Generalized Fourier transforms, their nonlinearization and the imaging of the brain. (English) Zbl 1076.42004

Fokas, A. S.; Sung, L.-Y.

Notices Am. Math. Soc. 52, No. 10, 1178-1192 (2005).

Summary: The authors show how the Fourier transform can be generalized and nonlinearized and how these in turn can be used to solve certain nonlinear partial differential equations as well as to analytically invert certain integrals. As an application of the latter, the authors consider single particle emission computerized tomography (SPECT) as used in brain imaging.

MSC:

42A38	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
68U01	General topics in computing methodologies
92C55	Biomedical imaging and signal processing
35Q55	NLS equations (nonlinear Schrödinger equations)

Keywords:

linear evolution PDEs; Cauchy problem; nonlinear Schrödinger equation; computer tomography; PET; SPECT