Contemporary Engineering Sciences, Vol. 11, 2018, no. 88, 4357 - 4368

HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ces.2018.88459

3D Modeling of Natural Heterogeneous Objects:

An Introduction to Methods and Algorithms

Miller Gómez-Mora and Rocío Rodríguez-Guerrero

Telematics Engineering Program, Research Group IMAGINET

Universidad Distrital Francisco José de Caldas

Bogotá, Colombia

Copyright © 2018 Miller Gómez-Mora and Rocío Rodriguez-Guerrero. This article is distributed

under the Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Abstract

The field of three-dimensional (3D) modeling of natural heterogeneous object is

relatively new and its results have many applications in science and engineering.

This process starts by taking medical images through several parallel slices. These

slices are then stacked in volume data and reconstructed into volume models in

order to visualize the 3D object structure. This document provides a general

description of the most well-known modeling methods and algorithms with a

discussion of relative advantages and disadvantages for using them in generation of

high-fidelity models built from medical image data such as computerized axial

tomography (CAT).

Keywords: 3D modeling, heterogeneous objects, implicit surfaces, surface meshes,

volumetric meshes

1 Introduction

Most natural objects contain multiple regions within complex volumetric

structures. These structures are usually organized in complicated geometric

configurations. Such objects are referred to as natural heterogeneous objects or

simply heterogeneous objects. In the context of 3D modeling of heterogeneous

objects, the most common way of producing a 3D model suitable for both

visualization and applications in engineering is by the following methodology (see

Figure 1):

a) Acquire volumetric images of the heterogeneous object using medical

images, for example, CAT images.

4358 Miller Gómez-Mora and Rocío Rodriguez-Guerrero

b) Segment parts or regions from medical images that make up the

heterogeneous object.

c) Reconstruct surface of the heterogeneous object and refine surface mesh

until obtaining a model with desired quality.

d) Add nodes inside model to produce three-dimensional elements that will

make up the final volumetric mesh.

Figure 1: Methodology for 3D modeling of a human liver.

In this article, methods and algorithms to modeling natural heterogeneous objects

are analyzed and discussed based on the assumption that steps a) and b) have been

carried out especially segmentation of all regions that make up the heterogeneous

object.

2 Surface reconstruction

2.1 Parametric methods

In the context of parametric representations, the problem of surface reconstruction

involves calculation of a surface S defined by a function 𝐹(𝑢, 𝑣) which

approximates, as much as possible, each point of the point cloud given in ℝ3, where

F belongs to a specific linear space of functions. Bezier curves and surfaces are the

first techniques used to solve surface reconstruction problems by adjusting local

surface patches [1]. However, due to Bezier curves cannot be modified locally and

movement of control points affect shape of the whole curve, the B-Spline method

was developed, which allows reaching a C2 continuity. For B-Spline surface at

parameters (u, v) a control point can be defined as:

𝑃(𝑢, 𝑣) = ∑ ∑ 𝐵𝑖,𝑗𝑁𝑖,𝑘(𝑢)𝑀𝑗,𝑙(𝑣)

𝑚

𝑗=0

𝑛

𝑖=0

a)

b)

c)

d)

3D modeling of natural heterogeneous objects 4359

Where, 𝐵𝑖,𝑗 are vertices of a polygonal control mesh, with i in the range between

0 and n, while j in the range between 0 and m. 𝑁𝑖,𝑘(𝑢) and 𝑀𝑗,𝑙(𝑣) are basic B-

Spline functions given by equation:

𝑁𝑖,𝑘(𝑢) =𝑢 − 𝑢𝑖

𝑢𝑖+𝑘−1 − 𝑢𝑖𝑁𝑖,𝑘−1(𝑢) +

𝑢𝑖+𝑘 − 𝑢

𝑢𝑖+𝑘 − 𝑢𝑖+1𝑁𝑖+1,𝑘−1(𝑢)

𝑀𝑗,𝑙(𝑣) =𝑣 − 𝑣𝑗

𝑣𝑗+𝑙−1 − 𝑣𝑗𝑀𝑗,𝑙−1(𝑣) +

𝑣𝑗+𝑙 − 𝑣

𝑣𝑗+𝑙 − 𝑣𝑗+1𝑀𝑗+1,𝑙−1(𝑣)

