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Imaging and Modelling from Serial Microscopic Sections for the Study of Anatomy

Authors:

Abstract

A system is considered for segmenting noisy intensity images and consequent three-dimensional object reconstruction from a set of planar contours. A new semi-automatic method for the extraction of contours from a sequence of cross-sectional images based on an active contour model (ACM) is proposed. The dynamic ACM proceeds along the sequence of cross-sections following a non-rigid motion, in accordance with the organ boundary. Image texture information is also employed in the model. Problems associated with topological reconstruction from planar contours are addressed, and several criteria promoting semi-automatic topological reconstruction are introduced. The proposed system is successfully applied to the processing of real data related to animal embryonic organs, proving that the system allows detailed modelling of irregular objects. The reconstructed models can be observed in wire-frame, solid, transparent or stereoscopic semi-transparent format. The human-computer interaction implemented in the procedure assists with problems of feature identification and object manipulation about an arbitrary axis.
lmaging
and
modelling
from
serial
microscopic
sections
for
the study of
anatomy
R. Durikovid* K. Kaneda H. Yamashita
Electric Machinery Laboratory, Faculty of Engineering, Hiroshima University, Japan
Abstract-A system
is considered
for
segmenting noisy intensity images
and consequent
three-dimensional object reconstruction
from a set of planar contours. A new semi-
automatic method for the extraction of contours from a seouence of cross-sectional
images based on an active contour model (ACM) ls proposed.
The dynamic ACM
proceeds
along
the sequence of cross-sections
following
a non-rigid motion, in accordance
with the organ boundary.
lmage texture
information is also employed in the model.
Problems
associated
with
topological
reconstruction
from
planar
contours are addressed,
and several criteria
promoting
semi-automatic topological reconstruction
are introduced.
The proposed
system is successfully applied to the processing
of real data related
to
animal embryonic organs,
proving
that the system allows detailed
modelling
of irregular
objects. The reconstructed models
can be observed
in wireJrame, solid,
transparent or
stereoscopic semi-transparent format. The human--computer
interaction implemented
in
the
procedure
assists with
problems
of feature
identification
and object
manipulation
about
an arbitrary
axis.
Keywords-Biological
structures,
Three-dimensional modelling, Topological reconstruction,
F
e atu re ide ntif i catio n
Med. Biol. Eng.
Comput.,
1998,36, 276-284
1 Introduction
ANAToMISTS oFTEN undertake the task of observing the shape
of a specific embryo organ contaminated by a test drug. The
primary source of information
during this observation
proce-
dure is a series of thin microscopic slices that render the inner
organs visible. For observing the overall shape of organs,
however, the slices
prove
rather limited, as
they provide
only
2D information.
Conversely, 3D models,
give more complete
information about the shape of the structure. Reconstruction of
a 3D surface or volume model from 2D information is there-
fore an indispensable technique for better understanding
the
shape and the position
of an organ inside an organism.
Although 3D reconstruction is widely used in computer
tomography (CT) and magnetic resonance imaging (MRI), the
methods do not fulfil all the needs in the field of anatomy.
Anatomists seek
information about the exact overall shape
and
try to ascertain the features that build it. The resolution in the
CT and MRI methods is not high enough for the process
of
scanning the small structures used in embryology (around
4.5 mm height in our applications). For this reason, manual
sectioning followed by serial microscopy scanning is used. In
anatomical fields that implement
serial microscopy, an image-
registration method is necessary.
However, the type of hard-
Correspondence should be addressed to Dr. Kaneda; email:
ki n@ e m l. h i rosh i m a- u. ac.j
p
*On leave from Department of Computer Graphics and lmage
Processing,
Faculty of Mathematics
and Physics,
Comenius
University,
Mlynskd dolina,
820'13 Bratislava, Slovakia.
First received
27 November 1995 and in final form 27 Januarv 1998
@ IFMBE:1998
276
ware used in the CT and MRI methods is designed
to minimise
the image-registration
process.
Moreover, in the case of CT
and MRI, voxel (volume) representation is used mainly
because
portion and volume are primary, whereas
surface
representation is most imporlant in dealing with the exact
shapes of anatomical structures.
2 Problems of 3D reconstruction and previous work
Several systems are available for reconstructing organs
from serial sections
(KRIETE
and Macoowsrr, 1990;
Tatsutr,rr
et al., 1990} STERECON is a system
(MARKo and LnttH,
1.992; PrmINs et al., 1993) with the capability of using
stereoscopic contour tracing with a digitising pad. Another
well-known package
based on manual contouring was devel-
oped at Columbia University (Ar-reN and Lrvrxrnar-, 1990).
