Reconstruction of a Differential Form from Doppler Transform

A vector …eld in n-space is reconstructed from a n-dimensional family of ray or line integrals. A similar problem is considered for di¤erential forms of higher order. Key words: Di¤erential form, ray integral, k-plane integral, reconstruction, support condition AMS subject classi…cation (2000): Primary, 53C65; Secondary, 65R32, 76M27 Abbreviated title: Doppler transform 1 Introduction Integrals of a vector …eld model measurements in tomographic imaging of moving mediums, in particular, imaging of liquid or gas ‡ows, tumor detection, optics and plasma physics etc.; see Braun and Hauck [1], Sparr et al [13], Osman and Prince [9], Wells et al [15], Sielschott [11] and references therein. In this context, the data of longitudinal line integrals of a vector …eld form is called the Doppler transform according to the Doppler shift law. See some basic mathematical properties in Natterer and Wübbeling [8]. The longitudinal measurement of a …eld can be interpreted as a line integral of the corresponding 1-di¤erential form f: The Doppler transform is invariant with respect to the gauge transformation f + da, since all line integrals vanish for the exact form da (irrotational part of the …eld). The di¤erential df of the form f is gauge invariant. By means the Helmholtz theorem reconstruction of the form df this is equivalent to reconstruction of the solenoidal part of the corresponding vector …eld. In 2D case the reconstrcution problem is immediately reduced to inversion of the Radon transform; see Norton [7], Howard [6].