EXPERIMENT

AL STUDY OF BOUNDING BOX ALGORITHMS

Darko Dimitrov

, Mathias Holst

, Christian Knauer

and Klaus Kriegel

Freie Universit

at Berlin, Institute of Computer Science

Takustraße 9, D-14195 Berlin, Germany

Universit

at Rostock, Institute of Computer Science

Albert Einstein Str. 21, D-18059 Rostock, Germany

Keywords:

Bounding Box Algorithms, Principal Component Analysis, Approximation Algorithms.

Abstract:

The computation of the minimum-volume bounding box of a point set in R

is a hard problem. The best known

exact algorithm requires O(n

) time, so several approximation algorithms and heuristics are preferred in prac-

tice. Among them, the algorithm based on PCA (Principal Component Analysis) plays an important role.

Recently, it has been shown that the discrete PCA algorithm may fail to approximate the minimum-volume

bounding box even for a large constant factor. Moreover, this happens only for some very special examples

with point clusters. As an alternative, it has been proven that the continuous version of PCA overcomes these

problems.

Here, we study the impact of the recent theoretical results on applications of several PCA variants in practice.

We give the closed form solutions for the case when the point set is a polyhedron or a polyhedral surface. To

the best of our knowledge, the continuous PCA over the volume of a 3D body is considered for the ﬁrst time.

We analyze the advantages and disadvantages of the different variants on realistic inputs, randomly generated

inputs, and specially constructed (worst case) instances. The results reveal that for most of the realistic inputs

the qualities of the discrete PCA and the continuous PCA bounding boxes are comparable. As it was expected

the discrete PCA versions are much faster, but behave bad on the clustered inputs. In addition, we evaluate

and compare the performances of several existing bounding box algorithms.

1 INTRODUCTION

Many computer graphics algorithms use bounding

boxes, as containers of point sets or complex objects,

to improve their performance. For example, bounding

boxes are used to maintain hierarchical data structures

for fast rendering of a scene or for collision detection.

Moreover, there are applications in shape analysis and

shape simpliﬁcation, or in statistics, for storing and

performing range-search queries on a large database

of samples.

A minimum-area bounding box of a set of n points in

can be computed in O(n logn) time, for example

with the rotating caliper algorithm (Toussaint, 1983).

(O’Rourke, 1985) presented a deterministic algo-

rithm, an elegant extension of the rotating caliper ap-

proach, for computing the minimum-volume bound-

ing box of a set of n points in R

. His algorithm re-

quires O(n

) time and O(n) space. Besides the high

run time, this algorithm is very difﬁcult to implement,

and therefore, its main contributions are more of the-

oretical interest.

(Barequet and Har-Peled, 2001) have contributed two

(1+ε)-approximation algorithms for computing the

minimum-volume bounding box for point sets in R

both with nearly linear time complexity. The run-

ning times of their algorithms are O(n + 1/ε

4.5

) and

O(nlogn + n/ε

), respectively. Although the above

mentioned algorithms have guaranties on the quality

of the approximation and are asymptotically fast, the

constant of proportionality hidden in the O-notation is

quite big, which makes them unpractical. An excep-

tion is a simpliﬁed variant of the second algorithm of

Barequet and Har-Peled that is used in this study.

Numerous heuristics have been proposed for com-

puting a box which encloses a given set of points.

The simplest heuristic is naturally to compute the

axis-aligned bounding box of the point set. Two-

Dimitrov D., Holst M., Knauer C. and Kriegel K. (2008).

EXPERIMENTAL STUDY OF BOUNDING BOX ALGORITHMS.

In Proceedings of the Third International Conference on Computer Graphics Theory and Applications, pages 15-22

DOI: 10.5220/0001096600150022

 SciTePress

dimensional variants of this heuristic include the well-

known R-tree, the packed R-tree (Roussopoulos and

Leifker, 1985), the R

∗

-tree (Beckmann et al., 1990),

the R

-tree (Sellis et al., 1987), etc. Further heuristics

of computing tight ﬁtting bounding boxes are based

on simulated annealing, or other optimization tech-

niques, for example Powell’s quadratic convergent

methods (Lahanas et al., 2000).

