Finite set of basis functions

Reflectances can be approximated using a finite set of basis functions (Brill 1978 Brill and West 1981 D Zmura and Lennie 1986 Finlayson et al. 1994a Funt and Drew 1988 Funt et al. 1991, 1992 Healey and Slater 1994 Ho et al. 1992 Maloney 1992 Maloney and Wandell 1992, 1986 Novak and Shafer 1992 Tominaga 1991 Wandell 1987). If we decompose the space of reflectances into a finite set of tir basis functions R, with ie l.nj then the reflectance RO.) for a given wavelength /. can be written as 

Funt et al. (1991, 1992) use a finite dimensional linear model to recover ambient illumination and the surface reflectance by examining mutual reflection between surfaces. Ho et al. (1992) show how a color signal spectrum can be separated into reflectance and illumination components. They compute the coefficients of the basis functions by finding a least squares solution, which best fits the given color signal. However, in order to do this, they require that the entire color spectrum and not only the measurements from the sensors is available. Ho et al. suggest to obtain the measured color spectrum from chromatic aberration. Novak and Shafer (1992) suggest to introduce a color chart with known spectral characteristics to estimate the spectral power distribution of an unknown illuminant. 

The standard model of color image formation was already described in the previous section. Let us now assume that the reflectances are approximated by a set of basis functions. 

If we also assume that the illuminant is uniform over the entire image and G(xobj) = L we obtain 

the measurements made by the sensor can be viewed as a linear transformation of the reflectances. Let us consider only the three wavelengths Xj with i e r, g, b], then the lighting matrix would be given by 

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At the beginning it is necessary to describe the unperturbed system very well, independently of the polarization functions Let us assume that the unperturbed system is reasonably well described by using some finite set of basis functions As... 

This representation permits analytic calculations, as opposed to fiiUy numerical solutions [47,48] of the Hartree-Fock equation. Variational SCF methods using finite expansions [Eq. (2.14)] yield optimal analytic Hartree-Fock-Roothaan orbitals, and their corresponding eigenvalues, within the subspace spanned by the finite set of basis functions. 

Usui and Nakauchi (1997) proposed the neural architecture shown in Figure 8.14. They assume that the reflectances and illuminants can be described by a finite set of basis functions. In their work, they assume that three basis functions are sufficient for both reflectances and illuminants. The neural architecture tries to estimate the coefficients of the reflectance basis functions. 

Another drawback of interaction energies obtained in the supermolecular approach is the so called basis-set-superposition-error (BSSE). This is a consequence of using a finite set of basis functions to describe the atomic orbitals the molecular orbitals are made of. A finite set of basis functions lead to an error. A, in the three energies that enter the rhs of equation (10). In fact, the BSSE can be defined as... 

Because computers can represent numbers, but not functions, the molecular orbitals at each stage of the SCF procedure have to be represented by an expansion in a finite set of basis functions < >j(r), i = 1,2,... N. If the set is mathematically complete, the result of the SCF procedure is termed the HF or KS limit otherwise the result is dependent on the basis set used. Many types of basis funtion have been explored, and several are currently used in routine applications. However, their interrelationships and relative strengths and weaknesses are not often clarified and it may be instructive to do so here. 

In order to solve the Hartree-Fock equations, one usually expands the solutions in some finite set of basis functions,... 

In the algebraic approximation, it is assumed that the large and small components of the wavefunction can be parametrized in terms of some finite set of basis functions. Thus for the large component, we put... 

To find the true Hartree-Fock orbitals, one must use a complete set in (1.295), which means using an infinite number of gk s. As a practical matter, one must use a finite number of basis functions, so that one gets approximations to the Hartree-Fock orbitals. However, with a well-chosen basis set, one can approach the true Hartree-Fock orbitals and energy quite closely with a not unreasonably large number of basis functions. Any MOs (or AOs) found by iterative solution of the Hartree-Fock-Roothaan equations are called self-consistent-field (SCF) orbitals, whether or not the basis set is large enough to give near-Hartree-Fock accuracy. 

The only method found so far which is flexible enough to yield ground and excited state wavefunctions, transition rates and other properties is based on expanding all wavefunctions and operators in a finite discrete set of basis functions. That is, a set of one-particle spin-orbitals < >. s-x are selected and the wavefunction is expanded in Slater determinants based on these orbitals. A direct expansion would require writing F as... 

To pose the optimal control problem as a nonlinear programming (NLP) problem the controls u t) are approximated by a finite dimensional representation. The time interval [t0, r r] is divided into a finite number of subintervals (Ns), each with a set of basis functions involving a finite number of parameters u(t) = t, zj), te[( tj.i, tj), j = 1,2,. . J ], where tj = tF. The functions switching time tpj = 1,2,..., J. The control constraints now become ... 