Manual segmentation
through the process
of tracking organ
contour
lines via mouse and cursor is a time-consuming
and
exhausting
task. Segmentation
algorithms for automating this
procedure are therefore desirable, but conventional image-
processing
methods (Pnas and VBNBrsANopouLos, 1990)
using only local information are often inadequate.
2.1 Registration
The most
obvious solution in embryology
is to set common
external reference
marks in the paraffin block before section-
ing. Four
marks
can be made with a laser
near
the object
to be
reconstructed.
The marks
on each
section
then determine
the
transformation
between
successive
sections
calculated
by an
automatic image-registration
technique
(BnoN and GRslarr-
Medical
& Biological
Engineering & Computing May 1998
LEr, 1992).ln a system
developed
by CanlaoH.r
et a\. (1994),
electron
microscopy images
are aligned using an interactive
digital blink comparator.
As one image is held stationary, the
user
translates
and
rotates the other image,
with the stationary
and moving images
shown alternately
on a graphics
screen.
An illusion of movement
is created
when the images
become
misaligned,
and
the movement
is minimised
as the images
are
once
again brought into alignment.
2.2 Segmentation
In the work of Leylaarue (1990),
the movement
of living
cells
on a planar
surface is observed
by processing
an anima-
tron
sequence.
Snakes
(KASS
et al.,1987) are
used to track the
moving cell and to follow the deformations
that occur as
the
cell moves.
CARTBoM
et al. (1994] introduced
the methodol-
ogy of tracking boundaries of neural dendrites from serial
electron
microscopy.
Here, the snakes
are used in an inter-
active technique consistent with manual tracking methods.
This interactive
technique
cannot
proceed
automatically from
slice to slice, nor can it follow highly concave structures
or
those whose topology changes (branching or merging)
between
two successive
sections.
Unfortunately,
because
of the complexity of microscopic
images
of embryo
sections,
the aforementioned
techniques
are
successful
only in reconstructing
such higher contrast regions
as the outer skin of an embryo.
The techniques
do not appear
to be useful in inner
areas. The
aim ofthis paper
is therefore to
introduce a semi-automatic method that can track both the
boundary
and the topology
ofan organ in a section-by-section
fashion.
2.3 Reconstruction
and visualisation
The traces
obtained
can be displayed as
a set ofcontours or
tessellated
by a triangulation technique to form a surface that
can be displayed as a solid object (KeNEDA et al., 1987).
Before the surface can be formed, it is essential to know how
the contours are grouped together (the contours must be
assigned
to each
other) in the formation ofthe object topology,
a step called
topological reconstruction.
The simple techniques
use the geometric relationships
between two contours,
such as
the overlapping area or the distance between them. Two
contours are then assigned if their distance is less then the
predefined threshold (Murr-nn and KrrNcBnr, 1993). Addi-
tional information available is the topological position of
contours on the slice (nesting of contours), formally repre-
sented
by a tree of nesting
(SurNacewA
et al., l99l). The root
of the tree corresponds
to an imaginary outer contour (image
boundary) that envelops
all the other contours. The successors
of a vertex of the tree correspond
to the contours immediately
contained in the contour of that vertex.
Nevertheless,
the above methods fail to reconstruct holes,
which are considered
as a particular characteristic
of branch-
ing but which connect the contours with different topological
positions (different levels of nesting, see Fig. 4a) within the
slice. In our work. we extend the above ideas to automatic
topological reconstruction
of branches
and holes.
Supposing that the topology has been previously recon-
structed, then the next step to be solved is the problem of
filling the space
between
planar contours. The surface is most
often approximated by a set of triangles based on either a
global criterion, minimum surface area (Fucus et al., 1977),
maximum volume (KnerEI-, 1975)
or a local one, minimum
lengths
of edges
(CunrsrnNsBN
and Saonneenc, 1978).
For a global criterion, the problem of finding the optimum
solution is reduced
to a problem of finding the minimum cost
cycles in a directed toroidal graph. As global methods are
Medical & Biological Engineering
& Computing May 1998
unable to deal with branches,
the
multi-branching
problem
has
been treated (EKouLE et al., l99l; Csor and panr. 1994)
through
the introduction
of an intermediate
contour.
Another
approach
is based
on reduoing
the problem
to a 2D interpola_
tion, by building as many intermediate
sections
us n...irury
for a reconstruction
of the object
surface
(Or-rve
et at., 1996j.
Several methods using local criteria were irnplemented
because
they can be adapted to solve the branchiirg problern,
and their calculation
time is negligible compared
with global
methods.
