LeGall 5/3 Wavelet
The LeGall 5/3 wavelet is a specific type of biorthogonal wavelet widely used in image compression (famously in the JPEG 2000 standard) because it is computationally efficient and provides a good balance between compression ratio and visual quality.
In signal processing, filter taps, coefficients, and delays are three ways of looking at the same structure, but they describe different physical or mathematical aspects of a filter.
To visualize this, think of a digital filter as a tapped delay line.
1. Taps vs. Coefficients
In most contexts, the number of taps is equal to the number of coefficients.
- A "Tap": This is a physical connection point in the delay line where the signal is "tapped" off to be processed.
- A "Coefficient": This is the mathematical weight (the number) you multiply by the signal at that specific tap.
If you have a 5-tap filter (like the low-pass side of the LeGall 5/3), it means you are looking at 5 different points in time of the signal. To process those 5 points, you need 5 specific multipliers (coefficients).
2. How Delays Fit In
The number of delays is almost always \(N-1\), where \(N\) is the number of taps.
- To compare the "current" sample with the "previous" sample, you need 1 delay element (\(z^{-1}\)), but you are using 2 taps (the signal before the delay and the signal after).
- In the LeGall 5/3 scheme, the "5" filter uses 4 delay units to hold the 5 samples needed for the calculation.
The Relationship Formula
For a standard FIR (Finite Impulse Response) filter:
3. When the Terms Diverge
While they usually correspond 1:1, there are technical scenarios where the terminology shifts:
Symmetric Filters (The LeGall Case)
The LeGall 5/3 is a symmetric filter. Because the coefficients are the same on both sides (e.g., the weights might be \([-1, 2, 6, 2, -1]\)), an engineer might say there are only 3 unique coefficients, even though it is a 5-tap filter.
Sparse Filters
If you have a filter where most weights are zero (e.g., you only care about the signal now and the signal 100ms ago), you might have:
- 2 Taps: (The two points you actually multiply).
- 100+ Delays: (The memory buffer required to "wait" for that 100ms to pass).
Hardware Implementation
In "Lifting" specifically, we often talk about Predict and Update steps rather than a single long delay line.
- In the Predict step of the 5/3, we use 2 taps (\(x_e[n]\) and \(x_e[n+1]\)) to calculate the high-pass value.
- In the Update step, we use 2 taps (\(d[n-1]\) and \(d[n]\)) to calculate the low-pass value.
- By combining these "short" lifting steps, we mathematically create the effect of a 5-tap and 3-tap filter without ever building a single 5-tap delay line.
Summary Table
| Term | What it represents | Example (5-tap filter) |
|---|---|---|
| Tap | A "window" into the signal's history | 5 Taps |
| Coefficient | The weight/multiplier at that window | 5 Coefficients |
| Delay | The memory storage between taps | 4 Delays |