Wavelet Toolbox authors are Michel Misiti, École Centrale de Lyon; Georges Oppenheim, Université de Marne-La-Vallée; Jean-Michel Poggi, Université René Descartes, Paris 5 Université; and Yves Misiti, Université Paris-Sud.
Wavelet Toolbox™ provides functions and apps to compute the continuous wavelet transform (CWT) of signals and images. You can analyze how the frequency content of a signal changes over time. You can also reconstruct time-frequency localized approximations of signals or filter out time-localized frequency components. Using wavelet coherence, you can reveal time-varying frequency content common in multiple signals. For images, continuous wavelet analysis shows how the frequency content of an image varies across the image and helps to reveal patterns in a noisy image.
Wavelet Toolbox provides functions and apps to analyze signals and images into progressively finer octave bands using decimated (downsampled) and nondecimated wavelet transforms, including the maximal overlap discrete wavelet transform (MODWT). It also supports wavelet packet transforms that partition the frequency content of signals and images into progressively finer equal-width intervals.
This multiresolution analysis enables you to detect patterns that are not visible in the raw data. For example, you can measure the multiscale correlation between two signals or obtain multiscale variance estimates of signals to detect changepoints. You can also reconstruct signal and image approximations that retain only desired features, and compare the distribution of energy in signals across frequency bands. Use the wavelet packet spectrum to obtain a time-frequency analysis of a signal.
Wavelet Toolbox provides functions to denoise and compress signals and images. Wavelet and wavelet packet denoising enable you to retain features in your data that are often removed or smoothed out by other denoising techniques. Wavelet Toolbox supports a variety of thresholding strategies you can apply to your data and use to compare results. Noise in a signal is not always uniform in time, so you can apply interval-dependent thresholds to denoise data with nonconstant variance.
You can denoise and compress collections of signals with wavelets by exploiting correlations between individual signals. You can also cluster groups of signals by filtering out unimportant details using sparse wavelet representations. You can compress data by setting perceptually unimportant wavelet and wavelet packet coefficients to zero and reconstructing the data. The toolbox offers the Wavelet Design and Analysis app, which you can use to explore denoising and compressing signals and images.
Wavelet Toolbox provides functions that enable you to use the most common orthogonal and biorthogonal wavelet filters, including: Daubechies, coiflets, Fejer-Korovkin, and biorthogonal spline filters. Filter banks are arrangements of lowpass, highpass, and bandpass filters that divide your data into subbands that you can process independently. The orthogonal and biorthogonal wavelet filters the toolbox provides are specially designed to decompose your data, enabling you to operate on the subbands at different rates. These filters can also reconstruct the data while cancelling any aliasing errors that occur. Orthogonal wavelet filters accomplish this with a single filter pair. Biorthogonal filters require two pairs of filters but have the added benefit of providing linear phase. In both cases, the wavelet filters have very few coefficients and therefore provide you with a computationally efficient, perfect reconstruction filter bank.
The toolbox offers functions to design your own perfect reconstruction filter bank with specific properties through elementary lifting steps. You can also add your own custom wavelet filters and use them with the toolbox functions and apps for discrete and continuous wavelet analysis.