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Description of object shapes by apparent boundary and convex hull
003) 3203.93 $6.00+.00 Pergamon Pre~ Ltd 1993 Pattern Recognitmn Socmty
Pattern Recoamuon, Vol 26, No. ), pp. 95 107, 1993 Printed in Great Britain
DESCRIPTION OF OBJECT SHAPES BY APPARENT BOUNDARY AND CONVEX HULL SHAN Ltu-Yu'l" and MONIQUE THONNATJ~§ "tCMA, ENSM P, Place Sophie Laflite, Sophia Antipolis, 06565 Valbonne, France INR1A, 2004 Route des Lucioles, Sophia Antipolis, 06565 Valbonne, France
(Received 3 May 1991; in revisedform 4 May 1992; receivedfor publication 28 May 1992) Abstract--The notions of the apparent boundary and the strict apparent boundary of an object arc introduced, which, together with the convex hull, provide an automatic approach for object shape analysis. Various properties of the apparent boundary and the strict apparent boundary are established which show that they are good approximations of the real object boundary. Applications of the apparent boundary and the convex hull to the automatic shape characterization of planktonic foraminifera are presented. Image analysis Shape description Planktonic foraminifera
Apparent boundary
Convex hull
Distance curve
hull and the convex deficiency, together with the apparent boundary for the shape description of planktonic foraminifera. The paper is organized as follows. In Section 2, we present two algorithms to define the apparent boundary and the strict apparent boundary of an object. We also show the geometric properties of these two boundaries and the algebraic properties of our operators which find these boundaries. In Section 3, we prove that the convex hull of the apparent boundary and that of the strict apparent boundary are both identical to the convex hull of the object. In Section 4, we present an application of the apparent boundary and the convex hull to the shape characterization of the planktonic foraminifera.
I. I N T R O D U C T I O N
Object recognition is one of the most important research subjects in computer vision. In this paper we propose an automatic approach for the shape description of objects, and we apply this method to the analysis of the planktonic foraminifera images. Our approach is based on information extracted from object boundaries, e.g. the perimeter, the diameter and the convexity of the boundary which are major descriptors of object shapes: t'2) The method presented here is applicable to various classes of objects of natural organisms, e.g. zooplanktons, fishes, and plants. The boundary of an object is usually detected by an edge detection algorithm followed by a linking process to associate edges into a meaningful boundary. Among the linking procedures, there are methods like searching near an approximate location, the Hough method for curve detection, graph searching, and dynamic programming. "~ In this paper, we introduce two new notions: the apparent boundary and the strict apparent boundary of an object. These two boundaries are shown to be good approximations of the real object boundary, and provide a simple and automatic approach for the boundary detection. Our approach is especially useful when the object boundary is noisy (broken or connected to the internal structure of the object). In addition to object boundaries, the convex hull of object boundaries is also a powerful tool for object shape recognition. The convex hull has found wide applications in pattern recognition and image processing. c3-6) It can be used for the normalization of patterns, the estimation of locations and the extraction of features, etc. These tasks are either performed directly by the use of convex hull or by finding the convex deficiency,t4) In this paper, we use the convex
2. A P P A R E N T B O U N D A R Y A N D S T R I C T A P P A R E N T BOUNDARY
In this section, we introduce the definitions of the apparent boundary and the strict apparent boundary of an object. We also show some geometric and algebraic properties of these boundaries. 2.1. Definition of apparent boundary Let ~ be an object, and C the set of points of this object. Denote by xp and yp the x and y coordinates of point p. We construct from C its apparent boundary A according to Algorithm 1 below.
AIaorithm 1. Apparent boundary o.f a set of points Input: Set of points C. Output: Apparent boundary A. (1) Lett x. t xm l t x ~ frr a i.n ' t rm a x be the extreme points such rain' that
§ Author to whom correspondence should be addressed.
x,~,° = mincexp,
xt~.. = maX~cx r
),,
.v~ = m a x y p .
m n
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96
SHANLtu-Yu and M. THONNAT
(2) If x I x = x ¢x or y t ~ = ~v y , then C contains t~ a l • m n . max . rain only vertical hnes or horizontal hnes, STOP. (3) Scan C horizontally and vertically and put the leftmost, rightmost, downmost and upmost points in C~i ., C~,~, C~i., C~,,, respectively, namely
2.2•
Definition of strict apparent boundary
C"rain = { plxp = p,~c.yp. min=ypXp.}
Now with the apparent boundary in hand, we can compute the strict apparent boundary ,4 of C by Algorithm 2 below.
C xm a x = { p l x p =
p'EC.yp, = yp
Algorithm 2. Strict apparent boundary Input: Apparent boundary A. Output: Strict apparent boundary ,4.
= tply p =
p,EC,xp. = x p
C ym l.n
C"m a x ={PlYp=
max
xp.}
min
y p , j*1
max
y
p.EC,xp, = x p p" ~
(4) Let T = C rain x. u C ~' u C'. ~ kC mya x . Partition T max rain into chains
the contour of P. Figure 1 illustrates the apparent boundary of a set of points.
li, l
(1) Without loss of generality, suppose that the apparent boundary A is oriented. (2) Scan A horizontally and vertically and put the leftmost, rightmost, downmost and upmost points in p~. pxm a x ' pr. pym a x ' respectively, namely ram' rnln'
P~. = {pJXp = p,EA,yp, min= y P Xp. l rain
! < i < j _< k, so that all the 8-connected neighbouring
points are in the same chain. (5) Let P be the polygon obtained by linking the chains in such a way that the extreme point of a chain is linked to one and only one extreme point of another chain and that the sum of the distances of the linking segments is minimized. (6) Let A be the polygon obtained from following the contour of P. This can be done by starting with t rxa i n and following the contour of P in the clockwise direction so that P is always on the right hand. The apparent boundary can now be defined as follows.
Definition 1. Let C be the set of points of an object t'l. The apparent boundary of C, denoted by A(C), or simply A, is obtained from applying Algorithm 1 to C. Notice that the apparent boundary A is a simple polygon due to the fact that A is obtained by following
{a)
f
p:, max
=}plxp= ~
max
p.EA.yp,=y p
x
/
P f
minx,,Y~,"} PY'm.,= { PIYp= p'~,t.~,, px max
=~ply~, = ~
max
v .'~
p'~A.Xp. =Xp "/p f "
(3) Let V= P~,i uPS,, wP>'miwP~,,, and let L be the list obtained by ordering the points of V in the direction of A. (4) The strict apparent boundary ,4 is the polygon with the vertices taken from the list L. For the sake of simplicity, Algorithms I and 2 will also be denoted by the operators .d~ and ,~/2, respectively. The strict apparent boundary can now be defined as follows.
(b)
Fig. I. (a) An arbitrary set of points; (b) its apparent boundary.
Description of object shapes
97
Suppose that p¢,C, and p is the self-intersection point of the segments s(a, b) and s(c, d), where a, b, c, d are the vertices of P. One has that d(a, b) + d(c, d) > d(a, c) + d(b, c), and that d(a,b) + d(c,d) >_d(a,d) + d(b,c), where d(a,b) denotes the distance between a and b. However, according to Algorithm 1, P is obtained from linking the extreme points of the chains in a way that the sum of the distances of the linking segments is minimized. Thus, p cannot be the self-intersection point of the segments s~a,b) and s(e,d). Therefore, peC • Fig. 2. The strict apparent boundary of Fig. 1.
