A 3D 6-subiteration thinning algorithm for ... - Semantic Scholar
Pattern Recognition Letters 19 Ž1998. 613–627
A 3D 6-subiteration thinning algorithm for extracting medial lines Kalman ´ ´ Palagyi ´ ) , Attila Kuba Department of Applied Informatics, Jozsef Attila UniÕersity, P.O. Box 652, H-6701 Szeged, Hungary ´ Received 24 March 1997; revised 16 January 1998
Abstract Thinning is a frequently used method for extracting skeletons in discrete spaces. This paper presents an efficient parallel thinning algorithm that directly extracts medial lines from elongated 3D binary objects Ži.e., without creating medial surface.. Our algorithm provides good results, preserves topology and it is easy to implement. q 1998 Elsevier Science B.V. All rights reserved. Keywords: 3D parallel thinning algorithms; Discrete topology; Topology preservation
1. Introduction Skeletonization provides shape features that are extracted from binary image data. It is a common preprocessing operation in raster-to-vector conversion or in pattern recognition. Its goal is to reduce the volume of elongated objects to their skeletons. In the 3D Euclidean space, the skeleton Žor the medial surface. of a geometry is the locus of the centers of all the maximal inscribed spheres ŽBlum, 1967.. In discrete spaces, the thinning procedure is frequently used method for generating an approximation to the ‘‘true’’ Euclidean skeleton in a topology preserving way ŽKong and Rosenfeld, 1989.. Border points of the binary object that satisfy certain topological and geometric constraints are deleted in iteration steps.
Deletion means that object elements are changed to background elements. The entire process is repeated until only the ‘‘skeleton’’ is left. In this paper, a new 3D thinning algorithm is proposed that produces medial lines Žconsisting of arcs andror curves instead of surfaces.. Producing medial lines is often useful in medical image analysis Že.g., identifying the 3D midline of vessels or airways.. Each iteration step contains 6 successive subiterations. Subiterations can be executed in parallel. That is, the object elements that satisfy the actual deletion conditions are to be changed to background elements simultaneously. The proof of topology preservation is given.
2. Basic notions and results
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Corresponding author.
In this section, some basic notions of 3D digital topology are presented Žsee Kong and Rosenfeld, 1989..
0167-8655r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 8 6 5 5 Ž 9 8 . 0 0 0 3 1 - 2
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Let x s Ž x 1 , x 2 , x 3 ., y s Ž y 1 , y 2 , y 3 . be two points with integer coordinates in the 3D digital space denoted by Z 3, and let us consider the Euclidean distance 5 xyy5s
(
3
Ý Ž x i y yi . 2 . is1
The points x and y are said to be 6-adjacent if 5 x y y 5 ( 1, 18-adjacent if 5 x y y 5 ( '2 , and 26adjacent if 5 x y y 5 ( '3 . Let us denote by Nj Ž p . the set of points j-adjacent to point p, for j s 6,18,26 Žsee Fig. 1.. The point p is said to be j-adjacent to the non-empty set of points X if there is point x g X, such that x g Nj Ž p .. The sequence of distinct points ² x 0 , x 1 , . . . , x n : is a j-path of length n 0 0 from point x 0 to point x n in a non-empty set of points X if each point of the sequence is in X and x i is j-adjacent to x iy1 for each 1 ( i ( n. ŽNote that a single point is a j-path of length 0.. Two points are j-connected in the set X if there is a j-path in X between them. A set of points X is j-connected if any two points in X are j-connected in X. The 3D binary Ž m,n. digital picture P is a quadruple P s ŽZ 3,m,n, B ., where Ž m,n. is a pair of adjacencies ŽŽ m,n. s Ž6,26., Ž26,6., Ž6,18., or Ž18,6. are generally considered.. Each element in Z 3 is called point or Õoxel of P. A point in B : Z 3 is called black point; a point in Z 3 _ B is called white point. Picture P is called finite if B is a finite set. Value 1 is assigned to each black point; value 0 is
Fig. 1. The frequently used adjacencies in Z 3. Points being in N6 Ž p . are marked U, N, E, S, W, and D; points being in N18 Ž p . but not in N6 Ž p . are marked ‘‘ Ø ’’; points being in N26 Ž p . but not in N18 Ž p . are marked ‘‘(’’. ŽNote that point p is in sets N6 Ž p ., N18 Ž p ., and N26 Ž p ...
assigned to each white point. Equivalence classes of B under m-connectivity are called black m-components or objects. A singleton black component is called isolated point. The n-connectiÕity relation and the white n-component can be defined in the same way. In a finite picture there is a unique infinite white component, which is said to be background. A finite white component is called caÕity in a picture. A black point is said to be border point if it is 6-adjacent to at least one white point. Note that this definition is correct only for the special cases m s 26 and m s 18. A border point p is called U-border point if the point marked by U in Fig. 1 is white. We can define N-, E-, S-, W-, and D-border points in the same way. There is an additional concept in 3D topology called hole or tunnel. A black component with a hole is topologically equivalent to a torus Ža doughnut-like object.. It is difficult to define the hole mathematically. Fortunately, the number of holes, what we need later, can be determined relatively easily. Let us suppose that the number of holes of a picture P denoted by H Ž P . is determined. Denote by O Ž P . and C Ž P . the number of objects Žblack components. and the number of cavities of P, respectively. The 3D Euler characteristic x Ž P . is defined by the global formula
x Ž P. sO Ž P. yH Ž P. qC Ž P. . Iterative thinning algorithms delete border points that satisfy certain conditions. The entire process is repeated until there are no more black points to be changed. The reduction is an operation in which only black points are deleted whereas white points should remain white. Accordingly, thinning is a reduction operation. Each thinning algorithm should satisfy the requirements of topology preservation. Kong and Rosenfeld Ž1989. discussed some definitions of topology preservation of 3D reduction operations. The most important is based on simple or deletable points introduced by Morgenthaler Ž1981.. Roughly speaking, a simple point is a point whose deletion does not change the topology of the picture. The following criterion has been stated: Criterion 1 ŽKong and Rosenfeld, 1989.. Let p be a black point of a 3D picture ŽZ 3 ,m,n, B .. The point p
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is simple if and only if all of the following conditions hold: 1. p is m-adjacent to only one m-component of N26 Ž p . l Ž B _ p4.. 2. p is n-adjacent to only one n-component of N26 Ž p . _ B. 3. x ŽŽZ 3 ,m,n, B l N26 Ž p ... s x ŽŽZ 3,m,n,Ž B _ p4. l N26 Ž p .... According to Criterion 1, simplicity can be decided locally by examining the 3 = 3 = 3 neighbourhood of black points. Saha and Chaudhuri Ž1994. proposed the following conditions for Ž26,6. pictures. Theorem 2. Black point p is simple in the picture P s ŽZ 3,26,6, B . if and only if all of the following three conditions hold: 1. The set Ž B _ p4. l N26 Ž p . contains exactly one 26-component. 2. The set ŽZ 3 _ B . l N6 Ž p . is not empty. 3. Any two points in ŽZ 3 _ B . l N6 Ž p . are 6-connected in the set ŽZ 3 _ B . l N18 Ž p .. It is easy to see that simultaneous removal of a set of simple points may not preserve topology. Let us consider a digital picture containing only a black cube of size 2 = 2 = 2. Every voxel of this cube is simple but their deletion will cause complete elimination of an object. This example shows that the simplicity of a set of points does not equal the set of simple points. Definition 3 ŽMa, 1994.. Let P be a picture. The set D s d1 , . . . ,d k 4 of black points is called a simple set of P if D can be arranged in a sequence ² d i , . . . ,d i : in which each d i is simple after 1 k j d i , . . . ,d i 4 is deleted from P, for j s 1, . . . ,k. 1 jy 1 ŽBy definition, let the empty set be simple.. Definition 4 ŽMa, 1994.. A 3D parallel reduction operation is topology preserÕing if, for all possible 3D pictures, the set of all points that are deleted simultaneously by the operation is simple. Fortunately, we do not have to test the algorithms for all possible 3D pictures. Ma Ž1994. has a signifi-
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cant result: we only need to check a rather small number of configurations to prove topology preserving. Theorem 5 ŽMa, 1994.. A 3D parallel reduction operation preserÕes topology for Ž26,6. pictures if all of the following conditions hold: 1. Only simple points can be deleted. 2. If two black corners, p and q, of a 2 = 2 square in Z 3 are deleted, then the set p,q4 is simple. 3. If three black corners, p, q and r, of a 2 = 2 square in Z 3 are deleted, then the set p,q,r 4 is simple. 4. If four black corners, p, q, r and s, of a 2 = 2 square in Z 3 are deleted, then the set p,q,r, s4 is simple. 5. No black component contained in a 2 = 2 = 2 cube in Z 3 can be deleted completely. Topology preservation is not the only requirement to be complied with. It is easy to see that any black component without a hole and a cavity can be shrunk into an isolated point by iterative deletions of simple points. Of course, this is not what we want. That is why, additional conditions are stated to supply the ‘‘skeleton’’. Criterion 6 ŽRosenfeld, 1975; Tsao and Fu, 1981.. A 3D reduction operation is a 3D thinning operation if it fulfills the following three conditions. 1. Topology is preserved. 2. Isolated points and curves remain unchanged. 3. An object of size 2 = 2 = 2 will be changed after applying the operation. ŽNote that a cube-shaped object is to be investigated and not a 2 = 2 = 2 part of an elongated object.. Existing 3D thinning algorithms can be classified from several points of view. One of them is the classification on the produced ‘‘skeletons’’: most of the developed algorithms result in medial surfaces ŽMorgenthaler, 1981; Reed, 1984. and a few produce medial lines Žsee Fig. 2.. The aim of the 3D thinning algorithms that are developed to extract medial lines is to shrink an object to a voxel list. A voxel list contains three kinds of points: line-end points, line
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Fig. 2. A 3D synthetic picture containing a character ‘‘A’’ Žleft., its medial surface Žcentre., and its medial lines Žright.. ŽCubes represent black points..
