Using an Adaptive Invasion–based Model
for Fast Range Image Registration
Ivanoe De Falco
Antonio Della Cioppa
Domenico Maisto
ICAR–CNR
Via P. Castellino, 111
Naples, Italy
University of Salerno
Via Ponte Don Melillo, 1
Fisciano (SA), Italy
ICAR–CNR
Via P. Castellino, 111
Naples, Italy
[email protected] [email protected]
[email protected]
Umberto Scafuri
Ernesto Tarantino
ICAR–CNR
Via P. Castellino, 111
Naples, Italy
ICAR–CNR
Via P. Castellino, 111
Naples, Italy
[email protected]
[email protected]
ABSTRACT
Keywords
This paper presents an adaptive model for automatically
pair–wise registering range images. Given two images
and set one as the model, the aim is to find the best
possible spatial transformation of the second image causing
3D reconstruction of the original object. Registration is
effected here by using a distributed Differential Evolution
algorithm characterized by a migration model inspired
by the phenomenon known as biological invasion, and
by applying a parallel Grid Closest Point algorithm.
The distributed algorithm is endowed with two adaptive
updating schemes to set the mutation and the crossover
parameters, whereas the subpopulation size is assumed to
be set in advance and kept fixed throughout the evolution
process. The adaptive procedure is tied to the migration
and is guided by a performance measure between two
consecutive migrations. Experimental results achieved by
our approach show the capability of this adaptive method
of picking up efficient transformations of images and are
compared with those of a recently proposed evolutionary
algorithm. This efficiency is evaluated in terms of both
quality and robustness of the reconstructed 3D image, and
of computational cost.
Distributed Differential Evolution, adaptive control parameter setting, range image registration.
1.
INTRODUCTION
Range Image Registration (RIR) is a crucial task in
computer vision used for integrating information acquired
under diverse viewing angles (multi–view analysis). Over
the years, several multi–view RIR techniques have been
developed [21, 29, 17, 30] to tackle many practical
applications, such as 3D modeling ranging from medical
imaging, remote sensing, digital archeology, restoration of
historic buildings, virtual museum, artificial vision, reverse
engineering and computer–aided design (CAD) [27].
Since a physical object cannot be completely scanned with
a single image due to the limited field of view of a sensor,
a set of images taken from different positions is required
to supply the information needed to build the whole 3D
model. To avoid manually producing such a model by means
of error–prone CAD–based techniques, multiple images are
acquired by using range scanners [2] and joined together
by a registration algorithm. The registration strategy can
differ according to whether all range views of the objects
are registered at the same time (multi–view registration)
or only a pair of adjacent range images is processed in
every execution (pair–wise registration). This paper is
focused on the pair–wise registration of range images. As
a consequence, starting from two views, i.e., the model and
the scene, the objective of our registration process consists in
finding the best spatial transformation that, when applied to
the scene, aligns it with the model in a common coordinate
system.
Image registration is usually formulated as an optimization problem solved by iterative procedures. Among these,
examples are Iterative Closest Point (ICP) methods [28],
centered on the point–to–point and point–to–plane correspondences. However, as a drawback, the majority of these
iterative algorithms requires to provide either a rough or
a near–optimal prealignment of the images to avoid being
trapped in local optima. Unfortunately, an exhaustive exploration of the search space of all the candidate solutions
becomes impracticable in case of absence of constraints for
Categories and Subject Descriptors
I.2.8 [Artificial Intelligence]: Problem Solving, Control
Methods, and Search—heuristic methods; I.4.3 [Image
Processing and Computer Vision]: Enhancement—
Registration
Permission to make digital or hard copies of all or part of this work for
personal or classroom use is granted without fee provided that copies are not
made or distributed for profit or commercial advantage and that copies bear
this notice and the full citation on the first page. Copyrights for components
of this work owned by others than ACM must be honored. Abstracting with
credit is permitted. To copy otherwise, or republish, to post on servers or to
redistribute to lists, requires prior specific permission and/or a fee. Request
permissions from [email protected].
GECCO’14, July 12–16, 2014, Vancouver, BC, Canada.
Copyright 2014 ACM 978-1-4503-2662-9/14/07 ...$15.00.
http://dx.doi.org/10.1145/2576768.2598340.
1095
the possible transformations. Moreover, most of the computation time is spent in finding the closest point in the model
image to every point in the aligned image. Thus stochastic
optimization algorithms, such as Evolutionary Algorithms
(EAs) [1], capable of providing a solution acceptably close
to the global optimum in a reasonable time, have been successfully applied to complex real–world problems in computer vision and image registration [3, 8, 11, 12, 20, 26,
30]. Although these methods constitute an interesting alternative because they do not necessitate a good prealignment of the views to converge to the global optimum, they
require a careful tuning of many control parameters. The
setting of these parameters can strongly affect the performance of the algorithms [15] and is a very time–consuming
task. Therefore, automatic procedures for efficiently setting
these crucial control parameters for dealing with complex
optimization problems are highly recommended. Moreover,
the computation time to find the closest point in the model
image can be significantly reduced by applying a Grid Closest Point (GCP) transformation [37] before the registration
process.
