Title: High-Performance Wavelet Lifting Transforms
Version: 0.9.0
Date: 2026-02-17
Description: Performs Wavelet Lifting Transforms focusing on signal denoising and functional data analysis (FDA). Implements a hybrid architecture with a zero-allocation 'C++' core for high-performance processing. Features include unified offline (batch) denoising, causal (real-time) filtering using a ring buffer engine, and adaptive recursive thresholding.
License: MIT + file LICENSE
URL: https://github.com/mkyou/rLifting
BugReports: https://github.com/mkyou/rLifting/issues
Encoding: UTF-8
LazyData: true
RoxygenNote: 7.3.3
VignetteBuilder: knitr
LinkingTo: Rcpp
Imports: Rcpp
Suggests: testthat (≥ 3.0.0), knitr, rmarkdown, dplyr, ggplot2, microbenchmark
Config/testthat/edition: 3
Depends: R (≥ 4.1.0)
NeedsCompilation: yes
Packaged: 2026-02-17 17:24:06 UTC; moise
Author: Moises da Silva [aut, cre]
Maintainer: Moises da Silva <moisesdff8@gmail.com>
Repository: CRAN
Date/Publication: 2026-02-20 10:00:10 UTC

rLifting: High-Performance Wavelet Lifting Transforms

Description

A unified framework for Wavelet Transforms using the Lifting Scheme. It provides robust tools for offline signal analysis and functional data analysis (FDA), while also enabling high-performance causal processing for real-time applications via a specialized 'C++' core.

Author(s)

Maintainer: Moises da Silva moisesdff8@gmail.com

See Also

Useful links:

Generate standard signals for wavelet testing (Donoho-Johnstone Benchmark)

Description

Creates synthetic classic signals used for wavelet validation. Formulas are based on Donoho and Johnstone and Liu et al.

Usage

.generate_signal(type, n = 512)

Arguments

type

Signal type: "const", "ramp", "poly2", "poly3", "random", "impulse", "sine", "doppler", "heavisine", "bumps".

n

Signal length (default 512).

Value

Numeric vector.

Implements factorizations based on Daubechies and Sweldens (1998).

Description

Implements factorizations based on Daubechies and Sweldens (1998).

Usage

.get_wavelet_config(name)

Causal Benchmark Results

Description

Comparison of execution time between rLifting's optimized causal mode and a naive sliding-window implementation using wavethresh.

Usage

data(benchmark_causal)

Format

A list containing:

rLifting_Time_Avg

Average time (seconds) for rLifting.

Wavethresh_Naive_Time

Time (seconds) for naive sliding window.

Speedup_Factor

Ratio of Naive Time to rLifting Time.

Offline Benchmark Results

Description

Comparison of execution time and reconstruction error (MSE) between rLifting and other packages (wavethresh, wavelets) using Haar wavelet.

Usage

data(benchmark_offline)

Format

A data frame with the following columns:

Pkg

Package name.

Time

Execution time in seconds.

MSE

Mean Squared Error.

Calculate Adaptive Threshold (Recursive)

Description

Estimates the optimal noise threshold based on current window statistics. Implements the recursive formula from Liu et al. (2014). Accelerated with 'C++'.

Usage

compute_adaptive_threshold(lwt_obj, alpha = 0.3, beta = 1.2)

Arguments

lwt_obj

Object returned by lwt().

alpha

Recursive adjustment parameter (Eq. 9).

beta

Initial threshold scale factor (Eq. 9).

Value

Object of class adaptive_thresholds (a list of thresholds).

Create a custom wavelet

Description

Wrapper to create a lifting_scheme object from manual steps.

Usage

custom_wavelet(name, steps, norm = c(1, 1))

Arguments

name

Identifier name for the wavelet.

steps

List of steps created via lift_step.

norm

Normalization vector c(K, 1/K).

Value

An object of class lifting_scheme.

Examples

p1 = lift_step("predict", c(1), position = "center")
u1 = lift_step("update", c(0.5), position = "center")
w = custom_wavelet("HaarManual", list(p1, u1), c(1.41, 0.707))

Causal Batch Denoising (Turbo Simulation)

Description

Processes a complete signal simulating the sequential arrival of data. Uses the specialized 'C++' class WaveletEngine to perform causal filtering efficiently on a historical dataset.