However, they failed in some carefully designed
cases. The proposal
offered in this research
is therefore
the
most effective
choice
among the available
local methods.
3 Data acquisition
Using routine
manual
processing,
an embryo
was
sectioned
with an ultramicrotome at an average
thickness of between 7
and
30
pm. The sections were further processed,
stained
with
haematoxylin-eosin
and then mounted
on glass.
Photographs
of a certain magnification were taken by a camera
attached
to
an optical microscope.
The images were then stored into
compact discs after they had been converted into digital
form with a resolution
of 720 x 580 pixels.
To reconstruct
a 3D strucfure
from the series of cross-
sections, it is necessary
for all sections
to share
a common
reference
co-ordinate
system;
in other words,
they should be
registered.
The most obvious solution is to use common
external reference marks set into the paraffin block before
sectioning
(BRoN
and Grcuu-rET, 1992).
However,
because
some important image data used in our experiments were
recorded some time ago, when no reference marks had been
adopted,
a manual
image registration process
was employed.
3.1 Image registration
Sections from serial microscopy are not only translated or
rotated. Often they are deformed by the heating required to
prepare
the tissue for display on microscopic slices. Thus,
serial
microscopy
requires a combination
of linear transforma-
tion processes
to bring the sections into approximate
align-
ment. We therefore adopted a combination of two methods:
the landmark method (BnowN, 1992) given by a user at
distinctive points; and the colour combination method, used
to conform quickly if the results are acceptable. These
methods have been employed separately
in different systems
with limited success.
-Howevei theii combination brings
together reference knowledge about sections and a quick
preview of results.
A pair of associated
sections
were registered
by using two
landmarks
selected on each section
at a distinctive point, such
as the point of a heart or the tip of a limb bud, indicated by the
symbols
X and * in Fig. l. The vectors
belonging
to each
cross-section, defined by the landmarks, can be used to
determine the translation and rotation of sections simply by
being aligned. For preview pu{poses,
initially, one of the full-
colour sections was tinted in green,
and the other was tinted in
magenta (see
Figs. la and b). While the user manipulated the
landmarks,
the colours of the two sections
were combined to
show the overlapping regions in some shade
of grey, as shown
in Fig. lc. The resultant
colour differences
were instructive
with regard to moving the landmarks
to achieve better section
registration. To provide better interactions with landmarks, a
preview transformation of colour-marked images was con-
ducted, but without using an anti-aliasing method. After
landmarks had been selected
correctly, the final, transformed
image was calculated
by taking into account the anti-aliasing
277
Fig. I Registration of two cross-sections of mouse embryo. (a), (b) Static image and its successive image with selected landmarks (symbols X
and t) placed at neural tube and anterior tip of embryo. (c) Colour merging of transformed and static images. (d) Best registration
obtained
of image rotation
(see
Fig. ld). The series of sections can be
registered
by repeating the registration of image
pairs from top
to bottom, or vice versa.
4 Gontour extraction: the ACM model
We used a semi-automatic approach
that extends the well-
known active-contour model (KASS
et al., 1987) to produce a
method that tracks both the boundary and the topology of an
organ in section-by-section
fashion. The ACM provided sig-
nificant assistance to the user in the accurate location of
structure boundaries.
The sequence of slice images obtained
was generalised in a video, where each frame had a time
component. The user quickly traced a contour to approximate
roughly the strucfure boundary for a few key frames from a
sequence. The dynamic simulation then conformed the con-
tour to the true structure boundary and proceeded through the
entire time sequence to extract quickly a sequence of profiles
of the same structure.
An ACM is a deformable curve composed
of abstract elastic
materials minimising its potential energy. Consider an ACM
o(s, t) : (x(s,
i), y(s, r)), with a spatial parameter
s and section
number i defined on spaces
O and N, respectively. Let l(x,y)
denote the image intensifi at position (x,y). The potential
energy function of the ACM, originally defined in Kess er a/.
(1987), consists of two components. First, the deformation
energy is written as
E*,(u): w1ft)lo'(s)12
* w2(s)la"(s)12 (l)
where the first and second derivatives of contour position z;
with respect to a parameter s are denoted a' and o. The
weighting functions 4(s) control the tension and rigidify of
the contour over the space O. Secondly, the image energy
computed from image intensities l,(x,y) is written as
Enoea(o)
: Go x li{u) (2)
where Gor. denotes
the convolution with a smoothing Gaus-
sian filter of width o.
In our previous work lDumr<ovre et al., 1995), we solved
four problems not addressed in the earlier work of K-a,ss et al.
