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:

dft = np.fft.fft2(image)
dft_shift = np.fft.fftshift(dft)

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

mask = np.zeros((rows, cols), np.uint8)
mask[crow-30:crow+30, ccol-30:ccol+30] = 1

• 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)
highpass_mask[crow-5:crow+5, ccol-5:ccol+5] = 0

• 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

dft_inverse = np.fft.ifftshift(dft_filtered) # Inverse shifting
image_filtered = np.fft.ifft2(dft_inverse) # Inverse transformation
image_filtered = np.abs(image_filtered) # Remove negative values

• 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.