NURBS (Non-uniform Rational B-Spline) surfaces are a generalization of Bezier

and B-splines surfaces. NURBS were developed because Bezier and B-splines

methods contain some limitations when modeling more complicated shapes that

require higher order Bezier curves, and the inability to accurately represent conic

curves. A NURBS surface with parameters (u, v) is defined as

𝑃 (𝑢, 𝑣) = ∑ ∑ 𝐵𝑖,𝑗𝑆𝑖,𝑗(𝑢, 𝑣)

𝑚

𝑗=0

𝑛

𝑖=0

Where, 𝐵𝑖,𝑗 are 3D points of the control mesh, and 𝑆𝑖,𝑗(𝑢, 𝑣) are rational B-Spline

surface functions, defined as:

𝑆𝑖,𝑗(𝑢, 𝑣) =ℎ𝑖,𝑗𝑁𝑖,𝑘(𝑢)𝑀𝑗,𝑙(𝑣)

∑ ∑ ℎ𝑖,𝑗𝑁𝑖,𝑘(𝑢)𝑀𝑗,𝑙(𝑣)𝑚+1𝑗=1

𝑛+1𝑖=1

Where ℎ𝑖,𝑗 is the weight, 𝑁𝑖,𝑘(𝑢) and 𝑀𝑗,𝑙(𝑣) are basic non-rational B-Spline

functions, given by the previously defined equations.

Currently, NURBS models are industrial standards for representation of surfaces

that are widely used in the field of reverse engineering [2]. NURBS can be used as

a method for approximation or interpolation of scattered data and is also

incorporated with most of the current geometric modeling systems for

reconstructing smooth surfaces and dealing with a non-uniform data set [3], [4].

2.2 Computational geometry methods

These methods depend on algorithms such as Delaunay triangulation and its dual,

Voronoi diagram. Alpha Shape algorithm [5] and Crust algorithm [6] are the two

most successful examples of this classification. In Alpha Shape algorithm, the shape

of the surface is carved by eliminating simplexes of Delaunay triangulation. A

simplex is eliminated if its circumscribed sphere is larger than the alpha ball. On

the other hand, Crust algorithm calculates Voronoi diagram from which Delaunay

4360 Miller Gómez-Mora and Rocío Rodriguez-Guerrero

Triangulation is calculated. The 3D mesh is then reconstructed by obtaining all

triangles that connect three points of Delaunay triangulation. This mesh

distinguishes triangles that are part of the object's surface from those that are inside

because the inner triangles have a Voronoi vertex as one of their vertices.

An improvement of Crust algorithm is Power Crust algorithm [7]. The main idea

behind this algorithm is to represent the surface as a polygon, obtained from polar

balls using Voronoi diagram. This algorithm is robust because it is expressed in

terms of cells on the surface of a solid object. No hole-filling mechanism is required

in this surface reconstruction algorithm, since it captures geometry of the surface.

However, noisy data causes scattering of points away from the surface.

An improvement and extension of Voronoi diagram is Tight Cocone algorithm

proposed by Dey and Giesen [8]. This algorithm is faster than Crust and Power

Crust algorithms, however, like other algorithms in this category, this algorithm is

not robust enough to handle noisy point clouds and point clouds with low density.

2.3 Implicit methods

Some researchers refer to implicit surfaces as volumetric representation or

approaches based on adjustment of functions. Implicit surfaces represent the surface

as a particular iso-contour of a scalar function. An implicit surface S in ℝ3 can be

defined as the zero-level set of some function 𝑓: ℝ3 → ℝ

𝑆 = {𝐱 ∈ ℝ3: 𝑓(𝐱) = 0}

An implicit surface 𝑓(𝑥) can be defined through data-based methods that take the

form of a signed distance field [9], radial base functions (RBF) [10], or an indicatrix

function [11]. These methods are divided into:

Global methods: These methods aim to build a unique function so that its zero-

level set interpolates or approximates the point cloud globally.

Local methods: In this case, the global function results from combination of

functions of local form, each of which interpolates or approaches a sub cloud of

points.

