Image Compression using Discrete Cosine Transform

Implementation of lossy image compression using the DCT method (JPEG standard), based on Ken Cabeen and Peter Gent document.

Table of Contents

• Introduction
• Requirements
• Installation
• Usage
• How It Works
• DCT Equation
• Output Files
• Code Review Notes
• References

Introduction

As our use of and reliance on computers continues to grow, so does the need for efficient ways of storing large amounts of data. This project implements JPEG-style lossy image compression using the Discrete Cosine Transform (DCT).

Key Concepts:

• Two kinds of compression algorithms: lossless and lossy image compression
• JPEG uses lossy compression with DCT to separate images into parts of different frequencies
• Through quantization, less important frequency information is discarded
• The reconstructed image contains some distortion, but file size is significantly reduced

Requirements

• Python 3.x
• NumPy
• OpenCV (cv2)
• Matplotlib

Installation

# Install required packages
pip install numpy opencv-python matplotlib

# Or using requirements.txt (recommended)
pip install -r requirements.txt

Note: Create a requirements.txt file with:

numpy>=1.19.0
opencv-python>=4.5.0
matplotlib>=3.3.0

Usage

Basic Usage

python DCT.py <input_file> <compression_level>

Examples

# Medium compression (quality 50)
python DCT.py Image/car1.jpg 50

# High quality (low compression)
python DCT.py Image/car2.jpg 90

# Low quality (high compression)
python DCT.py Image/cameraman.tif 10

Compression Level Guide:

• 1-30: High compression, lower quality
• 30-70: Moderate compression, good balance
• 70-100: Low compression, high quality (closer to original)

Important: The input image dimensions should be divisible by 8 for proper block processing. The script does not currently handle padding for non-conforming dimensions.

How It Works

Overview of the Process

1. Block Division: The image is broken into 8×8 blocks of pixels
2. DCT Application: Working from left to right, top to bottom, the DCT is applied to each block
3. Quantization: Each block is compressed through quantization (discarding high-frequency components)
4. Storage: The array of compressed blocks is stored with reduced size
5. Decompression: When desired, the image is reconstructed using Inverse DCT (IDCT)

Processing Steps (Technical)

1. Convert image to RGB channels and subtract 128 (center around zero)
2. Apply DCT to each 8×8 block using the DCT coefficient matrix
3. Quantize DCT coefficients using JPEG standard quantization tables
4. For decompression: multiply by quantization matrix and apply IDCT
5. Add 128 back and merge channels to reconstruct the image

DCT Equation

The Discrete Cosine Transform coefficient matrix is computed as:

Where the DCT coefficient matrix T is defined as:

• T[0,j] = 1/√8 for all j
• T[i,j] = √(2/8) × cos((2j+1)iπ/16) for i > 0

Output Files

The script generates three output files in the current directory:

1. DCT.jpg - Visualization of DCT coefficients
2. After_Quantiz.jpg - Quantized DCT coefficients
3. Decompressed.jpg - Final reconstructed image

Performance Metrics

The script outputs:

• Compression/Decompression Time - Processing time for each operation
• RMS Error - Root Mean Square error between original and decompressed image
• SNR - Signal-to-Noise Ratio (higher is better)

Code Review Notes

Known Issues

1. DCT_compress.py is incomplete:

   • Missing quantization_level() function definition
   • Import statements at the bottom (should be at top)
   • Contains Jupyter notebook artifacts

2. Limited error handling:

   • No validation for file existence
   • No check if image dimensions are divisible by 8
   • No exception handling for file I/O operations

3. Hardcoded outputs:

   • Output files are always written to the current directory
   • Output filenames cannot be customized

Recommendations for Improvement

• Add input validation (file existence, dimensions)
• Add error handling with try-except blocks
• Make output paths configurable via command-line arguments
• Add function docstrings
• Fix DCT_compress.py or remove if not needed
• Add padding for images not divisible by 8

References

For more details about JPEG compression and DCT:

• See Document/dct.pdf in this repository
• DCT Equation Image
• Ken Cabeen and Peter Gent's original documentation

License

See repository for license information.