points, and cross points. A black point p is line-end point if there is exactly one black point m-adjacent to p. A black point p is line point Žor curÕe point . if there are exactly two black points m-adjacent to p. A black point p is cross point if there are more than two black points m-adjacent to p. Three types of thinning algorithms have been developed. The first type examine the 3 = 3 = 3 neighbourhood of each border point. The iteration steps are divided into more subiterations. Only the border points having the prescribed 3 = 3 = 3 neighbourhood can be deleted during a subiteration. These are called border sequential algorithms. Generally, a
subiteration is executed in parallel ŽTsao and Fu, 1981; Gong and Bertrand, 1990; Lee et al., 1994; Bertrand, 1995; Palagyi ´ and Kuba, 1997.. It means that all the black points satisfying the deletion condition of the actual subiteration are deleted simultaneously. The second type of algorithms do not need subiterations but they investigate larger neighbourhood ŽMa, 1995; Ma and Sonka, 1996.. The third approach is the subfield sequential method. The picture is divided into more distinct subsets which are successively activated. At a given iteration, only black points being in the active subfield are meant to be deleted ŽBertrand and Aktouf, 1994..
Fig. 3. Base masks M1–M6 and their rotations around the vertical axis form the set of masks M belonging to the deletion direction U. ŽNote, that at least one point marked ‘‘x’’ is black in masks M1 and M5..
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Some algorithms have been developed to generate medial surface ŽGong and Bertrand, 1990; Bertrand, 1995; Ma, 1995; Bertrand, 1995. and others allow producing medial surface and medial lines, too ŽTsao and Fu, 1981; Lee et al., 1994; Bertrand and Aktouf, 1994..
3. The new thinning algorithm In this section, a new algorithm is described for thinning 3D binary Ž26,6. pictures. Our 6-subiteration border sequential 3D thinning algorithm directly creates medial lines, without ex-
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tracting medial surface. The algorithm is based on a ‘‘careful’’ Boolean characterization of simple points which are not line-end points, i.e., which are not extremities of curves. The new value of each voxel depends on its 3 = 3 = 3 neighbourhood. This dependence is given by a set of configurations of 3 = 3 = 3 lattice points. Only deletion conditions are used Ži.e., black points can be changed but white points cannot.. The deletion condition assigned to a subiteration is described by a set of masks Žor matching templates.. A black point is to be deleted if and only if its 3 = 3 = 3 neighbourhood matches at least one element of the given set of masks. The masks are constructed according to the six deletion directions U, D, N, E, S,
Fig. 4. Thinning of synthetic pictures of size 24 = 24 = 24 Žleft.; their medial lines extracted by using Algorithm 1 Žright..
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Fig. 4 Žcontinued..
W. These directions correspond to the 6-neighbours and deletion of the given type of border points are taken into consideration. The masks assigned to the direction U give the condition to delete certain U-border points. The masks M1–M6 assigned to the direction U are given by Fig. 3. Masks are usually described by three kinds of elements, ‘‘1’’ Žblack., ‘‘0’’ Žwhite., and ‘‘.’’ Ž‘‘don’t care’’., where ‘‘don’t care’’ matches either black or white point in a given picture. In order to reduce the number of masks we use the following notation: at least one point marked ‘‘x’’ is black. Additionally, all rotations around the vertical axis Žaccording to the U and D directions. of the base masks M1–M6 are masks, too. ŽThe rotation angles are 908, 1808, and 2708.. Let M denote the set of
masks assigned to the deletion direction U. Ž M contains the base masks M1–M6 and their rotated versions.. Deletion conditions of the other five subiterations Žassigned to the directions D, N, E, S, and W. can be derived from the appropriate rotations and reflections of the masks in M . Let P s ŽZ 3,26,6, B . be the finite 3D picture to be thinned. Since set B is finite, it can be stored in a finite 3D binary array X. ŽOutside of this array every voxel will be considered as 0 Žwhite... Let us suppose that a subiteration corresponding to the deletion direction D results in a picture P X s ŽZ 3,26,6, BX . from picture P. Denote by T Ž X, D . the binary array representing the picture PX , where D is one of the six deletion directions. We are now ready to present our thinning algorithm formally.
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Fig. 4 Žcontinued..
Algorithm 1 for thinning the picture P s ŽZ 3 ,26,6, B .: Input: binary array X representing the picture P Output: binary array Y representing the thinned picture begin Y s X; repeat Y s T Ž Y,U.; Y s T Ž Y,D.; Y s T Ž Y,N.; Y s T Ž Y,S.; Y s T Ž Y,E.; Y s T Ž Y,W.; until no points are deleted; end.
Note that choosing another order of the deletion directions yields another algorithm. Algorithm 1 terminates when there are no more black points to be deleted. Since all considered input pictures are finite, Algorithm 1 will terminate. The masks of our algorithm can be regarded as a ‘‘careful’’ characterization of simple points for 3D thinning. This is a Boolean characterization since each of our masks can be given by a Boolean condition. It makes easy implementation possible.