With the aim to overcome the above drawbacks, within
this paper we examine the ability of a recently proposed
adaptive model for distributed Differential Evolution (DE)
[25] to perform automatic pair–wise image registration by
exploiting a distributed implementation of GCP without
any a priori knowledge of the pose of the views. The
distributed adaptive model is characterized by a migration
model inspired by the phenomenon known as biological
invasion [34] and is endowed with two updating schemes for
automatically setting DE control parameters. Hereinafter,
this adaptive algorithm is referred to as Adaptive Invasion–
based migration Model for distributed DE (AIM–dDE) [7].
The adaptive procedure occurs at each migration time
and is based on two steps. Firstly a local performance
measure related to the average fitness improvement for
each subpopulation is computed between two consecutive
migration steps. Secondly, a specific updating scheme
based on these measures takes place to update the control
parameter values of some subpopulations.
The evolutionary system has been tested by means of
a set of eight widely used 3D range images. To further
estimate its effectiveness, it is also compared with a
Self–adaptive Evolutionary Optimization algorithm, named
SaEvO, recently proposed by Santamaria et al. [33] for
facing RIR problems. The experimental results demonstrate
the ability of the proposed adaptive approach in facing these
RIR problems.
Paper structure is as follows: Section 2 describes the state
of the art. Section 3 contains a brief description of AIM–
dDE algorithm and illustrates the application of our system
to the registration task defining the encoding and the fitness
of the related optimization problem. Section 4 reports the
results achieved by our tool. The last section contains final
remarks and future works.
applied to the image registration problems. Differently
from methods based on the ICP algorithm, EAs can
work with images without any kind of prealignment. An
extensive review of evolutionary image registration methods
is reported in [30].
He and Narayana [16] propose a real coding scheme that
makes use of arithmetic crossover and uniform mutation
operators within an elitist generational model including
a restart mechanism. The evolutionary method uses a
real–coded Genetic Algorithm (GA) to estimate the rigid
transformation and a local search procedure to refine the
obtained preliminary solution.
Chow et al. [3] still propose the use of real–coded GA
considering a rigid transformation but introduce a crossover
operator that randomly selects the number of genes to be
swapped and a different sophisticated restart mechanism.
Silva et al. [35] face the pair–wise registration problem of
range images captured by 3D laser range scanner by means
of a parameter–based approach for rigid transformations.
The proposed technique is inspired to the steady–state
evolutionary scheme of GAs. Tournament selection, uniform
crossover, and random selection mutation operator are used
together with a hill–climbing algorithm in order to improve
the precision of the results.
In [20] a new method for the pair–wise registration is
introduced. The novelty is represented by the inclusion of a
degree of overlapping parameter in the solution vector and
the utilization of the trimmed square metric as objective
function to be minimized.
Cordón et al. [4] present a Scatter Search [18] EA
adopting a matching–based approach while Santamaria et
al. propose different memetic–based image registration
techniques to deal with 3D reconstruction of forensic objects
[31, 32].
As regards the adaptive approaches, De Falco et al. [10]
propose a novel adaptive updating scheme based on chaos
theory for setting the control parameters for a DE algorithm
to tackle pair–wise registration problem.
Moreover,
Santamaria et al. [33] illustrate a self–adaptive memetic
evolutionary optimization algorithm, named SaEvO, for
facing real–world RIR problems.
3.
ADAPTIVE INVASION–BASED MODEL
Among EAs, DE has proven fast and reliable to
face several multivariable optimization tasks in many
applications [6, 24]. As underlined in Section 1, one of
the ways to improve the DE chance to generate solutions
with higher quality is to devise suitable control parameter
adaptive strategies. Based on these considerations we
introduced AIM–dDE, a novel adaptive distributed model
endowed with updating schemes for setting the values of
mutation and crossover parameters.
Our model is different from the others in literature relative
to dDEs in the following aspects:
• the control parameter updating is strictly tied to the
migration process in order to balance exploration and
exploitation in the parameter space;
2. STATE OF THE ART
Several surveys on RIR are available in literature.
A complete study focused on pair–wise registration is
presented in [5], while different techniques for both pair–
wise and multi–view registration, and a new classification,
can be found in [29].
In the last two decades, EAs have been extensively
• the choice of the subpopulations which update the
parameter values is driven by a performance measure
related to the average fitness improvements of each
subpopulation between two consecutive migration
times;
1096
• the subpopulations involved by the parameter
changing are unknown a priori, in that they are
selected based on the performance feedback coming
from the same subpopulations.
eters are sampled from two independent sequences of
an uniform distribution U (0.1, 1.0). This distribution
has been used in the following each time a parameter
is randomly generated.
AIM–dDE is based on IM–dDE [9], a previous distributed
invasion–based model proposed by us, with the addition
of an adaptive strategy for the DE control parameters.
Therefore, in the next subsections we firstly report a
brief description of our IM–dDE, and then we present the
associated adaptive schemes.