A frequently used heuristic for computing a

bounding box of a set of points is based on princi-

pal component analysis. The principal components

of the point set deﬁne the axes of the bounding box.

Once the axis directions are given, the spread of the

bounding box is easily found by the extreme values

of the projection of the points on the corresponding

axis. Two distinguished applications of this heuris-

tic are the OBB-tree (Gottschalk et al., 1996) and the

BOXTREE (Barequet et al., 1996). Both are hier-

archical bounding box structures which support efﬁ-

cient collision detection and ray tracing. Computing a

bounding box of a set of points in R

and R

by PCA

is simple and requires linear time. The popularity of

this heuristic, besides its speed, lies in its easy im-

plementation and in the fact that usually PCA bound-

ing boxes are tight ﬁtting. Recently, (Dimitrov et al.,

2007b) presented examples of discrete points sets in

the plane, showing that the worst case ratio of the vol-

ume of the PCA bounding box to the volume of the

minimum-volume bounding box tends to inﬁnity (see

Figure 1 for an illustration in R

). It has been shown

in (Dimitrov et al., 2007a) that the continuous PCA

version on convex point sets in R

guarantees a con-

stant approximation factor for the volume of the re-

sulting bounding box. However, in many applications

this guarantee has to be paid with an extra O(n logn)

run time for computing the convex hull of the input

point set.

In this paper, we study the impact of the rather the-

oretical results above on applications of several PCA

variants in practice. We analyze the advantages and

disadvantages of the different variants on realistic in-

puts, randomly generated inputs, and specially con-

structed (worst case) instances. The main issues of

our experimental study can be subsumed as follows:

• The traditional discrete PCA algorithm works

very well on most realistic inputs. It gives a bad

approximation ratio on special inputs with point

clusters.

• The continuous PCA version can not be fooled by

point clusters. In practice, for realistic and ran-

domly generated inputs, it achieves much better

approximations than the guaranteed bounds. The

only weakness arises from symmetries in the in-

put.

• To improve the performances of the algorithms

we apply two approaches. First, we combine the

run time advantages of PCA with the quality ad-

vantages of continuous PCA by a sampling tech-

nique. Second, we introduce a postprocessing

step to overcome most of the problems with spe-

cially constructed outliers.

The paper is organized as follows: In Section 2,

we review the basics of the principal component anal-

ysis. We also consider the continuous version of PCA

and give the closed form solutions for the case when

the point set is a polyhedron or a polyhedral surface.

To the best of our knowledge, this is the ﬁrst time that

the continuous PCA over the volume of the 3D body

has been considered. A few additional bounding box

algorithms and the experimental results are presented

in Section 3. The conclusion is given in Section 4.

2 PCA

The central idea and motivation of PCA (Jolliffe,

2002) (also known as the Karhunen-Loeve transform,

or the Hotelling transform) is to reduce the dimen-

sionality of a point set by identifying the most sig-

niﬁcant directions (principal components). Let X =

,...,x

}, where x

is a d-dimensional vector,

and c = (c

,...,c

) ∈ R

be the center of gravity

of X . For 1 ≤ k ≤ d, we use x

to denote the k-th

coordinate of the vector x

. Given two vectors u and

v, we use hu,vi to denote their inner product. For any

unit vector v ∈ R

, the variance of X in direction v is

var(X,v) =

∑

i=1

−c , vi

. (1)

The most signiﬁcant direction corresponds to the unit

vector v

such that var(X, v

) is maximum. In gen-

eral, after identifying the j most signiﬁcant directions

= {v

,...,v

}, the ( j + 1)-th most signiﬁcant di-

rection corresponds to the unit vector v

j+1

such that

var(X,v

j+1

) is maximum among all unit vectors per-

pendicular to v

,...,v

It can be veriﬁed that for any unit vector v ∈ R

var(X,v) = hCv, vi, (2)

where C is the covariance matrix of X. C is a sym-

metric d × d matrix where the (i, j)-th component,

i j

,1 ≤i, j ≤ d, is deﬁned as

i j

∑

k=1

−c

)(x

−c

). (3)

The procedure of ﬁnding the most signiﬁcant direc-

tions, in the sense mentioned above, can be formu-

lated as an eigenvalue problem. If χ

> χ

> ···> χ

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

1stP C

2ndP C

1stP C

2ndP C

Figure 1: Four points and its PCA bounding-box (left).