4. Discussion Algorithm 1 have been tested for several synthetic pictures. Here we present some examples. Thinning
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for six synthetic 3D objects can be seen in Fig. 4. However, our algorithm is border-sequential, it creates regular and symmetric skeletons from regular and symmetric objects. Lee et al. Ž1994. discussed different methods of characterization of 3D simple points. There are 2 26 Žs 67 108 864. different possible 3 = 3 = 3 configurations of voxels around a black point p. The point p is simple in 25 984 552 cases of these configurations. It is easy to see that the masks in M Žbelonging to the direction U. can delete more than 2 20 Žs 1 048 576. kinds of simple points. Note that the ‘‘goodness’’ of a thinning algorithm is not directly proportional to the number of the simple points that are recognized as points to be deleted. The thinning process may produce undesired branches due to noisy
boundary. Experimental results of Fig. 5 are to demonstrate the behavior of Algorithm 1 under noise. Skeleton generation methods have to take care of the following two aspects ŽSzekely, 1996.: ´ Ø the geometrical one, requiring that the ‘‘skeleton’’ is in its geometrically correct position, i.e., the ‘‘middle’’ of the object; Ø the topological one, forcing the ‘‘skeleton’’ to retain the topology of the original object, i.e., topology is to be preserved. Thinning based skeletonization concentrates on the topological aspect. Directional 6-subiteration 3D thinning algorithms ŽTsao and Fu, 1981; Gong and Bertrand, 1990; Lee et al., 1994; Bertrand, 1995. use the distance induced by the 6-adjacency, which cannot be regarded as a good approximation to the Euclidean distance. Therefore, these algorithms are
Fig. 5. Noise sensitiveness of Algorithm 1. Synthetic pictures and their noisy versions Žleft.; their medial lines Žright..
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Fig. 5 Žcontinued..
not invariant under rotation. The proposed algorithm is sensitive to object orientation, too. Two examples are presented in which Algorithm 1 produces relatively invariant results under object rotation, see Fig. 6.
ing condition 1 of Theorem 5 can be seen by using Theorem 2. In order to prove the conditions of Theorem 2 and Theorem 5 we state some properties of the set of masks M belonging to the deletion direction U Žsee Fig. 3.. We refer to the individual mask positions by assigning indices to N26 Ž p . Žwhere p g Z 3 ., see Fig. 7.
5. Verification In this section, we show that Algorithm 1 is a 3D thinning algorithm in the sense of Criterion 6. It is sufficient to prove that the first subiteration belonging to the deletion direction U fulfills all conditions of Criterion 6. The proof of condition 1 of Criterion 6 Ži.e., topology preserving. is based on Theorem 5. Satisfy-
Proposition 7. The position corresponding to the index 5 Ž see Fig. 7. is white in all masks in M . Ž In other words: masks in M can delete only some U-border points.. Proposition 8. At least one position corresponding to the set of indices 18, . . . ,264 is black in all masks in M .
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Fig. 5 Žcontinued..
Proposition 9. All of the following four statements hold for each mask in M : Ø if the mask position haÕing the index 2 may be black, then the mask position haÕing the index 11 must be black, too, Ø if the mask position haÕing the index 4 may be black, then the mask position haÕing the index 13 must be black, too, Ø if the mask position haÕing the index 6 may be black, then the mask position haÕing the index 14 must be black, too, and Ø if the mask position haÕing the index 8 may be black, then the mask position haÕing the index 16 must be black, too. Let us classify the mask positions now. A mask position of a mask in M is called black position if it
is always black. A mask position is called white position if it is always white. The other mask positions which are not white and not black, are called potentially black positions. A black or potentially black mask position is called non-white position. Proposition 10. If a black point p can be deleted by any mask in M and the black point q g Ž N18 Ž p . _ N6 Ž p .. coincides with a black position of this mask, then point q cannot be deleted by M . Ž It implies Propositions 1 and 3.. Proposition 11. If a black point p can be deleted by the base masks M2, M3, M4 or their all rotated Õersions and the black point q coincides with the mask position haÕing the index 22 Ž see Fig. 1., then point q cannot be deleted by any mask in M . Ž It implies Proposition 1..
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Proposition 12. If a black point p can be deleted by any mask in M from a picture ŽZ 3 ,26,6, B ., then the set Ž N26 Ž p . _ p4. l B contains at least two elements. The first thing to be proved is that any arbitrary black point deleted by the set of masks M is simple Žsee condition 1 of Theorem 5.. Lemma 13. Let p be a black point in the picture P s ŽZ 3,26,6, B .. If point p can be deleted by at least one mask in M , then p is simple. Ž In other words, each black point deleted by the first subiteration of Algorithm 1 from picture P is simple.. Proof. We show that all conditions of Theorem 2 are satisfied. These conditions will be proved only for
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the six base masks M1–M6, since the simplicity is a rotation invariant property. First we show that the set Ž B _ p4. l N26 Ž p . contains exactly one 26-component Žcondition 1 of Theorem 2. and any two points in ŽZ 3 _ B . l N6 Ž p . are 6-connected in the set ŽZ 3 _ B . l N18 Ž p . Žcondition 3 of Theorem 2. with the help of Fig. 8. Labels A, B, and C are assigned to the black positions. Labels a, b, and c are assigned to non-white positions 26-adjacent to the black position labelled by A, B, and C, respectively. Obviously, a position may be 26-adjacent to more black positions. In this case, it is multiply labelled. We show that there is a 26-path containing only non-white positions between any two non-white points. The only black position in base masks M1 and M5 is labelled by A. We can state that all potentially
Fig. 6. Invariance of the proposed algorithm under object rotation. The two original synthetic objects and their rotated versions Žleft.; their medial lines extracted by Algorithm 1 Žright..
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Fig. 6 Žcontinued..
black points are labelled by a in these masks. It is easy to see that any two non-white points are 26connected via an at most two-length 26-path. The
sequence of labels in this path can be ²a,A: or ²a,A,a:. Base masks M2 and M6 contain two black positions labelled by Ab and aB. All potentially black points are labelled in these masks. It is easy to see that any two non-white points 26-connected via an at most three-length 26-path. The sequence of labels in this path can be ²a?,Ab,a?: , ²a?,Ab,aB,?b: , ² ?b,aB,?b: ,
Fig. 7. Indices 0,1, . . . ,26 are assigned to N26 Ž p ..
where ’’?’’ can replace characters a, b, and the empty string. Note that any prefix of a path is a path, too.
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Fig. 8. Labels assigned to the non-white positions of the base masks M1–M6. Thick lines denote the required 6-paths in ŽZ 3 _ B . l N18 Ž p ..
Base masks M3 and M4 contain three black positions labelled by Abc, aBc, and abC. Each potentially black point is labelled in these masks. It is easy to see that any two non-white points are 26-connected via an at most three-length 26-path. The sequence of labels in this path can be ²a?,Abc,a?: , ²a?,Abc,aBc,?b?: ²a?,Abc,abC,?c: , ² ?b?,aBc,?b?: , ² ?b?,aBc,abC,?c: , ² ?c,abC,?c: , where ’’?’’ can replace any string over the alphabet a,b,c4 . It was shown with the help of Fig. 8 that for each base mask, any two non-white points are 26-connected. Proposition 12 shows that point p may not be an isolated point in P. Therefore, the set Ž B _ p4. l N26 Ž p . contains exactly one 26-component. Condition 3 of Theorem 5 can be proved with the help of Fig. 8, too. It is to be proved that any two points in ŽZ 3 _ B . l N6 Ž p . are 6-connected in the set ŽZ 3 _ B . l N18 Ž p .. To prove this property, we give the required 6-paths. Finally, masks in M satisfy condition 2 of Theorem 2 Ži.e., the set ŽZ 3 _ B . l N6 Ž p . is not empty. by Proposition 7. I
The proofs of Lemma 13 and Proposition 7 yield the following statement. Corollary 14. Let p and q be two black points in picture P s ŽZ 3 ,26,6, B . so that q g N6 Ž p .. If points p and q can be deleted by M , then any two points in ŽŽZ 3 _ B . j q4. l N6 Ž p . are 6-connected in the set ŽŽZ 3 _ B . j q4. l N18 Ž p .. With the help of Corollary 8 and the following two lemmas, the remaining conditions of Theorem 5 can be proved easily. Lemma 15. Let p be a black point in picture P s ŽZ 3 ,26,6, B . and let Q be any set of black points so that Q : Ž N18 Ž p . _ p4. l B. If point p can be deleted by at least one mask in M , then p is simple in picture PX s ŽZ 3,26,6, B _ Q .. Ž In other words, p is simple after the deletion of Q.. Proof. We know that it holds for Q sœ 0 by Lemma 13. Let us suppose that Q /œ 0 . We show that the proposition holds if point p can be deleted by one of the base masks M1–M6. If p is deleted by a mask in M1,M4,M5,M64 , then any black mask position that is 18-adjacent to point p may not coincide with any point q g Q by Proposition 10. In this case each point in Q must
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Fig. 9. The critical configuration if p is deleted by mask M2.
