## Documentation |

This release introduces a new example, Dual-Tree Wavelet Transforms, which demonstrates the advantages of the dual-tree discrete wavelet transform (DWT) over the critically sampled DWT. The example illustrates the approximate shift invariance and directional selectivity of the complex dual-tree wavelet transform. These properties enable the dual-tree wavelet transform to outperform the critically sampled DWT in a number of applications.

This release introduces the 2-D continuous wavelet transform (CWT) for images. The 2-D CWT provides information about images at specified scales, rotation angles, and positions in the plane. Applications of the 2-D CWT include:

Fault detection in images

Object recognition

Fringe pattern profilometry

For information on how to implement the 2-D CWT at the MATLAB^{®} command
line, see `cwtft2`.

To use `cwtft2` in the Wavelet Toolbox™ interactive
tool, enter

>> wavemenu

Then,
from the **Two-dimensional** tools section, select **Continuous
Wavelet Transform 2-D**. See 2-D
Continuous Wavelet Transform App for more information on the
2-D CWT app.

`cwtft2` supports both isotropic and anisotropic
2-D wavelets. Use isotropic wavelets to perform pointwise analysis
in images or when oriented features are not relevant. Use anisotropic
wavelets when your goal is to detect oriented features.

`cwtft2` implements the 2-D CWT using the
2-D discrete Fourier transform. Use `cwtftinfo2` to
obtain the 2-D Fourier transforms of the supported analyzing wavelets.

This release introduces two types of 1-D and 2-D oversampled
(frame) wavelet perfect reconstruction filter banks. For 1-D wavelet
analysis, use `dddtree` to
obtain the following wavelet transforms:

Complex dual-tree

Double-density

Complex dual-tree double-density

For 2-D wavelet analysis, use `dddtree2` to
obtain the following wavelet transforms:

Double-density

Real oriented dual-tree

Complex oriented dual-tree

Real oriented double-density dual-tree

Complex oriented double-density dual-tree

The dual-tree and double-density transforms mitigate a number of shortcomings of the critically sampled discrete wavelet transform. The double-density and dual-tree transforms achieve directional selectivity and approximate shift invariance with significantly less computational cost than the undecimated discrete wavelet transform.

In R2012a you can decompose a 1-D signal in a dictionary of time/frequency or time/scale atoms with matching pursuit.

Representing a signal in a union of time-frequency/time-scale bases can provide sparser signal representations than attainable with any single basis. Matching pursuit uses iterative greedy algorithms to reduce the computational complexity of searching through a redundant dictionary.

Wavelet Toolbox software supports basic matching pursuit,
orthogonal matching pursuit, and weak orthogonal matching pursuit
at the command line with `wmpdictionary` and `wmpalg`.
You can also perform matching pursuit with the interactive `wavemenu` tool.

You can build dictionaries using several internally supported options or provide your own custom dictionaries. See Matching Pursuit for background information and examples.

In R2011b, you can compute the Fourier transform based continuous
wavelet transform (CWT) and inverse CWT using the Wavelet Toolbox graphical
user interface `wavemenu`.
To access these graphical tools, enter `wavemenu` at
the command line, and select **Continuous Wavelet 1-D (using
FFT)**.

In R2011b, you can compute the inverse continuous wavelet transform
(CWT) for a wider class of analyzing wavelets using `icwtlin`. `icwtlin` returns
the inverse for CWT coefficients obtained at linearly spaced scales. `icwtlin` supports
the output of `cwtft` and
the output of `cwt` for
a select number of wavelets. See `icwtlin` for
detailed information.

In R2011b, you can generate MATLAB code for 1-D and 2-D discrete wavelet transforms (DWT), stationary wavelet transforms (SWT), and wavelet packet transforms. You can denoise or compress a signal or image in the GUI and export the MATLAB code to implement that operation at the command line. This approach allows you to set denoising thresholds or compression ratios aided by visualization tools and save the commands to reproduce those operations at the command line. See Generating MATLAB Code from Wavelet Toolbox GUI for examples.

