Gaussian Discrete Kernel
A Gaussian Discrete Kernel is a modified Bessel Function that is defined as [math]\displaystyle{ L(x, t) = \sum_{n=-\infty}^{\infty} f(x-n) \, T(n, t) }[/math].
- AKA: Discrete Gaussian Kernel Function.
- Contex:
- It is used in scale-space representation of discrete signals.
- …
- Counter-Example(s)
- See: Modified Bessel Function, Gaussian Error Function, Discrete-Time Fourier Transform, Discrete Fourier Transform.
References
2018
- (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Scale_space_implementation#The_discrete_Gaussian_kernel Retrieved:2018-4-8.
- A more refined approach is to convolve the original signal by the discrete Gaussian kernel T(n, t)[1] [2] [3] : [math]\displaystyle{ L(x, t) = \sum_{n=-\infty}^{\infty} f(x-n) \, T(n, t) }[/math] where : [math]\displaystyle{ T(n, t) = e^{-t} I_n(t) }[/math] and [math]\displaystyle{ I_n(t) }[/math] denotes the modified Bessel functions of integer order, n. This is the discrete counterpart of the continuous Gaussian in that it is the solution to the discrete diffusion equation (discrete space, continuous time), just as the continuous Gaussian is the solution to the continuous diffusion equation. [4] This filter can be truncated in the spatial domain as for the sampled Gaussian : [math]\displaystyle{ L(x, t) = \sum_{n=-M}^{M} f(x-n) \, T(n, t) }[/math] or can be implemented in the Fourier domain using a closed-form expression for its discrete-time Fourier transform: : [math]\displaystyle{ \widehat{T}(\theta, t) = \sum_{n=-\infty}^{\infty} T(n, t) \, e^{-i \theta n} = e^{t(\cos \theta - 1)}. }[/math] With this frequency-domain approach, the scale-space properties transfer exactly to the discrete domain, or with excellent approximation using periodic extension and a suitably long discrete Fourier transform to approximate the discrete-time Fourier transform of the signal being smoothed. Moreover, higher-order derivative approximations can be computed in a straightforward manner (and preserving scale-space properties) by applying small support central difference operators to the discrete scale space representation. [5]
As with the sampled Gaussian, a plain truncation of the infinite impulse response will in most cases be a sufficient approximation for small values of ε, while for larger values of ε it is better to use either a decomposition of the discrete Gaussian into a cascade of generalized binomial filters or alternatively to construct a finite approximate kernel by multiplying by a window function. If ε has been chosen too large such that effects of the truncation error begin to appear (for example as spurious extrema or spurious responses to higher-order derivative operators), then the options are to decrease the value of ε such that a larger finite kernel is used, with cutoff where the support is very small, or to use a tapered window.
- A more refined approach is to convolve the original signal by the discrete Gaussian kernel T(n, t)[1] [2] [3] : [math]\displaystyle{ L(x, t) = \sum_{n=-\infty}^{\infty} f(x-n) \, T(n, t) }[/math] where : [math]\displaystyle{ T(n, t) = e^{-t} I_n(t) }[/math] and [math]\displaystyle{ I_n(t) }[/math] denotes the modified Bessel functions of integer order, n. This is the discrete counterpart of the continuous Gaussian in that it is the solution to the discrete diffusion equation (discrete space, continuous time), just as the continuous Gaussian is the solution to the continuous diffusion equation. [4] This filter can be truncated in the spatial domain as for the sampled Gaussian : [math]\displaystyle{ L(x, t) = \sum_{n=-M}^{M} f(x-n) \, T(n, t) }[/math] or can be implemented in the Fourier domain using a closed-form expression for its discrete-time Fourier transform: : [math]\displaystyle{ \widehat{T}(\theta, t) = \sum_{n=-\infty}^{\infty} T(n, t) \, e^{-i \theta n} = e^{t(\cos \theta - 1)}. }[/math] With this frequency-domain approach, the scale-space properties transfer exactly to the discrete domain, or with excellent approximation using periodic extension and a suitably long discrete Fourier transform to approximate the discrete-time Fourier transform of the signal being smoothed. Moreover, higher-order derivative approximations can be computed in a straightforward manner (and preserving scale-space properties) by applying small support central difference operators to the discrete scale space representation. [5]
- ↑ Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234-254.
- ↑ Lindeberg, T., Scale-Space Theory in Computer Vision, Kluwer Academic Publishers, 1994,
- ↑ R.A. Haddad and A.N. Akansu, "A Class of Fast Gaussian Binomial Filters for Speech and Image Processing," IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 39, pp 723-727, March 1991.
- ↑ Campbell, J, 2007, The SMM model as a boundary value problem using the discrete diffusion equation, Theor Popul Biol. 2007 Dec;72(4):539-46.
- ↑ Lindeberg, T. Discrete derivative approximations with scale-space properties: A basis for low-level feature extraction, J. of Mathematical Imaging and Vision, 3(4), pp. 349--376, 1993.