3D Parallel Thinning Algorithms Based on Isthmuses - Springer Link
3D Parallel Thinning Algorithms Based on Isthmuses G´ abor N´emeth and K´ alm´an Pal´agyi Department of Image Processing and Computer Graphics, University of Szeged, Hungary {gnemeth,palagyi}@inf.u-szeged.hu
Abstract. Thinning is a widely used technique to obtain skeleton-like shape features (i.e., centerlines and medial surfaces) from digital binary objects. Conventional thinning algorithms preserve endpoints to provide important geometric information relative to the object to be represented. An alternative strategy is also proposed that preserves isthmuses (i.e., generalization of curve/surface interior points). In this paper we present ten 3D parallel isthmus-based thinning algorithm variants that are derived from some sufficient conditions for topology preserving reductions. Keywords: Shape analysis, Feature extraction, Skeletons, Thinning algorithms, Topology preservation.
1
Introduction
A skeleton is a region-based shape descriptor which represents the general shape of objects. Skeleton-like shape features (i.e., centerlines and medial surfaces) extracted from volumetric binary images play an important role in various applications in image processing, pattern recognition, and visualization [25]. Parallel thinning algorithms use parallel reduction operations: some object points in a binary image that satisfy certain topological and geometric constraints are deleted simultaneously, and an iteration step is repeated until stability is achieved [6]. Thinning has a major advantage over the alternative 3D skeletonization methods: it can produce both skeleton-like shape features. Surface-thinning algorithms can extract medial surfaces and curve-thinning algorithms can produce centerlines. Medial surfaces are generally extracted from general shapes and 3D tubular structures can be represented by their centerlines. Conventional 3D thinning algorithms preserve some points that provide relevant geometrical information with respect to the shape of the object. These points are called curve-endpoints or surface-endpoints. Bertrand and Couprie proposed an alternative strategy by accumulating some curve/surface interior points that are called isthmuses [4]. Characterizations of these isthmuses (for curve-thinning and surface-thinning) were defined first by Bertrand and Aktouf [3]. There are dozens of endpoint-based 3D thinning algorithms, but only five existing 3D algorithms use an isthmus-based thinning scheme [3,4,24]. J. Blanc-Talon et al. (Eds.): ACIVS 2012, LNCS 7517, pp. 325–335, 2012. c Springer-Verlag Berlin Heidelberg 2012
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Topology preservation [7] is an essential requirement for skeletonization algorithms. In order to verify that a given parallel 3D thinning algorithm preserves the topology, some sufficient conditions for topology preservation have been proposed [9,21]. Verifying these conditions usually means checking several configurations of points. Some previous algorithms [10,11] claim to be topology preserving, but they made a mistake in the proof. That is why Pal´agyi et al. [23] proposed a safe technique for designing topologically correct parallel 3D thinning algorithms. It is based on some sufficient conditions for topology preservation that consider individual points (instead of point configurations). The alternative sufficient conditions were combined with various parallel thinning strategies and various endpoint characterizations [23]. In this paper we present ten new algorithm variants that are based on our sufficient conditions for topology preservation and the isthmus-preserving thinning approach. Five of the new algorithms are capable of producing centerlines, and the remaining five are surface-thinning algorithms. This paper is organized as follows: Section 2 reviews the basic notions and results of 3D digital topology, and we present our sufficient conditions for topology preservation. In Section 3, we introduce the ten new parallel 3D isthmus-based thinning algorithms. Some illustrative results are presented in Section 4, and the new isthmus-based algorithms are compared with their corresponding variants that preserve endpoints. Finally, we round off the paper with some concluding remarks.
2
Basic Notions and Results
In this section, we outline some concepts of digital topology and related key results that will be used in the sequel. Let p be a point in the 3D digital space Z3 . Let us denote Nj (p) (for j = 6, 18, 26) the set of points that are j-adjacent to point p and let Nj∗ (p) = Nj (p)\{p} (see Fig. 1a). The sequence of distinct points x0 , x1 , . . . , xn is called a j-path (for j = 6, 18, 26) of length n from point x0 to point xn in a non-empty set of points X if each point of the sequence is in X and xi is j-adjacent to xi−1 for each 1 ≤ i ≤ n. Note that a single point is a j-path of length 0. Two points are said to be j-connected in the set X if there is a j-path in X between them (j = 6, 18, 26). A set of points X is j-connected in the set of points Y ⊇ X if any two points in X are j-connected in Y (j = 6, 18, 26). A 3D binary (26, 6) digital picture P is a quadruple P = (Z3 , 26, 6, B) [8]. Each element of Z3 is called a point of P. Each point in B ⊆ Z3 is called a black point and has a value of 1 assigned to it. Each point in Z3 \B is called a white point and has a value of 0 assigned to it. An object is a maximal 26-connected set of black points, while a white component is a maximal 6-connected set of white points. In a finite picture there is a unique infinite white component, which is called the background . A finite white component is called a cavity.