Definition 2. Let C be the set of points of an object t2. The strict apparent boundary of C, denoted by ,4(C), or simply ,4, is the resulting polygon of ~¢2o.~/~ applied to C: ,4(C)=,~¢2(,z/dC)). In other words, ,4(C) is obtained from applying Algorithms 1 and 2 successively to C. Observe that like the apparent boundary, the strict apparent boundary is also a simple polygon. Figure 2 illustrates the strict apparent boundary of Fig. 1, It is easy to check that the apparent boundary lies between the real object boundary and the strict apparent boundary. However, in many cases (like in planktonic foraminifera images), the apparent boundary and the strict apparent boundary coincide. 2.3. Geometric properties of the apparent boundary and the strict apparent boundary
Proof of Theorem 1. Let u,v be two arbitrary points of C. Denote by d(u, v) the Euclidean distance between u and v. Let ul,u2 and t,l,v2 be the extreme points of C such that )',~ = 3',
=
3'.2, xu~ <-"X. <_x.2,
Y,., = Y,. = Yt'2, x,,, <_x,, <_x,. 2. It is easy to see that d(u, v} __D{A). Hence D(C) = D(A). Observe now that all the vertices of A are the vertices of A, V(,4) g V(A). Therefore, D{,4)__D(AL so that D(,4) D(A). • =
2.3.1. Diameter. The diameter of an object is defined as the maximum Euclidean distance between the extreme points of the object] 2) The diameter of a polygon is defined as the maximum Euclidean distance between the vertices of the polygon. Theorem 1. Let C be the set of points of the edges of an object ~, the diameter of C, of its apparent boundary A, and of its strict apparent boundary A, denoted by D(C),D(A) and ~,4), respectively, are all identical: D(C) = D(A) = D(A). In order to prove this theorem, we need the following lemma. Lemma I. For any set of points C, the set of the vertices of A, denoted by V(A), is a subset of C: V(A) c_ C. Proof. Let P be the polygon obtained at step 5 of Algorithm 1. Since A is a simple polygon obtained from following the contour of P, the vertices of A are either leftmost, rightmost, downmost, upmost points in C or self-intersection points of P. Let p be an arbitrary self-intersection point. We show that p is one of the leftmost, rightmost, downmost or upmost points of C.
The diameter is an important parameter in shape characterization and classification for some sorts of objects. Theorem 1 allows us to compute the diameter of an object via its apparent boundary, or its strict apparent boundary. An application of this result is illustrated in Section 4. 2.3.2. Weak external visibility boundary. As we have pointed boundary and the strict apparent simple polygons. Within the class there is an important subclass externally visible polygons.
of strict apparent out, the apparent boundary are both of simple polygons, known as weakly
Definition 3. For a simple polygon P, let bd(P) denote its boundary. For any point x on the boundary, denote by r(x) an infinite half-line starting at point x and proceeding in any direction. A simple polygon is said to be weakly externally visible if, and only if, for every xebd(P), there exists an r(x) such that Pc~r(x) = {x I. Intuitively, consider a polygon P to be completely surrounded by a circle. If P is weakly externally visible, then the entire boundary of P is visible at one time or another as a guard patrols along the circle.
98
SHAN LIu-Yu and M. THONNAT
Theorem 2. For any set C of points, the strict apparent boundary .4(C) is weakly externally visible. Proof. Let V be the set of vertices of .4, namely, V = Pm~,w P=,, w P=~, w P ~ , (cf. step 3 of Algorithm 2). Let E be the set of internal points on the edges of .4. It is clear that VuE is the set of all (boundary) points of ,4. For all peV, there trivially exists a hairline ray r(p) starting from p such that r(p)r~.4 = {p}. For example, if peP~),,r(p) is the hairline ray starting with p and directing horizontally to the left. For all p~E, let p be on the segment s(u,v), where u, ve V. It is clear that u and v are two successive points in the list L (cf. step 4 of Algorithm 2), and that they are not 8-connected neighbours in the image plan, so we have y.=yo_+l
but
x.
or
x.>x~+l,
x.=x~_+l
but
y.
or
y.>y~+l.
or
Consider the case y. = y~ + 1 and x. < x ~ - I . The other cases can be treated analogously. Owing to the fact that y, = y~ + I and x. < x~ - 1, one sees that either u,v~P~,i, or u, vePX=.. Therefore, the hairline ray starting from p and directing horizontally to the left or to the right satisfies the relation r(p)c~A = {p}. Hence, ,4 is weakly externally visible. • The class of weakly externally visible simple polygons is of interest in computational geometry. For example, the simple linear convex hull algorithm of Sklansky ~) does not work for all simple polygons. (a~ However, it correctly finds the convex hull of any weakly externally visible polygon.(9> Making use of this property, we have implemented Sklansky's algorithm in order to find the convex hulls of planktonic foraminifera images (see Section 4 below). 2.4. Algebraic properties of operators Mt and Mz
Theorem 3. The strict apparent boundary ,4 is a fix point of operators MI and M2: ~'I(A) = A,
~z(A) = A-
Proof. It follows from the proof of Theorem 2 that all the points of ,4 (i.e. those in VuE) are visible from one of the four directions: left, right, up, down. As a consequence, when Algorithm I is applied to .4, the set T = C~t . w C=~,u x , . wm C y, , (step 4) contains only Cmi one closed chain which is composed by all the points of ,4. Therefore, steps 5 and 6 of Algorithm 1 do nothing for A, so that M1(,4) =/~. Similarly, in Algorithm 2, the set V =/~=l, w P~x w P~l.u P ~ . (step 3) contains all the points of//. Hence, a/2(A) = A. •
Corollary I. The operator • = ,af2o,aft is idempotent: ~o~
= 8.
Proof. Let C be the set of points of all the edges of an object ~ It is given by Definition 2 that ,~(C)= ~2o~/1(C) = ,4. From Theorem 3 we have that ato~(c) = ~(~) = d~o~/1(2 ) = ~ , ( 2 ) = 2 =~(c).
•
Theorem 3 and Corollary I show that the strict apparent boundary is stable in the sense that any iteration of the operators ~q/l and Mz results in the same boundary. Therefore, it is a good candidate for the automatic construction of object boundaries. Since the apparent boundary is between the original image and the strict apparent boundary, and since the original image, the apparent boundary and the strict apparent boundary have the same diameter (cf. Theorem 1) and the same convex hull (see Theorem 4 below), both the apparent boundary and the strict apparent boundary provide good approximations of the object boundary.
3. CONVEX HULL OF AN OBJECT AND OF ITS APPARENT BOUNDARY
The convex hull of an arbitrary set of points S or of a polygon P is defined as the smallest convex polygon containing S or P, respectively. The complement of $ or P with respect to its convex hull is called the convex deficiency of S or P. In computer vision, the set S is often an object region and P its boundary. It has been shown that the time complexity of finding the convex hull of n planar points is at least O(n log (n)).(I°- t a) Algorithms that find convex hull in O(n log (n)) time were proposed in various papers, see Day °4) for references. When the points in S are ordered or when P i~ a simple polygon, linear algorithms are proposed in references (I 5-18). Apart from sequential convex hull algorithms, one can also find parallel convex hull algorithms in the literature. "4' 19-23) Let C be the set of points of edges of an object f2. We are now going to show that the convex hull of C, denoted by CH(C), is the same as the convex hull of its apparent boundary A, denoted by CH(A), and is the same as that of its strict apparent boundary ,,], denoted by CH(,4). Recall that A and .~ are simple polygons. For any polygon P, we denote by V(P) the set of its vertices.
Theorem 4. The convex hulls of a set of points C, of its apparent boundary A, and of its strict apparent boundary A are all identical CH(C) = CH(A)= CH(.4). In order to prove this theorem, we need the following lemmas. /_,emma 2. For any polygon P, CH(P) = CH(V(P)).
Proof. Omitted.
•
Lemma 3. For any point veC, v is contained within CH(A), denoted by v-< CH(A).
Proof. Let pt,p2eC be the leftmost and rightmost
Description of object shapes points of C such that Yp, = Yp2= Y~, and
xp, < xv < xo2.
