^{1}

^{2}

^{1}

^{2}

We consider a curve reconstruction problem from unorganized point clouds with noise. In general, the result of curve reconstruction depends on how to select and order the representative points to resemble the shape of the clouds. We exploit a natural distance based on a property of one-dimensional Brownian motion to order sample points, which simultaneously reflect smoothness and nearness of points, so that our algorithm is able to reconstruct not only simple curves but also nonsimple curves. Numerous examples show that this algorithm is effective. The natural distance proposed in this paper is able to play an important role in a variety of fields of measuring the distance of points with considering direction.

In many engineering fields such as the reverse engineering of geometric models and image processing of medical images, great attention has been shown to the problems of reconstructing a shape from sample points. The curve reconstruction problem we want to discuss in this article plays an especially important role in shape reconstruction problems (see, e.g., [

Curve reconstruction problems can be regarded as the problem of fitting a curve or a family

Recently, there are two algorithms which use two important properties of human vision: proximity and smoothness [

Even though the second category, handling data points with noise, is closer to practical situations, relatively little research has been carried out on the second category (e.g., see [

Assume that the set

What we try to do in this paper is to expound a new persuasive measure (called a

In this section, we will discuss a naturalness for the distance

Suppose that

Suppose that

Two conditions (C.1) and (C.2) are similar to the first and second criterions of [

An easier way to understand Brownian motion is to think of it as a continuous version of the random walk process. As documented in an enormous amount of literature on stochastic processes such as [

For given time

Roughly speaking, (P.B) means that the set of original points on the real line at time

Before proving the naturalness for

The function

New coordinates for a scattered point

With the property (P.B) of Brownian motion, we further think of the

The distribution of Brownian motion after time

With this setting, we want to find a function

We adopt the function

Similarly thinking, one may take the function

The probabilities of the

However, the function

Standardizations of

Consequently, our problem is rewritten as the following, if

Choose

There are plenty of such increasing function

In this spirit, we define the

To prove the naturalness of

The

If

If

If

If

Some of these can be used to simplify numerical algorithms ordering the points. Here

The level curve passing through a point

Assume that

To show (C.2), it is enough to show that, for any fixed

The second equation of (

First, assume that

From Theorem

Now, we need to make a practical remark concerned with Theorem

Before closing this section, we remark on the effect of the subjective weights

Level curves for various subjective weights

In this section, we will describe a practical algorithm for ordering points in an unorganized point cloud as well as a sample point set.

Let

We explain a process for computing the set of candidates for the next point and choosing the next point among the candidates, called

The candidate points of

There is a main difference between our method and VICUR [

In order to deal with the first category problem, we assume that there is no noise in the point set

Compute Delaunay triangulation

Let

If

If

Step

The time complexity of our algorithm is

For the second category problem, we outline our algorithm as follows.

Compute Delaunay triangulation

Choose a point

Apply NextPointFinder with the last point

Apply NextPointFinder with the last point

In this section, we tested several reconstruction problems where each problem is focused on the following facts: (

Figure

Nonsimple curve reconstruction: (a) Delaunay triangulation, (b) enlarged Delaunay triangulation at the dense area, (c) reconstructed curve generated by [

Figure

Nonsimple curve reconstruction: (a) sample points and (b) result by NADIAS.

Figure

The process of NADIAS: (a) sample points, (b) NextPointFinder, and (c) bridging.

Examples with multicomponents.

There are two different results of point ordering for the same set of sample points by using Euclidean distance and Natural distance in Figure

Point orderings based on the different distance measures.

Euclidean distance [

Natural distance

In Figure

A complicated example using the natural point ordering method.

sample data

Figure

Extraction of the major configuration: sample points (a) and the result (b) with

Figure

Natural point orderings for point clouds: a point cloud generated by random process (a), a point cloud generated by drawing with mouse (b).

We have presented a new method for constructing curves from unorganized point clouds with noise. In general, the result of curve reconstruction depends on how to select and order the representative points to resemble the shape of the clouds. In this paper, in contradiction to the previous curve reconstruction algorithms based on Euclidean distance, we exploit a natural distance to reflect orientation to the ordering of sample points, so that our algorithm is able to reconstruct not only simple curves but also nonsimple curves. Moreover, for unorganized point clouds, this method efficiently extracts the skeletons of the clouds by cutting out the outliers, even though the result by our method is sensitive to the initial point and the initial direction. The method may be extended in two ways. The first thing is to extend the method to the surface reconstruction problem in three-dimensional space. We think that one should use the properties of two-dimensional Brownian motion in order to get a natural distance similar to that in this paper. Second, it is possible to improve our method which can control the subjective weight

The authors would like to express their sincere gratitude to reviewers for many helpful comments and valuable suggestions on the first draft of this paper. This work was supported by Korea Research Foundation Grant KRF-2003-041-C00039.