Erosion and Dilation on 2-D and 3-D Digital Images: A New Size-Independent Approach .

Abstract

This paper presents a new approach to achieve ele- mentary neighborhood operations on both 2D and 3D binary images by using the Extreme Vertices Model (EVM), a recent orthogonal polyhedra repre- sentation applied to digital images. The operations developed here are erosion and dilation. In contrast with previous techniques, this method do not use a voxel-based approach but deal with the inner sec- tions of the object. It allows to build an image size- independent algorithm. The proposed method also admits the use of structuring elements of arbitrary size and allows to treat 2D and 3D images in iden- tical way using the same algorithm.

Full text

Erosion and Dilation on 2D and 3D Digital Images: A new
size-independent approach
J. Rodriguez and D. Ayala
Universitat Polit`ecnica de Catalunya, Software Department (Departament de Llenguatges i Sistemes Inform`atics)
Av. Diagonal 647, 08028 Barcelona, Spain
Email: [jrodri,dolorsa]@lsi.upc.es
Abstract
This paper presents a new approach to achieve ele-
mentary neighborhood operations on both 2D and
3D binary images by using the Extreme Vertices
Model (EVM), a recent orthogonal polyhedra repre-
sentation applied to digital images. The operations
developed here are erosion and dilation. In contrast
with previous techniques, this method do not use a
voxel-based approach but deal with the inner sec-
tions of the object. It allows to build an image size-
independent algorithm. The proposed method also
admits the use of structuring elements of arbitrary
size and allows to treat 2D and 3D images in iden-
tical way using the same algorithm.
1 Introduction
A 3D digital image (or volume dataset) can be rep-
resented as a map , in such a
way that every point is assigned a value represent-
ing its color. In a binary image the image set is
. In [10] a 3D digital image is defined as an
union of voxels i.e. upright unit cubes whose ver-
tices have integer coordinates. Analogously, a 2D
digital image is defined as a union of pixels. In or-
der to generalize this term, we will call the elements
in a n-dimensional digital image n-voxels.
The most extensive class of binary images pro-
cessing operations is sometimes collectively de-
scribed as morphological operations [5]. This in-
cludes erosion and dilation which are the base of
most morphological operations such as opening,
closing or hit-or-miss transform. The operations are
used in several tasks such as elimination of small
spurious objects, smoothing of object boundaries,
This author has been supported by a grant from CONICIT
(VenezuelanCouncil of Research in Science and Technology)
object separation and elimination of small holes in
objects. All of the published algorithms for these
elementary operations have a voxel-based approach
and, therefore, their performance depends on the
size of the image.
In this work we propose a new approach to per-
form erosion and dilation operations using the Ex-
treme Vertices Model (EVM) as a representation
scheme of binary data. This representation model
allows to describe 2D and 3D binary images in a
very concise way and to build erosion and dilation
algorithms which are capable to process both 2D
and 3D digital images in identical way. Our pro-
posal admits the use of box-shaped structuring ele-
ments of arbitrary size. Finally, the algorithms de-
veloped have a performance which is independent
of the amount of n-voxels of the input set.
The section below surveys previous work to im-
prove erosion and dilation algorithms and intro-
duces coarsely our proposal. Then, in Section 3
we review the EVM, showing those definitions and
properties used in this work and discussing the abil-
ity and performance of the EVM as a concise de-
scription of digital images. The following section
describes our proposal for both erosion and dila-
tion operations, using the EVM. Finally, some ex-
perimental results, conclusions and future work are
presented.
2 Review of Erosion and Dilation
2.1 Erosion and dilation
Let A and B be sets in . Let de-
note the translation of B by x and let denote the
reflection of B with respect to its origin.
The erosion of A by B, , is defined as [5]:
(1)
VMV 2001 Stuttgart, Germany, November 21–23, 2001

