Controlled precision volume integration

Abstract

Traditional methods for evaluating the low-albedo volume rendering integral do not include bounds on the magnitude of approximation error. In this paper, we examine three techniques for solving this integral with error bounds: trapezoid rule, Simpson's rule, and a power series method. In each case, the expression for the error bound provides a mechanism for computing the integral to any specified precision. The formulations presented are appropriate for polynomial reconstruction from point samples; however, the approach is considerably mom general. The three techniques we present differ in relative efficiency for computing results to a given precision. The trapezoid rule and Simpson's rule are most efficient for low-to medium-precision solutions. The power series method converges rapidly to a machine precision solution, providing both an efficient means for high-accuracy volume rendering, and a reference standard by which other approximations may be measured.

Volume MLS Ray Casting

“…The TF must be smooth in order to compute the numerical integral efficiently and accurately as no general, closed-form solutions are available. For the case of a smooth TF, well-known adaptive numerical schemes can be employed [32], and theory from Monte Carlo integration can guide their design [7]. Also, interval arithmetic has been proposed to make the above approaches robust [33].…”

Second Order Pre-Integrated Volume Rendering

“…Among these, the Newton-Cotes formulas [14] are a group of formulas based on Lagrange polynomials for integrating discrete functions. In an early work [12], three methods have been proposed for controlling the quality of the approximation of the volume rendering integral, two of which being based on the Newton-Cotes formulas at respectively order 1 and 2. Given a ray segment contained in a voxel as well as a user defined error threshold, the remainder term of the chosen Newton-Cotes formula is used to solve for an appropriate subdivision step, the latter allowing to locally integrate the color and opacity values at the desired precision when reinjected in the Newton-Cotes formula.…”

Fast Wavelet Based Volume Rendering by Accumulation of Transparent Texture Maps

“…(18) and eq. (19). The advantage of this method is, that we can compute the slices in frequency space analytically and in some cases, the inverse FT as well.…”

“…Shading of the volume is figured out by estimations of the normal through the volume intensity gradient. Good mathematical analysis of the error bounds in volume rendering is given in [19]. Isosurfaces are treated in different ways.…”