2. At a generic invasion time t = kT , with k as above,
and before the new founding subpopulations have been
created on all the nodes, for each subpopulation the
local improvement ∆⟨Φp (t)⟩ of the average fitness
is evaluated as the difference between the average
fitness values after the last invasion and at the current
generation (for the first invasion time only, the initial
generation is taken into account as ‘last invasion’):
3.1 The IM–dDE algorithm
IM–dDE is a stepping–stone dDE using a migration
inspired by the biological invasion process instead of the
canonical exchange mechanism. The model makes use of
a locally connected topology where each node p hosts an
instance of a DE, and is connected to its neighboring nodes.
At each generation t, the subpopulation Pp (t) of each node
p performs a sequential DE until t equals tmax generations.
At every T generations, T being an algorithm parameter
named invasion interval, neighboring subpopulations
exchange individuals.
Hence, at a generic invasion
⌋} and tmax the
time t = kT , with k ∈ {1, . . . , ⌊ tmax
T
maximum number of generations, the set of individuals
each subpopulation Pp (t) sends to its neighbors, i.e., the
propagule, is indicated as MPp (t) and is determined by
collecting the individuals of Pp (t) which are fitter than its
current average fitness ⟨Φp (t)⟩:
∆⟨Φp (t)⟩ =⟨Φp (t − T )⟩ − ⟨Φp (t)⟩,
}
{
(2)
tmax
⌋ .
with t = kT , k ∈ 1, . . . , ⌊
T
3. According to a predefined updating scheme a set of
subpopulations is chosen and the values of their control
parameters are changed.
As concerns the step 3, several updating schemes can be
devised in order to select the set of subpopulations on which
the adaptive setting of control parameters has to take place.
To this end, we propose two updating schemes, i.e., RandAvg
and ChAvg.
• RandAvg. According to this scheme, the average of
all the local improvements, i.e., the global average
improvement ⟨∆⟨Φp (t)⟩⟩ is computed:
{
}
MPp (t) = xpi (t) ∈ Pp (t) Φ(xpi (t)) ≻ ⟨Φp (t)⟩
l
1∑
Φ(xpj (t)),
l j=1
{
}
tmax
t = kT , k ∈ 1, . . . , ⌊
⌋
T
with ⟨Φp (t)⟩ =
(1)
⟨∆⟨Φp (t)⟩⟩ =
N
1 ∑
∆⟨Φp (t)⟩,
N p=1
with t = kT , k ∈
where Φ(xpi (t)) is the fitness associated to the individual
xpi (t) and “≻” is a binary relation stating that the left
member is fitter than the right member.
All of the chosen individuals are sent to all of the
neighboring subpopulations Pp̃ (t), with p̃ belonging to the
set of neighbors of the node p. Then, at each invasion
∑ time,
each subpopulation receives a total number of p̃ MPp̃ (t) elements where MPp̃ (t) is the cardinality of the set MPp̃ (t) .
Afterwards, in each node, a founding subpopulation,
formed by both native and exogenous individuals, is
constructed by adding this massive number of invading
individuals to the local subpopulation.
Hence, the
founding subpopulation constitutes a source of heterogeneity
exploitable by the algorithm to improve its ability to search
new evolutionary paths.
Subsequently, a fitness–proportionate selection mechanism
is applied to the founding subpopulation to generate the new
subpopulation Pp′ (t).
{
}
tmax
1, . . . , ⌊
⌋ .
T
(3)
The control parameter values of each subpopulation
for which ∆⟨Φp (t)⟩ < ⟨∆⟨Φp (t)⟩⟩ are randomly
replaced. Then the control parameter values of all the
subpopulations are kept constant until next invasion
when the procedure takes place again.
• ChAvg. At each invasion time, the mutation and the
crossover parameter values of all the subpopulations
are generated according to a chaotic series. The
control parameter values of each subpopulation for
which ∆⟨Φp (t)⟩ < ⟨∆⟨Φp (t)⟩⟩ are updated as follows:
{
Ft+1
= µ · Ft · (1 − Ft )
(4)
CRt+1 = µ · CRt · (1 − CRt )
Then the control parameter values of all the
subpopulations are kept constant until next invasion
when the procedure takes place again. The ChAvg
scheme is based on one of the simplest and most
commonly used dynamic systems evidencing chaotic
behavior known as the logistic map [23] which
is described by the following quadratic recurrence
equation:
3.2 Adaptive Strategy
If, without loss of generality, minimization problems are
considered, the procedure on which the adaptive model is
based occurs in three steps as follows:
1. At the beginning of the evolution for each subpopulation random values for mutation and crossover param-
yt+1 = µ · yt · (1 − yt ), with t = 1, 2, 3, . . .