(1987) or CeRtBoN et al. (1994). First, we defined the area
274
energy Eo,"o, giving us a contour that tends to minimise its
delimited area against the length minimisation, as is found in
KASS
e/ al. (1987) and Te,Rzoporos
and MeraxRs (1991).
Secondly, the structures in our sections are expected
generally
to maintain average texture properties.
Therefore the problem
ofpoorly contrasting images is solved by a texture representa-
tion E,"rrr" of the image against the representation of the
image scale space only. Thirdly, the profile of a structure
that
consists of multiple regions is found using a contour-splitting
operation. Fourthly, the problem until now was to select model
parameters,
but, with our approach, the parameters
controlling
the elastic process can be set automatically, based on the
initial estimation of the structure boundary. The total energy
functional as proposed by Dururovte et al. (1995) is then
expressed by
tf
E,nokn: " l{Ei",@) * V{/1Ei*r"(a)
+ WrE,o,u,"1u1
z-l
O
* W1Eo,"o(u)lds (3)
The weight functions Wr, Wr and Wu control the image,
texture and area energy, respectively.
4.1 Contour operations
The most fundamental contour operations of adding and
deleting a single point are introduced- If the distance between
two neighbouring points is less or greater
than a given t}resh-
old, a single point from or between them is either deleted or
added" For the entire contour, dividing the contour into n parts
is proposed. Provided that the contour points are in a near-
equilibrium configuration with respect
to the internal energy,
the contour is divided into two parts using the segment
a,u,,
if
points z, and u, connected by vectors rii, rii, satisfu-the
followins conditions:
' lai
- oj1
- d*,
where dru, is a constant.
o The projections of vectors YEo,"o at points o, and o, onto
the vectors ru and 1r, respectively, have opposite direc-
tions (see
Fig. 2).
Medical & Biological Engineer-ng & Computing May 1998
Fig. 2 Conditions
for cutting ACM into two parts
Fig. 3 Automatic
tracking
of structure
boundary through
sequence
of sections
Moreover, a contour is divided recursively into n parts, if
n - 1 pairs of points obeying the above
conditions exist. Fig. 3
shows the significance of this property for speeding up the
tracing ofprofiles through the sequence ofslices ordered along
the time axis. The initialisation of a contour position for a
single (bottom) slice has been traced manually, whereas all
other contours were obtained automaticallv bv our ACM.
5 Data structure and topology
A hierarchical polygon data structure consists of objects,
sub-object surfaces, contours, and contour points. A simple
example is shown in Fig. 4a. To identify branches uniquely,
each contour line is associated with a sub-object surface. The
sub-object
surface is part of the structure
that includes neither
branch nor confluence. The object consists of several sub-
object surfaces,
but only the connections
between them create
the branching (see Fig. 46). Therefore, if a pair of contorrs
lying in the same
cross-section
belong to different sub-object
surfaces,
a branch must exit between the corresponding
cross-
sections.
A hole in the surface of the object is represented
as a
connected
branching (see Fig. 4). For example, representation
of a vein passing
through the hole is shown in Fig. 4b, where
the inner and outer surfaces of the vein form two different
objects Ob6 and, Ob6, each consisting only of a single sub-
object surface. On the other side, the object with the hole
consists
of five sub-object
surfaces
(Sos3,
...,,Sos7), forming
two branches
on the top and the bottom of the hole.
6 Automatic topological reconstruction
The problem of topological reconstruction is particularly
difficult if there are several contours on each
plane, and if it is
known that the coresponding sub-object
can have branches or
Medical & Biological
Engineering
& Computing May 1998
Sos, Sos,
Fig. 4 Hierarchical data structure and corresponding assignment
graphs. (a) Three objects reconstructed front four cross-
sectional images: Ob1 consisting of sub-object surfaces
Sos3,...,.Sos7;
Ob6 and Ob6, each consisting
of a single
sub-object surface Sosl and Sos2, respectively. (b) Assign-
ment graph for objects shown in (a), where solid lines
connect contours of a single sub-object surface, and
broken lines denote connection between
sub-obiect surfaces
forming hole
holes.
Both geometrical
and topological
relationships
between
two contours
are used to solve the problem of topological
reconstruction.
If there is a sufficient
number
of cross-sections
for geometrical reconstruction, the proposed topological
reconstruction
method also works well.
6.1 Assignment
graph
Let us consider a sequence
of parallel planes in space
consisting of a collection of non-intersecting, 2D contours.
