The wavelet transform modulus maxima (WTMM) is a method for detecting the fractal dimension of a signal.
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer.
More than this, the WTMM is capable of partitioning the time and scale domain of a signal into fractal dimension regions, and the method is sometimes referred to as a "mathematical microscope" due to its ability to inspect the multi-scale dimensional characteristics of a signal and possibly inform about the sources of these characteristics.
The WTMM method uses continuous wavelet transform rather than Fourier transforms to detect singularities singularity – that is discontinuities, areas in the signal that are not continuous at a particular derivative.
In mathematics, the continuous wavelet transform (CWT) is a formal tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously.
The Fourier transform (FT) decomposes a function of time into its constituent frequencies. This is similar to the way a musical chord can be expressed in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform of a function of time is itself a complex-valued function of frequency, whose magnitude (modulus) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation.
In particular, this method is useful when analyzing multifractal signals, that is, signals having multiple fractal dimensions.
Consider a signal that can be represented by the following equation:
where is close to and is a non-integer quantifying the local singularity. (Compare this to a Taylor series, where in practice only a limited number of low-order terms are used to approximate a continuous function.)
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
Generally, a continuous wavelet transform decomposes a signal as a function of time, rather than assuming the signal is stationary (For example, the Fourier transform). Any continuous wavelet can be used, though the first derivative of the Gaussian distribution and the Mexican hat wavelet (2nd derivative of Gaussian) are common. Choice of wavelet may depend on characteristics of the signal being investigated.
In mathematics and numerical analysis, the Ricker wavelet
Below we see one possible wavelet basis given by the first derivative of the Gaussian:
Once a "mother wavelet" is chosen, the continuous wavelet transform is carried out as a continuous, square-integrable function that can be scaled and translated. Let be the scaling constant and be the translation of the wavelet along the signal:
In mathematics, a square-integrable function, also called a quadratically integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line is defined as follows.
where is a continuous function in both the time domain and the frequency domain called the mother wavelet and represents the operation of complex conjugate.
By calculating for subsequent wavelets that are derivatives of the mother wavelet, singularities can be identified. Successive derivative wavelets remove the contribution of lower order terms in the signal, allowing the maximum to be detected. (Recall that when taking derivatives, lower order terms become 0.) This is the "modulus maxima".
Thus, this method identifies the singularity spectrum by convolving the signal with a wavelet at different scales and time offsets.
The WTMM is then capable of producing[ vague ] a "skeleton" that partitions the scale and time space by fractal dimension.
The WTMM was developed out of the larger field of continuous wavelet transforms, which arose in the 1980s, and its contemporary fractal dimension methods.
At its essence, it is a combination of fractal dimension "box counting" methods and continuous wavelet transforms, where wavelets at various scales are used instead of boxes.
WTMM was originally developed by Mallat and Hwang in 1992 and used for image processing.
Bacry, Muzy, and Arneodo were early users of this methodology. It has subsequently been used in fields related to signal processing.
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one recorded by a seismograph or heart monitor. Generally, wavelets are intentionally crafted to have specific properties that make them useful for signal processing. Using a "reverse, shift, multiply and integrate" technique called convolution, wavelets can be combined with known portions of a damaged signal to extract information from the unknown portions.
In mathematics, the Morlet wavelet is a wavelet composed of a complex exponential (carrier) multiplied by a Gaussian window (envelope). This wavelet is closely related to human perception, both hearing and vision.
Stéphane Georges Mallat is a French applied mathematician, Professor at College de France and Ecole Normale Superieure. He has made some fundamental contributions to the development of wavelet theory in the late 1980s and early 1990s. He has also done work in applied mathematics, signal processing, music synthesis and image segmentation.
In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information.
Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform. The Hermitian wavelet is defined as the derivative of a Gaussian distribution:
The Hermitian hat wavelet is a low-oscillation, complex-valued wavelet. The real and imaginary parts of this wavelet are defined to be the second and first derivatives of a Gaussian respectively:
In mathematics, a wavelet series is a representation of a square-integrable function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform.
The linear scale-space representation of an N-dimensional continuous signal,
The scale space representation of a signal obtained by Gaussian smoothing satisfies a number of special properties, scale-space axioms, which make it into a special form of multi-scale representation. There are, however, also other types of "multi-scale approaches" in the areas of computer vision, image processing and signal processing, in particular the notion of wavelets. The purpose of this article is to describe a few of these approaches:
A multifractal system is a generalization of a fractal system in which a single exponent is not enough to describe its dynamics; instead, a continuous spectrum of exponents is needed.
The Hurst exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. Studies involving the Hurst exponent were originally developed in hydrology for the practical matter of determining optimum dam sizing for the Nile river's volatile rain and drought conditions that had been observed over a long period of time. The name "Hurst exponent", or "Hurst coefficient", derives from Harold Edwin Hurst (1880–1978), who was the lead researcher in these studies; the use of the standard notation H for the coefficient also relates to his name.
WTMM may refer to:
In functional analysis, a Shannon wavelet may be either of real or complex type. Signal analysis by ideal bandpass filters defines a decomposition known as Shannon wavelets. The Haar and sinc systems are Fourier duals of each other.
In computer vision, speeded up robust features (SURF) is a patented local feature detector and descriptor. It can be used for tasks such as object recognition, image registration, classification or 3D reconstruction. It is partly inspired by the scale-invariant feature transform (SIFT) descriptor. The standard version of SURF is several times faster than SIFT and claimed by its authors to be more robust against different image transformations than SIFT.
In the fields of pattern recognition and machine learning the intrinsic dimension for a data set can be thought of as the number of variables needed in a minimal representation of the data. Similarly, in signal processing of multidimensional signals, the intrinsic dimension of the signal describes how many variables are needed to generate a good approximation of the signal.
In applied mathematics and mathematical analysis, the fractal derivative is a non-Newtonian type of derivative in which the variable such as t has been scaled according to tα. The derivative is defined in fractal geometry.