Analysis of difference fairing based on DFT-filter

Abstract

In this paper, even-order-difference fairing (EOD-fairing) of point-sequence curve (PS-curve) is analyzed using a discrete Fourier transform filter (DFT-filter). It is shown that EOD-fairing of a reflection-padded PS-curve is equivalent to DFT-filtering, and a DFT-filter corresponding to the EOD-fairing operation is derived. Based on the concept of filtering measure, a method of determining an “optimal” damping factor for EOD-fairing is proposed, and then the validity of the proposed method is demonstrated via experimental tests.

Introduction

A dense sequence of data points representing a smooth curve is often called a PS-curve (point-sequence curve). A PS-curve is usually obtained from an optical scanner or a coordinate measuring machine, but it may also be obtained as a result of surface–surface intersection or during NC cutter path generation [1]. PS-curves are subject to digitizing errors that need to be “filtered” by employing some fairing operations. One of the most straightforward PS-curve fairing methods is difference fairing.

A PS-curve representing a “$\text{y=f(x)}\text{type}$” curve is called an explicit PS-curve, which is given as a sequence of (x_k,y_k). A point-sequence obtained from a “line-type” laser scanner or coordinate measuring machine may better be treated as an explicit PS-curve (instead of 3D PS-curve) because only the “height-value” (y_k) at each pre-planned “measuring-location” (x_k) is measured. If Δ=x_k+1−x_k for all k, we have a uniformly spaced explicit PS-curve.

The subject of PS-curve difference fairing has been studied by a number of researchers. Practical procedures for 2nd- and 4th-difference fairing are presented in Choi and Jerard [1]. For a 2nd-difference fairing of periodic PS-curves, Taubin [2] proposes a DFT-filter in which “scale-factors” are employed to control the shrinkage effects. It is also postulated that finite difference fairing operations can be analyzed by using a discrete low-pass filter [3]. For fairing binary sequences, Legault and Suen [4] introduces a “local weighted averaging method” that covers difference fairing as a special case.

In this paper, (1) difference-fairing of explicit PS-curves is analyzed by using a DFT filter, (2) a method of determining “optimal” damping factors for use in difference-fairing is proposed, and (3) fairing experiments are performed to validate the damping-factor determination method. More specifically, it is shown that even-order difference fairing (EOD-fairing) of a reflection-padded [5] PS-curve is equivalent to a DFT filtering, and then a DFT filter (it is termed an EOD-filter) corresponding to the fairing formula is constructed. Based on the premise that an optimal EOD-filter is the one that is as close to an ideal low-pass filter as possible, a method of determining an “optimal” damping factor is proposed. A set of 2nd-difference fairing experiments has been carried out.

The rest of the paper is organized as follows. Preliminary definitions related to difference fairing are provided in Section 2, and major properties of difference fairing together with a DFT filter (called the EOD-filter) are presented in Section 3. Results of damping factor analysis based on the EOD filter are presented in Section 4, and experimental results are analyzed in Section 5, followed by concluding remarks in Section 6.