Spherical Harmonics
نویسنده
چکیده
5 Spherical Harmonics 7 5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2.1 Harmonic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.2.3 Rotational invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.2.4 Fast Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.2.5 Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.3 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
منابع مشابه
Determination of Fiber Direction in High Angular Resolution Diffusion Images using Spherical Harmonics Functions and Wiener Filter
Diffusion tensor imaging (DTI) MRI is a noninvasive imaging method of the cerebral tissues whose fibers directions are not evaluated correctly in the regions of the crossing fibers. For the same reason the high angular resolution diffusion images (HARDI) are used for estimation of the fiber direction in each voxel. One of the main methods to specify the direction of fibers is usage of the spher...
متن کاملSome Integral Identities for Spherical Harmonics in an Arbitrary Dimension
Spherical harmonics in an arbitrary dimension are employed widely in quantum theory. Spherical harmonics are also called hyperspherical harmonics when the dimension is larger than 3. In this paper, we derive some integral identities involving spherical harmonics in an arbitrary dimension.
متن کاملTraceless Symmetric Tensor Approach to Legendre Polynomials and Spherical Harmonics
In these notes I will describe the separation of variable technique for solving Laplace’s equation, using spherical polar coordinates. The solutions will involve Legendre polynomials for cases with azimuthal symmetry, and more generally they will involve spherical harmonics. I will construct these solutions using traceless symmetric tensors, but in Lecture Notes 8 I describe how the solutions i...
متن کاملFast Spin ±2 Spherical Harmonics Transforms
An exact fast algorithm is developed for the direct spin-weighted spherical harmonics transforms of bandlimited spin ±2 functions on the sphere. First, we define spin functions on the sphere and their decomposition in an orthonormal basis of spin-weighted spherical harmonics. Second, we discuss the a priori O(L4) asymptotic complexity of the spin ±2 spherical harmonics transforms, where 2L stan...
متن کاملSome Characterizations of the Spherical Harmonics Coefficients for Isotropic Random Fields
In this paper we provide some simple characterizations for the spherical harmonics coefficients of an isotropic random field on the sphere. The main result is a characterization of isotropic gaussian fields through independence of the coefficients of their development in spherical harmonics.
متن کاملEfficient and accurate rotation of finite spherical harmonics expansions
Spherical harmonics are employed in a wide range of applications in computational science and physics, and many of them require the rotation of functions. We present an efficient and accurate algorithm for the rotation of finite spherical harmonics expansions. Exploiting the pointwise action of the rotation group on functions on the sphere, we obtain the spherical harmonics expansion of a rotat...
متن کامل