BACKPROJECTION FILTERING FOR VARIABLE ORBIT FAN-BEAM TOMOGRAPHY

Backprojection filtering algorithms are presented for three variable o rbit fan-beam geometries. Expressions for the fan-beam projection and backprojection operators are given for a flat detector fan-beam geomet ry with fixed focal length, with variable focal length, and with fixed focal length and off-center focusing. Backprojection operators are de rived for each geometry using transformation of coordinates to transfo rm from a parallel geometry backprojector to a fan-beam backprojector for the appropriate geometry. The backprojection operator includes a f actor which is a function of the coordinates of the projection ray and the coordinates of the pixel in the backprojected image. The backproj ection filtering algorithm first backprojects the variable orbit fan-b eam projection data using the appropriately derived backprojector to o btain a 1/r blurring of the original image, then takes the two-dimensi onal (2D) Fast Fourier Transform (FFT) of the backprojected image, the n multiples the transformed image by the 2D ramp filter function, and finally takes the inverse 2D FFT to obtain the reconstructed image. Co mputer simulations verify that backprojectors with appropriate weighti ng give artifact free reconstructions of simulated line integral proje ctions. Also, it is shown that it is not necessary to assume a project ion model of line integrals, but the projector and backprojector can b e defined to model the physics of the imaging detection process. A bac kprojector for variable orbit fan-beam tomography with fixed focal len gth is derived which includes an additional factor which is a function of the flux density along the hat detector. It is shown that the impu lse response for the composite of the projection and backprojection op erations is equal to 1/r.