# M.Sc. thesis defense: A Discrete Fourier Transform Pair for Arbitrary Sampling Geometries with Applications to Frequency Domain Volume Rendering on the Body-Centered Cubic

**Presenter:** Ali Dornhofer

Frequency domain volume rendering (FDVR), also known as Fourier volume rendering (FVR), is currently the asymptotically fastest volume rendering method known. In accordance to the Fourier projection slice theorem this method needs the 3D spatial data to be transformed to the frequency domain in a preprocessing step by use of the fast Fourier transform (FFT). The per-frame rendering includes the user interaction part (the user chooses a viewing direction) and the computational part (resampling of a slice plane perpendicular to the viewing direction and an inverse 2D FFT) to generate an X-ray like image. My idea was to adapt the regular algorithm to the body-centered cubic (BCC) lattice (i. e., one additional sample is placed right into the center of eight neighboring samples making up a cube) as this special sampling geometry has the great advantage of needing 29.3 % fewer samples but maintaining the same information and hence the same quality compared to the regular Cartesian cubic (CC) lattice. For this to work I needed to develop a generalization of the regular discrete Fourier transform (DFT) formulae to build a solid theoretical foundation. The core of this work presents this environment which can readily be used to implement the new algorithm. In addition to the derivation of the new theory I will present some remarks about the implementation of the regular algorithm as well as some thoughts about the adaption to the body-centered cubic (BCC) lattice.