2D FFT (fft2) in SciPy - Mini Project: Build & Apply

2D FFT (fft2) Analysis of Image Data

📖 Scenario: You have a small 2D image represented as a grid of pixel brightness values. You want to analyze the image's frequency components using a 2D Fast Fourier Transform (FFT).

🎯 Goal: Build a Python program that creates a 2D array representing an image, sets up a configuration variable for the FFT, applies the 2D FFT using scipy.fft.fft2, and prints the transformed data.

📋 What You'll Learn

Create a 2D numpy array called image with exact values

Create a variable called norm_mode to configure normalization

Use scipy.fft.fft2 with image and norm=norm_mode

Print the resulting 2D FFT array

💡 Why This Matters

🌍 Real World

2D FFT is used in image processing to analyze patterns, compress images, and filter noise.

💼 Career

Understanding 2D FFT helps in roles like data analyst, image processing engineer, and machine learning specialist working with visual data.

1

Create the 2D image array

Create a 2D numpy array called image with these exact values: [[1, 2, 3], [4, 5, 6], [7, 8, 9]].

SciPy

import numpy as np
# Create the 2D numpy array called image with the specified values
# Your code here

Hint

Use np.array to create a 2D array with the exact nested list values.

2

Set the normalization mode for FFT

Create a variable called norm_mode and set it to the string 'ortho' to normalize the FFT output.

SciPy

import numpy as np
image = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
# Create the variable norm_mode and set it to 'ortho'
# Your code here

Hint

Assign the string 'ortho' to the variable norm_mode.

3

Apply the 2D FFT to the image

Import fft2 from scipy.fft and create a variable called fft_result by applying fft2 to image with norm=norm_mode.

SciPy

import numpy as np
image = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
norm_mode = 'ortho'
# Import fft2 and apply it to image with norm=norm_mode, store in fft_result
# Your code here

Hint

Use from scipy.fft import fft2 and call fft2(image, norm=norm_mode).

4

Print the 2D FFT result

Print the variable fft_result to display the 2D FFT output.

SciPy

import numpy as np
image = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
norm_mode = 'ortho'
from scipy.fft import fft2
fft_result = fft2(image, norm=norm_mode)
# Print the fft_result variable
# Your code here

Hint

Use print(fft_result) to show the FFT output.

Practice

1. What does the scipy.fft.fft2 function do with a 2D array?

easy

A. It calculates the sum of all elements in the 2D array.

B. It converts the 2D data from the spatial domain to the frequency domain.

C. It sorts the 2D array in ascending order.

D. It normalizes the 2D array values between 0 and 1.

Solution

1. Step 1: Understand the purpose of fft2

   The fft2 function computes the 2-dimensional discrete Fourier Transform, which changes data from spatial to frequency domain.

2. Step 2: Compare options with function behavior

   Only It converts the 2D data from the spatial domain to the frequency domain. correctly describes this transformation. Other options describe unrelated operations.

3. Final Answer:

   It converts the 2D data from the spatial domain to the frequency domain. -> Option B

4. Quick Check:

   2D FFT = spatial to frequency transform [OK]

Hint: Remember fft2 changes space data into frequency patterns [OK]

Common Mistakes:

• Confusing fft2 with sorting or summing functions
• Thinking fft2 normalizes data
• Assuming fft2 works only on 1D data

2. Which of the following is the correct way to import and use fft2 from scipy?

easy

A. import fft2 from scipy
result = fft2(data)

B. import scipy.fft2
result = scipy.fft2(data)

C. from scipy import fft2
result = fft2(data)

D. from scipy.fft import fft2
result = fft2(data)

Solution

1. Step 1: Recall correct import syntax for fft2

   The fft2 function is in the scipy.fft module, so the correct import is from scipy.fft import fft2.

2. Step 2: Check usage correctness

   Calling fft2(data) after import is correct. Other options have wrong import syntax or module names.

