NumPy, SciPy FFTs: distinct performance, real-valued optimizations.

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7 min readJan 15, 2024

PythonFleek: NumPy, SciPy FFTs: distinct performance, real-valued optimizations. — **PythonFleek, demo 3**: NumPy, SciPy FFTs: distinct performance, real-valued optimizations.

Hermitian, Standard FFT: SciPy Outperforms

The Fast Fourier Transform (FFT) is a fundamental tool in signal processing and data analysis, allowing us to transform data from the time domain into the frequency domain. Understanding the differences between various FFT methods provided by NumPy and SciPy is crucial for selecting the right approach for a given problem.
— NumPy and SciPy offer FFT methods for different types of data and dimensions.
— Standard FFTs work with complex or real-valued inputs.
— Real FFTs are optimized for real-valued inputs.
— Hermitian FFTs are specialized for inputs with Hermitian symmetry.

**PythonFleek, demo 2**: NumPy, SciPy FFTs: distinct performance, real-valued optimizations.

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CODE

import numpy as np
import scipy.fftpack as spfft

# ### Create sample data
data = np.random.random(1024)

# ### Standard FFT (NumPy)
fft_np = np.fft.fft(data)

# ### Standard FFT (SciPy)
fft_sp = spfft.fft(data)

# ### Real FFT (NumPy)
rfft_np = np.fft.rfft(data)

# ### Real FFT (SciPy)
rfft_sp = spfft.rfft(data)

# ### Hermitian FFT (NumPy)
hfft_np = np.fft.hfft(data[:512])

# ### Hermitian FFT (SciPy)
hfft_sp = spfft.hfft(data[:512])

EXPLANATION

Standard FFT (NumPy and SciPy):
— Used for complex or real-valued inputs.
— Computes the discrete Fourier Transform (DFT).
— np.fft.fft and spfft.fft are functionally similar but may have different performance characteristics.
Real FFT (NumPy and SciPy):
— Optimized for real-valued inputs.
— Faster than standard FFT for real data.
— np.fft.rfft and spfft.rfft exploit symmetry in the Fourier transform of real-valued inputs.
Hermitian FFT (NumPy and SciPy):
— Designed for data with Hermitian symmetry, typically representing a real spectrum.
— Efficient for certain types of real-valued signals.
— np.fft.hfft and spfft.hfft focus on the half-spectrum due to Hermitian symmetry.
Performance and Use Cases:
— NumPy is generally more popular and widely used, but SciPy’s FFT pack might offer better performance in some cases.
— Choice depends on data type, size, and specific application requirements.
Integration with Ecosystem: Both NumPy and SciPy FFT functions integrate well with other scientific computing tools in Python.
Precision and Accuracy: Both libraries offer high precision and accuracy, but minor differences may arise due to implementation details.
Flexibility:
— NumPy provides a simple and straightforward interface, suitable for most general purposes.
— SciPy might offer additional flexibility for advanced users.
Community Support and Documentation: Both libraries are well-supported with extensive documentation and community examples.
Computation Efficiency: Efficiency may vary based on the specific library versions and hardware used.
Compatibility: Both libraries are compatible with most Python distributions and work well in diverse environments.

Choosing between NumPy and SciPy FFT methods can make or break your analysis

Ever wondered why seasoned data scientists prefer one over the other for Fourier Transforms? Did you know that SciPy’s FFT methods can perform significantly faster than NumPy’s in complex computations?

Who: This topic is crucial for Scientists, Engineers, and anyone dealing with time-series data, image processing, or any domain where transforming data into the frequency domain is necessary.

What: FFT (Fast Fourier Transform) methods in NumPy and SciPy are algorithms for converting a signal from the time domain to the frequency domain. NumPy provides general FFT functionalities, while SciPy offers additional specialized methods, each with distinct characteristics and optimizations.

Why: Understanding these differences enables users to choose the most efficient and accurate method for their specific use case, particularly when dealing with large datasets, real-time processing, or requiring high precision in frequency domain analysis.

Install

# Conda
conda install numpy scipy

# Pip
pip install numpy scipy

# Poetry
poetry add numpy scipy

Algorithm

Python-Fleek: “Input Signal” -> “NumPy FFT” -> “Frequency Domain Representation” “Input Signal” -> “SciPy FFT” -> “Frequency Domain Representation” “Frequency Domain Representation” -> “Analysis/Application”

Demo 1

def demo_fft(input_data):
    print("Input Data:", input_data)
    fft_result = np.fft.fft(input_data)
    print("FFT Output:", fft_result)

demo_data = np.random.random(10)
demo_fft(demo_data)

Case Study

— Scenario: A Machine Learning Engineer at a tech company is developing a feature extraction pipeline for audio data to improve a speech recognition system.
— Application: The engineer uses SciPy’s FFT for preprocessing, as it offers specialized functions for real-valued audio signals, enhancing the accuracy of the feature extraction.
— Outcome: The optimized preprocessing leads to a significant improvement in the speech recognition model’s performance.

Pitfalls

— Aliasing Issues: Without proper sampling, FFT can lead to aliasing, distorting the frequency domain representation.
— Windowing Necessity: Failing to apply a window function can result in spectral leakage.
— Misinterpreting Output: The FFT output needs careful interpretation, especially understanding the Nyquist frequency and binning.
— Complexity of Inverse FFT: Not realizing that inverse FFT can return complex numbers, even for real inputs.
— Size Limitations: Overlooking the limitations in the size of the input array can lead to performance degradation.

