Gradient Estimation Revitalized

Abstract

Fig. 1. DVR images of the carp dataset. Left: Original Cartesian (256 3 ) dataset reconstructed using prefiltered tricubic B-splines and shaded using the analytic gradient. Middle: Downsampled Cartesian (128 3 ) dataset reconstructed using prefiltered tricubic B-splines and shaded by interpolating the gradient obtained through a fourth-order central differencing scheme (pFIR). Right: The middle dataset shaded using our proposed fourth-order shifted gradient estimation scheme (pFIR-s). Notice how the details in the bones and skull are much better preserved as compared to the central-differencing scheme even though the underlying scalar interpolation is the same. This gain in quality comes at no additional cost.Abstract-We investigate the use of a Fourier-domain derivative error kernel to quantify the error incurred while estimating the gradient of a function from scalar point samples on a regular lattice. We use the error kernel to show that gradient reconstruction quality is significantly enhanced merely by shifting the reconstruction kernel to the centers of the principal lattice directions. Additionally, we exploit the algebraic similarities between the scalar and derivative error kernels to design asymptotically optimal gradient estimation filters that can be factored into an infinite impulse response interpolation prefilter and a finite impulse response directional derivative filter. This leads to a significant performance gain both in terms of accuracy and computational efficiency. The interpolation prefilter provides an accurate scalar approximation and can be re-used to cheaply compute directional derivatives on-the-fly without the need to store gradients. We demonstrate the impact of our filters in the context of volume rendering of scalar data sampled on the Cartesian and Body-Centered Cubic lattices. Our results rival those obtained from other competitive gradient estimation methods while incurring no additional computational or storage overhead.

A Fresh Look at Generalized Sampling

“…(See section 9 for details.) In recent follow-up work, Alim et al [2010] used this additional freedom to derive FIR filters that can be applied locally, on demand, on the interpolating coefficients for f . This allows sharing the same coefficient vector for optimal-order approximations to f and its partial derivatives.…”

Quantitative Error Analysis for the Reconstruction of Derivatives

“…Thus, the minimization of the asymptotic constant, for a given approximation order, is a simple and efficient way of designing reconstruction methods of high quality. Applying this methodology to the reconstruction of derivatives is new and the authors are not aware, for instance, of results similar to (25) in the literature.…”

Compactly Supported Biorthogonal Wavelet Bases on the Body Centered Cubic Lattice

“…Alim et al . designed discrete derivative filters for the BCC lattice [AMC10, HAM11], and proposed a fast discrete Fourier transform for BCC data [AM09]. In terms of multiresolution approaches on the BCC lattice, there is not much work.…”