Radial basis functions are functions where the result is the invariant rotation

around a certain point 𝐱𝑖. They can be used to interpolate a function with n points

using n radial basis functions centered on these points. The interpolator 𝐹(𝐱) is the

result of the sum of n radial basis functions, each of which is associated with a

different center 𝐱𝑖, and weighted by an appropriate coefficient 𝑤𝑖.

𝐹(𝐱) = ∑ 𝑤𝑖𝜙(∥ 𝐱 − 𝐱𝑖 ∥)

𝑛

𝑖=1

Essentially, there are two main steps in a procedure to reconstruct a dispersed

(scattered) sampled surface with radial base functions [10]:

Construction of a distance function with a sign.

Adjusting a radial basis function to the resulting distance function.

3D modeling of natural heterogeneous objects 4361

Implicit methods based on radial base functions (RBF) are widely used because

they can reconstruct surfaces of any topology with almost any point distribution.

RBFs have long been known as one of the most accurate and stable methods to

solve the problem of interpolation of scattered data. Although this method of

interpolation provides the minimum curvature surface that passes through given

points, RBFs are a global technique, with the drawback that local characteristics

will be lost.

Ohtake [12] presents a method called multi-level partition of unity (MPU) for the

reconstruction of implicit surfaces using a local technique. The basic idea of this

method is to divide the input data domain Ω ∈ ℝ𝑑 into several overlapping

subdomains Ω𝑖 that cover Ω, i.e. Ω ⊆ ⋃ Ω𝑖𝑖 . Each subdomain approaches

independently of other subdomains using compact support RBF. These subdomains

share intervals in ℝ (see Figure 2), circles in ℝ2 and spheres in ℝ3 that overlap

near their limits.

Figure 2. Four local functions 𝑓𝑖 (𝑖 = 1,2,3,4) combined by weighting functions

𝑤𝑖 centered on points 𝐱𝑖 on the real line ℝ. The graph of the resulting function is

the solid red curve.

This partition of the unit is nothing more than a set of non-negative functions 𝜙𝑖

with compact support supp(𝜙𝑖) ⊆ Ω𝑖 that satisfy the condition ∑ 𝜙𝑖𝑖 = 1 in Ω. For

each subdomain {Ω𝑖}, the set of points 𝑃𝑖 is formed inside {Ω𝑖}, then calculating a

local function 𝑓𝑖 that adjusts the points of 𝑃𝑖. Global adjustment function F is then

the result of a combination of local functions 𝑓𝑖 weighted by partition functions 𝜙𝑖

as follows:

𝐹(𝐱) = ∑ 𝜙𝑖(𝐱)𝑓𝑖(𝐱)

𝑛

𝑖=1

For example, in Figure 2, the resulting function F (in red color) is constructed

from a combination of four local functions 𝑓𝑖, which are associated with four

weighting functions 𝑤𝑖. The condition ∑ 𝜙𝑖𝑖 = 1 is simply obtained from the

weight functions 𝑤𝑖 using the following normalization formula:

4362 Miller Gómez-Mora and Rocío Rodriguez-Guerrero

𝜙𝑖(𝐱) =𝑤𝑖(𝐱)

∑ 𝑤𝑗(𝐱)𝑗

Global adjustment function F can then be rewritten as:

𝐹(𝐱) =∑ 𝑤𝑖(𝐱)𝑓𝑖(𝐱)𝑖

∑ 𝑤𝑖(𝐱)𝑖

Last method analyzed in this category forms a Poisson equation to obtain the

surface that best fits a dense point cloud P. This approach called Poisson surface

reconstruction [11] requires as input a cloud of P points along with their oriented

normal vectors (see Figure 3). With this data, an indicatrix function 𝜒 whose value

is one inside and zero outside the reconstructed region Ω is defined. The gradient

of the indicatrix function is equated to a vector field, constructed from the normal

vectors of the point cloud.

∇𝜒𝛺 = W

Then, the Poisson equation is formed and it is solved to obtain the indicatrix

function.

∇ ⋅ ∇�̃�𝛺 = ∇ ⋅ V ⇒ �̃�𝛺 = ∇ ⋅ V This algorithm was extended in the work of Kazhdan and Hoppe [13] by

incorporating weights to points assigned for interpolation.

