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Let Pi(x) be the rectangle function, then the Fourier transform is F_x[Pi(x)](k)=sinc(pik), where sinc(x) is the sinc function.

There are three flavors of Hough transform. The first (original) was used to detect straight lines in bubble chamber images. The technique parameterizes pattern space, then ...

F_x[1/pi(1/2Gamma)/((x-x_0)^2+(1/2Gamma)^2)](k)=e^(-2piikx_0-Gammapi|k|). This transform arises in the computation of the characteristic function of the Cauchy distribution.

Let f(x) be a positive definite, measurable function on the interval (-infty,infty). Then there exists a monotone increasing, real-valued bounded function alpha(t) such that ...

The Fourier transform of the Heaviside step function H(x) is given by F_x[H(x)](k) = int_(-infty)^inftye^(-2piikx)H(x)dx (1) = 1/2[delta(k)-i/(pik)], (2) where delta(k) is ...

The integral transform defined by g(x)=int_1^inftyt^(1/4-nu/2)(t-1)^(1/4-nu/2)P_(-1/2+ix)^(nu-1/2)(2t-1)f(t)dt (Samko et al. 1993, p. 761) or ...

Let the two-dimensional cylinder function be defined by f(x,y)={1 for r<R; 0 for r>R. (1) Then the Radon transform is given by ...

The Fourier transform of a Gaussian function f(x)=e^(-ax^2) is given by F_x[e^(-ax^2)](k) = int_(-infty)^inftye^(-ax^2)e^(-2piikx)dx (1) = ...

Let R(x) be the ramp function, then the Fourier transform of R(x) is given by F_x[R(x)](k) = int_(-infty)^inftye^(-2piikx)R(x)dx (1) = i/(4pi)delta^'(k)-1/(4pi^2k^2), (2) ...

The Fourier transform of e^(-k_0|x|) is given by F_x[e^(-k_0|x|)](k)=int_(-infty)^inftye^(-k_0|x|)e^(-2piikx)dx = ...