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Rabu, 18 Juni 2014

Gabor–Wigner transform

10.15    No comments

The Gabor transform, named after Dennis Gabor, and the Wigner distribution function, named after Eugene Wigner, are both tools for time-frequency analysis. Since theGabor transform does not have high clarity, and the Wigner distribution function has a "cross term problem" (i.e. is non-linear), a 2007 study by S. C. Pei and J. J. Ding proposed a new combination of the two transforms that has high clarity and no cross term problem.[2] Since the cross term does not appear in the Gabor transform, the time frequency distribution of the Gabor transform can be used as a filter to filter out the cross term in the output of the Wigner distribution function.
  • Gabor transform
G_x(t,f) = \int_{-\infty}^\infty e^{-\pi(\tau-t)^2}e^{-j2\pi f\tau}x(\tau) \, d\tau
  • Wigner distribution function
W_x(t,f)=\int_{-\infty}^\infty x(t+\tau/2)x^*(t-\tau/2)e^{-j2\pi\tau\,f} \, d\tau
  • Gabor–Wigner transform
There are many different combinations to define the Gabor–Wigner transform. Here four different definitions are given.
  1. D_x(t,f)=G_x(t,f)\times W_x(t,f)
  2. D_x(t,f)=\min\left\{|G_x(t,f)|^2,|W_x(t,f)|\right\}
  3. D_x(t,f)=W_x(t,f)\times \{|G_x(t,f)|>0.25\}
  4. D_x(t,f)=G_x^{2.6}(t,f)W_x^{0.7}(t,f)

Sumber : http://en.wikipedia.org/wiki/Gabor%E2%80%93Wigner_transform

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Selasa, 17 Juni 2014

Gabor wavelet

10.13    No comments

Gabor wavelets are wavelets invented by Dennis Gabor using complex functions constructed to serve as a basis for Fourier transforms in information theory applications. They are very similar to Morlet wavelets. They are also closely related to Gabor filters (see Gabor filter#Wavelet space). The important property of the wavelet is that it minimizes the product of its standard deviations in the time and frequency domain. Put another way, the uncertainty in information carried by this wavelet is minimized. However they have the downside of being non-orthogonal, so efficient decomposition into the basis is difficult. Since their inception, various applications have appeared, from image processing to analyzing neurons in the human visual system. [1] [2]

The motivation for Gabor wavelets comes from finding some function  f(x)  which minimizes its standard deviation in the time and frequency domains. More formally, the variance in the position domain is:
(\Delta x)^2 = \frac {\int_{-\infty}^{\infty} (x-\mu)^2 f(x)f^{*}(x) \,dx} {\int_{-\infty}^{\infty} f(x)f^{*}(x) \, dx}
where f^{*}(x) is the complex conjugate of  f(x)  and  \mu  is the arithmetic mean, defined as:
\mu = \frac {\int_{-\infty}^{\infty} x f(x)f^{*}(x) \,dx} {\int_{-\infty}^{\infty} f(x)f^{*}(x)\,dx}
The variance in the wave number domain is:
(\Delta k)^2 = \frac {\int_{-\infty}^{\infty} (k-k_0)^2 F(k)F^{*}(k) \, dk} {\int_{-\infty}^{\infty} F(k)F^{*}(k) \, dk}
Where  k_0  is the arithmetic mean of the Fourier Transform of  f(x) ,  F(x) :
k_0 = \frac {\int_{-\infty}^{\infty} k F(k)F^{*}(k) \,dk} {\int_{-\infty}^{\infty} F(k)F^{*}(k) \,dk}
With these defined, the uncertainty is written as:
(\Delta x)(\Delta k)
This quantity has been shown to have a lower bound of  \frac12 . The quantum mechanics view is to interpret  (\Delta x)  as the uncertainty in position and  \hbar (\Delta k)  as uncertainty in momentum. A function  f(x)  that has the lowest theoretically possible uncertainty bound is the Gabor Wavelet.[3]

Sumber : http://en.wikipedia.org/wiki/Gabor_wavelet

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Senin, 16 Juni 2014

FBI transform

10.11    No comments

In mathematics, the FBI transform or Fourier–Bros–Iagolnitzer transform is a generalization of the Fourier transform developed by the French mathematical physicistsJacques Bros and Daniel Iagolnitzer in order to characterise the local analyticity of functions (or distributions) on Rn. The transform provides an alternative approach to analyticwave front sets of distributions, developed independently by the Japanese mathematicians Mikio Sato, Masaki Kashiwara and Takahiro Kawai in their approach to microlocal analysis. It can also be used to prove the analyticity of solutions of analytic elliptic partial differential equations as well as a version of the classical uniqueness theorem, strengthening the Cauchy–Kowalevski theorem, due to the Swedish mathematician Erik Holmgren (1873–1943).
The Fourier transform of a Schwartz function f in S(Rn) is defined by
({\mathcal F}f)(t) = (2\pi)^{-n / 2} \int_{{\mathbf R}^n}f(x) e^{-ix \cdot t}\, dx.
The FBI transform of f is defined for a ≥ 0 by
({\mathcal F}_a f)(t,y) = (2\pi)^{-n / 2} \int_{{\mathbf R}^n}f(x)e^{-a |x-y|^2/2} e^{-ix \cdot t}\, dx.
Thus, when a = 0, it essentially coincides with the Fourier transform.
The same formulas can be used to define the Fourier and FBI transforms of tempered distributions in S'(Rn).

