Low-pass and High-pass Filters

One of the key benefits of the Fourier Transformation is it enables us to do high-pass and low-pass filtering.

After we apply the Fourier Transformation:

we need to create a filtering mask

Low-Pass Filtering (Blurring)

A low-pass filter removes high-frequency components, which results in a blurred image. Low-pass mask:

rows, cols = image.shape: retrieves the number of rows and columns from the grayscale image;
crow, ccol = rows // 2, cols // 2: computes the center coordinates of the image;
mask = np.zeros((rows, cols), np.uint8): creates a mask of zeros with the same dimensions as the image;
mask[crow-30:crow+30, ccol-30:ccol+30] = 1: sets a 60×60 square region at the center of the mask to 1, allowing only the low-frequency components (near the center of the frequency domain) to pass through while filtering out high-frequency details.

High-Pass Filtering (Edge Detection)

A high-pass filter removes low-frequency components and enhances edges. High-pass mask:

highpass_mask = np.ones((rows, cols), np.uint8): initializes a mask of ones with the same dimensions as the image, meaning all frequencies are initially allowed to pass;
highpass_mask[crow-5:crow+5, ccol-5:ccol+5] = 0: sets a small 10×10 square at the center (low-frequency region) to zero, effectively blocking those frequencies.

Applying the filter

After creating the mask, we must apply a filter and transform our photo back to the spatial domain:

dft_filtered = dft_shift * mask: applies the frequency domain mask (e.g., low-pass or high-pass) by element-wise multiplication with the shifted DFT of the image;
dft_inverse = np.fft.ifftshift(dft_filtered): reverses the shift applied earlier to bring the frequency components back to their original positions;
image_filtered = np.fft.ifft2(dft_inverse): computes the Inverse Fourier Transform to convert the filtered frequency data back into the spatial domain;
image_filtered = np.abs(image_filtered): takes the absolute value to remove any residual imaginary components, resulting in a real-valued filtered image.

Task

Swipe to start coding

You are given an image of the sheep in the image variable and an image in the frequency domain in the dft_shift variable:

Get the shape of the image matrix and store it in rows and cols variables;
Calculate the center point and store in crow and ccol;
Define the low_mask as an array of zeros with (rows, cols) shape and np.uint8 type;
Choose the 20x20 center region of low_mask and fill it with 1.
Define the high_mask as an array of ones with (rows, cols) shape and np.uint8 type;
Choose the 20x20 center region of high_mask and fill it with 0;
Apply low_mask and high_mask to dft_shift and store the filtered frequences in lowpass_dft_filtered and highpass_dft_filtered accordingly;
Perform inverse transformation for lowpass_dft_filtered:
- Do inverse shifting and store in lowpass_dft_inverse;
- Do inverse transformation and store image in image_lowpass;
- Remove negative values for image_lowpass.
Perform inverse transformation for highpass_dft_inverse:
- Do inverse shifting and store in lowpass_dft_inverse;
- Do inverse transformation and store image in image_highpass;
- Remove negative values for image_highpass.

Solution

import io

import cv2

import numpy as np

import urllib.request

from PIL import Image

import matplotlib.pyplot as plt

url = "https://content-media-cdn.codefinity.com/courses/ef049f7b-ce21-45be-a9f2-5103360b0655/task_pictures/sheep.jpg"

with urllib.request.urlopen(url) as response:

image_data = response.read()

image = np.array(Image.open(io.BytesIO(image_data)))

image = cv2.cvtColor(image, cv2.COLOR_BGR2GRAY)

dft = np.fft.fft2(image)

dft_shift = np.fft.fftshift(dft)

# Write your code below

rows, cols = image.shape

crow, ccol = rows // 2, cols // 2

low_mask = np.zeros((rows, cols), np.uint8)

low_mask[crow-10:crow+10, ccol-10:ccol+10] = 1

high_mask = np.ones((rows, cols), np.uint8)

high_mask[crow-10:crow+10, ccol-10:ccol+10] = 0

lowpass_dft_filtered = dft_shift * low_mask

highpass_dft_filtered = dft_shift * high_mask

lowpass_dft_inverse = np.fft.ifftshift(lowpass_dft_filtered) # Inverse Shifting

image_lowpass = np.fft.ifft2(lowpass_dft_inverse) # Inverse Transformation

image_lowpass = np.abs(image_lowpass) # Remove negative values

highpass_dft_inverse = np.fft.ifftshift(highpass_dft_filtered) # Inverse Shifting

image_highpass = np.fft.ifft2(highpass_dft_inverse) # Inverse Transformation

image_highpass = np.abs(image_highpass) # Remove negative values

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Section 2. Chapter 3

single

import io

import cv2

import numpy as np

import urllib.request

from PIL import Image

import matplotlib.pyplot as plt

url = "https://content-media-cdn.codefinity.com/courses/ef049f7b-ce21-45be-a9f2-5103360b0655/task_pictures/sheep.jpg"

with urllib.request.urlopen(url) as response:

image_data = response.read()

image = np.array(Image.open(io.BytesIO(image_data)))

image = cv2.cvtColor(image, cv2.COLOR_BGR2GRAY)

dft = np.fft.fft2(image)

dft_shift = np.fft.fftshift(dft)

# Write your code below

rows, cols = ___

crow, ccol = ___

low_mask = ___

low_mask[___] = ___

high_mask = ___

high_mask[___] = ___

lowpass_dft_filtered = ___

highpass_dft_filtered = ___

lowpass_dft_inverse = ___ # Inverse Shifting

image_lowpass = ___ # Inverse Transformation

image_lowpass = ___ # Remove negative values

highpass_dft_inverse = ___ # Inverse Shifting

image_highpass = ___ # Inverse Transformation

image_highpass = ___ # Remove negative values

# Display the results

plt.figure(figsize=(10,10))

plt.subplot(3,1,1)

plt.imshow(image, cmap='gray')

plt.title('Original Image')

plt.subplot(3,1,2)

plt.imshow(image_lowpass, cmap='gray')

plt.title('Lowpass')

plt.subplot(3,1,3)

plt.imshow(image_highpass, cmap='gray')

plt.title('Highpass')

plt.tight_layout()

plt.show()

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