Lloyd-Max Quantizer for Gaussian Sources

This project implements and evaluates a scalar Lloyd-Max quantizer for samples drawn from a zero-mean Gaussian source. It iteratively learns decision boundaries and reproduction levels that reduce empirical mean squared quantization error.

Problem definition

For a bit depth b, the quantizer uses L = 2^b reproduction levels. Each input sample is assigned to one decision region and replaced by that region's centroid. The implementation measures distortion as

D = mean((x - Q(x))²)

and reports the signal-to-quantization-noise ratio as

SQNR = 10 log10(σ² / D).

Lloyd-Max algorithm

The implementation:

1. generates a deterministic Gaussian sample set;
2. initializes internal boundaries from sample quantiles;
3. assigns samples to regions;
4. updates each reproduction level to the mean of its assigned samples;
5. moves internal boundaries to the midpoint of adjacent reproduction levels; and
6. repeats until the distortion tolerance is reached or the iteration limit is exhausted.

Empty regions are handled explicitly by selecting a midpoint or a finite offset from the available boundary.

Experimental setup

The submitted experiment uses σ = 1, 300,000 samples, random seed 12345, quantile initialization, a visualization/update range of ±4σ, distortion tolerance 1e-9, and at most 200 iterations. The reproducibility script evaluates bit depths 2 through 5 with those parameters.

Results

The checked-in metrics were reproduced from the source code on the same deterministic sample set.

Bits	Levels	Iterations	MSE	SQNR (dB)
2	4	32	0.1170605606	9.315894
3	8	65	0.0342777468	14.649877
4	16	163	0.0093695110	20.282831
5	32	200	0.0025255458	25.976448

For this fixed sample set and configuration, increasing the bit depth reduced empirical distortion and increased SQNR. The 5-bit run reached the configured 200-iteration limit, so its values should be interpreted as the best state reached within that budget rather than proof of convergence.

Quantizer visualizations

2-bit quantizer

5-bit quantizer

How to run

Python 3.10 or newer is recommended.

python -m venv .venv
# Windows: .venv\Scripts\activate
# macOS/Linux: source .venv/bin/activate
python -m pip install -r requirements.txt
python generate_results.py

Running python quantization.py executes the original 5-bit experiment and opens an interactive visualization. Run the checks with:

python -m unittest discover -s tests

Project structure

.
├── quantization.py          # Canonical submitted implementation
├── generate_results.py      # Reproducible metrics and figure generation
├── requirements.txt
├── results/metrics.csv      # Deterministic results for 2–5 bits
├── docs/
│   ├── images/              # Selected generated visualizations
│   └── report-excerpt.pdf   # Privacy-sanitized report excerpt
└── tests/test_quantization.py

Limitations

• The optimization is based on a finite Monte Carlo sample rather than analytical integration of the Gaussian density.
• Results are specific to the documented seed, sample count, initialization, clipping parameter, and iteration budget.
• The source contains a level-spacing tolerance check; the distortion tolerance is the effective convergence condition in the reproduced runs.
• The project evaluates one zero-mean Gaussian source and does not implement a separate fixed uniform-quantizer benchmark.
• The sanitized report excerpt excludes the original cover and terminal screenshots because they contained personal or local-machine information.