Usage

denoise_signal_causal(
  signal,
  scheme,
  levels = 1,
  window_size = 256,
  alpha = 0.3,
  beta = 1.2,
  method = "semisoft",
  extension = "symmetric",
  update_freq = 1
)

Arguments

signal

Complete vector of the noisy signal.

scheme

lifting_scheme object.

levels

Decomposition levels.

window_size

Window size.

alpha

Threshold decay parameter (Eq 9).

beta

Threshold gain factor (Eq 9).

method

Thresholding method ("soft", "hard", "semisoft").

extension

Boundary treatment ('symmetric', 'periodic').

update_freq

Frequency of threshold updates.

Value

Filtered vector (same length as input).

Offline Denoising (Global Batch)

Description

Performs denoising on the entire signal at once using a non-causal approach. Uses global statistics for recursive threshold calculation (Eq. 9). This function is fully optimized in 'C++' (Zero-Allocation).

Usage

denoise_signal_offline(
  signal,
  scheme,
  alpha = 0.3,
  beta = 1.2,
  levels = 3,
  method = "semisoft",
  extension = "symmetric"
)

Arguments

signal

Numeric vector containing the complete signal.

scheme

A lifting_scheme object.

alpha

Recursive threshold parameter.

beta

Threshold scale factor.

levels

Number of decomposition levels.

method

Thresholding method ("hard", "soft", "semisoft").

extension

Extension mode ("symmetric", "periodic", "zero").

Value

Filtered numeric vector (same length as input).

Complete Wavelet Diagnosis

Description

Runs a battery of physical and mathematical tests on a wavelet.

Usage

diagnose_wavelet(wavelet_name, config, verbose = TRUE, plot = TRUE)

Arguments

wavelet_name

Name string or a lifting_scheme object.

config

Configuration list (is_ortho, vm_degrees, max_taps).

verbose

Print results to console handling? (Defaults to TRUE).

plot

Boolean. Visualize basis functions during diagnosis? (Defaults to TRUE).

Value

An object of class wavelet_diagnosis (S3), which is a list containing the results of each test. The object has a dedicated print method.

Noisy Doppler Signal Example

Description

A synthetic dataset containing a Doppler signal contaminated with Gaussian noise. Used in the "General Usage" vignette.

Usage

data(doppler_example)

Format

A data frame with 2048 rows and 3 columns:

index

Time index.

original

The pure Doppler signal.

noisy

The signal with added Gaussian noise (sd=0.5).

Inverse Lifting Wavelet Transform ('C++' Accelerated)

Description

Reconstructs the original signal from wavelet coefficients. Optimized with 'C++' backend.

Usage

ilwt(lwt_obj, scheme = NULL)

Arguments

lwt_obj

Object of class 'lwt' returned by lwt().

scheme

(Optional) lifting_scheme object. If NULL, uses the one from lwt_obj.

Value

Numeric vector containing the reconstructed signal.

Examples

s = c(1, 2, 3, 4)
sch = lifting_scheme("haar")
fwd = lwt(s, sch)
rec = ilwt(fwd)
print(rec) # Should match s

Leakage (Impulse Response) Results

Description

Measurement of energy leakage into the "past" when processing an impulse signal. Used to demonstrate the zero-lookahead property of the causal mode.

Usage

data(leakage_results)

Format

A data frame with:

Method

Method description (e.g. "rLifting causal").

Leakage

Sum of squared differences (leakage energy).

Create an individual Lifting Step

Description

Helper function to create prediction (P) or update (U) steps, abstracting the complexity of index management.

Usage

lift_step(
  type = c("predict", "update"),
  coeffs,
  start_idx = NULL,
  position = "center"
)

Arguments

type

Step type: "predict" (P) or "update" (U).

coeffs

Numeric vector containing the filter coefficients.

start_idx

(Optional) Manual start index. If provided, ignores the position parameter. Use this for fine-grained control.

position

Automatic index adjustment (used only if start_idx is NULL):

  • "center": Centers the filter (default).

