inst/extdata/SCALING_FUNCTIONS_GUIDE.md

In BioGSP: Biological Graph Signal Processing for Spatial Data Analysis

SGWT Scaling Functions Guide

This guide provides comprehensive information about the different scaling function options available in the Spectral Graph Wavelet Transform (SGWT) implementation.

Overview

Scaling functions in SGWT serve as low-pass filters that capture the smooth, low-frequency components of signals on graphs. The choice of scaling function affects the localization properties, smoothness, and reconstruction quality of the wavelet transform.

Available Scaling Functions

1. Gaussian (Default)

Formula: h(λ) = exp(-λ²/(2σ²)) - Properties: Smooth, infinite support, excellent localization in both vertex and spectral domains - Use case: General-purpose applications, when smoothness is important - Characteristics: Exponential decay, no compact support

sgwt_scaling_kernel(x, scale_param = 1, scaling_type = "gaussian")

2. Exponential

Formula: h(λ) = exp(-λ/σ) - Properties: Monotonic decay, infinite support, simple form - Use case: When you want faster decay than Gaussian but still smooth - Characteristics: Linear exponential decay, heavier tail than Gaussian

sgwt_scaling_kernel(x, scale_param = 1, scaling_type = "exponential")

3. Polynomial

Formula: h(λ) = (1 + (λ/σ)²)^(-p) where p=2 - Properties: Algebraic decay, infinite support, heavy tails - Use case: When you need polynomial decay behavior - Characteristics: Slower decay than exponential functions

sgwt_scaling_kernel(x, scale_param = 1, scaling_type = "polynomial")

4. Meyer

Formula: Piecewise function with smooth transitions - Properties: Compact support, smooth transitions, bandlimited-like behavior - Use case: When you need strict frequency band separation - Characteristics: - h(λ) = 1 for λ ≤ 0.5σ - h(λ) = 0 for λ ≥ σ - Smooth cosine transition in between

sgwt_scaling_kernel(x, scale_param = 1, scaling_type = "meyer")

5. Spline

Formula: B-spline based function - Properties: Compact support, piecewise polynomial, smooth - Use case: When you need localized support with polynomial smoothness - Characteristics: Cubic spline with support on [0, 2σ]

sgwt_scaling_kernel(x, scale_param = 1, scaling_type = "spline")

6. Hann

Formula: h(λ) = 0.5(1 + cos(πλ/σ)) for λ ≤ σ - Properties: Compact support, cosine-based, smooth - Use case: Signal processing applications, windowing - Characteristics: Hann window shape, support on [0, σ]

sgwt_scaling_kernel(x, scale_param = 1, scaling_type = "hann")

7. Tight Frame

Formula: Shifted Hann kernel designed for tight frame properties - Properties: Ensures perfect reconstruction, tight frame property - Use case: When perfect reconstruction is critical - Characteristics: Designed to satisfy tight frame conditions

sgwt_scaling_kernel(x, scale_param = 1, scaling_type = "tight_frame")

Comparison of Properties

| Scaling Type | Support | Smoothness | Decay Rate | Tight Frame | Best For | |--------------|---------|------------|------------|-------------|----------| | Gaussian | Infinite | C∞ | Exponential | No | General use, smooth signals | | Exponential | Infinite | C∞ | Exponential | No | Fast decay, simple form | | Polynomial | Infinite | C∞ | Algebraic | No | Heavy-tailed signals | | Meyer | Compact | C∞ | Sharp cutoff | No | Frequency separation | | Spline | Compact | C² | Polynomial | No | Local support needed | | Hann | Compact | C∞ | Cosine | No | Windowing applications | | Tight Frame | Compact | C∞ | Cosine | Yes | Perfect reconstruction |

Usage Examples

Basic Usage

# Load the SGWT functions
source("R/sgwt_core.R")

# Create sample data
eigenvals <- c(0, 0.1, 0.5, 1.0, 1.5, 2.0)
scales <- c(2, 1, 0.5)

# Use different scaling functions
filters_gaussian <- compute_sgwt_filters(eigenvals, scales, scaling_type = "gaussian")
filters_meyer <- compute_sgwt_filters(eigenvals, scales, scaling_type = "meyer")
filters_tight <- compute_sgwt_filters(eigenvals, scales, scaling_type = "tight_frame")

Forward Transform with Different Scaling Functions

# Assuming you have signal, eigenvectors, eigenvalues
result_gaussian <- sgwt_forward(signal, eigenvectors, eigenvalues, scales, 
                               scaling_type = "gaussian")
result_meyer <- sgwt_forward(signal, eigenvectors, eigenvalues, scales, 
                            scaling_type = "meyer")

Comparing Scaling Functions

# Visual comparison of all scaling functions
comparison <- compare_scaling_functions(x_range = c(0, 3), scale_param = 1)

# This will plot all scaling functions for comparison
# and return a data frame with the values

Choosing the Right Scaling Function

For General Applications

• Gaussian: Best all-around choice, smooth and well-localized
• Exponential: Good alternative when you want simpler form

For Specific Requirements

• Perfect Reconstruction: Use tight_frame
• Compact Support: Use meyer, spline, or hann
• Frequency Band Separation: Use meyer
• Heavy-tailed Signals: Use polynomial

For Signal Processing Applications

• Windowing: Use hann
• Spectral Analysis: Use meyer or tight_frame
• Denoising: Use gaussian or exponential

Technical Notes

Scale Parameter

The scale_param controls the width/spread of the scaling function: - Larger values → wider support, more low-pass filtering - Smaller values → narrower support, less smoothing

Computational Considerations

• Compact support functions (meyer, spline, hann, tight_frame) are computationally more efficient
• Infinite support functions (gaussian, exponential, polynomial) provide better theoretical properties but may require truncation

Reconstruction Quality

• Tight frame scaling functions guarantee perfect reconstruction
• Other functions may have small reconstruction errors depending on the graph and signal

References

1. Hammond, D. K., Vandergheynst, P., & Gribonval, R. (2011). Wavelets on graphs via spectral graph theory. Applied and computational harmonic analysis, 30(2), 129-150.

2. Shuman, D. I., Wiesmeyr, C., Holighaus, N., & Vandergheynst, P. (2015). Spectrum-adapted tight graph wavelet and vertex-frequency frames. IEEE Transactions on Signal Processing, 63(16), 4223-4235.

3. Leonardi, N., & Van De Ville, D. (2013). Tight wavelet frames on multislice graphs. IEEE Transactions on Signal Processing, 61(13), 3357-3367.

BioGSP documentation built on Feb. 2, 2026, 5:06 p.m.