3D Parallel Thinning Algorithms Based on Isthmuses a • ◦
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◦ N ◦ 1 0 1 0 0 1 2 1 7 6 2 6 p 1 0 1 3 0 3 4 0 4 W E 2 1 2 6 0 1 6 ◦ S• ◦◦ • 0 1 1 0 1 0 3 0 7 3 7 D ◦ 1 0 1 2 1 5 0 2 5 ◦ 0 1 3 0 3 7 1 0 7 • ◦ •
Fig. 1. Frequently used adjacencies in Z3 . The set N6 (p) contains point p and the six points marked U, D, N, E, S, and W. The set N18 (p) contains N6 (p) and the twelve points marked by “◦”. The set N26 (p) contains N18 (p) and the eight points marked by “•”. The usual divisions of Z3 into 2 (b), 4 (c), and 8 (d) subfields. If partitioning into k subfields is considered, then points marked “i” are in the subfield SFk (i) (k = 2, 4, 8, i = 0, 1, . . . , k − 1).
A black point is called a border point in (26, 6) pictures if it is 6-adjacent to at least one white point. A border point is called a U-border point if the point marked U in Fig. 1a is a white point. We can define D-, N-, E-, S-, and Wborder points in the same way. A black point is called an isolated point if it is not 26-adjacent to any other black point. The lexicographical order relation “≺” between two distinct points p = (px , py , pz ) and q = (qx , qy , qz ) in Z3 is defined as follows: p≺q
⇔
(pz < qz ) ∨ (pz = qz ∧ py < qy ) ∨ (pz = qz ∧ py = qy ∧ px < qx ).
Let X ⊆ Z3 be a set of points. Point p ∈ X is the smallest element of X if for any q ∈ X\{p}, p ≺ q. A unit lattice square is a set of four mutually 18-adjacent points in Z3 , while a unit lattice cube is a set of eight mutually 26-adjacent points in Z3 . An object is called a small object if it is contained in a unit lattice cube, but it is not contained in a unit lattice square. A reduction operation transforms a binary picture only by changing some black points to white ones (which is referred to as the deletion of black points). A reduction operation does not preserve topology [7] if any object in the input picture is split (into several ones) or is completely deleted, any cavity in the input picture is merged with the background or another cavity, or a cavity is created where there was none in the input picture. There is an additional concept called hole (which doughnuts have) in 3D pictures [8]. Topology preservation implies that eliminating or creating any hole is not allowed. A black point is simple in a (26, 6) picture if and only if its deletion is a topology preserving reduction operation [8]. A useful characterization of simple points on (26, 6) pictures is stated as follows: Theorem 1. [14] A black point p is simple in picture (Z3 , 26, 6, B) if and only if all of the following conditions hold:
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∗ 1. The set N26 (p) ∩ B contains exactly one 26–component. 2. The set N6 (p) \ B is not empty. 3. Any two points in N6 (p) \ B are 6–connected in the set N18 (p) \ B.
Based on Theorem 1, the simplicity of a point p can be decided by examining the set N26 (p). Parallel reduction operations delete a set of black points and not just a single simple point. Hence we need to consider what is meant by topology preservation when a number of black points are deleted simultaneously. Pal´agyi et al. gave some sufficient conditions for topology preservation as a basis for designing 3D parallel thinning algorithms [23]. Theorem 2. [23] The parallel reduction operation T is topology preserving for (26, 6) pictures if all of the following conditions hold for any black point p in any picture (Z3 , 26, 6, B) such that p is deleted by T . 1. Let Q ⊆ B be any set of simple points in (Z3 , 26, 6, B) such that p is the smallest element of Q, and Q is contained in a unit lattice square. Then point p is simple in picture (Z3 , 26, 6, B\(Q\{p})). 2. Point p is not the smallest element of any small object. In this paper we present ten parallel thinning algorithms that are based on the sufficient conditions of Theorem 2 combined with parallel thinning strategies and isthmus preservation. Our algorithms use the following characterizations of isthmuses. Definition 1. A border point p in a picture (Z3 , 26, 6, B) is an IC -isthmus ∗ (for curve-thinning) if the set N26 (p) ∩ B contains more than one 26–component (i.e., Condition 1 of Theorem 1 is violated). Definition 2. A border point p in a picture (Z3 , 26, 6, B) is an IS -isthmus (for surface-thinning) if p is not a simple point (i.e., Condition 1 of Theorem 1 or Condition 3 of Theorem 1 is violated). We can state that no isthmus point is simple. Note that these two characterizations correspond to the isthmuses proposed by Bertrand and Aktouf [3], with the exception that isolated (non-simple) points, which are not isthmuses by the terminology used in [3], are IS -isthmuses by Definition 2. Raynal and Couprie [24] used IC -isthmuses in their curve-thinning algorithm, but they consider an additional type of isthmuses: a border point is an isthmus in their surface-thinning algorithm if Condition 3 of Theorem 1 is violated. We want to produce medial surfaces that contain curves (i.e., 1D patches) for tubular parts. Hence we consider that each IC -isthmus point is an IS -isthmus, too.