Figure 3. Intuitive illustration of Poisson reconstruction in 2D. Taken from [11].

3 Geometric modeling of natural objects

The process of geometric modeling from segmented medical images begins with

the generation of a surface mesh using Marching Cubes (CM) algorithm [14]. The

principle behind MC algorithm is to subdivide the space into a series of small cubes

known as voxels (see Figure 4). Each point of the mesh is a vertex of a cubes corner.

3D modeling of natural heterogeneous objects 4363

Each vertex and each edge of the cube are indexed for searching in the search

tables. By determining which edges of the voxel intersect with the isosurface,

triangular patches are created that divide the cube into different regions that are

within the isosurface and regions that are outside it. The algorithm moves through

each of cubes as it evaluates the points of each corner and replaces the cube with

an appropriate set of polygons (see Figure 4-a)).

a) b)

Figure 4. a) 15 final combinations of Marching Cubes algorithm [14]. Yellow dots

indicate vertices that have been tested as "inside" the object. b) Step effect or

aliasing due to the segmented medical image.

As each of the eight vertices of a cube can be marked or not, there are 256 (28)

possible combinations of the state of the corner. These combinations are simplified,

taking into account the combinations of cells that are duplicated, to 15 final

combinations as shown in Figure 4-a). The final step of Marching Cubes algorithm

is to calculate the normal vector to faces of the triangulation to obtain a correct

representation of the surface.

Since segmented data of medical images are generally huge, irregular, and noisy,

the triangulation produced by Marching Cubes reflects these defects (see Figure 4-

b)), which prevents the use of this type of surfaces directly in engineering

applications. However, the initial mesh produced by Marching Cubes forms the

input data of many surface reconstruction algorithms and is also used to extract the

isosurface from implicit surfaces.

3.2 Parametric surfaces

Traditionally, parametric surface reconstruction algorithms consists of four main

steps:

Generation of the surface mesh from the not organized point cloud. This can be

done, for example, using Marching Cubes algorithm, Delaunay Triangulations,

or Alpha Shapes.

4364 Miller Gómez-Mora and Rocío Rodriguez-Guerrero

Partition of the mesh in homeomorphic patches to disks. These patches are also

known as maps (charts). The surface mesh partition becomes mandatory when

the surface is closed or has cavities (𝑔𝑒𝑛𝑢𝑠 > 0).

A local parameterization is built for each mesh patch. These local

parameterizations are performed together continuously, so that they collectively

form a globally continuous parameterization of the mesh.

Once the parameterization step has been completed, which generates a

collection of parameter pairs (𝑢𝑖, 𝑣𝑖) associated with points (𝑥𝑖, 𝑦𝑖 , 𝑧𝑖) of the

cloud, the problem of surface adjustment follows.

Surface adjustment consists of minimizing distance between each point (𝑥𝑖, 𝑦𝑖, 𝑧𝑖)

and its corresponding point 𝐹(𝑢𝑖 , 𝑣𝑖) on the surface. The standard approach of

surface adjustment is reduced to the following minimization problem:

min ∑ ∥ 𝐱𝑖 − 𝐹(𝑢𝑖 , 𝑣𝑖) ∥2

𝑖

Where 𝐱𝑖 is the i-th point of the input point cloud (𝑥𝑖, 𝑦𝑖, 𝑧𝑖), and ∥⋅∥ is the

Euclidean distance between 𝐱𝑖 and the corresponding point 𝐹(𝑢𝑖, 𝑣𝑖) on the surface

in the linear space of functions mentioned previously. The objective function of this

minimization problem is the square Euclidean norm whose calculation can be easily

done by least squares method.

3.3 Simplicial surfaces

Algorithms for surface reconstruction based on computational geometry surfaces

consist of two steps and end up having a spatial partition composed of tetrahedrons:

First, Delaunay triangulation (or Voronoi diagram) is constructed from a point

cloud, which consists of a partition of the convex hull of these points into a

finite set of tetrahedrons.

Once triangulation of the point cloud has been completed, all that remains is to

identify and extract triangles (simplexes) that belong to the surface.