The aim is to find the relationships
between the contours such
that they will group into the sub-object surfaces and, finally,
after determination of branches and holes, into the entire
object. The topological reconstruction creates an oulput in
the form of an assignment
graph. The vertices of the assign-
ment graph correspond to contours, and the edges connecting
the vertices form connected
parts of the object, as shown in
Fig. 4b. When considering the reconstruction between two
cross-sections,
three
different types
ofconnection are
possible.
These three comprise a connection between two contours (a
so-called
cylindrical connection), a connection between more
than two sub-object surfaces forming an open branch, and a
connection
between the sub-object surfaces forming a hole (a
connected
brarich see Fig. 4).
The proposed automatic construction of an assignment
graph can be summarised in two essential steps. First, the
nesting
tree (SnrNecAwA et al., l99l) was created
for each of
two successive sections. Secondly, pairs of contours, each
from different cross-sections,
were assigned
to each other if
they satisfied a connectivity criterion (see definition below)
and occupied the same level in the nesting tree. As multiple
cylindrical connections can generate branches, they were
further examined based on the criteria for open branches. At
this point, the cylindrical connections
and open branches were
determined. The connected branches (holes) were given the
lowest priority, and therefore the criterion for this kind of
branching was examined
as the last step. The criteria used for
cylindrical connections between contours and for the two
kinds ofbranch between sub-obiect surfaces
are as follows.
6.2 Connectivitlt criterion
Recently, MULLER
and KrrNceRr (1993) proposed criteria
for assigning two contours, based on the average distance
between contours P and,
Q, having ly'p and Nn number of
279
Sos, Sos6
ob^
/i\ril
liltll
t]/\11
Sos, Sos,
obB obc
points,
respectively.
The contours
are assumed to be planar
and belong to different parallel planes.
The asyrnnrctric
separation
between P and
Q is then expressed as
t Nr
d(P,
q: ;+ Itp, - plt
Llzl\ p :-
where the point Qi e Q is the nearest
point to P, e P, and A--
denotes the distance
between two sections.
Afterwards, two
contours are assigned
when l[J(P,9)]-' is greater
than a
threshold value.
Note that, for a particular
value
of i, Q, can
appear more than once in the summation. it is also worth
noting that 6(P,O is without dimension and not metric.
Unfortunately, ,X., .) satisfies only one of a totai of four
metric conditions, namely,
the non-negativity
condition. The
method
has
been used extensively
and works well for partially
overlapping
contours.
It is not sufficient, horvever, for con-
tours
having no common area
or of very different
shapes.
To solve this
problem,
two quasi-metrics
without
dimension
are proposed
in our approach,
one measuring the separation
and other measuring
the relative overlap between two con-
tours.
Defnition -1.'
Suppose two non-self-intersecting planar
con-
tours P and Q, having /y'p and ly'n number of points and
delimited areas A(P) and A(Q), respectively.
Their separation
is defined by
I
L(P,q:;Q*t6(P,Ol'-[a(0,p)] ') (4)
z
and their relative
overlao is measured bv
A(P,
e)
::ry* +:!9: (5)
2 A(P\ 2 A(Q)
where A(Int\ is the area of intersection
between
contours P
and Q.
Please
note that, for any P and Q, L(P,q e [0, 1] and
A(P,O [0, 1], respectively.
Both I and A are symmetric.
Separation I has a value
of0 when contours P and Qhavethe
same
shape and are close to each other. Similarly, the relative
overlap A is zero for no overlap and I for complete overlap.
When experimenting with Z and
A, we realised that, for
large contours,
the criterion
ofrelative overlap should be used
primarily, whereas, for small contours, the separation
criterion
is preferred. What contour is considered small is strongly
dependent
on image size and application area.
Defnition 2.' A non-self-intersecting
contour P lying inside
an image t having area A(P) and contour length i" is
considered
to be small relative
to the image 1, if
a(
p\t
^
,s(P)
:'# . T, (6)
A\I )I I
where
l(1) is the image
area, and
I is the length
of the image
border.
In the above definition, S(P) -+ 0 if the area
delimited by P
is small, similarly S(P) : I for the largest
contour,
i.e. the
contour identical with the image border. A threshold value
4 e [0, 1] was selected based on the application field by
selecting the representative
set of'small' contours and calcu-
lating the maximum of their respective value of S(P). The
following definition of a connectivity
criterion accommodates
the idea of selecting one of the quasi-metrics
A or L, based on
the relative size
of contours.