R2011b includes a new demo illustrating signal reconstruction using the continuous wavelet transform (CWT). The demo emphasizes the use of the CWT to analyze a signal and reconstruct a time- and scale-based approximation with select coefficients using the inverse CWT. See Signal Reconstruction from Continuous Wavelet Transform CoefficientsSignal Reconstruction from Continuous Wavelet Transform Coefficients for details.

In R2011b, the default values for the smallest scale, scale
increment, and number of scales have changed in `cwtft` for
the derivative of Gaussian (DOG) and Paul wavelets. The change in
the defaults also affects the Mexican hat wavelet, which is a special
case of the DOG wavelet. In R2011b, the default value of the smallest
scale for the Paul and DOG wavelets is `2*dt`, where `dt` is
the sampling period. The default scale increment, `ds`,
is 0.4875. The default number of scales is `fix(log2(length(sig))/ds)+1` for
the Paul wavelet and `max([fix(log2(length(sig))/ds),1])` for
the DOG wavelets, where `sig` is the input signal.

`cwtft` was
introduced in R2011a. In that release, the default smallest scales
for the DOG and Paul wavelets are `dt/8` and `dt` respectively,
where `dt` is the sampling interval. The default
scale increment is 0.5. The default number of scales is `fix(1.5*log2(length(sig))/ds)+1` for
the Paul wavelet. For DOG wavelets, the default number of scales is `fix(1.25*log2(length(sig))/ds)+1`,
where `sig` is the input signal. You can obtain results
in R2011b using `cwtft` with the DOG and Paul wavelets
identical to results in R2011a with the default values. To do so,
specify the smallest scale, scale increment, and number of scales
in a structure or cell array. See `cwtft` for
details.

In R2011a, you can compute the inverse continuous wavelet transform
(CWT) using an FFT-based algorithm. The inverse CWT allows you to
synthesize approximations to your 1D signal based on selected scales.
The inverse CWT is only supported for coefficients obtained using
the FFT-based CWT. See `icwtft` and `cwtft` for
details.

In R2011a, you can compute the continuous wavelet transform
(CWT) using an FFT-based algorithm with `cwtft`.
The CWT computed using an FFT algorithm supports the computation of
the inverse CWT. See `cwtft` and `icwtft` for
details. Only select wavelets are valid for use with `cwtft`.
See `cwtftinfo` for
a list of supported wavelets.

In R2011a there is a new demo using pattern adapted wavelets for signal detection. You can view this demo here Pattern adapted wavelets for signal detectionPattern adapted wavelets for signal detection. The Wavelet Toolbox software enables you to design admissible wavelets based on the pattern you wish to detect. Designing a valid wavelet based on your desired pattern allows you to exploit the optimality of matched filtering in the framework of the CWT. The demo illustrates this process on simulated data and human EEG recordings.

In R2010b, you can compute the cone of influence (COI) for the
continuous wavelet transform (CWT) of a signal. At each scale, the
COI determines the set of CWT coefficients influenced by the value
of the signal at a specified position. The COI provides an important
visual aid in interpreting the CWT. By overlaying the cone of influence
on the CWT image, you can determine which CWT coefficients each value
of the signal affects at every scale. See `conofinf` for
details.

In R2010b, you can estimate the wavelet cross spectrum and wavelet
coherence of two time series. The wavelet cross spectrum and coherence
provide wavelet-based alternatives for the Fourier-based cross spectrum
and coherence. These wavelet estimators are suitable for nonstationary
signals. Using a complex-valued analyzing wavelet, you can also examine
intervals in the time-scale plane where the two time series exhibit
common phase behavior. See `wcoher` and
the new demo Wavelet CoherenceWavelet Coherence for
details.

In R2010b, you can compute the wavelet packet spectrum with `wpspectrum`.
The wavelet packet spectrum provides a time-frequency analysis of
a time series. The wavelet packet spectrum is useful as wavelet-based
counterpart of the short-time Fourier transform.

In R2010b, you can order the wavelet packet transform terminal
nodes by natural (Payley) or frequency (sequency) order. See `otnodes` for
details.

