Operator algebra in the space of images

[摘要] A consistent description of images on the disk and of their transformations is given as elements of a vector space and of an operators algebra. The vector space of images on the disk is the Hilbert space L2(D) that has as a basis the Zernike functions. To construct the operator algebra that transforms the images, L2(D) must be complemented and the full rigged Hilbert space RHS(D) considered. Only this rigged Hilbert space allows indeed to write the operators of different cardinality we need to build the ladder operators on the Zernike functions that by inspection, belong to the representation D+1/2D+1/2of the algebra su(1, 1) ⊕ su(1, 1). Consequently the transformations of images are operators contained inside the universal enveloping algebra UEA[su(1, 1) ⊕ su(1, 1)]. Because of limited precision of experimental measures, physical states can be always described by vectors of the Schwartz space (D), dense in the L2(D) space where the manipulation of images is performed.

[发布机构] Dipartimento di Fisica, Università di Firenze, Sesto Fiorentino, Firenze; I50019, Italy^1;Depatamento de Fisica Téorica, Universidad de Valladolid, Valladolid; E-47005, Spain^2

[效力级别] 力学 [学科分类] 力学，机械学

[关键词] Cardinalities;Operator algebras;Physical state;Rigged Hilbert space;Universal enveloping algebras;Zernike function