Dense clusters of additional points signiﬁcantly affect the

orientation of the PCA bounding-box (right).

are the eigenvalues of C, then the unit eigenvector

for χ

is the j-th most signiﬁcant direction. All

s are non-negative and χ

= var(X, v

). Since the

matrix C is symmetric positive deﬁnite, its eigenvec-

tors are orthogonal. If the eigenvalues are not dis-

tinct, the eigenvectors are not unique. In this case,

for eigenspaces of dimension bigger than 1, the or-

thonormal eigenvector basis is chosen arbitrary. How-

ever, distinct eigenvalues can be achieved by a slight

perturbation of the point set. Since bounding boxes

of a point set P (with respect to any given orthog-

onal coordinate system) depend only on the convex

hull CH(P), the construction of the covariance ma-

trix should be based only on CH(P) and not on the

distribution of the points inside. Using the vertices,

i.e., the 0-dimensional faces of CH(P) to deﬁne the

covariance matrix C a bounding box BB

pca(d,0)

(P) is

obtained. Let λ

d,0

(P) denote the approximation factor

for the given point set P ⊆R

and let

d,0

= sup

d,0

(P) | P ⊆ R

,Vol(CH(P)) > 0

the approximation factor in general. The example in

Figure 1 shows that λ

2,0

(P) can be arbitrarily large if

the convex hull is nearly a thin rectangle, with a lot of

additional vertices in the middle of the two long sides.

This construction can be lifted into higher dimensions

that gives a general lower bound, namely λ

d,0

= ∞

for any d ≥2. To overcome this problem, one can ap-

ply a continuous version of PCA taking into account

the dense set of all points on the boundary of CH(P),

or even all points in CH(P). In this approach X is a

continuous set of d-dimensional vectors and the coef-

ﬁcients of the covariance matrix are deﬁned by inte-

grals instead of ﬁnite sums. The computation of the

coefﬁcients of the covariance matrix in the continuous

case can be done also in linear time, thus, the overall

complexity remains the same as in the discrete case.

2.1 Continuous PCA

Variants of the continuous PCA, applied on trian-

gulated surfaces of 3D objects, were presented by

(Gottschalk et al., 1996), (Lahanas et al., 2000) and

(Vrani

c et al., 2001). In what follows, we brieﬂy re-

view the basics of the continuous PCA in a general

setting.

Let X be a continuous set of d-dimensional vec-

tors with constant density. Then, the center of gravity

of X is

c =

x∈X

xdx

x∈X

. (4)

Here,

dx denotes either a line integral, an area inte-

gral, or a volume integral in higher dimensions. For

any unit vector v ∈ R

, the variance of X in direction

v is

var(X , v) =

x∈X

hx −c,vi

x∈X

. (5)

The covariance matrix of X has the form

C =

x∈X

(x −c)(x −c)

x∈X

, with (6)

i j

∈

−c

)(x

−c

)dx

x∈X

, (7)

where x

and x

are the i-th and j-th component of the

vector x, and c

and c

i-th and j-th component of the

center of gravity. The procedure of ﬁnding the most

signiﬁcant directions, can be also reformulated as an

eigenvalue problem.