3
Isthmus-Based Parallel 3D Thinning Algorithms
In this section, ten isthmus-based parallel 3D thinning algorithm variants are presented. These algorithms are composed of topology preserving parallel reduction operations, hence all algorithms are topologically correct.
3D Parallel Thinning Algorithms Based on Isthmuses
3.1
329
Fully Parallel Algorithms
In fully parallel [6] algorithms, the same parallel reduction operation is applied in each iteration step [1,10,11,15,19,22,23]. The scheme of the proposed two isthmus-based fully parallel thinning algorithms 3D-FP-IC and 3D-FP-IS are sketched in Algorithm 1. Algorithm 1. Algorithm 3D-FP-I 1: 2: 3: 4: 5: 6: 7: 8: 9: 10:
(I ∈ {IC , IS })
Input: picture (Z3 , 26, 6, X) Output: picture (Z3 , 26, 6, Y ) Y =X I=∅ repeat // one iteration step I = I ∪ { p | p ∈ Y \ I and p is an I-isthmus } D = { p | p ∈ Y \ I and p is 3D-FP-deletable in Y } Y =Y \D until D = ∅
3D-FP-deletable points are defined as follows: Definition 3. A black point is 3D-FP-deletable if all conditions of Theorem 2 hold. Deletable points of the proposed fully parallel algorithms (see Definition 3) are derived directly from conditions of Theorem 2. Hence, both algorithms 3D-FP-IC and 3D-FP-IS are topology preserving. 3.2
Subiteration-Based Algorithms
In subiteration-based (or frequently referred to as directional) thinning algorithms, an iteration step is decomposed into k successive parallel reduction operations according to k deletion directions [6]. If the current deletion direction is d, then a set of d-border points can be deleted by the parallel reduction operation assigned to it. Since there are six kinds of major directions in 3D cases, 6-subiteration algorithms were generally proposed [2,5,18,20,23,24,26,27]. In what follows, we present two parallel 3D 6-subiteration thinning algorithms 3D-6-SI-IC and 3D-6-SI-IS . These isthmus-based algorithms are described by Algorithm 2. The ordered list of deletion directions U, D, N, E, S, W [5,20] is considered in the proposed algorithms 3D-6-SI-IC and 3D-6-SI-IS . Note that subiterationbased thinning algorithms are not invariant under the order of deletion directions (i.e., choosing different orders may yield various results). In the first subiteration, the set of 3D-6-SI-U-deletable points are deleted simultaneously, and the set of 3D-6-SI-W-deletable points are deleted in the last (i.e., the 6th) subiteration. Now we define 3D-6-SI-U-deletable points.
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Algorithm 2. Algorithm 3D-6-SI-I 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:
(I ∈ {IC , IS })
Input: picture (Z3 , 26, 6, X) Output: picture (Z3 , 26, 6, Y ) Y =X I=∅ repeat // one iteration step for each d ∈ {U, D, N, E, S, W} do // subiteration for deleting some d-border points I = I ∪ { p | p ∈ Y \ I and p is an I-isthmus } D(d) = { p | p ∈ Y \ I and p is 3D-6-SI-d-deletable in Y } Y = Y \ D(d) end for until D(U) ∪ D(D) ∪ D(N) ∪ D(E) ∪ D(S) ∪ D(W) = ∅
Definition 4. A black point is 3D-6-SI-U-deletable if it is a U-border point, and all conditions of Theorem 2 hold. It can be readily seen that both subiteration-based algorithms 3D-6-SI-IC and 3D-6-SI-IS are topology preserving. Note that conditions of Theorem 2 can be simplified if d-border points (d ∈ {U, D, N, E, S, W}) are taken into consideration as potential deletable points [23]. 3.3
Subfield-Based Algorithms
The third kind of parallel thinning algorithm applies a subfield-based technique [6]. In existing subfield-based parallel 3D thinning algorithms, the digital space Z3 is partitioned into two [12,16,23], four [13,17,23], and eight [3,17,23] subfields which are alternatively activated. At a given iteration step of a k-subfield algorithm, k successive parallel reduction operations associated with the k subfields are performed. In each of them, some border points in the active subfield can be designated for deletion. Let us denote SFk (i) the i-th subfield if Z3 is partitioned into k subfields (k = 2, 4, 8; i = 0, . . . , k − 1). The considered divisions are illustrated in Fig. 1b-d. In order to reduce the noise sensitivity and the count of skeletal points, N´emeth et al. introduced a new subfield-based thinning scheme [16]. The iteration level border detection strategy takes the border points into consideration at the beginning of iteration steps as potential deletable points within the entire iteration. Next, we present our six new parallel 3D subfield-based thinning algorithms. The scheme of the subfield-based parallel thinning algorithm 3D-k-SF-I (k = 2, 4, 8; I ∈ {IC , IS }) with iteration-level checking is sketched in Algorithm 3. The 3D-k-SF-i-deletable points are defined as follows (k = 2, 4, 8; i = 0, . . . , k − 1):
3D Parallel Thinning Algorithms Based on Isthmuses
Algorithm 3. Algorithm 3D-k-SF-I 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14:
331
(k = 2, 4, 8; I ∈ {IC , IS })
Input: picture (Z3 , 26, 6, X) Output: picture (Z3 , 26, 6, Y ) Y =X I=∅ repeat // one iteration step E = { p | p is a border point in Y } I = I ∪ { p | p ∈ E and p is an I-isthmus } for i = 0 to k − 1 do // subfield SFk (i) is activated D(i) = { q | q ∈ E \ I and is 3D-k-SF-i-deletable in Y } Y = Y \ D(i) end for until D(0) ∪ D(1) ∪ . . . ∪ D(k − 1) = ∅
Definition 5. A black point p is 3D-k-SF-i-deletable if p ∈ SFk (i) and all conditions of Theorem 2 hold. It is easy to see that all of the six subfield-based algorithms 3D-k-SF-I (k = 2, 4, 8; I ∈ {IC , IS }) are topology preserving. Note that conditions of Theorem 2 can be simplified if elements of a subfield are considered as potential delatable points [23].