280
DeJinition
J: The connectivity criterion for two contours
p
and
p is a quasi-metric
defined as
C(P.q:
rfF#m,*-,
&e'q+llf;Ot |
-A(P,q)t
if .s(P),
s(o)
: 4
4ffiO {*ft"" P'
Q)
+
|
-
Ae'
a\l
itttP,Q't+t-A(P.q\ if .t(P), S(Q)
- r,
otherw'ise
(1)
6.2.1 C)tlindrical connection and open branch: Al1 pairs of
contours from consecutive
sections
both occupying the same
level in the nesting tree and satisfying C(P,q < Q were
assigned to each other in an assignment
graph.
These criteria
reconstruct both the cylindrical connections and open
branches.
6.2.2 Connected branch'. As the criteria for assigning the
connected branch (a hole), the small overlapping
area and
the distance between
the inner contour and the one on the
following or preceding
section are used
(see
Fig. 5). In Fig. 5,
the two contours A and C lying at different levels within the
nesting tree are connected to a single contour B of ttre
adjoining section. It is intuitively assumed
that the shape
of
the hole C is similar to the open part of .8. Based on this
reasoning, the criteria for assigning the connected
branch
to
the three contours A, B, and C are the mbasure
of the largest
overlapping area and the smallest distance between the hole
and the open
part.
Additionally, contours A and
B must satisfy
the criteria for cylindrical
connection. When there
are
multiple
open parts in C, all parts are compared with the hole
separately,
and the best match is selected.
A fully accurate
reconstruction from contours
is not always
possible
because
of many valid possibilities
that are unaccep-
table.
For example, ernploying automatic
topology reconstruc-
tion on our data sets of a mouse embryo and a frog's organs,
the program
correctly reconstructed 90-94% of all branches.
The remaining or effoneous
branches had to be corrected
manually by an expert
in the field.
7 Reconstruction from contour data
7.1 Cylindrical connection
Let us consider the problem of filling the space
between
a
pair of planar contours. In the proposed system, several
methods using local criteria were implemented to solve the
filling problem,
as they can be adapted to solve
the branching
problem and have negligible calculation time compared
with
global methods. However, they failed in some carefully
designed
cases.
The proposal
discussed in this Section
is the
most effective selection
among available local triangulation
methods.
The best selection can be made based
on one of the
orthogonal
open area
Fig. 5 Measure that assigns connected branch to three contours
based on similarity between open part and hole C
Medical & Biological Engineering
& Computing May 1998
global
optimum
criteria,
such as the minimisation
oftotal surface
area
or the maximisation of volume
of the reconstructed
object
portion.
We have considered
the shortest length of spans as a
good choice
for a selection criterion.
A span is the edge of a
triangle that does not belong to either planar
contour.
The filler triangular patches
were
generated
using two local
methods.
The first was the Christiansen local method
(Cunts-
TIANSEN
and SeoERenRc,
1978),
which is effective for con-
tours with similar shapes,
but ineffective when the shapes
vary
(see Figs. 6a--c). The second method is a modification of the
above method that was designed to reconstruct
rapid changes
in shape
(see
Figs. 6d-ll:
(i) Starting
from an initial base span ab, a new base
span is
sought.
(ii) For the two forward points
d and F6, the nearest
points
Nro and Nou in the neighbourhood
of the adjoined contour are
searched
(see
Fig. 6@.
(iii) Comparing the length of spans NooFo uni lyr,fi the
shorter span is taken as the new base span.
(iv) The triangles between the old and new base spans are
then generated
to minimise the total area of triangulation in
this area. Next, the new base span becomes the initial base
span, and the process
is repeated from step i.
Using this modification,
patches
such as those in Fig. 6e can
be generated.
The Christiansen
method fails in this type of
case, as
shown
in Fig. 6b. Conversely,
however, the modified
method fails in Fig. 6f, when two contours are thin, even if
they are of the same shape. In contrast, the Christiansen
method engenders
good results in this case. The combination
of the above methods represents a good solution, as shown in
Figs. 69 and ft. Using more than two local methods can greatly
improve results, a significant payoff for the additional time
that must be invested in computation.
7.2 Open branching
Tire multiple-branching
problem
is often
treated
by introdu-
cing an intennediate contour
and by splitting the
problem
into
several double-branching
problems.
Let the area between
the
multiple contour be called a channel
and the inner chain of
triangles filling this channel be a channel polygon. The
entering and exiting edges of the channel are called the
bridge segments. Our particular focus is qn_lhe 'opt1lq4ll
determination of bridge segments, denoted B2rB2, and BtrAIo
in Fig. 7. When the bridge is found,
reconstructing the channel
by horizontal triangular
patches
within bridge segments is the
simplest method.
