Discrete Wavelets

A Discrete Wavelet Transform (DWT) library for the web.

This library is well tested. Still, it may contain some errors. Therefore it is recommended to double check the results with another library such as PyWavelets. If you find any errors, please let me know by opening an issue or a pull request.

Importing this library

$3

- Run npm install discrete-wavelets
- Add an import to the npm package import wt from 'discrete-wavelets';
- Then you can use the library in your code.

$3

- Put the following script tag in the head of your HTML file.
- Then you can use the library in your code.

Types

The library uses the following types:

- PaddingMode: Signal extension modes.
- Wavelets: Wavelet bases.

$3

The following values for PaddingMode are supported at the moment:

Name | Value | Description
----------------------|-------------------|------------------------------------
Zero Padding | 'zero' | Adding zeros.
Constant Padding | 'constant' | Replication of border values.
Symmetric Padding | 'symmetric' | Mirroring of samples.
Reflect Padding | 'reflect' | Reflecting of samples.
Periodic Padding | 'periodic' | Treating signal as a periodic one.
Smooth Padding | 'smooth' | Signal extended as a straight line.
Antisymmetric Padding | 'antisymmetric' | Mirroring and negation of samples.

You can get a list of the supported signal extension modes:

``javascript
console.log(wt.Modes.modes);
// expected output: Array ['zero', 'constant', 'symmetric', 'periodic', 'smooth', 'reflect', 'antisymmetric']
`

$3

The following Wavelet types are supported at the moment:

Wavelet | Aliases
------------------------------------------------------------------|---------------------------
Daubechies 1 / Haar | 'db1', 'D2', 'haar'
Daubechies 2 | 'db2', 'D4'
Daubechies 3 | 'db3', 'D6'
Daubechies 4 | 'db4', 'D8'
Daubechies 5 | 'db5', 'D10'
Daubechies 6 | 'db6', 'D12'
Daubechies 7 | 'db7', 'D14'
Daubechies 8 | 'db8', 'D16'
Daubechies 9 | 'db9', 'D18'
Daubechies 10 | 'db10', 'D20'

API

The library offers the following functions:

- Discrete Wavelet Transform (DWT)
- dwt: Single level Discrete Wavelet Transform.
- wavedec: 1D wavelet decomposition. Transforms data by calculating coefficients from input data.
- Inverse Discrete Wavelet Transform (IDWT)
- idwt: Single level inverse Discrete Wavelet Transform.
- waverec: 1D wavelet reconstruction. Inverses a transform by calculating input data from coefficients.
- Other
- energy: Calculates the energy as sum of squares of an array of data or coefficients.
- maxLevel: Determines the maximum level of useful decomposition.
- pad: Extends a signal with a given padding mode.

$3

Single level Discrete Wavelet Transform.

#### Arguments

- data (number[]): Input data.
- wavelet (Wavelet): Wavelet to use.
- mode (PaddingMode): Signal extension mode. Defaults to 'symmetric'.

#### Return

coeffs (number[][]): Approximation and detail coefficients as result of the transform.

#### Example

`javascript
var coeffs = wt.dwt([1, 2, 3, 4], 'haar');

console.log(coeffs);
// expected output: Array [[2.1213203435596425, 4.9497474683058326], [-0.7071067811865475, -0.7071067811865475]]
`

$3

1D wavelet decomposition. Transforms data by calculating coefficients from input data.

#### Arguments

- data (number[]): Input data.
- wavelet (Wavelet): Wavelet to use.
- mode (PaddingMode): Signal extension mode. Defaults to 'symmetric'.
- level (number): Decomposition level. Defaults to level calculated by maxLevel function.

#### Return

coeffs (number[][]): Coefficients as result of the transform.