It is easy to see that v -< s(p t, P2). Since p ~,P2e A(C), we have that s(pl,p2).< CH(A). Therefore, v.< CH(A). • Lemma4. For any simple polygon P, CH(P)=
CH(s42(P)). Proof. For any point p (vertex or internal point on an edge) of P, if p is not on M2(P), then one can see from the scanning procedure of Algorithm 2 that p is within M2(P). Hence, we have that CH(P)-< CH(M2(P)). On the other hand, we have that V(~/2(P))c_ V(P), which implies that V(Mz(P))-< V(P). Therefore, CH(P) =
critic(P)).
99
of oceanic deep sea drilling. Foraminifera (about 2000 genera and 30,000 species) are presently the best known and the most comprehensively studied of all the calcareous microfossiis. By the end of the Cretaceous times (60-90Ma), planktonic foraminifera are essentially represented by the globotruncanids, a group which includes very distinctive forms characterized by trochospiral and keeled tests. These foraminifera present a succession of chambers coiled spirally. Within the globotruncanids, differentiation of species is based on test shape, number and shape of chambers in the last whorl, apertural and accessory structures, keels, etc. (see Figs 4, 1l(a), 12(a), and 13(a)).
•
Proof of Theorem 4. It follows from the definition of convex hull and Lemmas I and 2 that CH(A)~< CH(C). On the other hand, according to Lemma 3, we have CH(C) ~ CH(A). Therefore, CH(A) = CH(C). Applying Lemma 4 to the apparent boundary A yields that CH(A)= CH(~4), which completes the proof. •
4.2. Characterization of foraminifera by means of
apparent boundary and convex hull
Generally, we study a planktonic foraminiferon from its three sides, namely, the spiral side, the profile side, and the umbilical side. Here we limit our discussion to the spiral side of planktonic foraminifera (or simply called foraminifera) of group giobotruncanids. We Note that Theorem I can also be considered as a study their features directly on the scanning electronic consequence of Theorem 4 due to the fact that a microscope (SEM) photographs. (z4) polygon and its convex hull have the same diameter. We use the apparent boundary, the convex hull and the distance curve (see below) to extract information on the boundary shape, the number of chambers in the 4. APPLICATIONOF APPARENTBOUNDARY AND CONVEXnULL TO "rug SHAPE last whorl and the diameter of the foraminifera. CHARACTERIZATIONOF PLANKTONIC The scheme of our process is shown in Fig. 3. We FORAMINIFERA first detect the edges of the foraminiferon by the method of Deriche, (2s) followed by the hysteresis 4.1. A brief presentation of planktonic foraminifera thresholding (see Figs 4-6). We notice that the boundary The foraminifera are single-celled protists which live of the foraminiferon in Fig. 6 is not only broken but either on the sea floor (benthic foraminifera) or among also connected to the internal structures of the forathe marine plankton (planktonic foraminifera). The miniferon. We thus compute the apparent boundary protoplasmic mass of the foraminiferid cell is enclosed of the foraminiferon (see Fig. 7). Note that in most cases within a mineralized test (or shell, usually less than of our study, the strict apparent boundary of a foraI mm in diameter) which may consist of one or more miniferon remains the same as its apparent boundary, cavities termed chambers. therefore we here use only the notion of the apparent Foraminiferid tests are abundant in marine sedi- boundary. We then compute the convex hull (Fig. 8). ments and are known from early Cambrian (600 Ma) With the apparent boundary and the convex hull in through Recent times. They constitute invaluable hand, the convex deficiency can easily be obtained (see indicators in the age determination as well as the Fig. 9). We further compute the distances between the depositional environment of sedimentary strata. The apparent boundary and the convex hull, and we obtain impetus given by oil companies accelerated the study a distance curve (see Fig. 10). of these micro-organisms in the 1950s. Recently, microFigures I 1-13 are three other examples. Note that paleontology expanded greatly with the development the scales of all the distance curves are not the same.
image of a H foraminiferon
edge detection
Fig. 3. Diagram of the processing of foraminiferid images.
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Fig. 5. The result of edge detection of Fig. 4 by the method of Deriche.
Fig. 7. The apparent boundary of the foraminiferon shown in Fig. 4.
4.2.1. Recognition of boundary shapes. One of the most important criteria in the identification of foraminifera is their boundary shapes. Seen from the spiral side, the foraminifera commonly have four types of boundary, namely, lobate boundary, spinal boundary, circular or subcircular boundary, and polygonal boundary (see Figs 4, l l(a), 12(a), and 13(a), respectively). These shapes can be identified by using the distance curve and the convex hull as follows (see Fig. 14). Observe first that a distance curve with important and nearly equally separated peaks represents a lobate boundary or a spinal boundary; while a distance curve
with possibly one important peak and many other small and irregular peaks represents a circular or a polygonal boundary. Thus lobate boundaries and spinal boundaries can be separated from circular boundaries and polygonal ones. Observe also that spinal boundaries correspond to convex peaks; whereas lobate boundaries correspond to concave peaks. Therefore, we just need to detect the concavity of the peaks in the distance curves to distinguish lobate and spinal boundaries. Circular boundaries can be separated from polygonal ones by the fact that the convex hull of a circular boundary has many more vertices than that of a polygonal boundary.
Description of object shapes
101
j
.
Fig. 8. TheconvexhulloftheforaminiferonshowninFig. 4.
Fig. 9. The convex deficiency of the foraminiferon shown in Fig. 4.
4.2.2. Number of chambers. Another important measure of foraminifera is the number of chambers in the last whorl. For the foraminifera of lobate and spinal boundaries, this can be computed by the number of peaks in their distance curves. Indeed, the peaks in the distance curve indicate the junction points of neighbouring chambers for foraminifera of lobate boundaries: whereas for the foraminifera of spinal boundaries, it is the valleys that indicate the junction points of neighbouring chambers. A confidence coefficient should also be calculated in order to validate the counting process. The above method cannot be used to count chamber numbers for foraminifera of circular boundaries or
of polygonal boundaries. The counting of chamber numbers of these foraminifera is performed by another method, which is outside the scope of this paper.
4.2.3. Diameter. Owing to Theorem 1. the diameter of a foraminiferon can be obtained from its apparent boundary. Moreover, since the diameter of a polygon is the same as that of its convex hull, ~Sjthe diameter of a foraminiferon can be computed efficiently from the convex hull of its apparent boundary. There are many proposed algorithms which compute a two-dimensional (2D) diameter with linear time complexity in terms of the number of verticesJ 26'2~)
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Fig. 11. (a) A specimen of foraminiferon of Globotruncanita calcarata; (b) the result of Deriche's edge detector followed by hysteresis thresholding; (c) the apparent boundary and the convex hull; (d) the distance Gurve.
Description of object shapes
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Fig. 13. (a) A specimen of foraminiferon of Rosita contusa; (b) the result of Deriche's edge detector followed by hysteresis thresholding; (c) the apparent boundary and the convex hull; (d) the distance curve.
dist~ic~ curve
for lobateboundary
discriminafio~n b y J
shapeof peats N ~ distan~ cul~c
by
for spinal boundary
numberof peaksN'~ digtane~ curve
for circularbc.md-,~-y
discriminationby < number of vertices of the convex hull
distatXT~ curve for polygonalboundary Fig. 14. Recognition of boundary shapes by distance curve and convex hull.
Description of object shapes
105
(a)
(b)
Fig. 15. (a) Normal scanning; (b) enhanced scanning at the pocket.