which, in words, says that the erosion of A by B is
the set of all points x such that B, translated by x, is
contained in A.
The dilation of A by B, , is defined as [5]:
(2)
Thus, the dilation of A by B is the set of all x
displacements of the origin of such that and A
overlap by at least one nonzero element.
Set B is commonly referred to the structuring ele-
ment in erosion, dilation and in other morphological
operations. These operations can also be expressed
in terms of the Minkowski operators [11].
From the above definitions, erosion and dilation
can be described simply in terms of adding or re-
moving n-voxels from the binary image, according
to certain rules which depend on the pattern of the
neighboring n-voxels and the size and shape of the
structuring element. Therefore, erosion can be ex-
plained as a process where the interior of the object
is shrunk as much as the size of the structuring el-
ement, whereas dilation is a process that elongates
the interior of the object.
2.2 Related work
A number of different approaches to improve 2D
algorithms have been reported. All of them use
a more suitable representation of the image data
to improve the performance. Boomgaard and van
Balen [14] use a bitmap representation of binary im-
ages because the CPU operates on 32 pixels in par-
allel. Verwer and van Vliet [15] use a queue of the
contour pixels to avoid traversing all the image each
time. Parker [8] uses maps of distance and Young
et al. [16] used a run-length representation of the
image.
There are also 3D approaches. Zuiderveld [17]
proposes a method for successive dilation opera-
tions of 3D images by using a queue of contour
voxels, following the idea of Verwer and Vliet [15].
Nevertheless, this proposal demands anoverhead of
memory. Thurfjell et al. [12] proposes a method for
fast erode and dilate using the semiboundary repre-
sentation [13]. These two proposals only traverse
the contour n-voxels in the 3D data set. However,
they can not use structuring elements of arbitrary
size. Furthermore, the algorithms for 2D and 3D
data sets vary radically. In other words, they need
different algorithms for each case. Finally, in all of
these methods the cost, both in terms of memory
requirements and in terms of algorithmic efficiency,
increases when the binary data size increases. In
[7] the authors present an approach to improve the
efficiency of erosion and dilation algorithms. The
cost of these algorithms increases with square (cu-
bic) complexity with respect to the size of the struc-
turing element in the 2D (3D) case. Their approach,
then, reduces the complexity by decomposing the
structuring element into several smaller ones.
2.3 Our proposal
The conventional techniques and their improve-
ments perform the process using a voxel-based ap-
proach. In contrast, our proposal identifies the inner
sections of the object directly from the EVM rep-
resentation, then it operates (shrinks or elongates)
these sections and, afterwards, computes the new
EVM representation of the transformed object.
In contrast with previous implementations of ero-
sion and dilation, our method treats 2D and 3D im-
ages with the same algorithm. The algorithm is a
recursive process where the recursion is done over
the dimension. It decomposes the 3D image into 2D
sections which are themselves considered 2D im-
ages and, then, decomposes these 2D images into
1D sections. This is the base case of the recursion
and the algorithm actually performs 1-dimensional
transformations. Our approach admits the use of
box-shaped structuring elements of arbitrary size.
The visualization of the resulting object is as easy
as the visualization of the original one, with no
overhead of memory. Finally, the cost of these op-
erations is independent of the amount of n-voxels of
the input data set. Instead, it is associated with the
orthogonal shape of the object allowing to operate
large sets of co-planar adjacent n-voxels in a single
step. Moreover, our approach is alsoindependent of
the size of the structuring element.
3 Review of the EVM
Orthogonal polyhedra (OP) are polyhedra with all
their faces oriented in three orthogonal directions
and are also named rectangular or isothetic. Or-
thogonal Pseudo-polyhedra (OPP) will refer to reg-
ular and orthogonal polyhedra with a non-manifold
boundary. Figure 1 shows an OPP with a non-
manifold vertex and three non-manifold edges.
144