Finding the bridge as a convex hull of the convex
hulls of
two contours, as
proposed
by CHot and Panr (1994),
is often
inadequate
as the channel area is larger than actually necessa-
rily. First, in this approach, the convex
hulls of two contours
are identified, and then the convex hull of those convex
hulls
is calculated. There are two new segments
(bridge segments)
having end points in the different convex hulls of polygons.
We call this solution the maximum bridge because
of its
maximum channel area. The minimum segment between two
contours determines
the minimum bridge. Examples of a
maximum bridge denoted by segmetts M/M2o and MIML
and of a minimum bridge noted
by mg, mD are shown in Fig.
7. The 'optimum' selection of the bridge lies somewhere
between the rninimum and maximum bridges. The candidates
for bridge segments are all segments (Xc, Yd having
X, e lM/M[] and Y, etu[tr'l'o]. To speed up the search of
bridge segments, it is sufficient to search only the set of
segrnents filling the channel determined by the maximum
bridges generated
by the Christiansen method. The optimum
bridge is one in which the contour joined through the bridge
segments is similar to the contour in the next section. Two
contours were considered similar (GoNzelez and WINTZ,
Christiansen
method
total length=4.557
modi{ied
method
b=NFu
Fb
proposed method
selection
.-+
total lengith=6.815
WryW
c total lenglh=3.625 t total length=8.961
Fig. 6 Generation of triangles: (a) Christiansen algorithm. (b), (c) Failure and success of Christiansen algorithm- (d) Modifed algorithm. (e)
,
(fl Success and failure of modified algorithm. (d, @ Proposed method: selection based on span length between Christiansen and
modified method
Medical & Biological Engineering & Computing May 1998 281
maximum
bridge
Fig. 7 Findinq o{ bridge points on brctnch
seen
fi.on ubot'e.
B2rB2o
and Bt Bto are automoticall,v generated
bridge segntents
1987)
when the factor 12
I@EA) had a similar value for both
contours, where / and A are the perimeter and area, respec-
tively, of the surrounded
region.
8 Results
The system described for the reconstruction
and analysis of
optical microscope sections reduced the effort and time
required to reconstruct embryological structures from a few
months to only several days. As an example, a mouse embryo
(4.45mm height) was sectioned with an average section
thickness of about 7 pm using an ultramicrotome. The 636
cross-sections
obtained were further processed.
The proposed ACM was applied by the authors to the
location of target organs within the microscopic images and
to the three-dimensional reconstruction of a mouse embryo.
The new ACM offers a way to reduce the sensitivity of active
contours to localised noise. The method is shown to be
effective for refining estimates of an object's shape and
location and tracking any topology changes. The outlines of
organs that consist of multiple branches or holes can be
followed automatically from slice to slice. This phenomenon
occurs within the series
of cross-sections
as the splitting of a
contour when there is a different number of contours in two
consecutive
cross-sections.
The result of automatic topology reconstruction is demon-
strated in Fig. 8. These data contain several open and con-
nected
branches, and, in the worst case,90Yo
(total on average
94%) of them were reconstructed
correctly when measuring
for each organ separately.
Parameter
f for successful topo-
logical reconstruction
was calculated
as the maximum value of
S(P) on a representative
set of small contours; the result was
4 : 0.03113.
Another set of sections
used in the study was that of a frog's
cranial
area. This set was
selected
because ofthe complicated
topology
of a frog's semicircular
canals. The set consisted of
64 cross-sections with a thickness
of 35
pm. Outlines of the
semicircular canals were extracted from the sections by
initialising outlines for only a single section within the
entire set. With methods using only geometrical information,
reconstructing the topology of the obtained contour data is
extremely difficult, if not impossible.
However,
the proposed
reconstruction method reconstructed the topology of the
282
Fig. 8 Reconstructed topolog,t of 54 objects oJ'mouse embryts. Qy",
structure is neural tube, light brown stntcture in middle is
heart tyith blood vessels coloured dat* brown. Stomach is
structure behind heart, coloured mapenta
strucfure without any misassignments.
as seen in Fig. 9b.
Furthermore,
when the triangular patches
are generated,
trans-
parent and semi-transparent
(KANEDA
et al., 1987) visualisa-
tion techniques can be applied. The transparent
visualisation
of the heart and blood vessels
is shown in Fie. 9a.
9 Conclusions
In this paper,
a system for extracting contours from multiple
cross-sections and techniques for reconstruction of three-
dimensional structures have been proposed. A new adaptive
ACM was implemented in the proposed
system. The proposed
ACM can automatically follow slice to slice and can adapt its
shape to highly concave
structures
or to those whose topology
changes (branching or merging) between fwo successive
sections. The process of topology reconstruction was
improved by incorporating information about the nesting of
contours, and by the definition of quasi-metric criteria regard-
ing when to assign two contours to each other in an assign-
ment graph.