3. Final Answer:

   from scipy.fft import fft2 result = fft2(data) -> Option D

4. Quick Check:

   Use 'from scipy.fft import fft2' [OK]

Hint: Import fft2 from scipy.fft module directly [OK]

Common Mistakes:

• Trying to import fft2 directly from scipy
• Using incorrect import syntax like 'import fft2 from scipy'
• Assuming fft2 is a standalone package

3. What is the output of this code snippet?

import numpy as np
from scipy.fft import fft2

arr = np.array([[1, 2], [3, 4]])
result = fft2(arr)
print(np.round(result, 2))

medium

A. [[1.+0.j 2.+0.j]
[3.+0.j 4.+0.j]]

B. [[10.+0.j 2.+0.j]
[ 4.+0.j 0.+0.j]]

C. [[10.+0.j -2.+0.j]
[-4.+0.j 0.+0.j]]

D. [[0.+0.j 0.+0.j]
[0.+0.j 0.+0.j]]

Solution

1. Step 1: Calculate 2D FFT of the array

   The 2D FFT of [[1,2],[3,4]] results in [[10, -2], [-4, 0]] with complex zeros in imaginary parts.

2. Step 2: Round and check output

   Rounding to 2 decimals keeps values same. [[10.+0.j -2.+0.j] [-4.+0.j 0.+0.j]] matches the expected complex array output.

3. Final Answer:

   [[10.+0.j -2.+0.j] [-4.+0.j 0.+0.j]] -> Option C

4. Quick Check:

   fft2([[1,2],[3,4]]) = [[10,-2],[-4,0]] [OK]

Hint: Remember fft2 sums all elements for [0,0] frequency [OK]

Common Mistakes:

• Confusing fft2 output with original array
• Ignoring complex number format
• Mixing up signs in output

4. Identify the error in this code that tries to compute the 2D FFT:

import numpy as np
from scipy.fft import fft2

arr = [[1, 2], [3]]
result = fft2(arr)
print(result)

medium

A. The array shape is invalid for fft2.

B. The input array must be a NumPy array, not a list of lists.

C. The print statement syntax is incorrect.

D. fft2 function is not imported correctly.

Solution

1. Step 1: Check the input array construction

   The arr = [[1, 2], [3]] creates a ragged array (rows have different lengths: 2 and 1), which cannot be converted to a proper NumPy 2D array.

2. Step 2: Identify the error cause

   fft2(arr) fails because np.asarray internally raises ValueError due to invalid shape.

3. Final Answer:

   The array shape is invalid for fft2. -> Option A

4. Quick Check:

   ragged lists cause shape error in fft2 [OK]

Hint: Use consistent lengths in 2D lists before fft2 [OK]

Common Mistakes:

• Creating lists with uneven row lengths
• Not validating array shape before FFT
• Assuming fft2 accepts any list structure

5. You have a 2D image array and want to filter out high-frequency noise using fft2. Which sequence of steps correctly applies this?

hard

A. Apply fft2 to image, zero out high-frequency components, then apply ifft2 to get filtered image.

B. Apply ifft2 to image, zero out low-frequency components, then apply fft2 to get filtered image.

C. Apply fft2 to image, multiply all frequencies by zero, then apply ifft2 to get filtered image.

D. Apply fft2 to image, add random noise to frequency components, then apply ifft2.

Solution

1. Step 1: Understand filtering in frequency domain

   To remove high-frequency noise, transform image with fft2, then zero out high-frequency parts (usually edges of frequency matrix).

2. Step 2: Reconstruct filtered image

   After modifying frequency data, apply inverse fft2 (ifft2) to get back filtered spatial image.

3. Final Answer:

   Apply fft2 to image, zero out high-frequency components, then apply ifft2 to get filtered image. -> Option A

4. Quick Check:

   Filter noise by zeroing high frequencies in fft2 domain [OK]

Hint: Filter noise by zeroing high frequencies, then inverse fft2 [OK]

Common Mistakes:

• Applying ifft2 before fft2
• Zeroing all frequencies removes entire image
• Adding noise instead of removing it