The Extreme Vertices Model (EVM) is a very con-
cise representation scheme in which any OP or OPP
can be described using only a subset of its vertices.
A digital image is geometrically analogue to an or-
thogonal polyhedron and can therefore be repre-
sented by the EVM. The EVM is actually a com-
plete (non-ambiguous) solid model: it is an implicit
boundary representation (B-Rep) model, i. e., all
the geometry and topological relations concerning
faces, edges and vertices of the represented OPP can
be obtained from the EVM.
In the following we review some definitions and
properties that are needed in this paper. All con-
cepts are more extensively explained and proved in
a previous paper [2].
3.1 Definitions
Let be an orthogonal pseudo-polyhedron (OPP).
Abrink is the maximal uninterrupted segment built
out of a sequence of collinear and contiguous two-
manifold edges of . The ending vertices of a brink
are called extreme vertices (EV). Figure 1 shows an
OPP with a brink from vertex A to vertex E (both
extreme vertices). In this brink, vertices B, C and D
are non-extreme vertices.
The EVM is a representation scheme in which
any OPP is described by its (an only its) set of EV.
Aplane of vertices () is the set of vertices ly-
ing on a plane perpendicular to a main axis of .A
slice is the region between two consecutive planes
of vertices. A section () is the resulting polygon
from the intersection between and an orthogonal
plane. Each slice has its representing section. Fig-
ure 1 shows an OPP with its planes of vertices and
sections perpendicular to the axis.
All these definitions can be extended to any di-
mension [4]. In this paper we are concerned with
dimension . Planes of vertices and sections
obtained from a 3D object are, then, 2D orthogo-
nal polygons. From them, we can obtain their 1D
lines of vertices and their 2D slices with their corre-
sponding 1D sections. Finally, lines of vertices and
1D sections are 1D objects which are composed of
one or several brinks.
In the EVM model the set of EV can be ordered
in six possible ways depending on the coordinate
values: XYZ, XZY, YXZ, YZX, ZXY, and ZYX.
In an XYZ ordered EVM, planes of vertices per-
pendicular to the X axis appear ordered from low
values to high values and, in each of them, lines
of vertices parallel to the Y axis appear also ordered
from low values to high values.
X
Y
Z
S1 S2
Plv1 Plv2 S3 S4 S5 S6 = φS0 = φPlv3 Plv4 Plv5 Plv6
a)
ABCD E
Figure 1: An OPP with a marked brink from vertex
A to vertex E. Its planes of vertices and sections
perpendicular to the X axis are shown in dark and
light grey respectively.
3.2 Properties of the EVM
The first property concerning the EVM is that co-
ordinate values of non-extreme vertices may be ob-
tained from EV coordinates.
The second property allow sections to be com-
puted from planes of vertices and vice-versa:
(3)
(4)
where and denote the projections of
and onto a main plane parallel to ,
is the number of planes of vertices and de-
notes the regularized XOR operation. Note that in
order to operate with the projections we need not
take into account the coordinate of the extreme ver-
tices that corresponds to the projecting plane.
Applying the definition of the operation, this
last equation can be expressed as:
(5)
145

and, thus, we can decompose any plane of vertices
into two terms that we will call forward difference
and backward difference (FD and BD for short).
(6)
(7)
The following property, based on and ,
guarantees that the correct orientation of all faces of
can be obtained from its EVM: is the set
of faces on whose normal vector points to
one side of the main axis perpendicular to ,
while is the set of faces whose normal vec-
tor points to the opposite side. Figure 2 shows an
OPP with its sections and with its FD and BD. This
property together with the second one (equations 3
and 4) provide proof that the EVM is a complete
B-Rep model.
Plv1 Plv2 Plv3 Plv4 Plv5 Plv6
b)
Figure 2: Sections are shown in dark grey, FD in
black (pointing to the right) and BD in light grey
(pointing to the left).
The following property concerns the XOR oper-
ation: Let and be two d-D (d 3) OPP, having
and as their respective mod-
els, then,
(8)
This property means that the XOR operation
works in 0D, because it is applied to the EV of
the model. Therefore, sections are obtained from
planes of vertices and vice-versa by applying the
XOR operation to the extreme vertices.
General Boolean operations between OPP can
be carried out by applying recursively the same
Boolean operation over the corresponding OPP sec-
tions. This algorithm is presented in [1] and con-
sists in a geometric merge between the EVM of
both operands which involve as a basic operation
the XOR between EV.
3.3 Conversion algorithms
In a previous work [9] a conversion algorithm from
a 3D digital image to the EVM is presented. This
algorithm consists in a traversal of the set of voxels
in the image detecting in a very simple way all the
EV, which are stored in the EVM.
The EVM is a B-Rep model which contains the
information of all the boundary in an implicit way.
Therefore, we can compute the boundary of an
EVM representad object from the EVM. In [3] a
conversion algorithm from the EVM to a hierar-
chical B-Rep is presented. A hierarchical B-Rep
contains the geometric information of all vertices
(coordinates) and faces (oriented normal vectors)
and the topological relations and
. The obtained B-Rep nodel
is composed by a relatively few large orthogonal
faces instead of a large number of little triangular
faces, as in marching cubes like approaches [6], or
a large number of little quadrangular faces, as in
other block-form approaches as the semiboundary
representation (see Figures 3, 4 and 5).
3.4 Performance of the EVM
EVM is a very concise model but, for storage
and transmission purposes, it can be further com-
pressed. In [9] several compression techniques
applied to EVM have been presented and evalu-
ated. We compared the EVM with the semibound-
ary representation (SB) [13] because SB was the
block-form representation that best performed as
we concluded from the consulted literature. Now
we present two tables that sum up the comparison
between EVM and SB showing that EVM is more
concise than SB. The models referred in these tables
are shown in Figures 3, 4 and 5.
Table 1 compares the SB and EVM models in
terms of memory (bytes).
Table 2 compares the number of faces obtained
when the EVM and the SB models are converted
to a B-Rep model. In this table also is shown the
number of triangular faces for the marching cubes
representation (MC).
146