Use of a complete (necessarily infinite) set of basis orbitals in the expansion for the molecular orbitals would insure absolute convergence to the Hartree-Fock limit. In practice, this is both impossible and unnecessary, and only a finite number of basis functions is employed in the expansion. The selection of the finite basis set is, therefore, of crucial importance in determining how closely one approximates the true Hartree-Fock solution. 

We can discretize the model, in this case the anomalous conductivity distribution, Ad (r), by introducing a set of basis functions, V l (r), V 2 (r), / /v (r) in the finite dimensional Hilbert space M, which is a subspace of the complex Hilbert space Mo M/v C Mo- Let us approximate the anomalous conductivity by its projection over the basis functions ... 

For practical reasons, the number of LOS measurements is finite, and the tomographic reconstruction problem is ill-posed. Two reconstruction methods have been developed for cases where optical access is restricted, and the number of measurement LOSs is limited. One method, adaptive FDDI, requires 100 or more LOSs [1-3], while the other method. Tomographic Reconstruction via a Karhunen-Loeve Basis, requires far fewer [4, 5]. Because it requires very few LOSs, the authors believe that this latter method has potential for use in sensing for feedback control of combustion systems where optical access is limited however, it requires considerable a priori information in the form of a set of expected distributions, the training set. This set is analyzed via POD to yield a set of basis functions, the Karhunen-Loeve eigenfunctions, that are used for reconstruction. These training sets could come from measurements on prototype equipment or from computational combustion simulations. 

A mathematical series is a sum of terms. A series can have a finite number of terms or can have an infinite number of terms. If a series has an infinite number of terms, an important question is whether it approaches a finite limit as more and more terms of the series are included (in which case we say that it converges) or whether it becomes infinite in magnitude or oscillates endlessly (in which case we say that it diverges). A constant series has terms that are constants, so that it equals a constant if it converges. Afunctional series has terms that are functions of one or more independent variables, so that the series is a function of the same independent variables if it converges. Each term of a functional series contains a constant coefficient that multiplies a function from a set of basis functions. The process of constructing a functional series to represent a specific function is the process of determining the coefficients. We discuss two common types of functional series, power series and Fourier series. 

Now that we have a suitable set of basis functions, we can find the finite element approximation to our 3D problem. Our orginal problem can be formulated as... 

To do an MP electron-correlation calculation, one first chooses a basis set and carries out an SCF calculation to obtain o, hf> and virtual orbitals. One then evaluates EP (and perhaps higher corrections) by evaluating the integrals over spin-orbitals in (15.87) in terms of integrals over the basis functions. One ought to use a complete set of basis functions to expand the spin-orbitals. The SCF calculation will then produce the exact Hartree-Fock energy and will yield an infinite number of virtual orbitals. The first two sums in (15.87) will then contain an infinite number of terms. Of course, one always uses a finite, incomplete basis set, which yields a finite number of virtual orbitals, and the sums in (15.87) contain only a finite number of terms. One thus has a basis-set truncation error in addition to the error due to truncation of the MP perturbation energy at E or E or whatever. 

To apply the CC method, two approximations are made. First, instead of using a complete, and hence infinite, set of basis functions, one uses a finite basis set to express the spin-orbitals in the SCF wave function. One thus has available only a finite number of virtual orbitals to use in forming excited determinants. As usual, we have a basis-set truncation error. Second, instead of including all the operators fj, 72,..., r , one approximates the operator f by including only some of these operators. Theory shows (Wilson, p. 222) that the most important contribution to T is made by r2.The approximation T T 2 gives... 

Because of the complicated nature of biomolecular geometries and charge distributions, the PB equation (PBE) is usually solved numerically by a variety of computational methods. These methods typically discretize the (exact) continuous solution to the PBE via a finite-dimensional set of basis functions. In the case of the linearized PBE, the resulting discretized equations transform the partial differential equation into a linear matrix-vector form that can be solved directly. However, the nonlinear equations obtained from the full PBE require more specialized techniques, such as Newton methods, to determine the solution to the discretized algebraic equation. ... 

LCAO Scheme. A basis set is a set of one-electron functions, which are combined to form the molecular orbitals of the chemical species. This is known as the Linear Combination of Atomic Orbitals (LCAO) scheme. To approach the exact solution to the Schrodinger equation, an infinite set of basis fiinctions would be required, as this would introduce sufficient mathematical flexibility to allow for a complete description of the molecular orbitals. In practical calculations, we must use a finite number of basis functions, and it is thus important to choose basis functions that allow for the most likely distribution of electrons within the system. This is achieved using basis functions that are based on the atomic orbitals of the constituent atoms of the molecule. For example, if a chemical system contained an oxygen atom, the chosen basis set would include functions describing each of the Is, 2s, and three 2p orbitals of an oxygen atom. 

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