Methods for topology reconstruction of both open and
connected
branching have been proposed.
At the reconstruc-
tion step, the space between two contours was filled by
triangular patches resulting from the optimum solution, i.e.
the most effective solution among several local optimum
methods. The reconstruction of open branches can be pro-
cessed more accurately using discrete optimisation of the
bridge location limited to the minimum and maximum
bridges.
The proposed system was successfully applied to the
processing of actual data related to human and animal
embryonic organs, proving that the system allows detailed
modelling of complicated
objects.
Acknov,ledgments-Ihe
authors wish to thank Professor Mineo
Yasuda and Professor
Akinao G. Sato for their advice and for
providing
the sample data of a mouse
emtrryo.
We also wish to
thank Professor Yasuo
Harada for inspiration
regarding
the
visualisa-
tion of a seniicircular
canal. which has
comolicated
shaoes
in both
planar
contours
and 3D slnrctures.
and also ior providing
the cross-
section images.
Our thanks
also to the reviewers for their helpful
comments.
Medical & Biological Engineering
& Computing May 1998
Fig.9 Transparent visualisation ofreconstructed structures. (a) Mouse embryo heart and blood vessels. (b) Contplicated topologlt ofleft and
right frog semicircular canals
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Authors' biographies
Roman Durikovid graduated from Comenius University Bratislava,
Slovakia, where he received MD and MS degrees in Numerical
Analysis, in 1989. He obtained his PhD from the Faculty of
Engineering of Hiroshima University, Japan, in 1996. He was a
visiting scholar at the Groningen University, Netherlands, in 1991.
He is currently a visiting researcher
and member of the computer
graphics group at Hiroshima University. His research interests
include image processing, 3D reconstnrction, artificial life and
physical
modelling.
He is a member
of the IEEE.
Kazufumi Kaneda is an Associate Professor in the Faculty of
Engineering at Hiroshima University. He worked at the Chugoku
Electric
Power Company Ltd, Japan, from 1984 to 1986. He joined
Hiroshima University in 1986. He was a visiting researcher in the
Engineering Computer Graphics Laboratory at Brigham Young Uni-
versity in 1991. His research
interests
include computer graphics
and
image processing.
Kaneda
received
his BE, ME, and DE, in 1982,
1984,
and 1991, respectively,
from Hiroshima University. He is a
member of the ACM, IEE of Japan, IPS of Japan and IEICE of Japan.
283
Hideo
Yamashita
is a Professor
at Hiroshinra University. Depaftment
of Engineering,
E,lectric Machinery Laboratory, where he teaches
and
carries
out research
into the visualisation
of rra.gnetic
fields
and
linite element ar-ralysis.
He received
his BE and ME. degrees
in
Electrical Er-rgineering
from Hiroshima University, Japan,
in 1964
and 1968,
and Engineering
degree
in 19l'1
, from Waseda
Universitv.
Tokyo, Japan.
He was appointed
Research
Assistant,
in 196g,
and
Associate
Professor,
in 1978,
of the Faculty
of Engineering,
Hiro_
shirra {Jnive
rsity. He was an Associate Researcher
at Clarkson
University,
Potsdanr,
Ncr.v
York in l98l-1982. He is a member
of
the IEEE, ACM. the
IEE of Japan, the IECE of Japan,
and
the IpS of
Japan.
244 Medical
& Biological
Engineering
& Computing May 1998
... Computer-automated extraction of contours from biological objects of interest has been attempted by others to reduce the labor and subjectivity of manual contour tracing. Popular methods of selecting areas of interest use segmentation or edge detection algorithms (Toga, 1990;Montgomery and Ross, 1993;Durikovic et al., 1998). These algorithms attempt to select objects of interest based on their specific and uniform pixel intensity or by their distinct boundaries from the surrounding tissue. ...
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Book
Principles of Optics is one of the classic science books of the twentieth century, and probably the most influential book in optics published in the past forty years. This edition has been thoroughly revised and updated, with new material covering the CAT scan, interference with broad-band light and the so-called Rayleigh-Sommerfeld diffraction theory. This edition also details scattering from inhomogeneous media and presents an account of the principles of diffraction tomography to which Emil Wolf has made a basic contribution. Several new appendices are also included. This new edition will be invaluable to advanced undergraduates, graduate students and researchers working in most areas of optics.