"A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times."
A method for time-frequency analysis that uses wavelets to decompose a signal into its frequency components.
Fourier Transform: A mathematical technique used to transform signals from the time domain to the frequency domain, which is the foundation of wavelet analysis.
Signal Processing: The science of analyzing, modifying, and extracting information from signals.
Wavelet Basics: Wavelets are small waves that are used to analyze and decompose signals into various frequency components.
Discrete Wavelet Transform: An algorithm that decomposes a signal into a set of coefficients representing different frequency bands.
Continuous Wavelet Transform: A mathematical algorithm that uses wavelets to transform a signal into its time-frequency signal representation.
Stationary Wavelet Transform: An extension of the discrete wavelet transform that is used to analyze non-stationary signals.
Multiresolution Analysis: A type of signal processing technique that involves analyzing a signal at different time scales.
Wavelet Packet Transform: A type of wavelet transform that uses wavelet packets to decompose signals into even smaller frequency bands.
Time-Frequency Analysis: A technique used to analyze signals in both time and frequency domains.
Wavelet Denoising: A process of removing noise from a signal using a wavelet transform.
Feature Extraction: The process of extracting informative features from a signal for further analysis.
Applications of Wavelet Analysis: Wavelet analysis finds applications in several fields, including image processing, audio signal analysis, and data compression.
Continuous Wavelet Transform (CWT): This type of wavelet analysis is used to determine the time-varying frequency components of a signal by convolving it with a continuously scaled and shifted wavelet.
Discrete Wavelet Transform (DWT): This type of wavelet analysis involves decomposing a signal into different frequency bands using filters that vary in scale and shift. This technique is commonly used for signal compression and denoising.
Wavelet Packet Transform (WPT): This is an extension of the DWT technique and involves further decomposing a signal into sub-bands using a tree structure.
Stationary Wavelet Transform (SWT): Similar to DWT, this technique also involves decomposing a signal into frequency bands, but uses a non-decimated filter bank to avoid aliasing.
Dual-Tree Wavelet Transform (DTWT): This is an extension of SWT and uses a pair of non-decimated filter banks to achieve directional selectivity.
Discrete Cosine Transform (DCT): This is a transform similar to the DWT, but uses cosine functions instead of wavelets. It is commonly used for image and video compression.
Gabor Transform: This is a type of wavelet analysis that uses a Gaussian window and a sinusoidal carrier wave to analyze a signal's time-frequency characteristics.
Empirical Mode Decomposition (EMD): This technique decomposes a signal into intrinsic mode functions (IMFs) that capture the underlying oscillatory modes of the signal.
Synchrosqueezing Transform: This technique involves decomposing a signal into frequency sub-bands and then reassigning the energy of each sub-band to its dominant frequency component.
Ridgelet Transform: This technique involves decomposing a signal into its ridges or trajectories instead of its frequency components, using a non-linear filtering approach. It is commonly used for the analysis of geometric structures in images.
"Wavelets are imbued with specific properties that make them useful for signal processing."
"As a mathematical tool, wavelets can be used to extract information from many different kinds of data, including – but not limited to – audio signals and images."
"Sets of complementary wavelets decompose a signal without gaps or overlaps so that the decomposition process is mathematically reversible."
"It is desirable to recover the original information with minimal loss."
"This representation is a wavelet series representation of a square-integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set or frame of a vector space, for the Hilbert space of square-integrable functions."
"The diffraction phenomenon is described by the Huygens–Fresnel principle."
"The Huygens–Fresnel principle treats each point in a propagating wavefront as a collection of individual spherical wavelets."
"A wave from a coherent source (such as a laser) encounters a slit/aperture that is comparable in size to its wavelength."
"Multiple, closely spaced openings (e.g., a diffraction grating), can result in a complex pattern of varying intensity."
"Mathematically, a wavelet correlates with a signal if a portion of the signal is similar."
"Wavelets are imbued with specific properties that make them useful for signal processing."
"Sets of wavelets are needed to analyze data fully."
"Sets of complementary wavelets decompose a signal without gaps or overlaps so that the decomposition process is mathematically reversible."
"As a mathematical tool, wavelets can be used to extract information from many different kinds of data, including – but not limited to – audio signals and images."
"It is desirable to recover the original information with minimal loss."
"This is accomplished through coherent states."
"The characteristic bending pattern is most pronounced when a wave from a coherent source (such as a laser) encounters a slit/aperture that is comparable in size to its wavelength."
"The addition, or interference, of different points on the wavefront (or, equivalently, each wavelet) that travel by paths of different lengths to the registering surface."
"Multiple, closely spaced openings (e.g., a diffraction grating), can result in a complex pattern of varying intensity."