5. C O N C L U S I O N AND DISCUSSIONS
In this paper, we have introduced the notions of the apparent boundary and the strict apparent boundary of an object. We have shown some geometric properties of these two boundaries and the algebraic properties of our operators which find these boundaries. We have proven that the convex hulls of an object, of its apparent boundary, and of its strict apparent boundary are all identical. We have used the apparent boundary, the convex hull and the distance curve for the shape characterization of planktonic foraminifera. It is worthwhile noticing that the apparent boundary and the strict apparent boundary provide an automatic approach of boundary detection. They are particularly useful when edge detection algorithms yield boundaries that are broken and/or connected to the internal structures of the objects. Owing to the properties that we have established in this paper, these boundaries are good approximations of the object boundary. Note also that the convex hull as well as the distance curve are useful tools for object shape analysis, in particular, those of natural organisms, e.g. zooplanktons, fishes, and plants. We can solve the problem of shape recognition of these objects in an analogous way to the one presented in this paper. Like some other methods in computer vision, our method does not work for all objects. For instance, for objects possessing deeply folded "pockets", the apparent boundary may introduce some modifications to the shapes of the "pockets" (cf. Fig. 15(a)). This is due to the fact that in our algorithms, we scan the object only in two perpendicular directions. However, it is possible, at the expense of the computation time, to scan the object in all the directions so that all the weakly
externally visible boundary points of the object are included in the apparent boundary. In order to reduce the computational cost, one can perform such "enhanced scanning" only at the position of the pockets (which can trivially be detected by the distance curve), as illustrated in Fig. 15(b). Finally, we would like to point out that the identification of foraminifera has always been a tedious and time-consuming work for paleontologists. There exist a few expert systems to assist the identification of foraminifera. However, the extraction of the descriptive parameters is carried out by human beings. Thus the results of this paper, together with other parameters we have detected by different methods, can be used as the input to these expert systems. The next step of our research work is to design an expert system based on the shell CLASSIC, as was used in the automatic classification of zooplanktons and fish.~8~ Our work is aimed at automating both the extraction of the descriptive features and the identification of the specimens of foraminifera which play an important role in oil exploration. 6. SUMMARY
In this paper, we introduce the notions of the apparent boundary and the strict apparent boundary. These two boundaries are used to approximate the boundary of an object when boundary detection algorithms do not yield a satisfactory result, especially when the detected boundary is broken or connected to the internal structure of the object. The following properties are established in the paper: (I) an object, its apparent boundary and its strict apparent boundary have the same diameter; (2) an object, its apparent boundary and its strict apparent boundary have the same convex hull; (3) the strict
106
SHAN LIu-Yu and M. THONNAT
apparent boundary is a fixed point of the two operators that compute the apparent boundary and the strict apparent boundary; (4) the strict apparent boundary is weakly externally visible. The properties (1), (2) and (3) indicate that both the apparent boundary and the strict apparent boundary provide good approximations of the real object boundary. Moreover, the diameter and the convex hull of an object can be computed from its apparent boundary or from its strict apparent boundary. Property (4) allows us to find the convex hull of an object by the simple linear algorithm of Sklansky. As an application of our method, we use the apparent boundary and the convex hull for the shape characterization of planktonic foraminifera. Based on the apparent boundary and the convex hull, we construct a distance curve which allows us to recognize different b o u n d a r y shapes of foraminifera. Other information about the shape of foraminifera is also obtained from the convex hull and the distance curve, such as the number of chambers in the last whorl and the diameter of foraminifera. Acknowledgements--The authors are grateful to Professor Pierre Saint-Marc of Universit6 de Nice, France, for having provided us with images of planktonic foraminifera, and for helpful discussions on the characteristics and the classifications of planktonic foraminifera. The authors would also like to thank Professor Pierre Saint-Marc and the referee for their useful comments on a draft of the paper.
REFERENCES
1. D. H. Ballard and C. M. Brown, Computer Vision. PrenticeHall, New Jersey (1982). 2. R.C. Gonzalcz and P. Wintz, Dioital Image ProcessinO. Addison-Wesley, Reading, Massachusetts (1987). 3. S.G. Akl and G.T. Toussaint, Efficient convex hull algorithms for pattern recognition applications, Proc. 4th Int. Jt Conf. Pattern Recognition, Kyoto, Japan, pp. 483-487, 7-10 November (1978). 4. R.O. Duda and P. E. Hurt, Pattern Classification and Scene Analysis. Wiley, New York (1973). 5. G.T. Toussaint, Pattern recognition and geometrical complexity, Proc. 5th int. Conf. Pattern Recognition, Miami Beach, pp. 1324-1347 (1980). 6. G. T. Toussaint, Computational geometric problems in pattern recognition, Pattern Recognition Theory Applic., J. Kittler, K. S. Fu and L. F. Pau, eds. pp. 73-91. Reidel, New York (1982). 7. J. Sklansky, Measuring concavity on a rectangular mosaic, IEEE Trans. Comput. 21(12), 1355-1364 (1972).
8. A. Bykat, Convex hull of a finite set of points in two dimensions, lnf. Process. Lett. 7(6), 296-298 (1978). 9. G. T. Toussaint and D. Avis, On a convex hull algorithm for polygons and its application to triangulation problems, Pattern Recognition 15, 23-29 (1982). 10. P.V. Erode Boas, On the D(nlogn) lower bound for convex hull and maximal vector determination, Inf. Process. Lett. 10(3), 132-136 (1980). 11. F. P. Preparata and S. J. Hong, Convex hulls of finite sets of points in two and three dimensions, CACM 87-93 (1977). 12. M. I. Shamos and D. Hoey, Closest point problems, Proc. 16th A. Syrup. Found. Comput. Sci., IEEE, New York, pp. 151-162 (1975). 13. A. Yao, A lower bound to finding convex hulls, J. Ass. Comput. Mach. 28(4), 780-787 (1981). 14. A. M. Day, Planar convex hull algorithms in theory and practice, Computer Graphics Forum 7, pp. 177-193. NorthHolland, Amsterdam (1988). 15. S. K. Ghosh, A linear time algorithm for computing the convex hull of an ordered crossing polygon, Pattern Recognition 17, 351-358 (1984). 16. R. L. Graham and F. F. Yao, Finding the convex hull of a simple polygon, J. Algorithms No. 4, 324-331 (1983). 17. D. McCallum and D. Avis, A linear algorithm for finding the convex hull of a simple polygon, Inf. Process. Lett. 9(5), 201-206 (1979). 18. A. A. Melkman, On-line construction of the convex hull of a simple polygon, Inf. Process. Lett. 25, l l-12 (April 1987). 19. Z. Hussain, A fast approximation to a convex hull, Pattern Recognition Left. 8(5), 289-294 (1988). 20. C. S. Jeong and D. T. 1.,¢¢,Parallel convex hull algorithms in 2D and 3D on mesh-connected computers, Int. Conf. Comput. Vision Display, pp. 64-76 (1988). 21. R. Miller and Q. F. Stout, Efficient parallel convex hull algorithms, IEEE Trans. Comput. 37(12), 1605-1618 (1988). 22. J. Sklansky, L.P. Cordella and S. Levialdi, Parallel detection of concavities in cellular blobs, IEEE Trans. Comput. 25(2), 187-195 (1976). 23. I. Stojmenovic and D.J. Evans, Comments on two parallel algorithms for the planar convex hull problem, Parallel Computing, Vol. 5, pp. 373-375. North-Holland Amsterdam (1987). 24. F. Robaszynski, M. Caron, J. M. Gonzalez Donoso and A. H. Wonders, Atlas of Late Cretaceous GIobotruncanids, Revue Micropal~ontologie26(3-4), 145- 302 (1984). 25. R. Deriche, Optimal edge detection using recursive filtering, ICCV, pp. 501-505, London (1987). 26. M. A. Fischler, Fast algorithms for two maximal distance problems with applications to image analysis, Pattern Recognition 12, 35-40 (1980). 27. M. I. Shamos, Geometric complexity, Proc. 7th ACM Syrup. Theory Comput., pp. 224-233, May (1975). 28. M. Thonnat and M.-H. Gandelin, An expert system for the automatic classification and description of zooplanktons from monocular images, ICPR9, Roma, Italy, pp. 114-118, November (1988).
