Orthogonal moments

A Preliminary Study of Compression Efficiency and Noise Robustness of Orthogonal Moments on Medical X-Ray Images... 

Keywords— Orthogonal moments, Legendre, Tchebichef, compression, noise robustness. 

Among all the moments that have been reviewed, orthogonal moments are chosen for this study, as they provide no information redundancy and efficient image reconstruction computation. Two orthogonal moments have been chosen for this study, one from the continuous family, called the Legendre moments, another from the discrete family, called the Tchebichef moments. 

The organization of this paper is as follows. The first section gives some brief introduction about medical image compression, motivation and aim of this study. The mathematical background of the orthogonal moments used in this study is... 

State averaging gives a wave function that describes the first few electronic states equally well. This is done by computing several states at once with the same orbitals. It also keeps the wave functions strictly orthogonal. This is necessary to accurately compute the transition dipole moments. 

Intensities for electronic transitions are computed as transition dipole moments between states. This is most accurate if the states are orthogonal. Some of the best results are obtained from the CIS, MCSCF, and ZINDO methods. The CASPT2 method can be very accurate, but it often requires some manual manipulation in order to obtain the correct configurations in the reference space. 

Therefore, the orthogonal TOF mass spectrum is a snapshot of all the ions in the sampled ion beam at any one moment in time. The arrangement has advantages over magnetic sectors alone and TOF instruments alone (see Chapter 20 for further discussion). 

The fii st term is zero because I and its derivatives are orthogonal. The fourth term involves second moments and we use the coupled Hartree-Fock procedure to find the terms requiring the first derivative of the wavefunction. 

For a nonlinear molecule the rotational energy levels are a function of three principal moments of inertia /A, /B and /c- These are moments of inertia around three mutually orthogonal axes that have their origin (or intersection) at the center of mass of the molecule. They are oriented so that the products of inertia are zero. The relationship between the three moments of inertia, and hence the energy levels, depends upon the geometry of the molecules. 

Now, consider the special case of p(r) = 1 for the moment. Then the correlatoion function Eq. (197) is just a simple overlap of < > t) and x t). With use of the orthogonality and completeness of frozen Gaussian basis, this overlap can be simply expressed as... 

All three states were described by a single set of SCF molecular orbitals based on the occupied canonical orbitals of the X Z- state and a transformation of the canonical virtual space known as "K-orbitals" [10] which, among other properties, approximate the set of natural orbitals. Transition moments within orthogonal basis functions are easier to derive. For the X state the composition of the reference space was obtained by performing two Hartree-Fock single and double excitations (HFSD-CI) calculations at two typical intemuclear distances, i.e. R. (equilibrium geometry) and about 3Re,and adding to the HF... 

The emission dipole moment must not be orthogonal to the acceptor absorption dipole moment (see Chapter 1). 

As shown in the Appendix (in Section V), in the C2h point group, the 1Ag - 1Bll (i.e. the monoelectronic r - jr excitation) possesses only an electric dipole moment, while in the C2V structure the electric and magnetic dipole moments, both non-vanishing, are orthogonal. In both cases the product in equation 1 leads to zero rotational strength. 

Since the transition moments of the antisymmetric and symmetric CH2 stretching vibrations and the methylene chain axis are mutually perpendicular, the average orientation angle y of the hydrocarbon chain axis around the surface normal is obtained to be 27° by the orthogonal relation... 

The mean angular moment (in units of %) is fit = dqjdv and the mean dipole-dipole energy is ED — — dq/dp. In the limit of high temperatures we may expand the exponential and retain only the linear terms, v and /9 may then be considered as the coefficients of the expansion of p in orthogonal operators, since Tr(VmMa) — 0. In this limiting case it then becomes evident that v and / are independent. It is easily shown that the thermodynamical temperature, defined by... 

---