Memory (bytes) SB EVM
boxes(16x16x10) 3438 720
skull(96x96x69) 105979 60924
ctHead(256x256x94) 453016 291136
Table 1: Memory comparisons
BRep(faces) MC-BRep SB-BRep EVM-BRep
boxes(16x16x10) 1349 1236 36
skull(96x96x69) 97467 49030 16407
ctHead(256x256x94) 336283 180288 54570
Table 2: Number of faces
4 Erosion and Dilation using the EVM
Let be the EVM representation of an
OPP, . Let and be the plane of ver-
tices and section of respectively , .
The erosion (dilation) of using the EVM is a
sequential process that, for each coordinate axis,
traverses all the planes of vertices of
and computes all its sections of ; then,
it shrinks (or elongates) these sections and recom-
putes the new planes of vertices obtaining the EVM
of the eroded (dilated) .
The shrink and elongate operations are recur-
sive processes which transform the sections only
when they become one-dimensional, i.e., a list of
collinear brinks. The shrink operation, Shrink(P),is
defined by:
where dim is the dimension of P, is the sec-
tion of P and Shrink-1D (P) consists in shrinking
each brink of the one-dimensional . The shrink-
Figure 3: SB-BRep and EVM-BRep of boxes
Figure 4: SB-BRep and EVM-BRep of skull
Figure 5: SB-BRep and EVM-BRep of ctHead
ing operation depends on the size of structuring ele-
ment. An identical recursive process can be defined
for Elongate(P).
Applying the properties described by equations 3
to 5 (see 3.2), it is clear that Shrink(P) recomputes
all the planes of vertices of P in terms of its trans-
formed sections. All of these recomputed planes
of vertices together define the eroded . Figure 6
illustrates the recursion for the erosion of a single
polyhedron in one of the three needed directions.
Figure 6: Single-direction erosion of a polyhedron
In this process the structuring element is simply
the measure to shorten (or elongate) the brinks at the
lowest dimension. The structuring element is not a
mask as in the voxel-based approaches. Instead, it
is one measure for each dimension of the object.
Thus, we can define structuring elements of arbi-
trary size and also, with different ratios for each di-
147