coincide with a potentially black mask position. Simplicity of p does not depend on the deletion of any potentially black position Žsee the proof of Lemma 7.. If p is deleted by the mask M2, then the black position having the index 22 may not be deleted by Proposition 11. Therefore, that point is not in Q. Two cases are to be distinguished: Ža. the black point that coincides with the other black position Žhaving the index 11. is not in set Q, and Žb. the black point q that coincides with the other black mask position is in set Q. Let us consider the case Ža.. Each point in Q must coincide with a potentially black position, therefore, the simplicity of point p is not altered by the deletion of the set Q. In case Žb. the point coincides with the mask position having the index 2 is white by Proposition 7. There are two positions being not 26-adjacent to the black position having the index 22 and not being 18-adjacent to point p. These points are denoted by r and s in Fig. 9. They may not be in Q because each point in Q is 18-adjacent to p. Let us consider other two points denoted by t and u in Fig. 9. There are four cases: Žba. r s 0 and s s 0, Žbb. r s 0 and s s 1, Žbc. r s 1 and s s 0, and Žbd. r s 1 and s s 1. In case Žba., there is no problem. If case Žbb. holds, then u s 1 by Proposition 9 Žbecause point q can be deleted.. It is known that u f Q by Proposition 7. Since u is 26-connected to the black point that coincides with the black position
having the index 22, then s is not separated by the deletion of q. Therefore, condition 1 of Theorem 2 is satisfied. For axial symmetry, it holds for case Žbc., too. In case Žbd., t s 1 and u s 1 by Proposition 9, and t f Q and u f Q by Proposition 7. Since t and u are 26-connected to the black point that coincides with the black position having the index 22, r and s are not separated by the deletion of q. Therefore, condition 1 of Theorem 2 holds. Conditions 2 and 3 of Theorem 2 hold by Proposition 7 and Corollary 14, respectively. Therefore, p is simple in picture PX in case Žb.. We can prove in the same way that point p remains simple after deletion Q if p is deleted by the mask M3. It was shown that the proposition holds if point p is deleted by the base masks M1–M6. Obviously, it is also true for their rotated versions. I Note that the set Q j p4 in Lemma 15 contains any kinds of two or three or four black corners of a 2 = 2 lattice square. Therefore, we proved a more general property than conditions 2, 3, and 4 of Theorem 5. Lemma 16. Let C be an object in picture P s ŽZ 3,26,6, B .. If object C is a subset of eight grid points that forms a 2 = 2 = 2 cube, then C cannot be deleted completely by the set of masks M . Proof. Let us consider eight grid points forming a 2 = 2 = 2 cube as it illustrated in Fig. 10. The eight grid points are divided into two groups: the set of four points U s u1 ,u 2 ,u 3 ,u 4 4 forms the upper face of the cube; the set of the other four
Fig. 10. The upper and lower horizontal faces of a 2=2=2 cube, determined by the sets of points Us u1 ,u 2 ,u 3 ,u 4 4 and Ls l1 ,l 2 ,l 3 ,l 4 4, respectively.
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points L s l 1 ,l 2 ,l 3 ,l 4 4 forms the lower face of the cube. We can distinguish two cases: Ža. C l U sœ 0 Žb. C l U /œ 0 In case Ža., C l L /œ 0 , since object C contains at least one point. We know that any point in set L cannot be deleted by Proposition 8. Therefore, object C cannot be deleted completely. In case Žb., C l L /œ 0 by Proposition 8, too. Neither point in set L can be deleted by Proposition 8, therefore, object C cannot be deleted completely. I We are ready to prove the main theorem. Theorem 17. Algorithm 1 is a 3D thinning operation in the sense of Criterion 6. Proof. Condition 1 of Criterion 6 says, that the topology is to be preserved. The conditions of Theorem 5 are to be proved. Condition 1 holds by Lemma 13. The set Q j p4 in Lemma 15 covers each kind of set of points containing a 2 = 2 square, therefore, conditions 2, 3, and 4 hold. Condition 5 is satisfied by Lemma 16. Thus, the first subiteration of Algorithm 1 preserves topology. The set of masks M does not delete isolated points and line-end points by Proposition 12. Therefore, condition 2 of Criterion 6 holds, too. Condition 3 of Criterion 6 is also satisfied, because the set of masks M deletes the upper face U s u1 ,u 2 ,u 3 ,u 4 4 of the black cube of size 2 = 2 = 2 as it is illustrated in Fig. 10. It was shown that the first subiteration of Algorithm 1 is a 3D thinning operation. Therefore, the entire algorithm is a 3D thinning operation in the sense of Criterion 6, too. I
Acknowledgements This work was supported by OTKA T023804 and FKFP 0908r1997 grants.
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References Bertrand, G., 1995. A parallel thinning algorithm for medial surfaces. Pattern Recognition Lett. 16, 979–986. Bertrand, G., Aktouf, Z., 1994. A 3D thinning algorithms using subfields. In: Proc. SPIE Conf. on Vision Geometry III, vol. 2356, pp. 113–124. Blum, H., 1967. A transformation for extracting new descriptors of shape. In: Models for the Perception of Speech and Visual Form. MIT Press, Cambridge, MA, pp. 362–380. Gong, W.X., Bertrand, G., 1990. A simple parallel 3D thinning algorithm. In: Proc. 10th Internat. Conf. on Pattern Recognition, pp. 188–190. Kong, T.Y., Rosenfeld, A., 1989. Digital topology: Introduction and survey. Comput. Vision Graphics Image Process. 48, 357–393. Lee, T., Kashyap, R.L., Chu, C., 1994. Building skeleton models via 3-D medial surfaceraxis thinning algorithms. CVGIP: Graphical Models Image Process. 56, 462–478. Ma, C.M., 1994. On topology preservation in 3D thinning. CVGIP: Image Understanding 59, 328–339. Ma, C.M., 1995. A 3D fully parallel thinning algorithm for generating medial faces. Pattern Recognition Lett. 16, 83–87. Ma, C.M., Sonka, M., 1996. A fully parallel 3D thinning algorithm and its applications. Comput. Vision Image Understanding 64, 420–433. Morgenthaler, D.G., 1981. Three-dimensional simple points: Serial erosion, parallel thinning and skeletonization. Tech. Rept. TR-1005,. Computer Vision Laboratory, Computer Science Center, University of Maryland, College Park, MD. Palagyi, K., Kuba, A., 1997. A parallel 12-subiteration 3D thin´ ning algorithm to extract medial lines. In: Proc. 7th Internat. Conf., CAIP’97, Lecture Notes in Computer Science, vol. 1296. Springer, Berlin, pp. 400–407. Reed, G.M., 1984. On the characterization of simple closed surfaces in three-dimensional digital images. Comput. Graphics Image Process. 25, 226–235 . Rosenfeld, A., 1975. A characterization of parallel thinning algorithms. Inform. and Control 29, 286–291. Saha, P.K., Chaudhuri, B.B., 1994. Detection of 3-D simple points for topology preserving transformations with application to thinning. IEEE Trans. Pattern Anal. Machine Intell. 16, 1028– 1032. Szekely, G., 1996. Shape characterization by local symmetries. ´ Habilitationsschrift. ETH Zurich. ¨ Tsao, Y.F., Fu, K.S., 1981. A parallel thinning algorithm for 3-D pictures. Comput. Graphics Image Process. 17, 315–331.