For point sets P in R

we are especially inter-

ested in the cases when X represents the boundary

of CH(P), or all points in CH(P). Since the ﬁrst

case corresponds to the 1-dimensional faces of CH(P)

and the second case to the only 2-dimensional face of

CH(P), the generalization to a dimension d > 2 leads

to a series of d −1 continuous PCA versions. For a

point set P ∈ R

, C(P,i) denotes the covariance ma-

trix deﬁned by the points on the i-dimensional faces

of CH(P), and BB

pca(d,i)

(P), denotes the correspond-

ing bounding box. The approximation factors λ

d,i

(P)

and λ

d,i

are deﬁned as

d,i

(P) =

Vol(BB

pca(d,i)

(P))

Vol(BB

opt

(P))

, and

d,i

= sup

d,i

(P) | P ⊆ R

,Vol(CH(P)) > 0

In (Dimitrov et al., 2007b), it was shown that λ

d,i

∞ for any d ≥ 4 and any 1 ≤ i < d −1. This way,

there remain only two interesting cases for a given d:

the factor λ

d,d−1

corresponding to the boundary of the

convex hull, and the factor λ

d,d

corresponding to the

full convex hull.

The following lower and upper bounds on the

quality of the PCA bounding boxes were shown in

(Dimitrov et al., 2007a) and (Dimitrov et al., 2007b).

Theorem 2.1

• λ

3,2

≥ 4 and λ

3,3

≥ 4.

EXPERIMENTAL STUDY OF BOUNDING BOX ALGORITHMS

• If d is a power of two, then λ

d,d−1

≥

√

and

d,d

≥

√

• λ

2,1

≤ 2.737.

• λ

2,2

≤ 2.104.

• λ

3,3

≤ 7.72.

The thorough tests on the realistic and synthetic

inputs revealed that the quality of the resulting bound-

ing boxes was better than the theoretically guaranteed

quality.

2.2 Evaluation of the Expressions for

Continuous PCA

Although the continuous PCA approach is based on

integrals, it is possible to reduce the formulas to ordi-

nary sums if the point set X is a polyhedron or a poly-

hedral surface. Due to space limitation, we present

here only the closed formulas for a convex polytope

and leave the polyhedral surface case and correspond-

ing cases in R

to the extended version of the paper.

Continuous PCA over a convex polytope. Let

X be a convex polytope in R

. We assume that the

boundary of X is triangulated (if it is not, we can

triangulate it in preprocessing). We choose an arbi-

trary point ~o in the interior of X , for example, we can

choose that ~o is the center of gravity of the bound-

ary of X. Each triangle from the boundary together

with ~o forms a tetrahedron. Let the number of such

formed tetrahedra be n. The i-th tetrahedron, with

vertices ~x

1,i

,~x

2,i

,~x

3,i

,~x

4,i

= ~o, can be represented in a

parametric form by

(s,t,u) =~x

4,i

+ s (~x

1,i

−~x

4,i

) +

t (~x

2,i

−~x

4,i

) + u (~x

3,i

−~x

4,i

), for 0 ≤ s,t, u ≤ 1, and

s + t + u ≤1.

The center of gravity of the i-th tetrahedron is

1−s

1−s−t

ρ(

(s,t))

(s,t)dudt ds

1−s

1−s−t

ρ(

(s,t))dudt ds

where ρ(

(s,t)) is a mass density at a point

(s,t).

Since, we can assume ρ(

(s,t)) = 1, we have

1−s

1−s−t

(s,t)dudt ds

1−s

1−s−t

du dt ds

1,i

+~x

2,i

+~x

3,i

+~x

4,i

The contributions of each tetrahedron to the center of

gravity of X is proportional to its volume. If M

the 3 ×3 matrix whose k-th row is ~x

k,i

−~x

4,i

, for k =

1...3, then the volume of the i-th tetrahedron is

= volume(Q

) =

|det(M

We introduce a weight to each tetrahedron that is pro-

portional with its volume, deﬁne as

∑

i=1

Then, the center of gravity of X is

~c =

∑

i=1

The covariance matrix of the i-th tetrahedron is

1−s

1−s−t

(

(s,t,u)−~c) (

(s,t,u)−~c)

du dt ds

1−s

1−s−t

du dt ds

∑

j=1

∑

k=1

(~x

j,i

−~c)(~x

k,i

−~c)

∑

j=1

(~x

j,i

−~c)(~x

j,i

−~c)

The element C

of C

, where a,b ∈{1, 2,3} is

∑

j=1

∑

k=1

j,i

−c

)(x

k,i

−c

∑

j=1

j,i

−c

)(x

j,i

−c

)

with ~c = (c

). Finally, the covariance matrix of

X is

C =

∑

i=1

We would like to note that the above expressions hold

also for a star-shape object, where ~o is the kernel of

the object.