4
Results
In experiments the proposed algorithms were tested on objects of various images. Due to the lack of space, here we present two illustrative examples below (see Figs. 2-3). The new algorithms were compared with the corresponding endpointpreserving 3D parallel thinning algorithms presented in [23]. We can state that the isthmus-based algorithms produce fewer unwanted side branches or surface patches than the thinning algorithms that preserve curve-endpoints of type CE or surface-endpoints of type SE [23]. Figure 2 presents ten kinds of centerlines produced by the five new isthmus-based curve-thinning algorithms (3D-FP-IC , 3D-6-SI-IC , 3D-2-SF-IC , 3D-4-SF-IC , and 3D-8-SF-IC ) and the corresponding five algorithms (3D-FP-CE, 3D-6-SI-CE, 3D-2-SF-CE, 3D-4-SF-CE, and 3D8-SF-CE) that use curve-endpoints as geometrical constraint [23]. Ten kinds of medial surfaces are presented in Fig. 3: five of them are produced by the proposed isthmus-based surface-thinning algorithms (3D-FP-IS , 3D-6-SI-IS , 3D-2-SF-IS , 3D-4-SF-IS , and 3D-8-SF-IS ), and these medial surfaces can be compared with the results of the five endpoint-preserving surface thinning algorithms (3D-FPSE, 3D-6-SI-SE, 3D-2-SF-SE, 3D-4-SF-SE, and 3D-8-SF-SE) [23]. Note that the reported algorithms are not time consuming and it is easy to implement them on conventional sequential computers by adapting the efficient implementation method presented in [22]. Centerlines and medial surfaces can be extracted from large 3D shapes within one second on a usual PC.
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original image (273 743)
3D-FP-IC (1 297)
3D-FP-CE (1 399)
3D-6-SI-IC (1 332)
3D-6-SI-CE (1 360)
3D-2-SF-IC (1 353)
3D-2-SF-CE (1 387)
3D-4-SF-IC (1 344)
3D-4-SF-CE (1 377)
3D-8-SF-IC (1 353)
3D-8-SF-CE (1 357)
Fig. 2. A 102 × 381 × 255 image of a helicopter and its centerlines produced by the five new isthmus-based curve-thinning algorithms (left column) and the centerlines produced by the five existing endpoint-based curve-thinning algorithms (right column). (Numbers in parentheses mean the count of black points.)
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original image (2 150 151)
3D-FP-IS (70 386)
3D-FP-SE (94 114)
3D-6-SI-IS (58 097)
3D-6-SI-SE (107 028)
3D-2-SF-IS (68 367)
3D-2-SF-SE (82 539)
3D-4-SF-IS (67 136)
3D-4-SF-SE (81 853)
3D-8-SF-IS (66 869)
3D-8-SF-SE (82 179)
Fig. 3. A 515 × 455 × 110 image of an airplane and its medial surfaces produced by the five new isthmus-based surface-thinning algorithms (left column) and the medial surfaces produced by the five existing endpoint-based surface-thinning algorithms (right column). (Numbers in parentheses mean the count of black points.)
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Conclusions
Fast and reliable extraction of skeleton-like 3D shape features (i.e., centerlines and medial surfaces) is extremely important in numerous applications for large shapes (binary objects). This paper presents ten new variations for parallel thinning algorithms: five of them are curve-thinning algorithms (3D-FP-IC , 3D6-SI-IC , 3D-2-SF-IC , 3D-4-SF-IC , and 3D-8-SF-IC ), and the remaining five (3D-FP-IS , 3D-6-SI-IS , 3D-2-SF-IS , 3D-4-SF-IS , and 3D-8-SF-IS ) are capable of producing medial surfaces. Deletion rules of the proposed algorithms were not given by matching templates (as is usual), they were derived from some sufficient conditions for topology preserving parallel reductions. Hence their topological correctness is guaranteed. All of the reported algorithms are based on isthmuses, and they can produce fewer unwanted branches or surface patches than the corresponding conventional algorithms that preserve endpoints. Acknowledgements. This research was supported by the European Union and ´ the European Regional Development Fund under the grant agreement TAMOP4.2.1/B-09/1/KONV-2010-0005, and the grant CNK80370 of the National Office for Research and Technology (NKTH) & the Hungarian Scientific Research Fund (OTKA).