A 3D 6-subiteration thinning algorithm for extracting medial lines Kalman ´ ´ Palagyi ´ ) , Attila Kuba Department of Applied Informatics, Jozsef Attila UniÕersity, P.O. Box 652, H-6701 Szeged, Hungary ´ Received 24 March 1997; revised 16 January 1998
Abstract Thinning is a frequently used method for extracting skeletons in discrete spaces. This paper presents an efficient parallel thinning algorithm that directly extracts medial lines from elongated 3D binary objects Ži.e., without creating medial surface.. Our algorithm provides good results, preserves topology and it is easy to implement. q 1998 Elsevier Science B.V. All rights reserved. Keywords: 3D parallel thinning algorithms; Discrete topology; Topology preservation
1. Introduction Skeletonization provides shape features that are extracted from binary image data. It is a common preprocessing operation in raster-to-vector conversion or in pattern recognition. Its goal is to reduce the volume of elongated objects to their skeletons. In the 3D Euclidean space, the skeleton Žor the medial surface. of a geometry is the locus of the centers of all the maximal inscribed spheres ŽBlum, 1967.. In discrete spaces, the thinning procedure is frequently used method for generating an approximation to the ‘‘true’’ Euclidean skeleton in a topology preserving way ŽKong and Rosenfeld, 1989.. Border points of the binary object that satisfy certain topological and geometric constraints are deleted in iteration steps.
Deletion means that object elements are changed to background elements. The entire process is repeated until only the ‘‘skeleton’’ is left. In this paper, a new 3D thinning algorithm is proposed that produces medial lines Žconsisting of arcs andror curves instead of surfaces.. Producing medial lines is often useful in medical image analysis Že.g., identifying the 3D midline of vessels or airways.. Each iteration step contains 6 successive subiterations. Subiterations can be executed in parallel. That is, the object elements that satisfy the actual deletion conditions are to be changed to background elements simultaneously. The proof of topology preservation is given.
2. Basic notions and results
)
Corresponding author.
In this section, some basic notions of 3D digital topology are presented Žsee Kong and Rosenfeld, 1989..
0167-8655r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 8 6 5 5 Ž 9 8 . 0 0 0 3 1 - 2
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Let x s Ž x 1 , x 2 , x 3 ., y s Ž y 1 , y 2 , y 3 . be two points with integer coordinates in the 3D digital space denoted by Z 3, and let us consider the Euclidean distance 5 xyy5s
(
3
Ý Ž x i y yi . 2 . is1
The points x and y are said to be 6-adjacent if 5 x y y 5 ( 1, 18-adjacent if 5 x y y 5 ( '2 , and 26adjacent if 5 x y y 5 ( '3 . Let us denote by Nj Ž p . the set of points j-adjacent to point p, for j s 6,18,26 Žsee Fig. 1.. The point p is said to be j-adjacent to the non-empty set of points X if there is point x g X, such that x g Nj Ž p .. The sequence of distinct points ² x 0 , x 1 , . . . , x n : is a j-path of length n 0 0 from point x 0 to point x n in a non-empty set of points X if each point of the sequence is in X and x i is j-adjacent to x iy1 for each 1 ( i ( n. ŽNote that a single point is a j-path of length 0.. Two points are j-connected in the set X if there is a j-path in X between them. A set of points X is j-connected if any two points in X are j-connected in X. The 3D binary Ž m,n. digital picture P is a quadruple P s ŽZ 3,m,n, B ., where Ž m,n. is a pair of adjacencies ŽŽ m,n. s Ž6,26., Ž26,6., Ž6,18., or Ž18,6. are generally considered.. Each element in Z 3 is called point or Õoxel of P. A point in B : Z 3 is called black point; a point in Z 3 _ B is called white point. Picture P is called finite if B is a finite set. Value 1 is assigned to each black point; value 0 is
Fig. 1. The frequently used adjacencies in Z 3. Points being in N6 Ž p . are marked U, N, E, S, W, and D; points being in N18 Ž p . but not in N6 Ž p . are marked ‘‘ Ø ’’; points being in N26 Ž p . but not in N18 Ž p . are marked ‘‘(’’. ŽNote that point p is in sets N6 Ž p ., N18 Ž p ., and N26 Ž p ...
assigned to each white point. Equivalence classes of B under m-connectivity are called black m-components or objects. A singleton black component is called isolated point. The n-connectiÕity relation and the white n-component can be defined in the same way. In a finite picture there is a unique infinite white component, which is said to be background. A finite white component is called caÕity in a picture. A black point is said to be border point if it is 6-adjacent to at least one white point. Note that this definition is correct only for the special cases m s 26 and m s 18. A border point p is called U-border point if the point marked by U in Fig. 1 is white. We can define N-, E-, S-, W-, and D-border points in the same way. There is an additional concept in 3D topology called hole or tunnel. A black component with a hole is topologically equivalent to a torus Ža doughnut-like object.. It is difficult to define the hole mathematically. Fortunately, the number of holes, what we need later, can be determined relatively easily. Let us suppose that the number of holes of a picture P denoted by H Ž P . is determined. Denote by O Ž P . and C Ž P . the number of objects Žblack components. and the number of cavities of P, respectively. The 3D Euler characteristic x Ž P . is defined by the global formula
x Ž P. sO Ž P. yH Ž P. qC Ž P. . Iterative thinning algorithms delete border points that satisfy certain conditions. The entire process is repeated until there are no more black points to be changed. The reduction is an operation in which only black points are deleted whereas white points should remain white. Accordingly, thinning is a reduction operation. Each thinning algorithm should satisfy the requirements of topology preservation. Kong and Rosenfeld Ž1989. discussed some definitions of topology preservation of 3D reduction operations. The most important is based on simple or deletable points introduced by Morgenthaler Ž1981.. Roughly speaking, a simple point is a point whose deletion does not change the topology of the picture. The following criterion has been stated: Criterion 1 ŽKong and Rosenfeld, 1989.. Let p be a black point of a 3D picture ŽZ 3 ,m,n, B .. The point p
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is simple if and only if all of the following conditions hold: 1. p is m-adjacent to only one m-component of N26 Ž p . l Ž B _ p4.. 2. p is n-adjacent to only one n-component of N26 Ž p . _ B. 3. x ŽŽZ 3 ,m,n, B l N26 Ž p ... s x ŽŽZ 3,m,n,Ž B _ p4. l N26 Ž p .... According to Criterion 1, simplicity can be decided locally by examining the 3 = 3 = 3 neighbourhood of black points. Saha and Chaudhuri Ž1994. proposed the following conditions for Ž26,6. pictures. Theorem 2. Black point p is simple in the picture P s ŽZ 3,26,6, B . if and only if all of the following three conditions hold: 1. The set Ž B _ p4. l N26 Ž p . contains exactly one 26-component. 2. The set ŽZ 3 _ B . l N6 Ž p . is not empty. 3. Any two points in ŽZ 3 _ B . l N6 Ž p . are 6-connected in the set ŽZ 3 _ B . l N18 Ž p .. It is easy to see that simultaneous removal of a set of simple points may not preserve topology. Let us consider a digital picture containing only a black cube of size 2 = 2 = 2. Every voxel of this cube is simple but their deletion will cause complete elimination of an object. This example shows that the simplicity of a set of points does not equal the set of simple points. Definition 3 ŽMa, 1994.. Let P be a picture. The set D s d1 , . . . ,d k 4 of black points is called a simple set of P if D can be arranged in a sequence ² d i , . . . ,d i : in which each d i is simple after 1 k j d i , . . . ,d i 4 is deleted from P, for j s 1, . . . ,k. 1 jy 1 ŽBy definition, let the empty set be simple.. Definition 4 ŽMa, 1994.. A 3D parallel reduction operation is topology preserÕing if, for all possible 3D pictures, the set of all points that are deleted simultaneously by the operation is simple. Fortunately, we do not have to test the algorithms for all possible 3D pictures. Ma Ž1994. has a signifi-
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cant result: we only need to check a rather small number of configurations to prove topology preserving. Theorem 5 ŽMa, 1994.. A 3D parallel reduction operation preserÕes topology for Ž26,6. pictures if all of the following conditions hold: 1. Only simple points can be deleted. 2. If two black corners, p and q, of a 2 = 2 square in Z 3 are deleted, then the set p,q4 is simple. 3. If three black corners, p, q and r, of a 2 = 2 square in Z 3 are deleted, then the set p,q,r 4 is simple. 4. If four black corners, p, q, r and s, of a 2 = 2 square in Z 3 are deleted, then the set p,q,r, s4 is simple. 5. No black component contained in a 2 = 2 = 2 cube in Z 3 can be deleted completely. Topology preservation is not the only requirement to be complied with. It is easy to see that any black component without a hole and a cavity can be shrunk into an isolated point by iterative deletions of simple points. Of course, this is not what we want. That is why, additional conditions are stated to supply the ‘‘skeleton’’. Criterion 6 ŽRosenfeld, 1975; Tsao and Fu, 1981.. A 3D reduction operation is a 3D thinning operation if it fulfills the following three conditions. 1. Topology is preserved. 2. Isolated points and curves remain unchanged. 3. An object of size 2 = 2 = 2 will be changed after applying the operation. ŽNote that a cube-shaped object is to be investigated and not a 2 = 2 = 2 part of an elongated object.. Existing 3D thinning algorithms can be classified from several points of view. One of them is the classification on the produced ‘‘skeletons’’: most of the developed algorithms result in medial surfaces ŽMorgenthaler, 1981; Reed, 1984. and a few produce medial lines Žsee Fig. 2.. The aim of the 3D thinning algorithms that are developed to extract medial lines is to shrink an object to a voxel list. A voxel list contains three kinds of points: line-end points, line
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Fig. 2. A 3D synthetic picture containing a character ‘‘A’’ Žleft., its medial surface Žcentre., and its medial lines Žright.. ŽCubes represent black points..
