2D Fourier transform for images in Signal Processing

Introduction

The 2D Fourier transform helps us see the hidden patterns and frequencies inside an image. It changes the image from space view to frequency view.

To find repeating patterns or textures in a photo.

To remove noise or blur from an image.

To compress images by focusing on important frequencies.

To analyze shapes and edges in medical scans.

To detect periodic signals in satellite images.

Syntax

Signal Processing

F = fft2(image)
F_shifted = fftshift(F)
magnitude_spectrum = abs(F_shifted)

fft2 computes the 2D Fourier transform of the image.

fftshift moves the zero frequency to the center for easier viewing.

Examples

Compute the 2D Fourier transform of the image.

Signal Processing

F = fft2(image)

Shift the zero frequency component to the center of the spectrum.

Signal Processing

F_shifted = fftshift(F)

Get the magnitude (strength) of each frequency component.

Signal Processing

magnitude_spectrum = abs(F_shifted)

Sample Program

This code creates a simple 8x8 image with a small white square in the center. It then computes the 2D Fourier transform, shifts the zero frequency to the center, and calculates the magnitude spectrum. Finally, it prints the arrays and shows the images.

Signal Processing

import numpy as np
import matplotlib.pyplot as plt
from scipy.fft import fft2, fftshift

# Create a simple 8x8 image with a white square in the center
image = np.zeros((8, 8))
image[3:5, 3:5] = 1

# Compute the 2D Fourier transform
F = fft2(image)
F_shifted = fftshift(F)

# Compute magnitude spectrum
magnitude_spectrum = np.abs(F_shifted)

# Print the original image and magnitude spectrum
print('Original Image:')
print(image)
print('\nMagnitude Spectrum:')
print(np.round(magnitude_spectrum, 2))

# Plot for visualization
plt.subplot(1, 2, 1)
plt.title('Original Image')
plt.imshow(image, cmap='gray')
plt.axis('off')

plt.subplot(1, 2, 2)
plt.title('Magnitude Spectrum')
plt.imshow(np.log1p(magnitude_spectrum), cmap='gray')
plt.axis('off')
plt.show()

Important Notes

The magnitude spectrum shows how strong each frequency is in the image.

Using np.log1p helps to see details in the magnitude spectrum by scaling values.

High values in the center mean low-frequency components dominate the image.

Summary

The 2D Fourier transform changes an image from space to frequency view.

It helps find patterns, edges, and textures inside images.

Shifting the zero frequency to the center makes the frequency image easier to understand.

Practice

1. What does the 2D Fourier transform of an image primarily represent?

easy

A. The frequency components of the image

B. The color distribution of the image

C. The brightness levels of the image

D. The pixel coordinates of the image

Solution

1. Step 1: Understand the purpose of 2D Fourier transform

   The 2D Fourier transform converts spatial data into frequency data, showing how much of each frequency is present.

2. Step 2: Relate to image analysis

   In images, this means representing patterns and textures as frequency components rather than pixel values.

3. Final Answer:

   The frequency components of the image -> Option A

4. Quick Check:

   2D Fourier transform = frequency components [OK]

Hint: 2D Fourier transform shows frequencies, not colors or pixels [OK]

Common Mistakes:

• Confusing frequency with color or brightness
• Thinking it shows pixel positions
• Assuming it changes image colors

2. Which Python function is used to compute the 2D Fourier transform of an image array?

easy

A. np.fft.fft

B. np.fft.fft2

C. np.fft.ifft2

D. np.fft.fftshift

Solution

1. Step 1: Identify the function for 2D Fourier transform

   In numpy, fft2 computes the 2D Fourier transform of a 2D array (like an image).

2. Step 2: Differentiate from other functions

   fft is 1D, ifft2 is inverse transform, and fftshift centers the zero frequency.

3. Final Answer:

   np.fft.fft2 -> Option B

4. Quick Check:

   2D Fourier transform function = fft2 [OK]

Hint: Use fft2 for 2D transform, fftshift only re-centers [OK]

Common Mistakes:

• Using fft instead of fft2 for 2D data
• Confusing fft2 with inverse ifft2
• Using fftshift to compute transform

3. Given the code below, what is the shape of F_shifted?

import numpy as np
image = np.random.rand(64, 64)
F = np.fft.fft2(image)
F_shifted = np.fft.fftshift(F)

medium

A. (64, 64)

B. (32, 32)

C. (128, 128)

D. (64,)

Solution

1. Step 1: Check input image shape

   The input image has shape (64, 64), a 2D array with 64 rows and 64 columns.

2. Step 2: Understand fft2 and fftshift output shapes

   The 2D Fourier transform fft2 returns an array of the same shape as input. fftshift only reorders the frequencies but does not change shape.

3. Final Answer:

   (64, 64) -> Option A

4. Quick Check:

   fft2 and fftshift keep shape same as input [OK]

Hint: fft2 and fftshift keep the original image shape [OK]

Common Mistakes:

• Assuming fftshift changes shape
• Confusing shape with half or double size
• Thinking fft2 flattens the array

4. Identify the error in this code snippet for computing and displaying the magnitude spectrum of an image's 2D Fourier transform:

import numpy as np
import matplotlib.pyplot as plt
image = np.random.rand(128, 128)
F = np.fft.fft2(image)
magnitude = np.abs(F)
plt.imshow(magnitude, cmap='gray')
plt.show()

medium

A. Using np.abs instead of np.angle

B. Image size should be square

C. Missing fftshift to center zero frequency

D. plt.imshow requires RGB image

Solution

1. Step 1: Understand the purpose of fftshift

   After fft2, the zero frequency is at the corners. fftshift centers it for better visualization.

2. Step 2: Check code for fftshift usage

   The code computes magnitude but does not apply fftshift, so the spectrum will look off-centered.

3. Final Answer:

   Missing fftshift to center zero frequency -> Option C

4. Quick Check:

   fftshift centers spectrum for display [OK]

Hint: Always use fftshift before displaying magnitude spectrum [OK]

Common Mistakes:

• Skipping fftshift causes off-center spectrum
• Confusing magnitude with phase
• Thinking image must be RGB for imshow

5. You want to remove high-frequency noise from a grayscale image using its 2D Fourier transform. Which approach is best?

hard

A. Set low-frequency components to zero and inverse transform

B. Apply fftshift and keep all frequencies unchanged

C. Use np.abs on the transform and display it

D. Set high-frequency components to zero and inverse transform

Solution

1. Step 1: Identify noise frequency type

   High-frequency components correspond to noise and sharp changes; removing them smooths the image.

2. Step 2: Apply filtering in frequency domain

   Setting high-frequency components to zero removes noise; then inverse transform recovers the cleaned image.

3. Step 3: Understand why other options fail

   Setting low frequencies to zero removes main image info; fftshift only centers frequencies; np.abs just shows magnitude, no filtering.

4. Final Answer:

   Set high-frequency components to zero and inverse transform -> Option D

5. Quick Check:

   Remove noise by zeroing high frequencies [OK]

Hint: Zero high frequencies to remove noise, then inverse transform [OK]

Common Mistakes:

• Removing low frequencies loses main image details
• Thinking fftshift filters noise
• Confusing magnitude display with filtering