Therefore, reconstruction of a simplicial surfaces consists in finding the sub-plot

of Delaunay triangulation of the initial set of points [5]. Identification of triangles

on the surface varies from one algorithm to another.

3.4 Implicit surfaces

Implicit functions given as the zero-level set 𝑑 = 0 of an implicit function

𝑓(𝑥, 𝑦, 𝑧) are usually sampled in an underlying grid, where the reconstructed

surface is found through an isocontour with an appropriate isovalue. For a regular

grid, MC algorithm can be used [9], [10]. To adapt the resolution of the grid to a

point sampling density other methods use grids such as octrees [11], [13] and

adaptive 3D triangulations [15].

3D modeling of natural heterogeneous objects 4365

4 Geometric modeling of natural heterogeneous objects

Modeling natural heterogeneous objects from segmented medical images can be

done directly from segmented medical images using some method that extends MC

algorithm to multiple regions and refine the mesh to obtain the desired quality [16].

However, these methods may suffer some problems such as ambiguity in labeling,

and topological. As result inconsistent meshes can be created. Several solutions to

these problems have been proposed, including repairing of ambiguous voxels

before extracting meshes [17], subdivision of domain [18] and assignment and

interpolation of probabilities into vertices to extract topologically correct surface

elements [19].

Another option for modeling heterogeneous objects from medical images is to

model independently parts that make up the heterogeneous object and then apply

an assembly method. In this case, the heterogeneous object Ω can be described as

Ω = ⋃ 𝛺𝑖𝑛𝑖=1 , where Ω𝑖 is the ith region belonging to Ω and n is the number of

regions (see Figure 5-a)). This means that the object forms a space partition.

a) b)

Figure 5. a) Artificial heterogeneous object composed of 3 regions Ω𝑖. b) A

human liver together with two liver tumors embedded in it.

Based on the idea of spatial partitions, some researchers [20], [21], [22], [23]

propose several assembly strategies to model heterogeneous structures, all of them

based on computer-aided design software (CAD). However, creation of CAD-based

3D models is a completely manual process making the modeling of heterogeneous

objects extremely difficult, mainly due to irregular geometries of both the

heterogeneous structures being modeled and their constituent parts.

An alternative is to use implicit functions [24]. This is done by isolating each

region first, then extracting a cloud of points from each region using the Marching

Cubes algorithm, and finally constructing an implicit representation from the point

clouds from which the surface can be extracted. The RAM method (Region-aware

modeling) described in [24] organizes implicit functions, which represent different

regions of the heterogeneous object, into a vector of implicit functions by means of

4366 Miller Gómez-Mora and Rocío Rodriguez-Guerrero

which the computational model is manipulated. This representation allows regions

to be considered as a whole instead of isolated parts of the heterogeneous object,

for example, in the model of Figure 5-b) liver tumors are modeled as an integral

part of a human liver.

5 Conclusions

In this paper, a general review on methods and algorithms to modeling surface

and volume of natural heterogeneous objects has been presented. One of the first

reference works on the problem of surface reconstruction from point clouds is the

one published by Hoppe [9]. Since then, surface reconstruction has become a

subject of active research in computer graphics and computer vision. Three main

approaches have been used to reconstruct surfaces from point clouds (see section

2): parametric surface, simplicial surfaces and implicit surfaces. Implicit methods

include RBF, MPU, and Poisson surface reconstruction. These methods do not

require a uniform tessellation of the function domain, and are known as non-mesh

methods or Galerkin methods.

Formulating the reconstruction of an implicit surface as a Poisson problem offers

a number of advantages. Many methods of implicit surface adjustment first divide

data into regions for local adjustment and then combine these local approximations

using fusion functions. By the contrary, Poisson reconstruction is a global solution

that considers all data points at once, without resorting to heuristic partitions and

combinations. Therefore, just like RBFs, Poisson reconstruction creates smooth

surfaces that adapt robustly to noisy data. But, while ideal RBFs are globally

supported and do not decay, Poisson reconstruction supports a hierarchy of local

support functions, and therefore their solution is reduced to a well-conditioned

dispersed linear system.

Acknowledgements. This work has been partially supported by the Universidad

Distrital Francisco José de Caldas under research project code. 3307857717.

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Received: September 3, 2018; Published: October 11, 2018