Toward high-quality gradient estimation on regular lattices

In this paper, we present two methods for accurate gradient estimation from scalar field data sampled on regular lattices. The first method is based on the multidimensional Taylor series expansion of the convolution sum and allows us to specify design criteria such as compactness and approximation power. The second method is based on a Hilbert space framework and provides a minimum error solution in the form of an orthogonal projection operating between two approximation spaces. Both methods lead to discrete filters, which can be combined with continuous reconstruction kernels to yield highly accurate estimators as compared to the current state of the art. We demonstrate the advantages of our methods in the context of volume rendering of data sampled on Cartesian and Body-Centered Cubic lattices. Our results show significant qualitative and quantitative improvements for both synthetic and real data, while incurring a moderate preprocessing and storage overhead.