3 EXPERIMENTAL RESULTS

We have implemented and integrated in our testing

environment a number of bounding box algorithms

for a point set in R

. The algorithms were imple-

mented using C++ and Qt, and tested on a Core Duo

2.33GHz with 2GB memory. Below we detail the al-

gorithms used in this study. The tests were performed

on real graphics models and synthetic data. The real

graphics models were taken from various publicly

available sources (Standford 3D scanning repository,

3D Cafe). The synthetic test data were obtained in

several manners (see Figure 2):

• uniformly generated point set on the unit sphere;

• randomly generated point set in the unit cube;

• randomly generated clustered point set in a box

with arbitrary spread.

To evaluate the inﬂuence of the clusters on the quality

of the bounding boxes obtained by discrete PCA, we

also generated clusters on the boundary of the real ob-

jects. The volume of a computed bounding box very

often can be ”locally” improved (decreased) by pro-

jecting the point set into a plane perpendicular to one

of the directions of the bounding box, followed by

computing a minimum-area bounding rectangle of the

projected set in that plane, and using this rectangle as

the base of an improving bounding box. This heuris-

tic converges to a local minimum. We encountered

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

Figure 2: Bounding boxes of four spatial point sets: a) real data (Igea model) b) randomly generated point set in the unit cube

c) uniformly generated point set on the unit sphere d) randomly generated clusters point set in a box with an arbitrary spread.

Figure 3: Extension of the example from Figure 1 in R

Dense collection of additional points (the red clusters) sig-

niﬁcantly affect the orientation of the PCA bounding-box

of the cuboid. The outer box is the PCA bounding box, and

the inner box is the CPCA bounding box.

many examples when the reached local minimum was

not the global one. Each experiment was performed

twice, with and without this improving heuristic. The

parameter #iter in the tables below shows how many

times the computation of the minimum-area bounding

rectangle was performed to reach a local minimum.

3.1 Evaluation of the PCA and CPCA

Bounding Box Algorithms

We have implemented and tested the following PCA

and continuous PCA bounding box algorithms:

• PCA - computes the PCA bounding box of a dis-

crete point set.

• PCA-CH - computes the PCA bounding box of

the vertices of the convex hull of a point set.

• CPCA-area - computes the PCA bounding box of

a polyhedral surface.

• CPCA-area-CH - computes the PCA bounding

box of the boundary of the convex hull of an ob-

ject.

• CPCA-volume - computes the PCA bounding

box of a convex or a star-shaped object.

We have tested the above algorithms on a large

number of real and synthetic objects. Typical samples

of the results are given in Table 1 and Table 2. Due

to space limitations, we give more detailed results for

some of the tested data sets in the extended version

of the paper. For many of the tested data sets, the

volumes of the boxes obtained by CPCA algorithms

were slightly smaller than the volumes of the boxes

obtained by PCA, but usually the differences were

negligible. However, the CPCA methods have much

larger running times due to computing the convex

hull. Some of the synthetic data with clusters justiﬁes

the theoretical results that favors the CPCA bounding

boxes over PCA bounding boxes. Figure 3 is a typi-

EXPERIMENTAL STUDY OF BOUNDING BOX ALGORITHMS

Figure 4: The dypiramid in the ﬁgure has two equal eigen-

values. a) The PCA bounding box and its top and side pro-

jections. b) The improved PCA bounding box and its top

and side projections.

cal example and indicates that the PCA bounding box

can be arbitrarily bad.

