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Image Reconstruction by Gengsheng Lawrence Zeng

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Let u = Φ +t x ;d y = | t | 3 d x . The above expression becomes

G( Φ , β )= 0 f( y ) e 2πi t ( y Φ ) β 1 | t | 3 dtd y .

Let s= 1 t ;ds= 1 t 2 dt. We then have

G( Φ , β )= f( y ) e 2πis( y Φ ) β | s | 3 1 s 2 d y ds , G( Φ , β )= ( f( y ) e 2πi y ( s β ) d y )| s | e 2πis( Φ β ) ds .

Recognizing that the inner triple integral is the 3D Fourier transform of f, that is,

F( s β )= f( y

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