About the Autbor--SHAN LIu-Yu received the diploma of Bachelor of Engineering in biomedical instrumentation from Shanghai Jiao Tong University, People's Republic of China, in 1985. From 1985 to 1986, she was a graduate student in medical imaging in Shanghai Jiao Tong University. She is currently preparing a Ph.D. thesis at CMA, Ecole Nationale Sup~:rieure des Mines de Paris, France. Her research interests include image processing, pattern recognition, and medical imaging.
Description of object shapes About the Author-- MONIQUETHONNATgraduated from the Ecole Nationale Sup6rieure de Physique in Marseille in 1980. She received her Ph.D. in optics and signal processing from the University of St J6rome, Marseille, France, in 1982, having prepared her thesis at the Laboratorie d'Astronomie Spatiale of Centre National de la Recherche Scientifique, Marseille, France, in 1981 and 1982. In 1983, she joined the lnstitut National de Recherche en Informatique et Automatique as a research scientist. Her research interests include image understanding, image processing and artificial intelligence. She has been involved in the building of several vision expert systems in astronomy, biology and robotics.
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Pattern Recoamuon, Vol 26, No. ), pp. 95 107, 1993 Printed in Great Britain
DESCRIPTION OF OBJECT SHAPES BY APPARENT BOUNDARY AND CONVEX HULL SHAN Ltu-Yu'l" and MONIQUE THONNATJ~§ "tCMA, ENSM P, Place Sophie Laflite, Sophia Antipolis, 06565 Valbonne, France INR1A, 2004 Route des Lucioles, Sophia Antipolis, 06565 Valbonne, France
(Received 3 May 1991; in revisedform 4 May 1992; receivedfor publication 28 May 1992) Abstract--The notions of the apparent boundary and the strict apparent boundary of an object arc introduced, which, together with the convex hull, provide an automatic approach for object shape analysis. Various properties of the apparent boundary and the strict apparent boundary are established which show that they are good approximations of the real object boundary. Applications of the apparent boundary and the convex hull to the automatic shape characterization of planktonic foraminifera are presented. Image analysis Shape description Planktonic foraminifera
Apparent boundary
Convex hull
Distance curve
hull and the convex deficiency, together with the apparent boundary for the shape description of planktonic foraminifera. The paper is organized as follows. In Section 2, we present two algorithms to define the apparent boundary and the strict apparent boundary of an object. We also show the geometric properties of these two boundaries and the algebraic properties of our operators which find these boundaries. In Section 3, we prove that the convex hull of the apparent boundary and that of the strict apparent boundary are both identical to the convex hull of the object. In Section 4, we present an application of the apparent boundary and the convex hull to the shape characterization of the planktonic foraminifera.
I. I N T R O D U C T I O N
Object recognition is one of the most important research subjects in computer vision. In this paper we propose an automatic approach for the shape description of objects, and we apply this method to the analysis of the planktonic foraminifera images. Our approach is based on information extracted from object boundaries, e.g. the perimeter, the diameter and the convexity of the boundary which are major descriptors of object shapes: t'2) The method presented here is applicable to various classes of objects of natural organisms, e.g. zooplanktons, fishes, and plants. The boundary of an object is usually detected by an edge detection algorithm followed by a linking process to associate edges into a meaningful boundary. Among the linking procedures, there are methods like searching near an approximate location, the Hough method for curve detection, graph searching, and dynamic programming. "~ In this paper, we introduce two new notions: the apparent boundary and the strict apparent boundary of an object. These two boundaries are shown to be good approximations of the real object boundary, and provide a simple and automatic approach for the boundary detection. Our approach is especially useful when the object boundary is noisy (broken or connected to the internal structure of the object). In addition to object boundaries, the convex hull of object boundaries is also a powerful tool for object shape recognition. The convex hull has found wide applications in pattern recognition and image processing. c3-6) It can be used for the normalization of patterns, the estimation of locations and the extraction of features, etc. These tasks are either performed directly by the use of convex hull or by finding the convex deficiency,t4) In this paper, we use the convex
2. A P P A R E N T B O U N D A R Y A N D S T R I C T A P P A R E N T BOUNDARY
In this section, we introduce the definitions of the apparent boundary and the strict apparent boundary of an object. We also show some geometric and algebraic properties of these boundaries. 2.1. Definition of apparent boundary Let ~ be an object, and C the set of points of this object. Denote by xp and yp the x and y coordinates of point p. We construct from C its apparent boundary A according to Algorithm 1 below.
AIaorithm 1. Apparent boundary o.f a set of points Input: Set of points C. Output: Apparent boundary A. (1) Lett x. t xm l t x ~ frr a i.n ' t rm a x be the extreme points such rain' that
§ Author to whom correspondence should be addressed.
x,~,° = mincexp,
xt~.. = maX~cx r
),,
.v~ = m a x y p .
m n
95
-=minyp, /~(-"
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96
SHANLtu-Yu and M. THONNAT
(2) If x I x = x ¢x or y t ~ = ~v y , then C contains t~ a l • m n . max . rain only vertical hnes or horizontal hnes, STOP. (3) Scan C horizontally and vertically and put the leftmost, rightmost, downmost and upmost points in C~i ., C~,~, C~i., C~,,, respectively, namely
2.2•
Definition of strict apparent boundary
C"rain = { plxp = p,~c.yp. min=ypXp.}
Now with the apparent boundary in hand, we can compute the strict apparent boundary ,4 of C by Algorithm 2 below.
C xm a x = { p l x p =
p'EC.yp, = yp
Algorithm 2. Strict apparent boundary Input: Apparent boundary A. Output: Strict apparent boundary ,4.
= tply p =
p,EC,xp. = x p
C ym l.n
C"m a x ={PlYp=
max
xp.}
min
y p , j*1
max
y
p.EC,xp, = x p p" ~
(4) Let T = C rain x. u C ~' u C'. ~ kC mya x . Partition T max rain into chains
the contour of P. Figure 1 illustrates the apparent boundary of a set of points.
li, l
(1) Without loss of generality, suppose that the apparent boundary A is oriented. (2) Scan A horizontally and vertically and put the leftmost, rightmost, downmost and upmost points in p~. pxm a x ' pr. pym a x ' respectively, namely ram' rnln'
P~. = {pJXp = p,EA,yp, min= y P Xp. l rain
! < i < j _< k, so that all the 8-connected neighbouring
points are in the same chain. (5) Let P be the polygon obtained by linking the chains in such a way that the extreme point of a chain is linked to one and only one extreme point of another chain and that the sum of the distances of the linking segments is minimized. (6) Let A be the polygon obtained from following the contour of P. This can be done by starting with t rxa i n and following the contour of P in the clockwise direction so that P is always on the right hand. The apparent boundary can now be defined as follows.
Definition 1. Let C be the set of points of an object t'l. The apparent boundary of C, denoted by A(C), or simply A, is obtained from applying Algorithm 1 to C. Notice that the apparent boundary A is a simple polygon due to the fact that A is obtained by following
{a)
f
p:, max
=}plxp= ~
max
p.EA.yp,=y p
x
/
P f
minx,,Y~,"} PY'm.,= { PIYp= p'~,t.~,, px max
=~ply~, = ~
max
v .'~
p'~A.Xp. =Xp "/p f "
(3) Let V= P~,i uPS,, wP>'miwP~,,, and let L be the list obtained by ordering the points of V in the direction of A. (4) The strict apparent boundary ,4 is the polygon with the vertices taken from the list L. For the sake of simplicity, Algorithms I and 2 will also be denoted by the operators .d~ and ,~/2, respectively. The strict apparent boundary can now be defined as follows.