Table 1: Performance of the PCA bounding box algorithms

on a real data (Lucy: 262909 vertices, 525814 triangles, 622

convex hull vertices, 1240 convex hull triangles).

Lucy

algorithm volume #iter time[s]

PCA 756184 - 0.23635

improved 702004 2 2.90236

PCA-CH 736099 - 6.470

improved 704615 3 6.52645

CPCA-area 731496 - 0.533937

improved 692082 2 3.1634

CPCA-area-CH 726545 - 4.38215

improved 696356 2 4.47444

CPCA-volume 729131 - 5.31306

improved 699059 3 5.34954

As previously mentioned, for eigenspaces of dimen-

sion bigger than 1, the orthonormal basis of eigen-

vectors is chosen arbitrarily. This can result in un-

predictable and large bounding boxes, see Figure 4

for an illustration. We solve this problem by comput-

ing bounding boxes that are aligned with one principal

component. The other two directions are determined

by computing the exact minimum-area bounding rect-

angle of the projections of the points into a plane or-

thogonal to the ﬁrst chosen direction.

If the input is given as a (triangulated) surface,

then we can improve the run time of the PCA and

PCA-area methods, without decreasing the quality of

the bounding boxes, by sampling the surface and ap-

plying the PCA on the sampled points. Once the

principal components are determined, we compute

Table 2: Performance of the PCA bounding box algorithms

on the clustered point set with 10000 points.

clustered point set

algorithm volume #iter time[s]

PCA 31.3084 - 0.036038

improved 17.4366 6 0.285556

PCA-CH 33.4428 - 1.93812

improved 17.4593 9 2.18226

CPCA-area-CH 21.0176 - 1.5961

improved 17.4559 3 1.66884

CPCA-volume 19.4125 - 1.32058

improved 17.4591 5 1.39327

Table 3: Performance of the sampling approach on a real

data (Igea: 134345 vertices, 268688 triangles). The values

in the table are the average of the results of 100 runs of

the algorithms, each time with a newly generated sampling

point set.

Igea

algorithm #sampling pnts volume time[s]

PCA - 6.73373 0.189644

PCA-area - 6.70684 0.297377

PCA-sample 50 6.81354 0.122567

PCA-sample 100 6.6936 0.123895

PCA-sample 1000 6.69176 0.131753

PCA-sample 10000 6.70855 0.13825

PCA-sample 50000 6.70546 0.178306

PCA-sample 60000 6.70629 0.173158

PCA-sample 70000 6.70525 0.188299

the bounding box of the original surface. We do the

sampling uniformly, in the sense that the number of

the sampled points on the particular triangle is pro-

portional to the relative area of the triangle. Table

3 shows the performance of this sampling approach

(denoted by PCA-sample) on a real model. The re-

sults reveal that even for a small number of sampling

points, the resulting bounding boxes are comparable

with the PCA and CPCA-area bounding boxes. Also,

if the number of the sampling points is smaller than

half of the original point set the sampling approach is

faster than PCA approach.

3.2 Evaluation of other Bounding Box

Algorithms

Next, we describe a few additional bounding box al-

gorithms, whose performance we have analyzed.

• AABB - computes the axis parallel bounding box

of the input point set. This algorithm reads the

points only once and as such is a good reference

in comparing the running times of the other algo-

rithms.

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

• BHP - this algorithm is based on the (1 +

ε)-approximation algorithm from (Barequet and

Har-Peled, 2001), with run time complexity

O(nlog n + n/ε

). It is an exhaustive grid-base

search, and gives by far the best results among

all the algorithms. In many cases, that we were

able to veriﬁed, it outputs bounding boxes that are

the minimum-volume or close to the minimum-

volume bounding boxes. However, due to the ex-

haustive search it is also the slowest one.

• BHP-CH - same as BHP, but on the convex hull

vertices.

• DiameterBB - computes a bounding box based

on the diameter of the point set. First, (1 −ε) - ap-

proximation of the diameter of P that determines

the longest side of the bounding box is computed.