1097
(5)
transformation f ∗ achieving the best alignment of both
f (IS ) and IM based on the chosen similarity metric Φ to
optimize:
Algorithm 1 Pseudo-code of AIM–dDE on a generic node
p with p ∈ {1, ..., PN }
t=0
randomly
initialize
a
subpopulation
Pp (t)
=
{xp1 (t), · · · , xpl (t)}
evaluate the fitness Φ(xpi (t)), ∀i ∈ {1, . . . , l}
randomly initialize the values for F and CR from two
independent U (0.1, 1.0)
while halting conditions are not satisfied do
t=t+1
update the subpopulation Pp (t) using the evolutionary
operators of the DE
evaluate the fitness Φ(xpi (t)), ∀i ∈ {1, . . . , l}
if ((t mod T ) = 0) then
create the propagule MPp (t) for the current subpopulation
send the propagule MPp (t) to the neighboring subpopulations
receive the propagules MPp̃ (t) from the neighboring
subpopulations
construct the founding subpopulation Πp (t)
select l individuals in Πp (t) through a fitness–
proportionate selection mechanism
insert the l selected individuals into a new subpopulation
Pp′ (t)
replace the subpopulation Pp (t) with Pp′ (t)
set the values for F and CR according to the chosen
updating scheme
end if
end while
f ∗ = arg min Φ(IS , IM ; f )
f
(6)
Due to its robustness in presence of outliers (i.e., acquired
noisy range images), the similarity metric Φ usually
considered in 3D modeling is the median square error
(MedSE ) [27]. It can be formulated as:
Φ(IS , IM ; f ) = M edSE(d2i ), ∀i = {1, . . . , n}
(7)
where MedSE corresponds to the median value of all the
2
squared Euclidean distances, d2i = ||f (⃗
pi ) − p
⃗ ′j || (j =
1, . . . , m), between the transformed scene point, f (⃗
pi ), and
its corresponding closest point, p
⃗ ′j , in the model view IM .
To speed up the computation of the closest point the
GCP transform is used. It is important to remark here
that, although performed only once before the registration,
such a computation is the most time–consuming part of the
RIR problem. In fact, the amount of time needed for this
computation strongly depends on the discretization of the
physical space into which the object is, and on the number of
points composing the two images to register. In a sequential
environment this computation can last even hours, before
evolution can start and a suitable transformation is found.
Yet, the computation of the GCP can highly profit from
a distributed approach as the one we are proposing. In
fact, it can be easily and completely parallelized without
introducing any parallel overhead: each image can be
divided into slices along one of the axes, and each slice can
be assigned to a different core. In this way, this task can be
completed in an amount of time depending on the quantity
of hardware available, the higher the number of processing
units the higher the number of slices and, consequently, the
lower the computation time.
where µ is a positive constant sometimes known as
biotic potential. The behavior of the system is greatly
affected by the value of µ which determines whether
y stabilizes at a constant size, oscillates among a
limited sequence of sizes, or behaves chaotically in
an unpredictable pattern. A very small difference
in the initial value y1 causes large differences in the
long–time dynamic behavior [19]. Eq. (5) exhibits
chaotic dynamics with values within the range [0, 1]
at µ approximately 3.57. Beyond µ = 4 and y1 ̸=
0, 0.25, 0.50, 0.75, 1, the values eventually leave the
interval [0, 1] and diverge for almost all initial values.
4.
EXPERIMENTAL RESULTS
To investigate the behavior of AIM–dDE in the RIR field,
a set of benchmarks from the image repository [22] collected
at Signal Analysis and Machine Perceptron Laboratory
(SAMPL) at the Ohio State University has been taken into
account. From that repository, the following eight objects
have been considered: Angel, Bird, Buddha, Bunny, Duck,
Frog, Lobster, and Teletubby. For each of them the pair of
images taken at angles 0 and 40 degrees has been chosen as
exemplary instances for pair–wise RIR.
The parallel algorithms, which use the Message Passing
Interface (MPI) [36], are written in C language. All the
experiments have been carried out on an Apple iMac that is
equipped with an Intel Core i7 Quad–core, each core running
at 3.4 GHz.
Throughout the experiments we have made use of a
DE/rand /1/bin strategy. After a preliminary tuning, for the
whole experimental phase the parameter setting has been
chosen as follows: number of subpopulations N equal to 16,
size of each subpopulation l = 10, the value for the migration
interval T equal to 30. Finally, AIM–dDE has been executed
for each of the two updating schemes. For the logistic map,
we have used µ = 4 and y(1) =]0, 0.5[−{0.25}.
As concerns the number of generations for our distributed
algorithm, its exact value has been set after making a great
deal of careful consideration. In fact, to assess in a fair way
Depending on the specific scheme selected, we have two
algorithms for AIM–dDE, namely AIM–dDE–RandAvg and
AIM–dDE–ChAvg.
The pseudo–code of the AIM–dDE model for a generic
node, regardless of the specific updating scheme chosen, is
reported in Algorithm 1.
3.3 Encoding and Fitness
Given two input images, named scene IS = {⃗
p1 , . . . , p
⃗n }
and model IM = {⃗
p ′1 , . . . , p
⃗ ′m } with n and m points
respectively, RIR aims to find the best possible Euclidean
motion f for IS determined by the rotation Rθ of an angle
θ around an axis A and the translation ⃗t = (tx , ty , tz ).
