27. Fidelity Evaluation Functions

Lossy Compression Needs a Distortion Measure

So far, all coding has been lossless — perfect reconstruction is required. But many applications (audio, video, images) can tolerate some distortion. The question becomes: how much can we compress if we allow a small amount of loss?

This requires measuring "how close" the reconstructed signal is to the original.

The Fidelity Criterion

A fidelity evaluation function \(d(x, y)\) measures the distortion between original \(x\) and reconstruction \(y\).

For discrete symbols: \(d(x_i, y_j)\) is a per-symbol distortion.

For continuous functions: \(d\) could be integrated over time/space.

Common Distortion Measures

1. R.M.S. Criterion (Mean Squared Error)

\[ d(x, y) = \frac{1}{T} \int_0^T (x(t) - y(t))^2 \, dt \]

Most common in signal processing. Mathematically tractable, corresponds to energy difference.

2. Frequency-Weighted R.M.S.

Apply different weights to different frequencies (matching human perception):

\[ d(x, y) = \int_0^W W(f) |X(f) - Y(f)|^2 \, df \]

Where \(W(f)\) emphasizes frequencies the ear is sensitive to. Used in audio coding (MP3, AAC).

3. Absolute Error

\[ d(x, y) = \frac{1}{T} \int_0^T |x(t) - y(t)| \, dt \]

More robust to outliers than squared error. Used in image coding and robust statistics.

4. Perceptual Criteria

The ear and brain implicitly define evaluation functions. For example:
- Masking: loud sounds hide quieter ones at nearby frequencies
- Critical bands: the ear groups frequencies into ~24 bands
- Temporal masking: loud sounds mask preceding/following quiet sounds

Modern audio codecs (MP3, AAC, Opus) explicitly model these perceptual criteria.

The Discrete Case as Specialization

The discrete case (Part I–II) implicitly used a Hamming-like fidelity criterion:

\[ d(x, y) = \begin{cases} 0 & x = y \\ 1 & x \neq y \end{cases} \]

Zero distortion for exact match, constant distortion for any error. Rate-distortion with this criterion is equivalent to lossy source coding with symbol error rate constraint.

Fidelity vs. Rate Trade-off

graph LR
    A["High Fidelity<br/>Low Distortion"] -->|"more bits"| B["Low Rate<br/>High Compression"]
    B -->|"fewer bits"| A

The fundamental question: for a given source and distortion measure, what is the minimum rate needed to achieve distortion \(\leq D\)?

Next: §28 — The Rate for a Source Relative to a Fidelity Evaluation