References 1. Arcelli, C., di Baja, G.S., Serino, L.: New Removal Operators for Surface Skeletonization. In: Kuba, A., Ny´ ul, L.G., Pal´ agyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 555–566. Springer, Heidelberg (2006) 2. Bertrand, G.: A parallel thinning algorithm for medial surfaces. Pattern Recognition Letters 16, 979–986 (1995) 3. Bertrand, G., Aktouf, Z.: A 3D thinning algorithm using subfields. In: SPIE Proc. of Conf. on Vision Geometry, pp. 113–124 (1994) 4. Bertrand, G., Couprie, M.: Transformations topologiques discr`etes. In: Coeurjolly, D., Montanvert, A., Chassery, J. (eds.) G´eom´etrie Discr`ete et Images Num´eriques, Herm`es, pp. 187–209 (2007) 5. Gong, W.X., Bertrand, G.: A simple parallel 3D thinning algorithm. In: Proc. 10th IEEE Internat. Conf. on Pattern Recognition, ICPR 1990, pp. 188–190 (1990) 6. Hall, R.W.: Parallel connectivity-preserving thinning algorithms. In: Kong, T.Y., Rosenfeld, A. (eds.) Topological Algorithms for Digital Image Processing, pp. 145– 179. Elsevier Science (1996) 7. Kong, T.Y.: On topology preservation in 2–d and 3–d thinning. International Journal of Pattern Recognition and Artificial Intelligence 9, 813–844 (1995) 8. Kong, T.Y., Rosenfeld, A.: Digital topology: Introduction and survey. Computer Vision, Graphics, and Image Processing 48, 357–393 (1989) 9. Ma, C.M.: On topology preservation in 3D thinning. CVGIP: Image Understanding 59, 328–339 (1994) 10. Ma, C.M.: A 3D fully parallel thinning algorithm for generating medial faces. Pattern Recognition Letters 16, 83–87 (1995)
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11. Ma, C.M., Sonka, M.: A fully parallel 3D thinning algorithm and its applications. Computer Vision and Image Understanding 64, 420–433 (1996) 12. Ma, C.M., Wan, S.Y., Chang, H.K.: Extracting medial curves on 3D images. Pattern Recognition Letters 23, 895–904 (2002) 13. Ma, C.M., Wan, S.Y., Lee, J.D.: Three-dimensional topology preserving reduction on the 4-subfields. IEEE Transaction on Pattern Analysis and Machine Intelligence 24, 1594–1605 (2002) 14. Malandain, G., Bertrand, G.: Fast characterization of 3D simple points. In: Proc. 11th IEEE Internat. Conf. on Pattern Recognition, ICPR 1992, pp. 232–235 (1992) 15. Manzanera, A., Bernard, T.M., Prˆeteux, F., Longuet, B.: Medial faces from a concuise 3D thinning algorithm. In: Proc. 7th IEEE Int. Conf. on Computer Vision, pp. 337–343 (1999) 16. N´emeth, G., Kardos, P., Pal´ agyi, K.: Topology preserving 2-subfield 3D thinning algorithms. In: Proc. 7th IASTED Int. Conf. Signal Processing, Pattern Recognition and Applications, pp. 310–316 (2009) 17. N´emeth, G., Kardos, P., Pal´ agyi, K.: Topology Preserving 3D Thinning Algorithms Using Four and Eight Subfields. In: Campilho, A., Kamel, M. (eds.) ICIAR 2010. LNCS, vol. 6111, pp. 316–325. Springer, Heidelberg (2010) 18. N´emeth, G., Kardos, P., Pal´ agyi, K.: A Family of Topology–Preserving 3D Parallel 6–Subiteration Thinning Algorithms. In: Aggarwal, J.K., Barneva, R.P., Brimkov, V.E., Koroutchev, K.N., Korutcheva, E.R. (eds.) IWCIA 2011. LNCS, vol. 6636, pp. 17–30. Springer, Heidelberg (2011) 19. Pal´ agyi, K.: A 3D fully parallel surface-thinning algorithm. Theoretical Computer Science 406, 119–135 (2008) 20. Pal´ agyi, K., Kuba, A.: A 3D 6–subiteration thinning algorithm for extracting medial lines. Pattern Recognition Letters 19, 613–627 (1998) 21. Pal´ agyi, K., Kuba, A.: A parallel 3D 12-subiteration thinning algorithm. Graphical Models and Image Processing 61, 199–221 (1999) 22. Pal´ agyi, K., N´emeth, G.: Fully Parallel 3D Thinning Algorithms Based on Sufficient Conditions for Topology Preservation. In: Brlek, S., Reutenauer, C., Proven¸cal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 481–492. Springer, Heidelberg (2009) 23. Pal´ agyi, K., N´emeth, G., Kardos, P.: Topology preserving parallel 3D thinning algorithms. In: Barneva, R., Brimkov, V. (eds.) Digital Geometry Algorithms. Theoretical Foundations and Applications to Computational Imaging, pp. 165–188. Springer (2012) 24. Raynal, B., Couprie, M.: Isthmus-based 6-directional parallel thinning algorithms. In: Domenjoud, E. (ed.) DGCI 2011. LNCS, vol. 6607, pp. 175–186. Springer, Heidelberg (2011) 25. Siddiqi, K., Pizer, S. (eds.): Medial representations – Mathematics, algorithms and applications. Computational Imaging and Vision, vol. 37. Springer, New York (2008) 26. Tsao, Y.F., Fu, K.S.: A parallel thinning algorithm for 3–D pictures. Computer Graphics and Image Processing 17, 315–331 (1981) 27. Xie, W., Thompson, R.P., Perucchio, R.: A topology-preserving parallel 3D thinning algorithm for extracting the curve skeleton. Pattern Recognition 36, 1529–1544 (2003)