Then the transformed points are denoted as: f (⃗
pi ) =
Rθ (⃗
pi ) + ⃗t, i = 1, . . . , n. Actually unit quaternions are
used to manage rotations in order to avoid singularities
and discontinuities, e.g. gimbal lock. In this notation
Rθ = (θ, Ax , Ay , Az ).
The pair–wise RIR problem can then be seen as a
numerical optimization problem in which solutions are
encoded as seven–dimensional vectors of real values x =
(θ, Ax , Ay , Az , tx , ty , tz ). The aim is to search the Euclidean
1098
Table 1: RIR results in terms of best and average values, and standard deviations obtained over 25 runs.
Algorithm
AIM–dDE–RandAvg
AIM–dDE–ChAvg
SaEvO
Φb
⟨Φ⟩
σΦ
Φb
⟨Φ⟩
σΦ
Φb
⟨Φ⟩
σΦ
Angel
0.5550
0.5622
0.0045
0.5524
0.5611
0.0049
0.3493
0.4983
0.2175
Bird
0.2274
0.2279
0.0005
0.2274
0.2280
0.0005
0.2028
0.4451
0.3052
Buddha
0.6571
0.6592
0.0046
0.6524
0.6541
0.0018
0.3990
0.6120
0.1224
the quality of the results achieved, in this paper we also
wish to compare our results against those obtained by the
best–so–far algorithm used in the literature to perform range
image registration over the same eight objects, i.e., SaEvO
described in [33]. The fitness function used in that paper
is the same as in ours, so comparisons in terms of averaged
final fitness values are sensible.
Unfortunately, in that paper the exact number of fitness
evaluations performed to achieve those results is not given,
rather their sequential algorithm is allowed an execution
time of 20 seconds for evolution over an Intel Pentium IV
2.6 GHz processor, and then evolution is stopped.
Therefore, to carry out an as–fair–as–possible comparison,
we need to carefully set for our distributed algorithm an
execution time for evolution that is equivalent to that of
their sequential one, and we should also take into account
the differences in the number and types of processors used
and in their clock rates.
Since we use four 3.4 Ghz cores, the following equality
has to hold: 20 · 2.6 = (4 · τ ) · 3.4 where τ is the
execution time for the evolution of our distributed program
over each core. This equality yields τ = 3.824 seconds.
Furthermore, we wish to run our distributed algorithm over
16 subpopulations, that means that each core should run
four subpopulations. Therefore, the actual time for the
evolution of each subpopulation is 0.956 seconds.
The images of the eight test objects contain different
numbers of points, ranging from a minimum of 5, 612 up to
a maximum of 14, 258. Depending on the different numbers
of points, different times would be needed to carry out the
computation of the quality of each registration proposed
by the evolutionary algorithm over the objects, so different
numbers of fitness evaluations could be carried out for the
different problems in the 0.956 seconds allowed.
Following [33] we randomly sample the points composing
the images by means of a uniform distribution, and choose
5, 000 points for each image, exactly as in that paper.
In this way, by running our distributed code we see that
over each core we can execute 30 generations for each of
the four there allocated subpopulations in a time equal
to 42.61 milliseconds. The number of 30 generations has
been chosen due to the value assigned to the migration
interval, so as to include in this time also that needed to
carry out migration. Therefore, in the time allowed to each
subpopulation, we can carry out a number of generations
equal to 0, 956 · 30/0.04261 = 673.08. This, multiplied by
the population size and by the number of subpopulations,
yields a total number of fitness evaluations equal to 107, 693.
Actually, we have decided to use a number of fitness
evaluations lower than the one computed above, because
we wish to take into account also other issues, including,
among others, the fact that the Intel Pentium IV used in
Bunny
0.2015
0.2025
0.0007
0.2015
0.2030
0.0008
0.0712
0.1800
0.2052
Duck
0.2245
0.2262
0.0012
0.2240
0.2261
0.0012
0.1585
0.2355
0.1135
Frog
0.2364
0.2399
0.0025
0.2362
0.2396
0.0026
0.2536
0.3991
0.1963
Lobster
0.3886
0.3949
0.0034
0.3926
0.3950
0.0021
0.2505
0.3787
0.1984
Teletubby
0.1723
0.1746
0.0022
0.1707
0.1732
0.0017
0.1050
0.1911
0.1667
the reference paper has several features that are no longer
up–to–date, which are reflected in an execution a bit slower
than the one that can be achieved with our more modern
processors. Therefore, we have limited that number to
100, 000. This latter figure results in 625 generations.
As regards the GCP, our distributed approach allows
strongly reducing the computation time to just some
minutes. As examples, for the image with the lowest number
of points, i.e., Teletubby, the sequential time for the GCP is
3,077 s, whereas the distributed is 87 s. The corresponding
times for the image with highest number of points, i.e.,
Buddha, are 8,489 s and 272 s respectively. This is a
noticeable by-product offered by a distributed approach such
as ours.