This can be done efﬁciently in O(n +

log

)

time. See (Har-Peled, 2001) for more details. The

diameter of the projection of P onto the plain or-

thogonal to longest side of the bounding box de-

termines the second side of the bounding box. The

third side is determined by the direction orthogo-

nal to the ﬁrst two sides. This idea is old, and can

be traced back to (Macbeath, 1951).

Note that DiameterBB applied on convex hull ver-

tices gives the same bounding box as applied on the

original point set.

Typical samples of the results are given in Table 4

and Table 5. Due to space limitation, we present more

results in the extended version of the paper.

Table 4: Performance of the additional bounding box algo-

rithms on a real data.

Lucy

algorithm volume #iter time[s]

AABB 789279 - 0.016606

improved 705152 3 4.92674

BHP 743677 - 3.25255

improved 705648 1 5.91975

BHP-CH 687723 - 8.63365

improved 687695 1 8.72115

DiameterBB 1504660 - 0.123361

improved 790190 4 4.89671

An improvement for a convex-hull method requires

less additional time than an improvement for a non-

convex-hull method. This is due to the fact that the

convex hull of a point set P in general has less than

|P| vertices. Once the convex hull in R

is computed,

it sufﬁces to project it to the plane of projection to ob-

tain the convex hull in R

. It should be observed that

the number of iterations needed for the improvement

of the AABB method, as well as its initial quality, de-

pends heavily on the orientation of the point set.

Table 5: Performance of the additional bounding box algo-

rithms on the clustered point set with 10000 points. The

results were obtained on the same point set as those from

Table 2.

clustered point set

algorithm volume #iter time[s]

AABB 30.2574 - 0.000624

improved 16.4563 7 0.247101

BHP 15.5662 - 3.13794

improved 15.5662 0 3.13794

BHP-CH 15.5662 - 3.13335

improved 15.5662 0 3.13345

DiameterBB 31.5521 - 0.013173

improved 16.6952 4 0.205163

4 CONCLUSIONS

In short, we can draw the following conclusions:

• The traditional discrete PCA algorithm can be

easily fooled by inputs with point clusters. In con-

trast, the continuous PCA variants are not sensi-

tive to the clustered inputs.

• The continuous PCA version on convex point sets

guarantees a constant approximation factor for the

volume of the resulting bounding box. However,

in many applications this guarantee has to be paid

with an extra O(n logn) run time for computing

the convex hull of the input instance. The tests on

the realistic and synthetic inputs revealed that the

quality of the resulting bounding boxes was better

than the theoretically guaranteed quality.

• For most of the real world inputs the qualities of

the discrete PCA and the continuous PCA bound-

ing boxes are comparable.

• The run time of the discrete PCA and continu-

ous PCA (CPCA-area) heuristics can be improved

without decreasing the quality of the resulting

bounding boxes by sampling the surface and ap-

plying the discrete PCA on the sampled points.

This approach assumes that an input is given as a

(triangulated) surface. If this is not a case, a sur-

face reconstruction must be performed, which is

usually slower than the computation of the con-

vex hull.

• Both the discrete and the continuous PCA are sen-

sitive to symmetries in the input.

EXPERIMENTAL STUDY OF BOUNDING BOX ALGORITHMS

• The diameter based heuristic is not sensitive to

clusters and can be used as an alternative to con-

tinuous PCA approaches.

• An improvement step, performed by computing

the minimum-area bounding rectangle of the pro-

jected point set, is a powerful technique that of-

ten signiﬁcantly decreases the existing bounding

boxes. This technique can be also used by PCA

approaches when the eigenvectors are not unique.

• The experiments show that the sizes of the bound-

ing boxes obtained by CPCA-area and CPCA-

volume are comparable. This indicates that the

upper bound of λ

3,2

, that is an open problem,

should be similar to that of λ

3,3

Future work includes obtaining closed form so-

lutions for the continuous PCA over non-polyhedral

objects. A practical and fast (1 + ε)-approximation

algorithm for the minimum-volume bounding box of

a point set in R

is also of general interest.

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