rection. Nevertheless, the structuring elements are
box-shaped.
The following algorithm shows this recursive
process for the erosion operation along one of
the main axes (the C axis). Dilation only differs
in the implementation of the 1D-transformation,
changing the call to Shrink-1D for Elongate-1D.
PROCEDURE C-shrink (INPUT p: EVM, dim: int, ratio: int
OUTPUT q: EVM)
VAR
plv, S0, S1, ShrunkS0, ShrunkS1, New-plv: EVM; nc: int
ENDVAR
IF (dim = 1) THEN
Shrink-1D (p, ratio, ShrunkS1);
ELSE
dim:= dim - 1;
S0:= ;
ShrunkS0:= S0;
DOplv:= ReadPlv (p, dim);
nc:= GetCoord (plv, dim);
S1:= S0 plv;
C-shrink (S1, dim, ratio, ShrunkS1);
New-plv:= ShrunkS0 ShrunkS1;
SetCoord (New-plv, nc);
PutPlv (New-plv, q, dim);
S0:= S1;
ShrunkS0:= ShrunkS1;
WHILE(NOT(EndEVM(p)))
ENDIF
ENDPROC
This algorithm uses the basic operations of the
EVM: ReadPlv (p, dim) returns the current plane of
vertices of p.PutPlv (p, q, dim) adds the plane or
line of vertices pto the EVM q.EndEVM () detects
the end of the model. The operation between two
sections produces a plane or line of vertices and be-
tween a section and a plane or line of vertices pro-
duces the next section (see 3.2). Finally,GetCoord
(p, dim) gets the constant coordinate of ,
which is a plane or a line of vertices, according to
its dimension, dim, and SetCoord (plv, nc) sets the
corresponding coordinate of as nc.
It is important to note that the recursive algorithm
above is called once for each ABC ordering (ABX,
ABY and ABZ) of the EVM. Thus, the trans-
formation of the 1D sections done by the proce-
dure Shrink-1D (or Elongate-1D), shrinks (or elon-
gates) only the corresponding C-coordinate each
time. Procedure Shrink-1D shrinks a 1D section
which consists of one or more brinks. All brinks
of this line are shortened in both extreme vertices
by the specified ratio and some of them may be-
come null. Alternatively, in the C-elongate proce-
dure, Elongate-1D, elongates all brinks and some
of them can merge (see Figure 7).
Figure 7: Procedures (a) Shrink-1D and (b)
Elongate-1D applied to a 1D section composed by
three brinks. In (a) the first brink becomes null and
in (b) the two last brinks are merged
5 Experimental Results
Now we present some results obtained with our
EVM oriented approach. Figure 8 shows an orig-
inal image with its corresponding eroded image and
Figure 9 shows a progressive erosion of a picture
(magallanes). Figure 10 shows another example of
erosion and dilation using a square structuring el-
ement (test2D) in which the erosion operation has
eliminated a thin part. Figure 11 presents a non-
uniform dilation of a CT slice called testCT. The di-
lation is applied only in the direction and, there-
fore, the object appears more and more broader in
each dilation step but its height remains unchanged.
Finally, Figure 12 shows dilation of the frontal part
of a skull obtained from CT data (skull).
Figure 8: Original image (above) with the corre-
sponding eroded one (below). In the middle, gray
pixels would be the eliminated pixels in a voxel-
oriented approach
148

Figure 9: Progressive erosion (Magallanes)
Figure 10: Erosion and dilation of test2D
Table 3 presents, for each data set, its resolu-
tion, the number of boundary n-voxels of the im-
age, the number of brinks within its EVM represen-
tation and the processing time for both erosion and
dilation operations. We can note that the process-
ing time increases only when the number of brinks
grows.
Table 4 shows a double-size copy of the former
data sets. For each one, we have obtained the EVM
representation again. We can note that the number
of boundary elements has increased a lot. However
the processing time remains almost the same be-
cause the number of brinks is almost the same too.
It proves that our algorithm’s performance is image
Figure 11: Non-uniform dilation testCT
Figure 12: Two views of the dilation of a skull ex-
tracted from a CT data set (skull)
Data set #bound #brinks Erosion Dilation
Magallanes
400x159 9692 1173 0.19 0.19
test2D
100x100 2516 14 0.01 0.01
testCT
96x96 2158 224 0.04 0.02
skull
96x41x69 36184 6839 12.39 13.42
Table 3: Processing time (in seconds)
size-independent.
6 Conclusions and future work
Previous authors have presented important im-
provements of the morphological operations ero-
sion and dilation by using suitable representation
of the data, but they must to use different schemes
to represent 2D and 3D images. Therefore, they
had to build different algorithms for each case. We
have developed an algorithm which deals with both
cases in identical way because of its recursive be-
havior. Our proposal, also admits the using of ar-
bitrary size structuring elements and works directly
on the EVM representation without using any aux-
iliary data structure. Furthermore, these operations
become image-size independent in our approach.
The EVM representation, in our experience, has
become a useful representation scheme for digital
149

Doubles #bound #brinks Eros. Dilat.
Magallanes
(767x318) 37124 1173 0.22 0.21
test2D
(200x200) 10064 14 0.01 0.01
testCT
(192x192) 8632 224 0.04 0.03
skull
(192x82x138) 187592 6859 13.9 14.78
Table 4: Processing time for double-sized images
2D and 3D images and the developed morphologi-
cal operations increase the potential of this model.
We hope, in the future, to complete this work by
studying operations as connected component label-
ing, thinning, detection of non-manifold zones and
holes and computing the genus of 2D and 3D digital
images, using the EVM model.
Acknowledgements
This work has been partially supported by a CICYT
grant TIC99-1230-C02-02.
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