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Continuous Fourier Volume Rendering of Irregularly Sampled Data Using Anisotropic RBFs

We describe a Fourier Volume Rendering (FVR) algorithm for datasets that are irregularly sampled and require anisotropic (e.g., elliptical) kernels for reconstruction. We sample the continuous frequency spectrum of such datasets by computing the continuous Fourier transform of the spatial interpolation kernel which is a radially symmetric basis function (RBF) that may be anisotropically scaled. While in the frequency domain, we can apply low, band, and high-pass filters and arbitrary magnification and minification of the dataset before performing an inverse 2D Fourier transform to obtain the X-ray projection. Our algorithm is particularly amenable to implementation on commodity programmable graphics boards, and can interactively render X-rays for datasets on the order of tens of thousands of points. We describe the theoretical considerations to properly sample the frequency spectrum of anisotropic RBFs to avoid aliasing in the resulting X-ray and present a practical method for datasets with high sampling requirements. A significant benefit of our algorithm is that it can be applied to anisotropic RBFs that have been fitted to data through optimization techniques, allowing the incorporation of advanced data-sensitive constraints, such as smoothness, sharpness, and feature preservation.