  • "left": Causal filter (looks into the past).

  • "right": Anti-causal filter (looks into the future).

Value

A list formatted for the internal lifting engine.

Lifting Scheme Constructor

Description

Creates an S3 object containing the prediction (P) and update (U) steps required for the Lifting Transform.

Usage

lifting_scheme(wavelet = "haar", custom_steps = NULL, custom_norm = NULL)

Arguments

wavelet

Wavelet name (string). Options: "haar", "db2", "cdf53", "cdf97", "dd4", "lazy".

custom_steps

List of custom steps (optional). If provided, ignores internal lookup.

custom_norm

Normalization vector (optional).

Value

An object of class lifting_scheme.

Lifting Wavelet Transform (Forward)

Description

Performs the Forward Wavelet Transform using the Lifting Scheme. Optimized with 'C++' backend.

Usage

lwt(signal, scheme, levels = 1, extension = "symmetric")

Arguments

signal

Numeric vector containing the input signal.

scheme

A lifting_scheme object.

levels

Integer. Number of decomposition levels.

extension

Boundary extension mode: "symmetric" (default), "periodic", or "zero".

Value

An object of class 'lwt'. It is a list containing 'coeffs' (list of details d1..dn and approximation an) and 'scheme' (the scheme object used).

Examples

data = c(1, 2, 3, 4, 5, 6, 7, 8)
sch = lifting_scheme("haar")
res = lwt(data, sch, levels = 2)
print(res)

Create an Adaptive Wavelet Stream Processor ('C++' Core)

Description

Generates a stateful function backed by a high-performance 'C++' Ring Buffer engine. It implements Sliding Window + Lifting Decomposition + Adaptive Thresholding in constant amortized time (O(1)) per sample.

Usage

new_wavelet_stream(
  scheme,
  window_size = 256,
  levels = 1,
  alpha = 0.3,
  beta = 1.2,
  method = "semisoft",
  extension = "symmetric",
  update_freq = 1
)

Arguments

scheme

A lifting_scheme object.

window_size

Sliding window size (W). Must be > 8.

levels

Decomposition levels (default 1).

alpha

Threshold decay parameter (Eq 9).

beta

Threshold gain factor (Eq 9).

method

Shrinkage method: "hard", "soft", "semisoft".

extension

Boundary handling ('symmetric', 'periodic', 'zero').

update_freq

How often to recompute threshold statistics (default 1).

Value

A closure function processor(new_sample) that accepts a single numeric value and returns the filtered value immediately.

Plot method for Adaptive Thresholds

Description

Plot method for Adaptive Thresholds

Usage

## S3 method for class 'adaptive_thresholds'
plot(x, ...)

Arguments

x

Object of class adaptive_thresholds.

...

Additional arguments.

Value

Invisibly returns NULL.

Plot method for Lifting Scheme

Description

Plot method for Lifting Scheme

Usage

## S3 method for class 'lifting_scheme'
plot(x, ...)

Arguments

x

An object of class lifting_scheme.

...

Additional arguments passed to visualize_wavelet_basis.

Value

Invisibly returns NULL.

Plot method for LWT Decomposition

Description

Plot method for LWT Decomposition

Usage

## S3 method for class 'lwt'
plot(x, ...)

Arguments

x

An object of class lwt.

...

Additional arguments.

Value

Invisibly returns NULL.

Print method for Adaptive Thresholds

Description

Print method for Adaptive Thresholds

Usage

## S3 method for class 'adaptive_thresholds'
print(x, ...)

Arguments

x

Object of class adaptive_thresholds.

...

Additional arguments.

Value

Invisibly returns x.

Print method

Description

Print method

Usage

## S3 method for class 'lifting_scheme'
print(x, ...)

Arguments

x

object of class lifting_scheme.

...

additional arguments.

Value

Invisibly returns NULL. Called for side effects (printing).

Print method for LWT

Description

Print method for LWT

Usage

## S3 method for class 'lwt'
print(x, ...)

Arguments

x

An object of class lwt.

...

Additional arguments.