points, and cross points. A black point p is line-end point if there is exactly one black point m-adjacent to p. A black point p is line point Žor curÕe point . if there are exactly two black points m-adjacent to p. A black point p is cross point if there are more than two black points m-adjacent to p. Three types of thinning algorithms have been developed. The first type examine the 3 = 3 = 3 neighbourhood of each border point. The iteration steps are divided into more subiterations. Only the border points having the prescribed 3 = 3 = 3 neighbourhood can be deleted during a subiteration. These are called border sequential algorithms. Generally, a
subiteration is executed in parallel ŽTsao and Fu, 1981; Gong and Bertrand, 1990; Lee et al., 1994; Bertrand, 1995; Palagyi ´ and Kuba, 1997.. It means that all the black points satisfying the deletion condition of the actual subiteration are deleted simultaneously. The second type of algorithms do not need subiterations but they investigate larger neighbourhood ŽMa, 1995; Ma and Sonka, 1996.. The third approach is the subfield sequential method. The picture is divided into more distinct subsets which are successively activated. At a given iteration, only black points being in the active subfield are meant to be deleted ŽBertrand and Aktouf, 1994..
Fig. 3. Base masks M1–M6 and their rotations around the vertical axis form the set of masks M belonging to the deletion direction U. ŽNote, that at least one point marked ‘‘x’’ is black in masks M1 and M5..
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Some algorithms have been developed to generate medial surface ŽGong and Bertrand, 1990; Bertrand, 1995; Ma, 1995; Bertrand, 1995. and others allow producing medial surface and medial lines, too ŽTsao and Fu, 1981; Lee et al., 1994; Bertrand and Aktouf, 1994..
3. The new thinning algorithm In this section, a new algorithm is described for thinning 3D binary Ž26,6. pictures. Our 6-subiteration border sequential 3D thinning algorithm directly creates medial lines, without ex-
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tracting medial surface. The algorithm is based on a ‘‘careful’’ Boolean characterization of simple points which are not line-end points, i.e., which are not extremities of curves. The new value of each voxel depends on its 3 = 3 = 3 neighbourhood. This dependence is given by a set of configurations of 3 = 3 = 3 lattice points. Only deletion conditions are used Ži.e., black points can be changed but white points cannot.. The deletion condition assigned to a subiteration is described by a set of masks Žor matching templates.. A black point is to be deleted if and only if its 3 = 3 = 3 neighbourhood matches at least one element of the given set of masks. The masks are constructed according to the six deletion directions U, D, N, E, S,
Fig. 4. Thinning of synthetic pictures of size 24 = 24 = 24 Žleft.; their medial lines extracted by using Algorithm 1 Žright..
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Fig. 4 Žcontinued..
W. These directions correspond to the 6-neighbours and deletion of the given type of border points are taken into consideration. The masks assigned to the direction U give the condition to delete certain U-border points. The masks M1–M6 assigned to the direction U are given by Fig. 3. Masks are usually described by three kinds of elements, ‘‘1’’ Žblack., ‘‘0’’ Žwhite., and ‘‘.’’ Ž‘‘don’t care’’., where ‘‘don’t care’’ matches either black or white point in a given picture. In order to reduce the number of masks we use the following notation: at least one point marked ‘‘x’’ is black. Additionally, all rotations around the vertical axis Žaccording to the U and D directions. of the base masks M1–M6 are masks, too. ŽThe rotation angles are 908, 1808, and 2708.. Let M denote the set of
masks assigned to the deletion direction U. Ž M contains the base masks M1–M6 and their rotated versions.. Deletion conditions of the other five subiterations Žassigned to the directions D, N, E, S, and W. can be derived from the appropriate rotations and reflections of the masks in M . Let P s ŽZ 3,26,6, B . be the finite 3D picture to be thinned. Since set B is finite, it can be stored in a finite 3D binary array X. ŽOutside of this array every voxel will be considered as 0 Žwhite... Let us suppose that a subiteration corresponding to the deletion direction D results in a picture P X s ŽZ 3,26,6, BX . from picture P. Denote by T Ž X, D . the binary array representing the picture PX , where D is one of the six deletion directions. We are now ready to present our thinning algorithm formally.
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Fig. 4 Žcontinued..
Algorithm 1 for thinning the picture P s ŽZ 3 ,26,6, B .: Input: binary array X representing the picture P Output: binary array Y representing the thinned picture begin Y s X; repeat Y s T Ž Y,U.; Y s T Ž Y,D.; Y s T Ž Y,N.; Y s T Ž Y,S.; Y s T Ž Y,E.; Y s T Ž Y,W.; until no points are deleted; end.
Note that choosing another order of the deletion directions yields another algorithm. Algorithm 1 terminates when there are no more black points to be deleted. Since all considered input pictures are finite, Algorithm 1 will terminate. The masks of our algorithm can be regarded as a ‘‘careful’’ characterization of simple points for 3D thinning. This is a Boolean characterization since each of our masks can be given by a Boolean condition. It makes easy implementation possible.