Sumber : http://en.wikipedia.org/wiki/FBI_transform

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Minggu, 15 Juni 2014

Discrete Fourier transform

10.09    No comments

In mathematics, the discrete Fourier transform (DFT) converts a finite list of equally spaced samples of a function into the list of coefficients of a finite combination of complex sinusoids, ordered by their frequencies, that has those same sample values. It can be said to convert the sampled function from its original domain (often time or position along a line) to the frequency domain.
The input samples are complex numbers (in practice, usually real numbers), and the output coefficients are complex as well. The frequencies of the output sinusoids are integer multiples of a fundamental frequency, whose corresponding period is the length of the sampling interval. The combination of sinusoids obtained through the DFT is therefore periodic with that same period. The DFT differs from the discrete-time Fourier transform (DTFT) in that its input and output sequences are both finite; it is therefore said to be the Fourier analysis of finite-domain (or periodic) discrete-time functions.
The DFT is the most important discrete transform, used to perform Fourier analysis in many practical applications.[1] In digital signal processing, the function is any quantity or signal that varies over time, such as the pressure of a sound wave, a radio signal, or daily temperature readings, sampled over a finite time interval (often defined by a window function). In image processing, the samples can be the values of pixels along a row or column of a raster image. The DFT is also used to efficiently solve partial differential equations, and to perform other operations such as convolutions or multiplying large integers.
Since it deals with a finite amount of data, it can be implemented in computers by numerical algorithms or even dedicated hardware. These implementations usually employ efficient fast Fourier transform (FFT) algorithms;[2] so much so that the terms "FFT" and "DFT" are often used interchangeably. Prior to its current usage, the "FFT" initialism may have also been used for the ambiguous term finite Fourier transform.
The sequence of N complex numbers x_0, x_1, \ldots, x_{N-1} is transformed into an N-periodic sequence of complex numbers:




(Eq.1)
Each X_k is a complex number that encodes both amplitude and phase of a sinusoidal component of function x_n. The sinusoid's frequency is k/N cycles per sample.  Its amplitude and phase are:
|X_k|/N = \sqrt{\operatorname{Re}(X_k)^2 + \operatorname{Im}(X_k)^2}/N
\arg(X_k) = \operatorname{atan2}\big( \operatorname{Im}(X_k), \operatorname{Re}(X_k) \big),
where atan2 is the two-argument form of the arctan function. Due to periodicity (see Periodicity), the customary domain of k actually computed is [0, N-1]. That is always the case when the DFT is implemented via the Fast Fourier transform algorithm. But other common domains are  [-N/2, N/2-1]  (N even)  and  [-(N-1)/2, (N-1)/2]  (N odd), as when the left and right halves of an FFT output sequence are swapped.[3]
The transform is sometimes denoted by the symbol \mathcal{F}, as in \mathbf{X} = \mathcal{F} \left \{ \mathbf{x} \right \}  or \mathcal{F} \left ( \mathbf{x} \right ) or \mathcal{F} \mathbf{x}.[note 2]
Eq.1 can be interpreted or derived in various ways, for example:
x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \cdot e^{i 2 \pi k n / N}, \quad n\in\mathbb{Z}\,




(Eq.2)
which is also N-periodic. In the domain  \scriptstyle n\ \in\ [0,\ N-1],  this is the inverse transform of Eq.1.
The normalization factor multiplying the DFT and IDFT (here 1 and 1/N) and the signs of the exponents are merely conventions, and differ in some treatments. The only requirements of these conventions are that the DFT and IDFT have opposite-sign exponents and that the product of their normalization factors be 1/N.  A normalization of \scriptstyle \sqrt{1/N}for both the DFT and IDFT, for instance, makes the transforms unitary.
In the following discussion the terms "sequence" and "vector" will be considered interchangeable.

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Sabtu, 14 Juni 2014

Fourier transform

10.06    No comments

The Fourier transform (English pronunciation: /ˈfɔərieɪ/), named after Joseph Fourier, is a mathematical transformation employed to transform signals between time (or spatial) domain and frequency domain, which has many applications in physics and engineering. It is reversible, being able to transform from either domain to the other. The term itself refers to both the transform operation and to the function it produces.
In the case of a periodic function over time (for example, a continuous but not necessarily sinusoidal musical sound), the Fourier transform can be simplified to the calculation of a discrete set of complex amplitudes, called Fourier series coefficients. They represent the frequency spectrum of the original time-domain signal. Also, when a time-domain function is sampled to facilitate storage or computer-processing, it is still possible to recreate a version of the original Fourier transform according to the Poisson summation formula, also known as discrete-time Fourier transform. See alsoFourier analysis and List of Fourier-related transforms.
There are several common conventions for defining the Fourier transform \hat{f} of an integrable function f : \mathbb R \rightarrow \mathbb C (Kaiser 1994, p. 29), (Rahman 2011, p. 11). This article will use the following definition:
\hat{f}(\xi) = \int_{-\infty}^\infty f(x)\ e^{- 2\pi i x \xi}\,dx,   for any real number ξ.
When the independent variable x represents time (with SI unit of seconds), the transform variable ξ represents frequency (in hertz). Under suitable conditions, f is determined by \hat f via the inverse transform:
f(x) = \int_{-\infty}^\infty \hat f(\xi)\ e^{2 \pi i \xi x}\,d\xi,   for any real number x.
The statement that f can be reconstructed from \hat f is known as the Fourier inversion theorem, and was first introduced in Fourier's Analytical Theory of Heat (Fourier 1822, p. 525), (Fourier & Freeman 1878, p. 408), although what would be considered a proof by modern standards was not given until much later (Titchmarsh 1948, p. 1). The functions f and \hat{f} often are referred to as a Fourier integral pair or Fourier transform pair (Rahman 2011, p. 10).
For other common conventions and notations, including using the angular frequency ω instead of the frequency ξ, see Other conventions and Other notations below. The Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum.

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