Tips for Production

— Vectorization: Utilize NumPy’s vectorized operations to speed up FFT calculations.
— Use Real FFT for Real Data: When working with real-valued data, use rfft to halve computation time.
— Efficient Memory Usage: Be mindful of memory usage, especially for large datasets.
— Parallel Processing: Leverage parallel processing capabilities for large-scale FFT computations.
— Appropriate Library Choice: Choose between NumPy and SciPy based on the specific needs of the application.

Demo 2

import matplotlib.pyplot as plt
import numpy as np
from scipy import fftpack
from skimage import io, color

def plot_fft_demo(image, axes, title=''):
    # Check if the image is grayscale; if not, convert it
    if len(image.shape) == 3 and image.shape[2] == 3:
        image = color.rgb2gray(image)

    # Compute FFT
    fft_image = fftpack.fft2(image)
    fft_shifted = fftpack.fftshift(fft_image)

    # Calculate magnitude spectrum and avoid log(0) issue
    magnitude_spectrum = np.log(np.abs(fft_shifted) + 1e-8)  # Small constant to avoid log(0)

    # Create masks for low and high pass filters
    rows, cols = image.shape
    crow, ccol = rows // 2, cols // 2
    mask_low = np.zeros((rows, cols), np.uint8)
    mask_low[crow-30:crow+30, ccol-30:ccol+30] = 1
    mask_high = np.ones((rows, cols), np.uint8)
    mask_high[crow-30:crow+30, ccol-30:ccol+30] = 0

    # Apply low and high pass filters
    fft_shifted_low_pass = fft_shifted * mask_low
    magnitude_spectrum_low_pass = np.log(np.abs(fft_shifted_low_pass) + 1e-8)
    image_low_pass = np.abs(fftpack.ifft2(fftpack.ifftshift(fft_shifted_low_pass)))
    fft_shifted_high_pass = fft_shifted * mask_high
    magnitude_spectrum_high_pass = np.log(np.abs(fft_shifted_high_pass) + 1e-8)
    image_high_pass = np.abs(fftpack.ifft2(fftpack.ifftshift(fft_shifted_high_pass)))

    # Determine the aspect ratio for the plots
    aspect_ratio = image.shape[1] / image.shape[0]

    # Plotting
#     fig, axes = plt.subplots(2, 2, figsize=(8 * aspect_ratio, 8))
    axes[0, 0].imshow(image, cmap='gray')
    axes[0, 0].set_title(f'Original Image - {title}')
    axes[0, 0].axis('off')
    axes[0, 1].imshow(magnitude_spectrum, cmap='prism')
    axes[0, 1].set_title('Normalized FFT (prism-banded colors)')
    axes[0, 1].axis('off')
    axes[1, 0].imshow(image_high_pass, cmap='flag')
    axes[1, 0].set_title('FFT High Pass (RWBK-banded colors)')
    axes[1, 0].axis('off')
    axes[1, 1].imshow(image_high_pass, cmap='gist_earth')
    axes[1, 1].set_title('FFT High Pass (Earth colors)')
    axes[1, 1].axis('off')
#     plt.show()

# Load the image
url = 'https://upload.wikimedia.org/wikipedia/commons/thumb/6/66/Louvre_Museum_Wikimedia_Commons.jpg/800px-Louvre_Museum_Wikimedia_Commons.jpg'
image = io.imread(url)

aspect_ratio = 2.2
fig, axes = plt.subplots(2, 2, figsize=(8 * aspect_ratio, 8))
plot_fft_demo(image, axes, title='Louvre Museum')

Demo 3


import numpy as np
import matplotlib.pyplot as plt
from scipy import fftpack
from skimage import color

def create_square_wave(size):
    image = np.zeros((size, size))
    image[size // 4:3 * size // 4, size // 4:3 * size // 4] = 1
    return image

def create_circle_pattern(size):
    y, x = np.ogrid[:size, :size]
    center = (size // 2, size // 2)
    radius = size // 4
    mask = (x - center[0])**2 + (y - center[1])**2 <= radius**2
    image = np.zeros((size, size))
    image[mask] = 1
    return image

def create_checkerboard(size):
    image = np.zeros((size, size))
    image[::3, 1::3] = 1.
    image[1::2, ::2] = 0.5
    return image

def create_radial_sine_wave(size):
    y, x = np.ogrid[-size/2:size/2, -size/2:size/2]
    r = np.sqrt(x**2 + y**2)
    sine_wave = np.sin(2 * np.pi * r / 20)
    return sine_wave

# Generate geometric images
size = 256  # Size of the images
square_wave_image = create_square_wave(size)
circle_pattern_image = create_circle_pattern(size)
checkerboard_image = create_checkerboard(size)
radial_sine_wave_image = create_radial_sine_wave(size)

# Plot FFTs for each image

aspect_ratio = 1
fig, axes = plt.subplots(4, 4, figsize=(18 * aspect_ratio, 18))

plot_fft_demo(square_wave_image, axes[:2,:2], title='Square Wave')

plot_fft_demo(circle_pattern_image, axes[2:,:2], title='Circle')

plot_fft_demo(checkerboard_image, axes[2:,2:], title='Checkerboard')

plot_fft_demo(radial_sine_wave_image, axes[:2,2:], title='Radial Sine Wave')