4. Discussion Algorithm 1 have been tested for several synthetic pictures. Here we present some examples. Thinning
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for six synthetic 3D objects can be seen in Fig. 4. However, our algorithm is border-sequential, it creates regular and symmetric skeletons from regular and symmetric objects. Lee et al. Ž1994. discussed different methods of characterization of 3D simple points. There are 2 26 Žs 67 108 864. different possible 3 = 3 = 3 configurations of voxels around a black point p. The point p is simple in 25 984 552 cases of these configurations. It is easy to see that the masks in M Žbelonging to the direction U. can delete more than 2 20 Žs 1 048 576. kinds of simple points. Note that the ‘‘goodness’’ of a thinning algorithm is not directly proportional to the number of the simple points that are recognized as points to be deleted. The thinning process may produce undesired branches due to noisy
boundary. Experimental results of Fig. 5 are to demonstrate the behavior of Algorithm 1 under noise. Skeleton generation methods have to take care of the following two aspects ŽSzekely, 1996.: ´ Ø the geometrical one, requiring that the ‘‘skeleton’’ is in its geometrically correct position, i.e., the ‘‘middle’’ of the object; Ø the topological one, forcing the ‘‘skeleton’’ to retain the topology of the original object, i.e., topology is to be preserved. Thinning based skeletonization concentrates on the topological aspect. Directional 6-subiteration 3D thinning algorithms ŽTsao and Fu, 1981; Gong and Bertrand, 1990; Lee et al., 1994; Bertrand, 1995. use the distance induced by the 6-adjacency, which cannot be regarded as a good approximation to the Euclidean distance. Therefore, these algorithms are
Fig. 5. Noise sensitiveness of Algorithm 1. Synthetic pictures and their noisy versions Žleft.; their medial lines Žright..
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Fig. 5 Žcontinued..
not invariant under rotation. The proposed algorithm is sensitive to object orientation, too. Two examples are presented in which Algorithm 1 produces relatively invariant results under object rotation, see Fig. 6.
ing condition 1 of Theorem 5 can be seen by using Theorem 2. In order to prove the conditions of Theorem 2 and Theorem 5 we state some properties of the set of masks M belonging to the deletion direction U Žsee Fig. 3.. We refer to the individual mask positions by assigning indices to N26 Ž p . Žwhere p g Z 3 ., see Fig. 7.
5. Verification In this section, we show that Algorithm 1 is a 3D thinning algorithm in the sense of Criterion 6. It is sufficient to prove that the first subiteration belonging to the deletion direction U fulfills all conditions of Criterion 6. The proof of condition 1 of Criterion 6 Ži.e., topology preserving. is based on Theorem 5. Satisfy-
Proposition 7. The position corresponding to the index 5 Ž see Fig. 7. is white in all masks in M . Ž In other words: masks in M can delete only some U-border points.. Proposition 8. At least one position corresponding to the set of indices 18, . . . ,264 is black in all masks in M .
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Fig. 5 Žcontinued..
Proposition 9. All of the following four statements hold for each mask in M : Ø if the mask position haÕing the index 2 may be black, then the mask position haÕing the index 11 must be black, too, Ø if the mask position haÕing the index 4 may be black, then the mask position haÕing the index 13 must be black, too, Ø if the mask position haÕing the index 6 may be black, then the mask position haÕing the index 14 must be black, too, and Ø if the mask position haÕing the index 8 may be black, then the mask position haÕing the index 16 must be black, too. Let us classify the mask positions now. A mask position of a mask in M is called black position if it
is always black. A mask position is called white position if it is always white. The other mask positions which are not white and not black, are called potentially black positions. A black or potentially black mask position is called non-white position. Proposition 10. If a black point p can be deleted by any mask in M and the black point q g Ž N18 Ž p . _ N6 Ž p .. coincides with a black position of this mask, then point q cannot be deleted by M . Ž It implies Propositions 1 and 3.. Proposition 11. If a black point p can be deleted by the base masks M2, M3, M4 or their all rotated Õersions and the black point q coincides with the mask position haÕing the index 22 Ž see Fig. 1., then point q cannot be deleted by any mask in M . Ž It implies Proposition 1..
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Proposition 12. If a black point p can be deleted by any mask in M from a picture ŽZ 3 ,26,6, B ., then the set Ž N26 Ž p . _ p4. l B contains at least two elements. The first thing to be proved is that any arbitrary black point deleted by the set of masks M is simple Žsee condition 1 of Theorem 5.. Lemma 13. Let p be a black point in the picture P s ŽZ 3,26,6, B .. If point p can be deleted by at least one mask in M , then p is simple. Ž In other words, each black point deleted by the first subiteration of Algorithm 1 from picture P is simple.. Proof. We show that all conditions of Theorem 2 are satisfied. These conditions will be proved only for
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the six base masks M1–M6, since the simplicity is a rotation invariant property. First we show that the set Ž B _ p4. l N26 Ž p . contains exactly one 26-component Žcondition 1 of Theorem 2. and any two points in ŽZ 3 _ B . l N6 Ž p . are 6-connected in the set ŽZ 3 _ B . l N18 Ž p . Žcondition 3 of Theorem 2. with the help of Fig. 8. Labels A, B, and C are assigned to the black positions. Labels a, b, and c are assigned to non-white positions 26-adjacent to the black position labelled by A, B, and C, respectively. Obviously, a position may be 26-adjacent to more black positions. In this case, it is multiply labelled. We show that there is a 26-path containing only non-white positions between any two non-white points. The only black position in base masks M1 and M5 is labelled by A. We can state that all potentially
Fig. 6. Invariance of the proposed algorithm under object rotation. The two original synthetic objects and their rotated versions Žleft.; their medial lines extracted by Algorithm 1 Žright..
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Fig. 6 Žcontinued..
black points are labelled by a in these masks. It is easy to see that any two non-white points are 26connected via an at most two-length 26-path. The
sequence of labels in this path can be ²a,A: or ²a,A,a:. Base masks M2 and M6 contain two black positions labelled by Ab and aB. All potentially black points are labelled in these masks. It is easy to see that any two non-white points 26-connected via an at most three-length 26-path. The sequence of labels in this path can be ²a?,Ab,a?: , ²a?,Ab,aB,?b: , ² ?b,aB,?b: ,
Fig. 7. Indices 0,1, . . . ,26 are assigned to N26 Ž p ..
where ’’?’’ can replace characters a, b, and the empty string. Note that any prefix of a path is a path, too.
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Fig. 8. Labels assigned to the non-white positions of the base masks M1–M6. Thick lines denote the required 6-paths in ŽZ 3 _ B . l N18 Ž p ..
Base masks M3 and M4 contain three black positions labelled by Abc, aBc, and abC. Each potentially black point is labelled in these masks. It is easy to see that any two non-white points are 26-connected via an at most three-length 26-path. The sequence of labels in this path can be ²a?,Abc,a?: , ²a?,Abc,aBc,?b?: ²a?,Abc,abC,?c: , ² ?b?,aBc,?b?: , ² ?b?,aBc,abC,?c: , ² ?c,abC,?c: , where ’’?’’ can replace any string over the alphabet a,b,c4 . It was shown with the help of Fig. 8 that for each base mask, any two non-white points are 26-connected. Proposition 12 shows that point p may not be an isolated point in P. Therefore, the set Ž B _ p4. l N26 Ž p . contains exactly one 26-component. Condition 3 of Theorem 5 can be proved with the help of Fig. 8, too. It is to be proved that any two points in ŽZ 3 _ B . l N6 Ž p . are 6-connected in the set ŽZ 3 _ B . l N18 Ž p .. To prove this property, we give the required 6-paths. Finally, masks in M satisfy condition 2 of Theorem 2 Ži.e., the set ŽZ 3 _ B . l N6 Ž p . is not empty. by Proposition 7. I
The proofs of Lemma 13 and Proposition 7 yield the following statement. Corollary 14. Let p and q be two black points in picture P s ŽZ 3 ,26,6, B . so that q g N6 Ž p .. If points p and q can be deleted by M , then any two points in ŽŽZ 3 _ B . j q4. l N6 Ž p . are 6-connected in the set ŽŽZ 3 _ B . j q4. l N18 Ž p .. With the help of Corollary 8 and the following two lemmas, the remaining conditions of Theorem 5 can be proved easily. Lemma 15. Let p be a black point in picture P s ŽZ 3 ,26,6, B . and let Q be any set of black points so that Q : Ž N18 Ž p . _ p4. l B. If point p can be deleted by at least one mask in M , then p is simple in picture PX s ŽZ 3,26,6, B _ Q .. Ž In other words, p is simple after the deletion of Q.. Proof. We know that it holds for Q sœ 0 by Lemma 13. Let us suppose that Q /œ 0 . We show that the proposition holds if point p can be deleted by one of the base masks M1–M6. If p is deleted by a mask in M1,M4,M5,M64 , then any black mask position that is 18-adjacent to point p may not coincide with any point q g Q by Proposition 10. In this case each point in Q must
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Fig. 9. The critical configuration if p is deleted by mask M2.
