🌍 Projet OMI — 2D Fourier Transform Applied to Image Processing

Frequency analysis of images using the 2D Fourier Transform: decomposition, filtering, and reconstruction of an Earth image through sinusoidal components.

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📌 Table of Contents

• Overview
• Mathematical Concepts
• Project Structure
• Notebook Content
• Results & Interpretations
• Real-World Applications
• Key Formulas
• How to Run
• Dependencies

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🔬 Overview

This project explores the 2D Fourier Transform (FT2D) and its applications to digital image processing. Through two progressive exercises, we analyze the decomposition of images into frequency components (sinusoids) and their reconstruction via the Inverse Fourier Transform (IFT).

The reference image used throughout the project is a photograph of the Earth (Earth.png), converted to grayscale for analysis.

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🧠 Mathematical Concepts

Concept	Description
2D Fourier Transform	Decomposes an image into spatial frequencies
Inverse Fourier Transform	Reconstructs the image from its frequency components
Sinusoidal grating	2D pattern whose amplitude varies sinusoidally
Amplitude	Contrast between light and dark regions of the grating
Phase	Lateral shift of the sinusoidal signal
Spatial frequency	Number of cycles per unit distance in the image
Point pairs (FT2D)	Symmetric points in Fourier space representing a sinusoid
fftshift	Recenters the Fourier spectrum around low frequencies

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📂 Project Structure

projet-OMI-transformee-fourier/
│
├── notebooks/                      # Jupyter / Colab notebooks
│   └── PROJET_OMI.ipynb            # Main notebook (Google Colab)
│
├── images/                         # Reference images
│   └── Earth.png                   # Earth image used in the project
│
├── docs/                           # Additional documentation
│   └── rapport.md                  # Supplementary notes and explanations
│
├── .gitignore                      # Files ignored by Git
├── requirements.txt                # Python dependencies
└── README.md                       # Project documentation

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📓 Notebook Content

Exercise 1 — Interactive Sinusoidal Grating Visualization

Generates an interactive 2D sinusoidal grating with sliders to vary in real time:

• Amplitude (0.1 to 50) — controls image contrast
• Phase (-π to π) — controls lateral shift of the grating

Key observations:

Parameter	Effect
High amplitude (e.g., 50)	Strong contrast, distinct black and white bands
Low amplitude (e.g., 0.1)	Nearly all-black image, total loss of contrast
Increasing phase	Grating shifts to the left
Decreasing phase	Grating shifts to the right

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Exercise 2 — Earth Image Analysis via FT2D

This exercise proceeds through five progressive steps:

Earth.png (RGB)
       │
       ▼
┌──────────────────────┐
│ Step 1: Load &       │──→ Convert to grayscale (np.mean on RGB)
│ Convert Image        │
└──────────┬───────────┘
           │
           ▼
┌──────────────────────┐
│ Step 2: Compute      │──→ np.fft.fft2 + fftshift
│ 2D Fourier Transform │    Log-scale spectrum display
└──────────┬───────────┘
           │
           ▼
┌──────────────────────┐
│ Step 3: Identify     │──→ Sort frequencies by amplitude
│ Point Pairs          │    Identify dominant sinusoidal components
└──────────┬───────────┘
           │
           ▼
┌──────────────────────┐
│ Step 4: Iterative    │──→ IFT per pair, cumulative reconstruction
│ IFT Reconstruction   │    Observe progressive image formation
└──────────┬───────────┘
           │
           ▼
┌──────────────────────┐
│ Step 5: Frequency    │──→ Reconstruct using % of frequencies
│ Percentage Filtering │    Compare low vs high frequency content
└──────────────────────┘

Step 1 — Image Loading & Conversion: Upload Earth.png into Google Colab, convert to grayscale using np.mean across RGB channels.

Step 2 — 2D Fourier Transform: Compute FT2D with np.fft.fft2, apply np.fft.fftshift to recenter the spectrum, and display in logarithmic scale (np.log(abs(...))) to visualize frequencies.

Step 3 — Point Pair Identification: Point pairs in the Fourier spectrum represent sinusoidal waves in the image. Frequencies are sorted by decreasing amplitude to identify the most significant components.

Step 4 — Iterative IFT Reconstruction: For each frequency pair, apply the Inverse Fourier Transform and accumulate sinusoids one by one to progressively reconstruct the image.

Step 5 — Frequency Percentage Filtering: Reconstruct the image using only a percentage of available frequencies:

Frequencies Used	Result
100%	Identical to the original image
10% lowest	Still recognizable (low freq = global contours)
10% highest	Grayish image (high freq = fine details, little useful info)

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🔬 Results & Interpretations

Why Identify Point Pairs?

In the FT2D, each symmetric pair of points corresponds to a 2D sinusoidal wave in the image. By modulating the amplitude, phase, and frequency of these waves, then applying the IFT, we can:

• Reconstruct the original image
• Filter specific frequencies (compression, denoising)
• Analyze the frequency structure of the image

High vs Low Frequencies

Frequencies	Content	Effect on Image
Low frequencies	Global shapes, main contours	Image recognizable even with few
High frequencies	Fine details, textures, noise	Weak contribution but needed for sharpness

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🌍 Real-World Applications

The Fourier Transform is a fundamental mathematical tool used across many fields:

Domain	Application
Signal Processing	Filtering, spectral analysis
Image Processing	JPEG compression, edge detection
Medicine	MRI (Magnetic Resonance Imaging), CT scanners
Telecommunications	Signal modulation, WiFi, 4G/5G
Astronomy	Light and radio wave analysis
Quantum Computing	Shor's algorithm for factorization

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📐 Key Formulas

Discrete 2D Fourier Transform

$$F(u,v) = \sum_{x=0}^{M-1} \sum_{y=0}^{N-1} f(x,y) \cdot e^{-2\pi i \left(\frac{ux}{M} + \frac{vy}{N}\right)}$$

Inverse Fourier Transform

$$f(x,y) = \frac{1}{MN} \sum_{u=0}^{M-1} \sum_{v=0}^{N-1} F(u,v) \cdot e^{2\pi i \left(\frac{ux}{M} + \frac{vy}{N}\right)}$$

2D Sinusoidal Grating

$$g(x,y) = A \cdot \sin\left(2\pi(f_x \cdot x + f_y \cdot y) + \phi\right)$$

Where:

• A = amplitude
• f_x, f_y = spatial frequencies in x and y
• φ = phase

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🚀 How to Run

Option 1 — Google Colab (recommended, no installation required)

1. Open the notebook in Colab
2. Click "Runtime" → "Run all"
3. When the files.upload() cell executes, upload the Earth.png file

Option 2 — Local (Jupyter Notebook)

Prerequisites: Python 3.8+, pip

# 1. Clone the repository
git clone https://github.com/YOUR_USERNAME/projet-OMI-transformee-fourier.git
cd projet-OMI-transformee-fourier

# 2. Install dependencies
pip install numpy matplotlib ipywidgets jupyter

# 3. Launch Jupyter
jupyter notebook notebooks/PROJET_OMI.ipynb

⚠️ For local execution, replace the from google.colab import files and files.upload() lines with:

image_filename = "../images/Earth.png"

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📦 Dependencies

Library	Purpose
numpy	Numerical computing, FFT (np.fft)
matplotlib	Image display and plotting
ipywidgets	Interactive sliders (amplitude, phase)
google.colab	File upload (Colab only)