#### Example

`javascript
var coeffs = wt.wavedec([1, 2, 3, 4], 'haar');

console.log(coeffs);
// expected output: Array [[4.999999999999999], [-1.9999999999999993], [-0.7071067811865475, -0.7071067811865475]]
`

Be aware that due to floating point imprecision the result diverges slightly from the analytical solution [[5], [-2], [-0.7071067811865475, -0.7071067811865475]]

$3

Single level inverse Discrete Wavelet Transform.

#### Arguments

- approx (number[]): Approximation coefficients. If undefined, it will be set to an array of zeros with length equal to the detail coefficients.
- detail (number[]): Detail coefficients. If undefined, it will be set to an array of zeros with length equal to the approximation coefficients.
- wavelet (Wavelet): Wavelet to use.

#### Return

rec (number[]): Approximation coefficients of previous level of transform.

#### Example

`javascript
var rec = wt.idwt(
[(1 + 2) / Math.SQRT2, (3 + 4) / Math.SQRT2],
[(1 - 2) / Math.SQRT2, (3 - 4) / Math.SQRT2],
'haar'
);

console.log(rec);
// expected output: Array [0.9999999999999999, 1.9999999999999996, 2.9999999999999996, 3.9999999999999996]
`

Be aware that due to floating point imprecision the result diverges slightly from the analytical solution [1, 2, 3, 4]

$3

1D wavelet reconstruction. Inverses a transform by calculating input data from coefficients.

#### Arguments

- coeffs (number[][]): Coefficients as result of a transform.
- wavelet (Wavelet): Wavelet to use.

#### Return

data (number[]): Input data as result of the inverse transform.

#### Example

`javascript
var data = wt.waverec(
[[5], [-2], [-1 / Math.SQRT2, -1 / Math.SQRT2]],
'haar'
);

console.log(data);
// expected output: Array [0.9999999999999999, 1.9999999999999996, 2.999999999999999, 3.999999999999999]
`

Be aware that due to floating point imprecision the result diverges slightly from the analytical solution [1, 2, 3, 4]

$3

Calculates the energy as sum of squares of an array of data or coefficients.

#### Argument

- values (number[] | number[][]): Array of data or coefficients.

#### Return

energy (number): Energy of values as the sum of squares.

#### Examples

`javascript
console.log(
wt.energy([-1, 2, 6, 1])
);
// expected output: 42

console.log(
wt.energy([[5], [-2], [-1 / Math.SQRT2, -1 / Math.SQRT2]])
);
// expected output: 30
`

$3

Determines the maximum level of useful decomposition.

#### Arguments

- dataLength (number): Length of input data.
- wavelet (Wavelet): Wavelet to use.

#### Return

maxLevel (number): Maximum useful level of decomposition.

#### Examples

`javascript
var maxLevel = wt.maxLevel(4, 'haar');

console.log(maxLevel);
// expected output: 2
`

`javascript
var maxLevel = wt.maxLevel(1024, 'haar');

console.log(maxLevel);
// expected output: 10
`

$3

Extends a signal with a given padding mode.

#### Arguments

- data (number[]): Input data.
- padWidths ([number, number]): Widths of padding at front and back.
- mode (PaddingMode): Signal extension mode.

#### Return

pad (number[]): Data with padding.

#### Example

`javascript
var pad = wt.pad([42, 51], [2, 1], 'zero');

console.log(pad);
// expected output: Array [0, 0, 42, 51, 0]
`

NPM scripts

- npm install: Install dependencies
- npm test: Run test suite
- npm start: Run npm run build in watch mode
- npm run test:watch: Run test suite in interactive watch mode
- npm run test:prod: Run linting and generate coverage
- npm run build: Generate bundles and typings, create docs
- npm run lint: Lints code

This library in action

An exemplary application with code using this library can be found at https://symmetronic.github.io/covid-19-dwt-analysis/

Related project

Symmetronic Scaleogram is a web component that allows to easily create a scaleogram visualization from wavelet coefficients.

Contributing

Pull requests are welcome! Please include new tests for your code and make sure that all tests succeed running npm test`.