coincide with a potentially black mask position. Simplicity of p does not depend on the deletion of any potentially black position Žsee the proof of Lemma 7.. If p is deleted by the mask M2, then the black position having the index 22 may not be deleted by Proposition 11. Therefore, that point is not in Q. Two cases are to be distinguished: Ža. the black point that coincides with the other black position Žhaving the index 11. is not in set Q, and Žb. the black point q that coincides with the other black mask position is in set Q. Let us consider the case Ža.. Each point in Q must coincide with a potentially black position, therefore, the simplicity of point p is not altered by the deletion of the set Q. In case Žb. the point coincides with the mask position having the index 2 is white by Proposition 7. There are two positions being not 26-adjacent to the black position having the index 22 and not being 18-adjacent to point p. These points are denoted by r and s in Fig. 9. They may not be in Q because each point in Q is 18-adjacent to p. Let us consider other two points denoted by t and u in Fig. 9. There are four cases: Žba. r s 0 and s s 0, Žbb. r s 0 and s s 1, Žbc. r s 1 and s s 0, and Žbd. r s 1 and s s 1. In case Žba., there is no problem. If case Žbb. holds, then u s 1 by Proposition 9 Žbecause point q can be deleted.. It is known that u f Q by Proposition 7. Since u is 26-connected to the black point that coincides with the black position
having the index 22, then s is not separated by the deletion of q. Therefore, condition 1 of Theorem 2 is satisfied. For axial symmetry, it holds for case Žbc., too. In case Žbd., t s 1 and u s 1 by Proposition 9, and t f Q and u f Q by Proposition 7. Since t and u are 26-connected to the black point that coincides with the black position having the index 22, r and s are not separated by the deletion of q. Therefore, condition 1 of Theorem 2 holds. Conditions 2 and 3 of Theorem 2 hold by Proposition 7 and Corollary 14, respectively. Therefore, p is simple in picture PX in case Žb.. We can prove in the same way that point p remains simple after deletion Q if p is deleted by the mask M3. It was shown that the proposition holds if point p is deleted by the base masks M1–M6. Obviously, it is also true for their rotated versions. I Note that the set Q j p4 in Lemma 15 contains any kinds of two or three or four black corners of a 2 = 2 lattice square. Therefore, we proved a more general property than conditions 2, 3, and 4 of Theorem 5. Lemma 16. Let C be an object in picture P s ŽZ 3,26,6, B .. If object C is a subset of eight grid points that forms a 2 = 2 = 2 cube, then C cannot be deleted completely by the set of masks M . Proof. Let us consider eight grid points forming a 2 = 2 = 2 cube as it illustrated in Fig. 10. The eight grid points are divided into two groups: the set of four points U s u1 ,u 2 ,u 3 ,u 4 4 forms the upper face of the cube; the set of the other four
Fig. 10. The upper and lower horizontal faces of a 2=2=2 cube, determined by the sets of points Us u1 ,u 2 ,u 3 ,u 4 4 and Ls l1 ,l 2 ,l 3 ,l 4 4, respectively.
K. Palagyi, A. Kubar Pattern Recognition Letters 19 (1998) 613–627 ´
points L s l 1 ,l 2 ,l 3 ,l 4 4 forms the lower face of the cube. We can distinguish two cases: Ža. C l U sœ 0 Žb. C l U /œ 0 In case Ža., C l L /œ 0 , since object C contains at least one point. We know that any point in set L cannot be deleted by Proposition 8. Therefore, object C cannot be deleted completely. In case Žb., C l L /œ 0 by Proposition 8, too. Neither point in set L can be deleted by Proposition 8, therefore, object C cannot be deleted completely. I We are ready to prove the main theorem. Theorem 17. Algorithm 1 is a 3D thinning operation in the sense of Criterion 6. Proof. Condition 1 of Criterion 6 says, that the topology is to be preserved. The conditions of Theorem 5 are to be proved. Condition 1 holds by Lemma 13. The set Q j p4 in Lemma 15 covers each kind of set of points containing a 2 = 2 square, therefore, conditions 2, 3, and 4 hold. Condition 5 is satisfied by Lemma 16. Thus, the first subiteration of Algorithm 1 preserves topology. The set of masks M does not delete isolated points and line-end points by Proposition 12. Therefore, condition 2 of Criterion 6 holds, too. Condition 3 of Criterion 6 is also satisfied, because the set of masks M deletes the upper face U s u1 ,u 2 ,u 3 ,u 4 4 of the black cube of size 2 = 2 = 2 as it is illustrated in Fig. 10. It was shown that the first subiteration of Algorithm 1 is a 3D thinning operation. Therefore, the entire algorithm is a 3D thinning operation in the sense of Criterion 6, too. I
Acknowledgements This work was supported by OTKA T023804 and FKFP 0908r1997 grants.
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References Bertrand, G., 1995. A parallel thinning algorithm for medial surfaces. Pattern Recognition Lett. 16, 979–986. Bertrand, G., Aktouf, Z., 1994. A 3D thinning algorithms using subfields. In: Proc. SPIE Conf. on Vision Geometry III, vol. 2356, pp. 113–124. Blum, H., 1967. A transformation for extracting new descriptors of shape. In: Models for the Perception of Speech and Visual Form. MIT Press, Cambridge, MA, pp. 362–380. Gong, W.X., Bertrand, G., 1990. A simple parallel 3D thinning algorithm. In: Proc. 10th Internat. Conf. on Pattern Recognition, pp. 188–190. Kong, T.Y., Rosenfeld, A., 1989. Digital topology: Introduction and survey. Comput. Vision Graphics Image Process. 48, 357–393. Lee, T., Kashyap, R.L., Chu, C., 1994. Building skeleton models via 3-D medial surfaceraxis thinning algorithms. CVGIP: Graphical Models Image Process. 56, 462–478. Ma, C.M., 1994. On topology preservation in 3D thinning. CVGIP: Image Understanding 59, 328–339. Ma, C.M., 1995. A 3D fully parallel thinning algorithm for generating medial faces. Pattern Recognition Lett. 16, 83–87. Ma, C.M., Sonka, M., 1996. A fully parallel 3D thinning algorithm and its applications. Comput. Vision Image Understanding 64, 420–433. Morgenthaler, D.G., 1981. Three-dimensional simple points: Serial erosion, parallel thinning and skeletonization. Tech. Rept. TR-1005,. Computer Vision Laboratory, Computer Science Center, University of Maryland, College Park, MD. Palagyi, K., Kuba, A., 1997. A parallel 12-subiteration 3D thin´ ning algorithm to extract medial lines. In: Proc. 7th Internat. Conf., CAIP’97, Lecture Notes in Computer Science, vol. 1296. Springer, Berlin, pp. 400–407. Reed, G.M., 1984. On the characterization of simple closed surfaces in three-dimensional digital images. Comput. Graphics Image Process. 25, 226–235 . Rosenfeld, A., 1975. A characterization of parallel thinning algorithms. Inform. and Control 29, 286–291. Saha, P.K., Chaudhuri, B.B., 1994. Detection of 3-D simple points for topology preserving transformations with application to thinning. IEEE Trans. Pattern Anal. Machine Intell. 16, 1028– 1032. Szekely, G., 1996. Shape characterization by local symmetries. ´ Habilitationsschrift. ETH Zurich. ¨ Tsao, Y.F., Fu, K.S., 1981. A parallel thinning algorithm for 3-D pictures. Comput. Graphics Image Process. 17, 315–331.
