Convolution kernel

To propagate the solvent-structuring effect induced by the presence of the hydrophobic spheres, we replace the position-dependent dielectric by an integral kernel convoluted with the electric field at position r to represent the correlations with the field at neighboring positions r. This prompts us to replace the classical Poisson equation by the rigorous relation (see Chap. 14) ... 

Various aspects of the model are too simple. An explicit expression hke Eq. (9.14) can only be derived for a point dipole at the center. For more comphcated charge distributions, only series expansions in terms of spherical harmonics are possible, and the frequency dependence becomes considerably more complicated. Nevertheless, it keeps its essential features the restoring force on the dipole is modified by equilibrium fluctuations in the solvent, and the dynamics leads to a friction kernel convoluting with the velocity of the oscillator. These are the features that we use in the next sections to investigate the effects of such an environment on the barrier transition. 

These equations are the coupled system of discrete equations that define the rigorous forward problem. Note that we can take advantage of the convolution form for indices (i — I) and (j — J). Then, by exciting the conductive media with a number N/ oi frequencies, one can obtain the multifrequency model. The kernels of the integral equations are described in [13] and [3j. 

This method is known as the marching method. The accuracy of the procedure and the correctness of the program can be verified by testing it with analytically soluble Volterra equations, for example, the test problems with nonsingular convolution kernels listed on pp. 505-507 of Brunner and van der Houwen s book (1986). 

Let us again consider the convolution integral. Equation (86) is an example of a Fredholm integral equation of the first kind. In such equations the kernel can be expressed as a more-general function of both x and x ... 

A chief advantage of all the filters described in Section IV is convenience. We have written the Fourier transfer function of each one. Certainly it is possible to perform the filtering in the Fourier space. It is also possible, however, and often even more convenient, to convolve the data with the filter function itself. This is especially true if the filter can be adequately approximated by a convolution kernel that vanishes except over a relatively small domain. 

Figure 10.3 The input images are shown in the first row, the result of a convolution with a Gaussian kernel is shown in the second row, and the result of a convolution with an exponential kernel is shown in the last row.

They may be obtained by means of the matrix IET but only together with the kernel E(f) = F(t) specified by its Laplace transformation (3.244), which is concentration-independent. However, from the more general point of view, Eqs. (3.707) are an implementation of the memory function formalism in chemical kinetics. The form of these equations shows the essentially non-Markovian character of the reversible reactions in solution the kernel holds the memory effect, and the convolution integrals entail the prehistoric evolution of the process. In the framework of ordinary chemical kinetics S(/j = d(t), so that the system (3.707) acquires the purely differential form. In fact, this is possible only in the limit when the reaction is entirely under kinetic control. 

We notice that the correlation function defined by Eq. (147) is stationary. Thus, it fits the Onsager principle [101], which establishes that the regression to equilibrium of an infinitely aged system is described by the unperturbed correlation function. The authors of Ref. 102 have successfully addressed this issue, using the following arguments. According to an earlier work [96] the GME of infinite age has the same time convoluted structure as Eq. (59), with the memory kernel T(t) replaced by (1>,XJ (f). They proved that the Laplace transform of Too is... 

We apply to this equation the same remarks as those adopted for the comparison among Eqs. (316)—(318). We note, first of all, that the structure of Eq. (325) is very attractive, because it implies a time convolution with a Lindblad form, thereby yielding the condition of positivity that many quantum GME violate. However, if we identify the memory kernel with the correlation function of the 1/2-spin operator ux, assumed to be identical to the dichotomous fluctuation E, studied in Section XIV, we get a reliable result only if this correlation function is exponential. In the non-Poisson case, this equation has the same weakness as the generalized diffusion equation (133). This structure is... 

If the deformations are small enough, the functional can be written in terms of linear differential equations with constant coefficients or, equivalently, in terms of convolution integrals with difference kernels. 

Note the difference in the 5-dependence of the two coefficients of the right-hand side of Eq. (132). The inverse Laplace transform of the first term yields the Mittag-Leffler function as found in the homogeneous case above. The inverse Laplace transform of the second term is the convolution of the random force and a stationary kernel. The kernel is given by the series... 

At this stage of development the theory is already mature to allow the extraction of 4>(t) from 8 t). In fact, since both kernels (14) and (15) do not depend on t and t separately, but only on the ratio t/r, the integral (17) can be reduced to a convolution integral which can be solved for 4> t) by standard techniques taken from the theory of Fourier transform. These rigorous methods, however, are not flexible and do not allow people to understand the physical meaning of the parameters contained in the experimental datum. 

Many edge detection operators on the discrete image I(xm,yn) can be expressed as a linear convolution of I(xm,yn) with a convolution kernel c... 

For example, the convolution kernel which corresponds to the operator in Eq. (15) is... 

The convolution linear operator met in linear input-response systems (Theorem 16.5) is of great importance in LSA and is encountered in many different contexts. This linear operator consists of a kernel function, g t), and a specific integration operation. 

The integration operation of the convolution operator is symmetric with respect to the kernel and the operand. Thus, L Qi(t)) = L,(g(t)). For this reason a convolution involving two functions, g and h, is often written g h (or h g). 

In terms of the usual (constant electric field) dieleetrie relaxation time Tp, we have = Xj s /s. Let us assume that the whole relaxation kineties is exponential (the Debye model). Gaussian proeess with exponential eorrelation is Markovian [297]. In this case, the n-point distribution function G can be factorized into a product of two-point distribution functions G2, and the corresponding convolution-type perturbation series for the rate kernel can be summed up exactly [88] leading to Eq. (9.37), that is. 

In the nonparametric approach, the input-output relation is represented either analytically (in convolutional form through Volterra-Wiener expansions where the unknown quantities are kernel functions). 

The multiple convolutions of the Volterra model involve kernel functions fc,(mi,..., m,) which constitute the descriptors of the system nonlinear dynamics. Consequently, the system identification task is to obtain estimates of these kernels from input-output data. These kernel functions are symmetric with respect to their arguments. 

Fig. 2.6 Forman phase correction method. The real and imaginary part of the spectrum corresponding to the transmission of the atmosphere from 0 to 42 cm has been distorted (top-left) with a linear phase error (top-right). The measured interferogram is not symmetric anymore (centre-left). After extracting the convolution kernel (centre-right) and applying the correction method 5 times, the interferogram symmetry is improved (bottom-left). Fourier transforming the corrected interferogram, the spectrum is recovered (bottom-right) and is real...

The convolution kernel depends on the apodization function, in this case a top-hat function has been used. By Fourier transforming the data in u and v, one obtains source images for various frequencies. 

The Forman phase correction algorithm, presented in Chap. 2, is shown in Fig. 3.6. Initially, the raw interferogram is cropped around the zero path difference (ZPD) to get a symmetric interferogram called subset. This subset is multiplied by a triangular apodization function and Fourier transformed. With the complex phase obtained from the FFT a convolution Kernel is obtained, which is used to filter the original interferogram and correct the phase. Finally the result of the last operation is Fourier transformed to get the phase corrected spectrum. This process is repeated until the convolution Kernel approximates to a Dirac delta function. 

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