Abstract. Thinning is a widely used technique to obtain skeleton-like shape features (i.e., centerlines and medial surfaces) from digital binary objects. Conventional thinning algorithms preserve endpoints to provide important geometric information relative to the object to be represented. An alternative strategy is also proposed that preserves isthmuses (i.e., generalization of curve/surface interior points). In this paper we present ten 3D parallel isthmus-based thinning algorithm variants that are derived from some sufficient conditions for topology preserving reductions. Keywords: Shape analysis, Feature extraction, Skeletons, Thinning algorithms, Topology preservation.
1
Introduction
A skeleton is a region-based shape descriptor which represents the general shape of objects. Skeleton-like shape features (i.e., centerlines and medial surfaces) extracted from volumetric binary images play an important role in various applications in image processing, pattern recognition, and visualization [25]. Parallel thinning algorithms use parallel reduction operations: some object points in a binary image that satisfy certain topological and geometric constraints are deleted simultaneously, and an iteration step is repeated until stability is achieved [6]. Thinning has a major advantage over the alternative 3D skeletonization methods: it can produce both skeleton-like shape features. Surface-thinning algorithms can extract medial surfaces and curve-thinning algorithms can produce centerlines. Medial surfaces are generally extracted from general shapes and 3D tubular structures can be represented by their centerlines. Conventional 3D thinning algorithms preserve some points that provide relevant geometrical information with respect to the shape of the object. These points are called curve-endpoints or surface-endpoints. Bertrand and Couprie proposed an alternative strategy by accumulating some curve/surface interior points that are called isthmuses [4]. Characterizations of these isthmuses (for curve-thinning and surface-thinning) were defined first by Bertrand and Aktouf [3]. There are dozens of endpoint-based 3D thinning algorithms, but only five existing 3D algorithms use an isthmus-based thinning scheme [3,4,24]. J. Blanc-Talon et al. (Eds.): ACIVS 2012, LNCS 7517, pp. 325–335, 2012. c Springer-Verlag Berlin Heidelberg 2012
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Topology preservation [7] is an essential requirement for skeletonization algorithms. In order to verify that a given parallel 3D thinning algorithm preserves the topology, some sufficient conditions for topology preservation have been proposed [9,21]. Verifying these conditions usually means checking several configurations of points. Some previous algorithms [10,11] claim to be topology preserving, but they made a mistake in the proof. That is why Pal´agyi et al. [23] proposed a safe technique for designing topologically correct parallel 3D thinning algorithms. It is based on some sufficient conditions for topology preservation that consider individual points (instead of point configurations). The alternative sufficient conditions were combined with various parallel thinning strategies and various endpoint characterizations [23]. In this paper we present ten new algorithm variants that are based on our sufficient conditions for topology preservation and the isthmus-preserving thinning approach. Five of the new algorithms are capable of producing centerlines, and the remaining five are surface-thinning algorithms. This paper is organized as follows: Section 2 reviews the basic notions and results of 3D digital topology, and we present our sufficient conditions for topology preservation. In Section 3, we introduce the ten new parallel 3D isthmus-based thinning algorithms. Some illustrative results are presented in Section 4, and the new isthmus-based algorithms are compared with their corresponding variants that preserve endpoints. Finally, we round off the paper with some concluding remarks.