(b)
Fig. I. (a) An arbitrary set of points; (b) its apparent boundary.
Description of object shapes
97
Suppose that p¢,C, and p is the self-intersection point of the segments s(a, b) and s(c, d), where a, b, c, d are the vertices of P. One has that d(a, b) + d(c, d) > d(a, c) + d(b, c), and that d(a,b) + d(c,d) >_d(a,d) + d(b,c), where d(a,b) denotes the distance between a and b. However, according to Algorithm 1, P is obtained from linking the extreme points of the chains in a way that the sum of the distances of the linking segments is minimized. Thus, p cannot be the self-intersection point of the segments s~a,b) and s(e,d). Therefore, peC • Fig. 2. The strict apparent boundary of Fig. 1.
Definition 2. Let C be the set of points of an object t2. The strict apparent boundary of C, denoted by ,4(C), or simply ,4, is the resulting polygon of ~¢2o.~/~ applied to C: ,4(C)=,~¢2(,z/dC)). In other words, ,4(C) is obtained from applying Algorithms 1 and 2 successively to C. Observe that like the apparent boundary, the strict apparent boundary is also a simple polygon. Figure 2 illustrates the strict apparent boundary of Fig. 1, It is easy to check that the apparent boundary lies between the real object boundary and the strict apparent boundary. However, in many cases (like in planktonic foraminifera images), the apparent boundary and the strict apparent boundary coincide. 2.3. Geometric properties of the apparent boundary and the strict apparent boundary
Proof of Theorem 1. Let u,v be two arbitrary points of C. Denote by d(u, v) the Euclidean distance between u and v. Let ul,u2 and t,l,v2 be the extreme points of C such that )',~ = 3',
=
3'.2, xu~ <-"X. <_x.2,
Y,., = Y,. = Yt'2, x,,, <_x,, <_x,. 2. It is easy to see that d(u, v} _
2.3.1. Diameter. The diameter of an object is defined as the maximum Euclidean distance between the extreme points of the object] 2) The diameter of a polygon is defined as the maximum Euclidean distance between the vertices of the polygon. Theorem 1. Let C be the set of points of the edges of an object ~, the diameter of C, of its apparent boundary A, and of its strict apparent boundary A, denoted by D(C),D(A) and ~,4), respectively, are all identical: D(C) = D(A) = D(A). In order to prove this theorem, we need the following lemma. Lemma I. For any set of points C, the set of the vertices of A, denoted by V(A), is a subset of C: V(A) c_ C. Proof. Let P be the polygon obtained at step 5 of Algorithm 1. Since A is a simple polygon obtained from following the contour of P, the vertices of A are either leftmost, rightmost, downmost, upmost points in C or self-intersection points of P. Let p be an arbitrary self-intersection point. We show that p is one of the leftmost, rightmost, downmost or upmost points of C.
The diameter is an important parameter in shape characterization and classification for some sorts of objects. Theorem 1 allows us to compute the diameter of an object via its apparent boundary, or its strict apparent boundary. An application of this result is illustrated in Section 4. 2.3.2. Weak external visibility boundary. As we have pointed boundary and the strict apparent simple polygons. Within the class there is an important subclass externally visible polygons.
of strict apparent out, the apparent boundary are both of simple polygons, known as weakly
Definition 3. For a simple polygon P, let bd(P) denote its boundary. For any point x on the boundary, denote by r(x) an infinite half-line starting at point x and proceeding in any direction. A simple polygon is said to be weakly externally visible if, and only if, for every xebd(P), there exists an r(x) such that Pc~r(x) = {x I. Intuitively, consider a polygon P to be completely surrounded by a circle. If P is weakly externally visible, then the entire boundary of P is visible at one time or another as a guard patrols along the circle.
98
SHAN LIu-Yu and M. THONNAT
Theorem 2. For any set C of points, the strict apparent boundary .4(C) is weakly externally visible. Proof. Let V be the set of vertices of .4, namely, V = Pm~,w P=,, w P=~, w P ~ , (cf. step 3 of Algorithm 2). Let E be the set of internal points on the edges of .4. It is clear that VuE is the set of all (boundary) points of ,4. For all peV, there trivially exists a hairline ray r(p) starting from p such that r(p)r~.4 = {p}. For example, if peP~),,r(p) is the hairline ray starting with p and directing horizontally to the left. For all p~E, let p be on the segment s(u,v), where u, ve V. It is clear that u and v are two successive points in the list L (cf. step 4 of Algorithm 2), and that they are not 8-connected neighbours in the image plan, so we have y.=yo_+l
but
x.
or
x.>x~+l,
x.=x~_+l
but
y.
or
y.>y~+l.
or
Consider the case y. = y~ + 1 and x. < x ~ - I . The other cases can be treated analogously. Owing to the fact that y, = y~ + I and x. < x~ - 1, one sees that either u,v~P~,i, or u, vePX=.. Therefore, the hairline ray starting from p and directing horizontally to the left or to the right satisfies the relation r(p)c~A = {p}. Hence, ,4 is weakly externally visible. • The class of weakly externally visible simple polygons is of interest in computational geometry. For example, the simple linear convex hull algorithm of Sklansky ~) does not work for all simple polygons. (a~ However, it correctly finds the convex hull of any weakly externally visible polygon.(9> Making use of this property, we have implemented Sklansky's algorithm in order to find the convex hulls of planktonic foraminifera images (see Section 4 below). 2.4. Algebraic properties of operators Mt and Mz
Theorem 3. The strict apparent boundary ,4 is a fix point of operators MI and M2: ~'I(A) = A,
~z(A) = A-
Proof. It follows from the proof of Theorem 2 that all the points of ,4 (i.e. those in VuE) are visible from one of the four directions: left, right, up, down. As a consequence, when Algorithm I is applied to .4, the set T = C~t . w C=~,u x , . wm C y, , (step 4) contains only Cmi one closed chain which is composed by all the points of ,4. Therefore, steps 5 and 6 of Algorithm 1 do nothing for A, so that M1(,4) =/~. Similarly, in Algorithm 2, the set V =/~=l, w P~x w P~l.u P ~ . (step 3) contains all the points of//. Hence, a/2(A) = A. •
Corollary I. The operator • = ,af2o,aft is idempotent: ~o~
= 8.
Proof. Let C be the set of points of all the edges of an object ~ It is given by Definition 2 that ,~(C)= ~2o~/1(C) = ,4. From Theorem 3 we have that ato~(c) = ~(~) = d~o~/1(2 ) = ~ , ( 2 ) = 2 =~(c).
•
Theorem 3 and Corollary I show that the strict apparent boundary is stable in the sense that any iteration of the operators ~q/l and Mz results in the same boundary. Therefore, it is a good candidate for the automatic construction of object boundaries. Since the apparent boundary is between the original image and the strict apparent boundary, and since the original image, the apparent boundary and the strict apparent boundary have the same diameter (cf. Theorem 1) and the same convex hull (see Theorem 4 below), both the apparent boundary and the strict apparent boundary provide good approximations of the object boundary.
3. CONVEX HULL OF AN OBJECT AND OF ITS APPARENT BOUNDARY
The convex hull of an arbitrary set of points S or of a polygon P is defined as the smallest convex polygon containing S or P, respectively. The complement of $ or P with respect to its convex hull is called the convex deficiency of S or P. In computer vision, the set S is often an object region and P its boundary. It has been shown that the time complexity of finding the convex hull of n planar points is at least O(n log (n)).(I°- t a) Algorithms that find convex hull in O(n log (n)) time were proposed in various papers, see Day °4) for references. When the points in S are ordered or when P i~ a simple polygon, linear algorithms are proposed in references (I 5-18). Apart from sequential convex hull algorithms, one can also find parallel convex hull algorithms in the literature. "4' 19-23) Let C be the set of points of edges of an object f2. We are now going to show that the convex hull of C, denoted by CH(C), is the same as the convex hull of its apparent boundary A, denoted by CH(A), and is the same as that of its strict apparent boundary ,,], denoted by CH(,4). Recall that A and .~ are simple polygons. For any polygon P, we denote by V(P) the set of its vertices.