Value

Invisibly returns NULL. Called for side effects (printing).

Print method for Wavelet Diagnosis

Description

Print method for Wavelet Diagnosis

Usage

## S3 method for class 'wavelet_diagnosis'
print(x, ...)

Arguments

x

Object of class wavelet_diagnosis.

...

Additional arguments.

Value

Invisibly returns x.

Print method for Wavelet Stream Processor

Description

Print method for Wavelet Stream Processor

Usage

## S3 method for class 'wavelet_stream'
print(x, ...)

Arguments

x

Object of class wavelet_stream.

...

Additional arguments.

Value

Invisibly returns x.

General Thresholding Wrapper

Description

General Thresholding Wrapper

Usage

threshold(x, lambda, method = "soft")

Arguments

x

Input vector.

lambda

Threshold value.

method

Method: "hard", "soft" or "semisoft".

Value

Numeric vector of the same length as x with thresholded coefficients.

Hard Thresholding

Description

Sets coefficients below the threshold to zero, keeping others unchanged. Known as the "keep or kill" policy.

Usage

threshold_hard(x, lambda)

Arguments

x

Vector of coefficients (details).

lambda

Positive threshold value.

Value

Processed vector.

Semisoft Shrinkage (Hyperbolic)

Description

Implementation based on Liu et al. (2014). Combines the stability of Soft Thresholding with the amplitude precision of Hard Thresholding. Function: sign(x) * sqrt(x^2 - lambda^2) for values above lambda.

Usage

threshold_semisoft(x, lambda)

Arguments

x

Vector of coefficients.

lambda

Positive threshold value.

Value

Processed vector.

Soft Thresholding

Description

Sets coefficients below the threshold to zero and shrinks others towards zero. Reduces noise variance but introduces amplitude bias.

Usage

threshold_soft(x, lambda)

Arguments

x

Vector of coefficients.

lambda

Positive threshold value.

Value

Processed vector.

Validate Compact Support (FIR Compliance)

Description

Verifies if the impulse response is finite (FIR Filter).

Usage

validate_compact_support(scheme, max_width)

Arguments

scheme

Object of class lifting_scheme.

max_width

Maximum expected width (number of taps).

Value

List with status and number of active taps.

Validate Orthogonality (Energy Conservation)

Description

Verifies Parseval's Theorem. Only applicable for orthogonal wavelets.

Usage

validate_orthogonality(scheme, expected = TRUE, tol = 1e-09)

Arguments

scheme

Object of class lifting_scheme.

expected

Boolean. If TRUE, expects orthogonality.

tol

Tolerance (default 1e-9).

Value

List with status and energy ratio (Out/In).

Validate Perfect Reconstruction (Stress Test)

Description

Verifies wavelet invertibility against a battery of signals.

Usage

validate_perfect_reconstruction(scheme, tol = 1e-09)

Arguments

scheme

Object of class lifting_scheme.

tol

Numerical error tolerance (default 1e-9).

Value

List with global status and maximum error found.

Validate Shift Sensitivity (Shift Variance)

Description

Decimated wavelets are not translation invariant. This test quantifies the variation in detail energy when shifting the input signal by 1 sample.

Usage

validate_shift_sensitivity(scheme)

Arguments

scheme

Object of class lifting_scheme.

Value

List with status and percentage variation.

Validate Vanishing Moments

Description

Verifies if the wavelet cancels polynomials of a specific degree.

Usage

validate_vanishing_moments(scheme, degree = 0, tol = 1e-09)

Arguments

scheme

Object of class lifting_scheme.

degree

Polynomial degree (0=Constant, 1=Ramp, 2=Parabola...).

tol

Residual energy tolerance (default 1e-9).

Value

List with status and residual energy.

Visualize Basis Functions (Scaling and Wavelet)

Description

Plots the waveform by iterating the reconstruction over several levels.

Usage

visualize_wavelet_basis(scheme, plot = TRUE, levels = 8)

Arguments

scheme

Object of class lifting_scheme.

plot

Boolean.

levels

Number of cascade levels.

Value

Invisibly returns NULL. Called for side effects (plotting).