IEEE Transactions on Pattern Analysis and Machine Intelligence

A Glimpse at Set Theory. The Real Numbers. The Topology of Cartesian Spaces. Convergence. Continuous Functions. Functions of One Variable. Infinite Series. Differentiation in RP Integration in RP.

The elements of real analysis (2nd edition), by Robert G. Bartle. Pp xv, 480. £10. 1976. SBM 0 471 05464 X (Wiley)

Theory of Equations. By J. V. Uspensky. Pp. vii, 353. 27s. 1948. (McGraw-Hill)

Three-dimensional (3D) image acquisition systems are rapidly becoming more affordable, especially systems based on commodity electronic cameras. At the same time, personal computers with graphics hardware capable of displaying complex 3D models are also becoming inexpensive enough to be available to a large population. As a result, there is potentially an opportunity to consider new virtual reality applications as diverse as cultural heritage and retail sales that will allow people to view realistic 3D objects on home computers. Although there are many physical techniques for acquiring 3D data—including laser scanners, structured light and time-of-flight—there is a basic pipeline of operations for taking the acquired data and producing a usable numerical model. We look at the fundamental problems of range image registration, line-of-sight errors, mesh integration, surface detail and color, and texture mapping. In the area of registration we consider both the problems of finding an initial global alignment using manual and automatic means, and refining this alignment with variations of the Iterative Closest Point methods. To account for scanner line-of-sight errors we compare several averaging approaches. In the area of mesh integration, that is finding a single mesh joining the data from all scans, we compare various methods for computing interpolating and approximating surfaces. We then look at various ways in which surface properties such as color (more properly, spectral reflectance) can be extracted from acquired imagery. Finally, we examine techniques for producing a final model representation that can be efficiently rendered using graphics hardware.

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The 3D Model Acquisition Pipeline

The 3D Model Acquisition Pipeline.

The Pythagorean hodograph (PH) curves are characterized by certain Pythagorean n-tuple identities in the polynomial ring, involving the derivatives of the curve coordinate functions. Such curves have many advantageous properties in computer aided geometric design. Thus far, PH curves have been studied in 2- or 3-dimensional Euclidean and Minkowski spaces. The characterization of PH curves in each of these contexts gives rise to different combinations of polynomials that satisfy further complicated identities. We present a novel approach to the Pythagorean hodograph curves, based on Clifford algebra methods, that unifies all known incarnations of PH curves into a single coherent framework. Furthermore, we discuss certain differential or algebraic geometric perspectives that arise from this new approach. In this paper, we present an approach that unifies all known incarnations of the socalled Pythagorean hodograph curves into a single coherent framework, through the use of Clifford algebra. As we shall see, Pythagorean hodograph (PH) curves are characterized by a certain Pythagorean n-tuple identity in the polynomial ring, relating derivatives of the curve coordinate functions. Thus far, the PH curves have been studied in 2- or 3-dimensional Euclidean and Minkowski spaces. The characterization of PH curves in each of these contexts involves rather different combinations of certain sets of polynomials. Currently, the diverse algebraic forms for PH curves in spaces with different dimensions and metrics are something of an enigma, that suggest the presence of a deeper underlying structure. The approach expounded in this paper reveals that all the different forms of PH curves can be expressed via a certain map, which we shall call the PH representation

Clifford algebra, spin representation, and rational parameterization of curves and surfaces ∗

Construction and shape analysis of PH Hermite interpolants

New algorithm for medial axis transform of plane domain

We present a new approach to medial axis transform and offset curve computation. Our algorithm is based on the domain decomposition scheme which reduces a complicated domain into a union of simple subdomains each of which is very easy to handle. This domain decomposition approach gives rise to the decomposition of the corresponding medial axis transform which is regarded as a geometric graph in the three dimensional Minkowski space R 2,1 . Each simple piece of the domain, called the fundamental domain, corresponds to a space-like curve in R 2,1 . Then using the new spline, called the Minkowski Pythagorean hodograph curve which was recently introduced, we approximate within the desired degree of accuracy the curve part of the medial axis transform with a G 1 cubic spline of Minkowski Pythagorean hodograph. This curve has the property of enabling us to write all offset curves as rational curves. Further, this Minkowski Pythagorean hodograph curve representation together with the domain decomposition lemma makes the trimming process essentially trivial. We give a simple procedure to obtain the trimmed offset curves in terms of the radius function of the MPH curve representing the medial axis transform.

Hyeong In Choi

Papers

Clifford algebra, spin representation, and rational parameterization of curves and surfaces ∗

Construction and shape analysis of PH Hermite interpolants

New algorithm for medial axis transform of plane domain

Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves

Euler–Rodrigues frames on spatial Pythagorean-hodograph curves