2
Basic Notions and Results
In this section, we outline some concepts of digital topology and related key results that will be used in the sequel. Let p be a point in the 3D digital space Z3 . Let us denote Nj (p) (for j = 6, 18, 26) the set of points that are j-adjacent to point p and let Nj∗ (p) = Nj (p)\{p} (see Fig. 1a). The sequence of distinct points x0 , x1 , . . . , xn is called a j-path (for j = 6, 18, 26) of length n from point x0 to point xn in a non-empty set of points X if each point of the sequence is in X and xi is j-adjacent to xi−1 for each 1 ≤ i ≤ n. Note that a single point is a j-path of length 0. Two points are said to be j-connected in the set X if there is a j-path in X between them (j = 6, 18, 26). A set of points X is j-connected in the set of points Y ⊇ X if any two points in X are j-connected in Y (j = 6, 18, 26). A 3D binary (26, 6) digital picture P is a quadruple P = (Z3 , 26, 6, B) [8]. Each element of Z3 is called a point of P. Each point in B ⊆ Z3 is called a black point and has a value of 1 assigned to it. Each point in Z3 \B is called a white point and has a value of 0 assigned to it. An object is a maximal 26-connected set of black points, while a white component is a maximal 6-connected set of white points. In a finite picture there is a unique infinite white component, which is called the background . A finite white component is called a cavity.
3D Parallel Thinning Algorithms Based on Isthmuses a • ◦
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•
◦ • ◦ • U
b 1 0
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1
c 0 2
3
2 0 3 1
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◦ N ◦ 1 0 1 0 0 1 2 1 7 6 2 6 p 1 0 1 3 0 3 4 0 4 W E 2 1 2 6 0 1 6 ◦ S• ◦◦ • 0 1 1 0 1 0 3 0 7 3 7 D ◦ 1 0 1 2 1 5 0 2 5 ◦ 0 1 3 0 3 7 1 0 7 • ◦ •
Fig. 1. Frequently used adjacencies in Z3 . The set N6 (p) contains point p and the six points marked U, D, N, E, S, and W. The set N18 (p) contains N6 (p) and the twelve points marked by “◦”. The set N26 (p) contains N18 (p) and the eight points marked by “•”. The usual divisions of Z3 into 2 (b), 4 (c), and 8 (d) subfields. If partitioning into k subfields is considered, then points marked “i” are in the subfield SFk (i) (k = 2, 4, 8, i = 0, 1, . . . , k − 1).
A black point is called a border point in (26, 6) pictures if it is 6-adjacent to at least one white point. A border point is called a U-border point if the point marked U in Fig. 1a is a white point. We can define D-, N-, E-, S-, and Wborder points in the same way. A black point is called an isolated point if it is not 26-adjacent to any other black point. The lexicographical order relation “≺” between two distinct points p = (px , py , pz ) and q = (qx , qy , qz ) in Z3 is defined as follows: p≺q
⇔
(pz < qz ) ∨ (pz = qz ∧ py < qy ) ∨ (pz = qz ∧ py = qy ∧ px < qx ).
Let X ⊆ Z3 be a set of points. Point p ∈ X is the smallest element of X if for any q ∈ X\{p}, p ≺ q. A unit lattice square is a set of four mutually 18-adjacent points in Z3 , while a unit lattice cube is a set of eight mutually 26-adjacent points in Z3 . An object is called a small object if it is contained in a unit lattice cube, but it is not contained in a unit lattice square. A reduction operation transforms a binary picture only by changing some black points to white ones (which is referred to as the deletion of black points). A reduction operation does not preserve topology [7] if any object in the input picture is split (into several ones) or is completely deleted, any cavity in the input picture is merged with the background or another cavity, or a cavity is created where there was none in the input picture. There is an additional concept called hole (which doughnuts have) in 3D pictures [8]. Topology preservation implies that eliminating or creating any hole is not allowed. A black point is simple in a (26, 6) picture if and only if its deletion is a topology preserving reduction operation [8]. A useful characterization of simple points on (26, 6) pictures is stated as follows: Theorem 1. [14] A black point p is simple in picture (Z3 , 26, 6, B) if and only if all of the following conditions hold:
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∗ 1. The set N26 (p) ∩ B contains exactly one 26–component. 2. The set N6 (p) \ B is not empty. 3. Any two points in N6 (p) \ B are 6–connected in the set N18 (p) \ B.
Based on Theorem 1, the simplicity of a point p can be decided by examining the set N26 (p). Parallel reduction operations delete a set of black points and not just a single simple point. Hence we need to consider what is meant by topology preservation when a number of black points are deleted simultaneously. Pal´agyi et al. gave some sufficient conditions for topology preservation as a basis for designing 3D parallel thinning algorithms [23]. Theorem 2. [23] The parallel reduction operation T is topology preserving for (26, 6) pictures if all of the following conditions hold for any black point p in any picture (Z3 , 26, 6, B) such that p is deleted by T . 1. Let Q ⊆ B be any set of simple points in (Z3 , 26, 6, B) such that p is the smallest element of Q, and Q is contained in a unit lattice square. Then point p is simple in picture (Z3 , 26, 6, B\(Q\{p})). 2. Point p is not the smallest element of any small object. In this paper we present ten parallel thinning algorithms that are based on the sufficient conditions of Theorem 2 combined with parallel thinning strategies and isthmus preservation. Our algorithms use the following characterizations of isthmuses. Definition 1. A border point p in a picture (Z3 , 26, 6, B) is an IC -isthmus ∗ (for curve-thinning) if the set N26 (p) ∩ B contains more than one 26–component (i.e., Condition 1 of Theorem 1 is violated). Definition 2. A border point p in a picture (Z3 , 26, 6, B) is an IS -isthmus (for surface-thinning) if p is not a simple point (i.e., Condition 1 of Theorem 1 or Condition 3 of Theorem 1 is violated). We can state that no isthmus point is simple. Note that these two characterizations correspond to the isthmuses proposed by Bertrand and Aktouf [3], with the exception that isolated (non-simple) points, which are not isthmuses by the terminology used in [3], are IS -isthmuses by Definition 2. Raynal and Couprie [24] used IC -isthmuses in their curve-thinning algorithm, but they consider an additional type of isthmuses: a border point is an isthmus in their surface-thinning algorithm if Condition 3 of Theorem 1 is violated. We want to produce medial surfaces that contain curves (i.e., 1D patches) for tubular parts. Hence we consider that each IC -isthmus point is an IS -isthmus, too.
