. 12. 16. 12. 8. 4.]
[24. 10. 4. 6. 8. 6. 4. 2.]]
Downsample the image by a factor of 2 and then upsample the image by a factor of 2.
The Laplacian filter can be get by using the original image minus the upsampled image.
matrix = generate_matrix(8)
matrix = np.array(matrix)
downsampled_matrix = downsample(matrix, G_x, G_y)
upsampled_matrix = upsample(downsampled_matrix, G)
laplacian_matrix = matrix - upsampled_matrix
print("\nLaplacian Matrix:")
print(laplacian_matrix)
Downsampled Matrix:
[[2.6953125 3.2421875 3.9609375 2.98828125]
[3.2265625 6.0859375 5.59375 3.59765625]
[3.34765625 5.125 5.5625 4.5703125 ]
[3.171875 2.96875 4.5859375 4.17578125]]
Upsampled Matrix (Insert Zeros):
[[2.6953125 0. 3.2421875 0. 3.9609375 0.
2.98828125 0. ]
[0. 0. 0. 0. 0. 0.
0. 0. ]
[3.2265625 0. 6.0859375 0. 5.59375 0.
3.59765625 0. ]
[0. 0. 0. 0. 0. 0.
0. 0. ]
[3.34765625 0. 5.125 0. 5.5625 0.
4.5703125 0. ]
[0. 0. 0. 0. 0. 0.
0. 0. ]
[3.171875 0. 2.96875 0. 4.5859375 0.
4.17578125 0. ]
[0. 0. 0. 0. 0. 0.
0. 0. ]]
Upsampled Matrix (After Applying Gaussian Filter):
[[24.2578125 17.8125 19.453125 21.609375 23.765625 20.84765625
17.9296875 8.96484375]
[17.765625 15.25 18.65625 18.8828125 19.109375 16.140625
13.171875 6.5859375 ]
[19.359375 18.625 24.34375 23.359375 22.375 18.3828125
14.390625 7.1953125 ]
[19.72265625 17.78515625 22.421875 22.3671875 22.3125 19.32421875
16.3359375 8.16796875]
[20.0859375 16.9453125 20.5 21.375 22.25 20.265625
18.28125 9.140625 ]
[19.55859375 14.61328125 16.1875 18.2421875 20.296875 18.89453125
17.4921875 8.74609375]
[19.03125 12.28125 11.875 15.109375 18.34375 17.5234375
16.703125 8.3515625 ]
[ 9.515625 6.140625 5.9375 7.5546875 9.171875 8.76171875
8.3515625 4.17578125]]
Laplacian Matrix:
[[-18.2578125 -11.8125 -19.453125 -17.609375 -15.765625
-13.84765625 -11.9296875 -4.96484375]
[-10.765625 -10.25 -9.65625 -15.8828125 -11.109375
-14.140625 -9.171875 -4.5859375 ]
[-18.359375 -9.625 -20.34375 -15.359375 -13.375
-16.3828125 -10.390625 -6.1953125 ]
[-18.72265625 -12.78515625 -15.421875 -14.3671875 -21.3125
-14.32421875 -10.3359375 -3.16796875]
[-11.0859375 -13.9453125 -12.5 -14.375 -15.25
-12.265625 -14.28125 -9.140625 ]
[-11.55859375 -14.61328125 -15.1875 -12.2421875 -20.296875
-9.89453125 -10.4921875 -3.74609375]
[-16.03125 -7.28125 -10.875 -12.109375 -9.34375
-14.5234375 -13.703125 -6.3515625 ]
[ -1.515625 0.859375 -4.9375 -6.5546875 -4.171875
-0.76171875 -1.3515625 -3.17578125]]