Theorem 4. The convex hulls of a set of points C, of its apparent boundary A, and of its strict apparent boundary A are all identical CH(C) = CH(A)= CH(.4). In order to prove this theorem, we need the following lemmas. /_,emma 2. For any polygon P, CH(P) = CH(V(P)).
Proof. Omitted.
•
Lemma 3. For any point veC, v is contained within CH(A), denoted by v-< CH(A).
Proof. Let pt,p2eC be the leftmost and rightmost
Description of object shapes points of C such that Yp, = Yp2= Y~, and
xp, < xv < xo2.
It is easy to see that v -< s(p t, P2). Since p ~,P2e A(C), we have that s(pl,p2).< CH(A). Therefore, v.< CH(A). • Lemma4. For any simple polygon P, CH(P)=
CH(s42(P)). Proof. For any point p (vertex or internal point on an edge) of P, if p is not on M2(P), then one can see from the scanning procedure of Algorithm 2 that p is within M2(P). Hence, we have that CH(P)-< CH(M2(P)). On the other hand, we have that V(~/2(P))c_ V(P), which implies that V(Mz(P))-< V(P). Therefore, CH(P) =
critic(P)).
99
of oceanic deep sea drilling. Foraminifera (about 2000 genera and 30,000 species) are presently the best known and the most comprehensively studied of all the calcareous microfossiis. By the end of the Cretaceous times (60-90Ma), planktonic foraminifera are essentially represented by the globotruncanids, a group which includes very distinctive forms characterized by trochospiral and keeled tests. These foraminifera present a succession of chambers coiled spirally. Within the globotruncanids, differentiation of species is based on test shape, number and shape of chambers in the last whorl, apertural and accessory structures, keels, etc. (see Figs 4, 1l(a), 12(a), and 13(a)).
•
Proof of Theorem 4. It follows from the definition of convex hull and Lemmas I and 2 that CH(A)~< CH(C). On the other hand, according to Lemma 3, we have CH(C) ~ CH(A). Therefore, CH(A) = CH(C). Applying Lemma 4 to the apparent boundary A yields that CH(A)= CH(~4), which completes the proof. •
4.2. Characterization of foraminifera by means of
apparent boundary and convex hull
Generally, we study a planktonic foraminiferon from its three sides, namely, the spiral side, the profile side, and the umbilical side. Here we limit our discussion to the spiral side of planktonic foraminifera (or simply called foraminifera) of group giobotruncanids. We Note that Theorem I can also be considered as a study their features directly on the scanning electronic consequence of Theorem 4 due to the fact that a microscope (SEM) photographs. (z4) polygon and its convex hull have the same diameter. We use the apparent boundary, the convex hull and the distance curve (see below) to extract information on the boundary shape, the number of chambers in the 4. APPLICATIONOF APPARENTBOUNDARY AND CONVEXnULL TO "rug SHAPE last whorl and the diameter of the foraminifera. CHARACTERIZATIONOF PLANKTONIC The scheme of our process is shown in Fig. 3. We FORAMINIFERA first detect the edges of the foraminiferon by the method of Deriche, (2s) followed by the hysteresis 4.1. A brief presentation of planktonic foraminifera thresholding (see Figs 4-6). We notice that the boundary The foraminifera are single-celled protists which live of the foraminiferon in Fig. 6 is not only broken but either on the sea floor (benthic foraminifera) or among also connected to the internal structures of the forathe marine plankton (planktonic foraminifera). The miniferon. We thus compute the apparent boundary protoplasmic mass of the foraminiferid cell is enclosed of the foraminiferon (see Fig. 7). Note that in most cases within a mineralized test (or shell, usually less than of our study, the strict apparent boundary of a foraI mm in diameter) which may consist of one or more miniferon remains the same as its apparent boundary, cavities termed chambers. therefore we here use only the notion of the apparent Foraminiferid tests are abundant in marine sedi- boundary. We then compute the convex hull (Fig. 8). ments and are known from early Cambrian (600 Ma) With the apparent boundary and the convex hull in through Recent times. They constitute invaluable hand, the convex deficiency can easily be obtained (see indicators in the age determination as well as the Fig. 9). We further compute the distances between the depositional environment of sedimentary strata. The apparent boundary and the convex hull, and we obtain impetus given by oil companies accelerated the study a distance curve (see Fig. 10). of these micro-organisms in the 1950s. Recently, microFigures I 1-13 are three other examples. Note that paleontology expanded greatly with the development the scales of all the distance curves are not the same.
image of a H foraminiferon
edge detection
Fig. 3. Diagram of the processing of foraminiferid images.
I00
SH^N LIu-Yu and M. THONNAT
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Fig. 7. The apparent boundary of the foraminiferon shown in Fig. 4.
4.2.1. Recognition of boundary shapes. One of the most important criteria in the identification of foraminifera is their boundary shapes. Seen from the spiral side, the foraminifera commonly have four types of boundary, namely, lobate boundary, spinal boundary, circular or subcircular boundary, and polygonal boundary (see Figs 4, l l(a), 12(a), and 13(a), respectively). These shapes can be identified by using the distance curve and the convex hull as follows (see Fig. 14). Observe first that a distance curve with important and nearly equally separated peaks represents a lobate boundary or a spinal boundary; while a distance curve
with possibly one important peak and many other small and irregular peaks represents a circular or a polygonal boundary. Thus lobate boundaries and spinal boundaries can be separated from circular boundaries and polygonal ones. Observe also that spinal boundaries correspond to convex peaks; whereas lobate boundaries correspond to concave peaks. Therefore, we just need to detect the concavity of the peaks in the distance curves to distinguish lobate and spinal boundaries. Circular boundaries can be separated from polygonal ones by the fact that the convex hull of a circular boundary has many more vertices than that of a polygonal boundary.
Description of object shapes
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Fig. 9. The convex deficiency of the foraminiferon shown in Fig. 4.
4.2.2. Number of chambers. Another important measure of foraminifera is the number of chambers in the last whorl. For the foraminifera of lobate and spinal boundaries, this can be computed by the number of peaks in their distance curves. Indeed, the peaks in the distance curve indicate the junction points of neighbouring chambers for foraminifera of lobate boundaries: whereas for the foraminifera of spinal boundaries, it is the valleys that indicate the junction points of neighbouring chambers. A confidence coefficient should also be calculated in order to validate the counting process. The above method cannot be used to count chamber numbers for foraminifera of circular boundaries or
of polygonal boundaries. The counting of chamber numbers of these foraminifera is performed by another method, which is outside the scope of this paper.
4.2.3. Diameter. Owing to Theorem 1. the diameter of a foraminiferon can be obtained from its apparent boundary. Moreover, since the diameter of a polygon is the same as that of its convex hull, ~Sjthe diameter of a foraminiferon can be computed efficiently from the convex hull of its apparent boundary. There are many proposed algorithms which compute a two-dimensional (2D) diameter with linear time complexity in terms of the number of verticesJ 26'2~)
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Description of object shapes
105
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Fig. 15. (a) Normal scanning; (b) enhanced scanning at the pocket.