For each algorithm and for each problem Table 1 reports
the best final value Φb achieved in 25 runs, the average
value ⟨Φ⟩ over the 25 final values, and the related standard
deviation σΦ . With reference to each such index, the table
shows in bold the algorithm with the best value for each
problem.
As a first remark, the superiority of one algorithm with
respect to the others in terms of performance is not evident.
In fact, SaEvO outperforms the other methods in terms of
Φb , whereas the average results achieved by all the methods
are at first glance comparable. On the other hand, AIM–
dDE shows an evident superiority as far as σΦ is accounted.
In fact, on all the eight problems both the adaptive versions
of AIM–dDE show lower values for σΦ with respect to
SaEvO, meaning that the adaptive invasion model is very
robust. Nevertheless, it should be noted that the standard
deviation values shown by SaEvO are very high and of the
same magnitude of the average values.
As an example of the results obtained by AIM–dDE,
Fig. 1 shows the outcome for Angel object, while Fig.
2 shows the registered images. It is worth noting that
the differences between the aligned images by AIM–dDE–
RndAvg and AIM–dDE–ChAvg are barely perceivable as
evidenced by the numerical results.
4.1
Statistical analysis
To compare the algorithms from a statistical point of
view, a classical approach based on nonparametric statistical
tests has been carried out, following [13, 14]. To do so,
the ControlTest package [14] has been used. It is a Java
package freely downloadable at http://sci2s.ugr.es/sicidm/,
developed to compute the rankings for these tests, and
to carry out the related post–hoc procedures and the
computation of the adjusted p–values.
The results for the one–to–all analysis are reported in the
following. Table 2 contains the results of the Friedman,
Aligned Friedman, and Quade tests in terms of average
rankings obtained by the three algorithms. The last two
1099
Figure 1: Some examples of results for Angel problem. Top left: original model image at zero degrees.
Top right: original scene image at forty degrees. Bottom left: the best transformation of the scene image
achieved by AIM–dDE–RandAvg. Bottom right: the best transformation of the scene image achieved by
AIM–dDE–ChAvg.
Figure 2: Reconstructed objects. Left pane: AIM–dDE–RndAvg. Right pane: AIM–dDE–ChAvg.
the other algorithms, SaEvO is the second in Friedman
and Quade, whereas AIM–dDE–RandAvg is the second in
Aligned Friedman.
Furthermore, with the aim to examine if some hypotheses
of equivalence between the best performing algorithm and
the other ones can be rejected, the complete statistical
analysis based on the post–hoc procedures ideated by Holm,
Hochberg, Hommel, Holland, Rom, Finner, and Li has been
carried out following [14]. Moreover, the adjusted p–values
have been computed by means of [14].
Table 3 reports the results of this analysis performed at
a level of significance α = 0.05. The level of significance
represents the maximum allowable probability of incorrectly
rejecting a given null hypothesis. In our case, α = 0.05
means that if an equivalence hypothesis is rejected, there is
a 5% probability of making a mistake in rejecting it, so a
95% that we are correctly rejecting it.
Table 2: Rankings of the algorithms.
Algorithm
AIM–dDE–ChAvg
AIM–dDE–RandAvg
SaEvO
test statistic
p–value
Friedman
1.875
2.125
2.000
0.250
0.896
Aligned Friedman
12.000
12.250
13.250
5.968
0.051
Quade
1.861
2.083
2.056
0.082
0.921
rows show the statistic and the p–value for each test,
respectively. For Friedman and Aligned Friedman tests the
statistic is distributed according to chi–square with 2 degrees
of freedom, whereas for Quade test it is distributed according
to F–distribution with 2 and 14 degrees of freedom.
In each of the three tests, the lower the value for an
algorithm, the better the algorithm. AIM–dDE–ChAvg
turns out to be the best in all of the three tests. Among
1100
In this table the other algorithms are ranked in terms of
distance from the best performing one, and each algorithm
is compared against this latter to investigate whether or
not the equivalence hypothesis can be rejected. For each
algorithm each table reports the z value, the unadjusted p–
value, and the adjusted p–values according to the different
post-hoc procedures. The variable z represents the test
statistic for comparing the algorithms, and its definition
depends on the main nonparametric test used. In [14] all
the different definitions for z, corresponding to the different
tests, are reported. The last row in each sub–table contains
for each procedure the threshold value T h such that the
procedure considered rejects those equivalence hypotheses
that have an adjusted p–value lower than or equal to T h.
Summarizing the results of these tables, in each of
the three tests AIM–dDE–ChAvg is for all the post–hoc
procedures statistically better than the third classified
algorithm, whereas statistical equivalence hypothesis cannot
be rejected with the runner up.
[5] Dalley, G., Flynn, P.: Pair-wise range image
registration: a study in outlier classification.