In R2010b, you can measure the quality of your signal or image
approximation using a number of widely-used quality metrics. These
metrics include: the peak signal-to-noise ratio (PSNR), the mean square
error (MSE), the maximum absolute error, and the energy ratio of the
approximation to the original. See `measerr` for
details.

This release adds new functions and a GUI to support the 3-D
discrete wavelet transform. This new functionality lets you decompose,
analyze, and display a 3-D object using a different wavelet for each
dimension. The new functions are: `dwt3`, `idwt3`, `wavedec3`,
and `waverec3`.
A demo (`wavelet3ddemo`) is also included.

New nondecimated wavelet transform functions support signals
of arbitrary size and different extension modes. Previous functionality
had two limitations: signal length had to equal a power of 2 and the
only allowable extension mode was periodized. The new functions are: `ndwt`, `indwt`, `ndwt2`,
and `indwt2`.
A demo (`ndwtdemo`) is also included.

The new `cmddenoise` function
uses interval-dependent denoising to compute the denoised signal and
coefficients. This allows you to apply different denoising thresholds
to different portions of the signal, which is typically nonuniform.
You can also export thresholds from the GUI and use them in the `cmddenoise` function.
The toolbox includes a denoising demo (`cmddenoise`).

The toolbox can now process true color images. All major toolbox GUIs and all of the 2D-oriented command line functions have been also updated and support true color images.

The new `cwtext` function
lets you calculate 1D continuious wavelet parameters using extension
parameters.

The Multisignal 1D GUI and other related GUIs now include 1-norm, 2-norm, and inf-norm calculations.

A new function, `waveletfamilies`,
displays all the available wavelet families and their properties.

You can now import data from the workspace to all toolbox GUIs
and export data from all toolbox GUIs to the workspace. Use **Import from Workspace** and **Export
to Workspace**, respectively, on the GUI's **File** menu.

The ability to compute scalograms of the wavelet coefficients
in continuous wavelet analysis has been added as an option to the `cwt` function.
You can also pass the structure produced by `cwt` directly
to the new `wscalogram` function.
Scalograms show the percentage of energy in each wavelet coefficient.

You can now construct clusters from hierarchical cluster trees
in multisignal analysis using the new `mdwtcluster` function.

The following command-line functions for 1D multisignal analysis, compression, and denoising have been added to the toolbox:

The following command-line functions for 1D multisignal wavelets and clustering have been added to the toolbox:

Change Multisignal 1D decomposition coeffs | |

Multisignal 1D wavelet decomposition | |

Multisignal 1D wavelet reconstruction. | |

Multisignal 1D decomposition energy repartition |

A new command-line function (`wmulden`)
and a new GUI (**Multivariate Denoising** from
the `wavemenu` initial
window) for de-noising a matrix of signals have been added. Both the
function and GUI take into account the signals themselves and the
correlations between the signals. A two-step process is used. First,
a change of basis is performed to deal with noise spatial correlation
de-noising in the new basis. Then, a principal component analysis
is performed to take advantage of the deterministic relationships
between the signals, leading to an additional de-noising effect.

A new command-line function (`wmspca`)
and a new GUI (**Multiscale Princ. Comp. Analysis** from
the `wavemenu` initial
window) for simplifying a matrix of signals have been added. Both
the function and GUI take into account the signals themselves and
the correlations between the signals. The multiscale principal component
analysis mixes wavelet decompositions and principal component analysis.

Release | Features or Changes with Compatibility Considerations |
---|---|

R2014b | None |

R2014a | None |

R2013b | None |

R2013a | None |

R2012b | None |

R2012a | None |

R2011b | Changes in Fourier Transform Based Continuous Wavelet Transform Defaults for Derivative of Gaussian (DOG) and Paul Wavelets |

R2011a | None |

R2010b | None |

R2010a | None |

R2009b | None |

R2009a | None |

R2008b | None |

R2008a | None |

R2007b | None |

R2007a | None |

R2006b | None |

R2006a | None |

R14SP3 | None |

R14SP2 | None |

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