5. C O N C L U S I O N AND DISCUSSIONS
In this paper, we have introduced the notions of the apparent boundary and the strict apparent boundary of an object. We have shown some geometric properties of these two boundaries and the algebraic properties of our operators which find these boundaries. We have proven that the convex hulls of an object, of its apparent boundary, and of its strict apparent boundary are all identical. We have used the apparent boundary, the convex hull and the distance curve for the shape characterization of planktonic foraminifera. It is worthwhile noticing that the apparent boundary and the strict apparent boundary provide an automatic approach of boundary detection. They are particularly useful when edge detection algorithms yield boundaries that are broken and/or connected to the internal structures of the objects. Owing to the properties that we have established in this paper, these boundaries are good approximations of the object boundary. Note also that the convex hull as well as the distance curve are useful tools for object shape analysis, in particular, those of natural organisms, e.g. zooplanktons, fishes, and plants. We can solve the problem of shape recognition of these objects in an analogous way to the one presented in this paper. Like some other methods in computer vision, our method does not work for all objects. For instance, for objects possessing deeply folded "pockets", the apparent boundary may introduce some modifications to the shapes of the "pockets" (cf. Fig. 15(a)). This is due to the fact that in our algorithms, we scan the object only in two perpendicular directions. However, it is possible, at the expense of the computation time, to scan the object in all the directions so that all the weakly
externally visible boundary points of the object are included in the apparent boundary. In order to reduce the computational cost, one can perform such "enhanced scanning" only at the position of the pockets (which can trivially be detected by the distance curve), as illustrated in Fig. 15(b). Finally, we would like to point out that the identification of foraminifera has always been a tedious and time-consuming work for paleontologists. There exist a few expert systems to assist the identification of foraminifera. However, the extraction of the descriptive parameters is carried out by human beings. Thus the results of this paper, together with other parameters we have detected by different methods, can be used as the input to these expert systems. The next step of our research work is to design an expert system based on the shell CLASSIC, as was used in the automatic classification of zooplanktons and fish.~8~ Our work is aimed at automating both the extraction of the descriptive features and the identification of the specimens of foraminifera which play an important role in oil exploration. 6. SUMMARY
In this paper, we introduce the notions of the apparent boundary and the strict apparent boundary. These two boundaries are used to approximate the boundary of an object when boundary detection algorithms do not yield a satisfactory result, especially when the detected boundary is broken or connected to the internal structure of the object. The following properties are established in the paper: (I) an object, its apparent boundary and its strict apparent boundary have the same diameter; (2) an object, its apparent boundary and its strict apparent boundary have the same convex hull; (3) the strict
106
SHAN LIu-Yu and M. THONNAT
apparent boundary is a fixed point of the two operators that compute the apparent boundary and the strict apparent boundary; (4) the strict apparent boundary is weakly externally visible. The properties (1), (2) and (3) indicate that both the apparent boundary and the strict apparent boundary provide good approximations of the real object boundary. Moreover, the diameter and the convex hull of an object can be computed from its apparent boundary or from its strict apparent boundary. Property (4) allows us to find the convex hull of an object by the simple linear algorithm of Sklansky. As an application of our method, we use the apparent boundary and the convex hull for the shape characterization of planktonic foraminifera. Based on the apparent boundary and the convex hull, we construct a distance curve which allows us to recognize different b o u n d a r y shapes of foraminifera. Other information about the shape of foraminifera is also obtained from the convex hull and the distance curve, such as the number of chambers in the last whorl and the diameter of foraminifera. Acknowledgements--The authors are grateful to Professor Pierre Saint-Marc of Universit6 de Nice, France, for having provided us with images of planktonic foraminifera, and for helpful discussions on the characteristics and the classifications of planktonic foraminifera. The authors would also like to thank Professor Pierre Saint-Marc and the referee for their useful comments on a draft of the paper.
REFERENCES
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8. A. Bykat, Convex hull of a finite set of points in two dimensions, lnf. Process. Lett. 7(6), 296-298 (1978). 9. G. T. Toussaint and D. Avis, On a convex hull algorithm for polygons and its application to triangulation problems, Pattern Recognition 15, 23-29 (1982). 10. P.V. Erode Boas, On the D(nlogn) lower bound for convex hull and maximal vector determination, Inf. Process. Lett. 10(3), 132-136 (1980). 11. F. P. Preparata and S. J. Hong, Convex hulls of finite sets of points in two and three dimensions, CACM 87-93 (1977). 12. M. I. Shamos and D. Hoey, Closest point problems, Proc. 16th A. Syrup. Found. Comput. Sci., IEEE, New York, pp. 151-162 (1975). 13. A. Yao, A lower bound to finding convex hulls, J. Ass. Comput. Mach. 28(4), 780-787 (1981). 14. A. M. Day, Planar convex hull algorithms in theory and practice, Computer Graphics Forum 7, pp. 177-193. NorthHolland, Amsterdam (1988). 15. S. K. Ghosh, A linear time algorithm for computing the convex hull of an ordered crossing polygon, Pattern Recognition 17, 351-358 (1984). 16. R. L. Graham and F. F. Yao, Finding the convex hull of a simple polygon, J. Algorithms No. 4, 324-331 (1983). 17. D. McCallum and D. Avis, A linear algorithm for finding the convex hull of a simple polygon, Inf. Process. Lett. 9(5), 201-206 (1979). 18. A. A. Melkman, On-line construction of the convex hull of a simple polygon, Inf. Process. Lett. 25, l l-12 (April 1987). 19. Z. Hussain, A fast approximation to a convex hull, Pattern Recognition Left. 8(5), 289-294 (1988). 20. C. S. Jeong and D. T. 1.,¢¢,Parallel convex hull algorithms in 2D and 3D on mesh-connected computers, Int. Conf. Comput. Vision Display, pp. 64-76 (1988). 21. R. Miller and Q. F. Stout, Efficient parallel convex hull algorithms, IEEE Trans. Comput. 37(12), 1605-1618 (1988). 22. J. Sklansky, L.P. Cordella and S. Levialdi, Parallel detection of concavities in cellular blobs, IEEE Trans. Comput. 25(2), 187-195 (1976). 23. I. Stojmenovic and D.J. Evans, Comments on two parallel algorithms for the planar convex hull problem, Parallel Computing, Vol. 5, pp. 373-375. North-Holland Amsterdam (1987). 24. F. Robaszynski, M. Caron, J. M. Gonzalez Donoso and A. H. Wonders, Atlas of Late Cretaceous GIobotruncanids, Revue Micropal~ontologie26(3-4), 145- 302 (1984). 25. R. Deriche, Optimal edge detection using recursive filtering, ICCV, pp. 501-505, London (1987). 26. M. A. Fischler, Fast algorithms for two maximal distance problems with applications to image analysis, Pattern Recognition 12, 35-40 (1980). 27. M. I. Shamos, Geometric complexity, Proc. 7th ACM Syrup. Theory Comput., pp. 224-233, May (1975). 28. M. Thonnat and M.-H. Gandelin, An expert system for the automatic classification and description of zooplanktons from monocular images, ICPR9, Roma, Italy, pp. 114-118, November (1988).
About the Autbor--SHAN LIu-Yu received the diploma of Bachelor of Engineering in biomedical instrumentation from Shanghai Jiao Tong University, People's Republic of China, in 1985. From 1985 to 1986, she was a graduate student in medical imaging in Shanghai Jiao Tong University. She is currently preparing a Ph.D. thesis at CMA, Ecole Nationale Sup~:rieure des Mines de Paris, France. Her research interests include image processing, pattern recognition, and medical imaging.
Description of object shapes About the Author-- MONIQUETHONNATgraduated from the Ecole Nationale Sup6rieure de Physique in Marseille in 1980. She received her Ph.D. in optics and signal processing from the University of St J6rome, Marseille, France, in 1982, having prepared her thesis at the Laboratorie d'Astronomie Spatiale of Centre National de la Recherche Scientifique, Marseille, France, in 1981 and 1982. In 1983, she joined the lnstitut National de Recherche en Informatique et Automatique as a research scientist. Her research interests include image understanding, image processing and artificial intelligence. She has been involved in the building of several vision expert systems in astronomy, biology and robotics.
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