Computer Vision and Image Understanding 87(1–3),
104–115 (2002)
[6] De Falco, I., Della Cioppa, A., Iazzetta, A., Tarantino,
E.: An evolutionary approach for automatically
extracting intelligible classification rules. Knowledge
and Information Systems 7(2), 179–201 (2005)
[7] De Falco, I., Della Cioppa, A., Maisto, D., Scafuri, U.,
Tarantino, E.: An adaptive invasion–based model for
distributed differential evolution. Information Sciences
(2014), http://dx.doi.org/10.1016/j.ins.2014.03.083
[8] De Falco, I., Della Cioppa, A., Maisto, D., Scafuri, U.,
Tarantino, E.: Satellite image registration by
distributed differential evolution. In: Lecture Notes in
Computer Science, vol. 4448, pp. 251–260.
Springer–Verlag (2007)
[9] De Falco, I., Della Cioppa, A., Maisto, D., Scafuri, U.,
Tarantino, E.: Biological invasion–inspired migration
in distributed evolutionary algorithms. Information
Sciences 207, 50–65 (2012)
[10] De Falco, I., Della Cioppa, A., Maisto, D., Scafuri, U.,
Tarantino, E.: Adding chaos to differential evolution
for range image registration. In: Lecture Notes in
Computer Science, vol. 7835, pp. 344–353.
Springer–Verlag (2013)
[11] De Falco, I., Della Cioppa, A., Maisto, D., Tarantino,
E.: Differential evolution as a viable tool for satellite
image registration. Applied Soft Computing 8(4),
1453–1462 (2008)
[12] De Falco, I., Maisto, D., Scafuri, U., Tarantino, E.,
Della Cioppa, A.: Distributed differential evolution for
the registration of remotely sensed images. In: 15th
EUROMICRO International Conference on Parallel,
Distributed and Network-based Processing. pp.
358–362. IEEE (2007)
[13] Dems̆ar, J.: Statistical comparisons of classifiers over
multiple data sets. Journal of Machine Learning
Research 7, 1–30 (2006)
[14] Derrac, J., Garcı́a, S., Molina, D., Herrera, F.: A
practical tutorial on the use of nonparametric
statistical tests as a methodology for comparing
evolutionary and swarm intelligence algorithms.
Swarm and Evolutionary Computation 1, 3–18 (2011)
[15] Eiben, A.E., Hinterding, R., Michalewicz, Z.:
Parameter control in evolutionary algorithms. IEEE
Transactions on Evolutionary Computation 3(2),
124–141 (1999)
[16] He, R., Narayana, P.A.: Global optimization of
mutual information: application to three–dimensional
retrospective registration of magnetic resonance
images. Comput. Med. Imag. Grap 26, 277–292 (2002)
[17] Kurogi, S., Nagi, T., Yoshinaga, S., Koya, H., Nishida,
T.: Multiview range image registration using
competitive associative net and leave-one-image-out
cross-validation error. In: Neural Information
Processing. pp. 621–628. Springer (2011)
[18] Laguna, M., Martı́, R., Martı́, R.C.: Scatter search:
methodology and implementations in C, vol. 24.
Springer (2003)
[19] Liu, J., Lampinen, J.: A fuzzy adaptive differential
evolution algorithm. Soft Computing-A Fusion of
5. CONCLUSIONS AND FUTURE WORKS
In this paper an adaptive distributed Differential Evolution technique based on invasions has been investigated to
optimize a 3D rigid transformation for automatic pair–wise
registration of range images without considering any previous knowledge of the pose of the view. A parallel version of
Grid Closest Point has been used to speed up the computation of the closest points. The experimental phase, carried
out on a set of benchmark range images, shows that our
AIM–dDE with the adaptive scheme based on chaos theory
is at least statistically equivalent the best–so–far algorithm
from literature in terms of performance, while it performs
better in terms of robustness of the reconstructed 3D image,
and of computational cost. If computational costs are considered, our approach has two positive features. Firstly, the
time for the computation of GCP has been strongly reduced,
which is very useful when large images are to be registered.
Secondly, the time for the evolution has been decreased from
20 s to 3.824 s.
Although promising, there is plenty of work still to do
to further evaluate the effectiveness of our system, and its
limitations as well.
Firstly, a wide tuning phase will be carried out to
investigate if other chaotic parameter settings for AIM–
dDE–ChAvg are, on average, more fruitful than others.
Moreover, the use of other chaotic maps will be investigated.
6. REFERENCES
[1] Bäck, T., Fogel, D., Michalewicz, Z.: Handbook of
Evolutionary Computation. IOP Publishing Ltd.
(1997)
[2] Campbell, R.J., Flynn, P.J.: A survey of free-form
object representation and recognition techniques.
Computer Vision and Image Understanding 81(2),
166–210 (2001)
[3] Chow, K.W., Tsui, T.: Surface registration using a
dynamic genetic algorithm. Pattern Recognition 37,
105–117 (2004)
[4] Cordón, O., Damas, S., Santamarı́a, J., Martı́, R.:
Scatter search for the point-matching problem in 3D
image registration. Informs J. on Computing 20(1),
55–68 (2008)
1101
Table 3: Results of post–hoc procedures for Friedman(top), Aligned Friedman (center), and Quade (bottom)
tests over all tools (at α = 0.05).
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
i
2
1
Th
Algorithm
AIM–dDE–RandAvg
SaEvO
z = (R0 − Ri )/SE
0.500
0.250
p
0.617
0.803
Holm/Hochberg/Hommel
0.025
0.050
0.025/–/0.025
Holland
0.025
0.050
0.025
Rom
0.025
0.050
0.025
Finner
0.025
0.050
0.025
Li
0.010
0.050
0.010
i
2
1
Th
Algorithm
SaEvO
AIM–dDE–RandAvg
z = (R0 − Ri )/SE
0.354
0.071
p
0.724
0.944
Holm/Hochberg/Hommel
0.025
0.050
0.025/–/0.025
Holland
0.025
0.050
0.025
Rom
0.025
0.050
0.025
Finner
0.025
0.050
0.025
Li
0.003
0.050
0.003
i
2
1
Th
Algorithm
AIM–dDE–RandAvg
SaEvO
z = (R0 − Ri )/SE
0.396
0.347
p
0.692
0.729
Holm/Hochberg/Hommel
0.025
0.050
0.025/–/0.025
Holland
0.025
0.050
0.025
Rom
0.025
0.050
0.025
Finner
0.025
0.050
0.025
Li
0.014
0.050
0.014
[31] Santamarı́a, J., Cordón, O., Damas, S., Aleman, I.,
Botella, M.: A scatter search-based technique for
pair-wise 3d range image registration in forensic
anthropology. Soft Computing-A Fusion of
Foundations, Methodologies and Applications 11(9),
819–828 (2007)
[32] Santamarı́a, J., Cordón, O., Damas, S., Garcı́a-Torres,
J.M., Quirin, A.: Performance evaluation of memetic
approaches in 3d reconstruction of forensic objects.
Soft Computing-A Fusion of Foundations,
Methodologies and Applications 13(8), 883–904 (2009)
[33] Santamarı́a, J., Damas, S., Cordón, O., Escámez, A.:
Self–adaptive evolution toward new parameter free
image registration methods. IEEE Transactions on
Evolutionary Computation 17(4), 545–557 (2013)
[34] Shigesada, N., Kawasaki, K.: Biological invasions:
theory and practice. Oxford University Press, USA
(1997)
[35] Silva, L., Bellon, O.R.P., Boyer, K.L.: Precision range
image registration using a robust surface
interpenetration measure and enhanced genetic
algorithms. IEEE Transactions on Pattern Analysis
27(5), 762–776 (2005)
[36] Snir, M., Otto S, Huss-Lederman, S., Walker, D.,
Dongarra, J.: MPI: The Complete Reference – The
MPI Core, vol. 1. MIT Press, Cambridge, MA, USA
(1998)
[37] Yamany, S.M., Ahmed, M.N., Heyamed, E.E., Farag,
A.A.: Novel surface registration using the grid closest
point (cgp) transform. In: Int. Conf. on Image
Processing. vol. 3, pp. 809–813. IEEE Press (1998)
Foundations, Methodologies and Applications 9(6),
448–462 (2005)
Lomonosov, E., Chetverikov, D., Ekárt, A.:
Pre-registration of arbitrarily oriented 3D surfaces
using a genetic algorithm. Pattern Recognition Lett.
27(11), 1201–1208 (2006)
Matsuda, T.: Log–polar height maps for multiple
range image registration. Computer Vision and Image
Understanding 87(1–3) (2009)
Ohio State University: SAMPL image repository,
http://sampl.eng.ohio-state.edu/ sampl/database.htm
(2009)
Peitgen, H.O., Jürgens, H., Saupe, D.: Chaos and
Fractals: New Frontiers of Science. Springer Verlag
(2004)
Price, K., Storn, R., Lampinen, J.: Differential
Evolution: A Practical Approach to Global
Optimization. Natural Computing Series,
Springer–Verlag (2005)
Price, K., Storn, R.: Differential evolution. Dr. Dobb’s
Journal 22(4), 18–24 (1997)
Robertson, C., Fisher, R.: Parallel evolutionary
registration of range data. Computer Vision and
Image Understanding 87(1), 39–50 (2002)
Rodrigues, M., Fisher, R., Liu, Y.: Introduction:
Special issue on registration and fusion of range
images. Computer Vision and Image Understanding
87, 1–7 (2002)
Rusinkiewicz, S., Levoy, M.: Efficient variant of the icp
algorithm. In: Third International Conference on 3–D
Digital Imaging and Modeling. pp. 145–152 (2001)
Salvi, J., Matabosch, C., Fofi, D., Forest, J.: A review
of recent range image registration methods with
accuracy evaluation. Image and Vision Computing
25(5), 578–596 (2007)
Santamarı́a, J., Cordón, O., Damas, S.: A
comparative study of state–of–the–art evolutionary
image registration methods for 3d modeling.
Computer Vision and Image Understanding 115(9),
1340–1354 (2011)
1102