3
Isthmus-Based Parallel 3D Thinning Algorithms
In this section, ten isthmus-based parallel 3D thinning algorithm variants are presented. These algorithms are composed of topology preserving parallel reduction operations, hence all algorithms are topologically correct.
3D Parallel Thinning Algorithms Based on Isthmuses
3.1
329
Fully Parallel Algorithms
In fully parallel [6] algorithms, the same parallel reduction operation is applied in each iteration step [1,10,11,15,19,22,23]. The scheme of the proposed two isthmus-based fully parallel thinning algorithms 3D-FP-IC and 3D-FP-IS are sketched in Algorithm 1. Algorithm 1. Algorithm 3D-FP-I 1: 2: 3: 4: 5: 6: 7: 8: 9: 10:
(I ∈ {IC , IS })
Input: picture (Z3 , 26, 6, X) Output: picture (Z3 , 26, 6, Y ) Y =X I=∅ repeat // one iteration step I = I ∪ { p | p ∈ Y \ I and p is an I-isthmus } D = { p | p ∈ Y \ I and p is 3D-FP-deletable in Y } Y =Y \D until D = ∅
3D-FP-deletable points are defined as follows: Definition 3. A black point is 3D-FP-deletable if all conditions of Theorem 2 hold. Deletable points of the proposed fully parallel algorithms (see Definition 3) are derived directly from conditions of Theorem 2. Hence, both algorithms 3D-FP-IC and 3D-FP-IS are topology preserving. 3.2
Subiteration-Based Algorithms
In subiteration-based (or frequently referred to as directional) thinning algorithms, an iteration step is decomposed into k successive parallel reduction operations according to k deletion directions [6]. If the current deletion direction is d, then a set of d-border points can be deleted by the parallel reduction operation assigned to it. Since there are six kinds of major directions in 3D cases, 6-subiteration algorithms were generally proposed [2,5,18,20,23,24,26,27]. In what follows, we present two parallel 3D 6-subiteration thinning algorithms 3D-6-SI-IC and 3D-6-SI-IS . These isthmus-based algorithms are described by Algorithm 2. The ordered list of deletion directions U, D, N, E, S, W [5,20] is considered in the proposed algorithms 3D-6-SI-IC and 3D-6-SI-IS . Note that subiterationbased thinning algorithms are not invariant under the order of deletion directions (i.e., choosing different orders may yield various results). In the first subiteration, the set of 3D-6-SI-U-deletable points are deleted simultaneously, and the set of 3D-6-SI-W-deletable points are deleted in the last (i.e., the 6th) subiteration. Now we define 3D-6-SI-U-deletable points.
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Algorithm 2. Algorithm 3D-6-SI-I 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:
(I ∈ {IC , IS })
Input: picture (Z3 , 26, 6, X) Output: picture (Z3 , 26, 6, Y ) Y =X I=∅ repeat // one iteration step for each d ∈ {U, D, N, E, S, W} do // subiteration for deleting some d-border points I = I ∪ { p | p ∈ Y \ I and p is an I-isthmus } D(d) = { p | p ∈ Y \ I and p is 3D-6-SI-d-deletable in Y } Y = Y \ D(d) end for until D(U) ∪ D(D) ∪ D(N) ∪ D(E) ∪ D(S) ∪ D(W) = ∅
Definition 4. A black point is 3D-6-SI-U-deletable if it is a U-border point, and all conditions of Theorem 2 hold. It can be readily seen that both subiteration-based algorithms 3D-6-SI-IC and 3D-6-SI-IS are topology preserving. Note that conditions of Theorem 2 can be simplified if d-border points (d ∈ {U, D, N, E, S, W}) are taken into consideration as potential deletable points [23]. 3.3
Subfield-Based Algorithms
The third kind of parallel thinning algorithm applies a subfield-based technique [6]. In existing subfield-based parallel 3D thinning algorithms, the digital space Z3 is partitioned into two [12,16,23], four [13,17,23], and eight [3,17,23] subfields which are alternatively activated. At a given iteration step of a k-subfield algorithm, k successive parallel reduction operations associated with the k subfields are performed. In each of them, some border points in the active subfield can be designated for deletion. Let us denote SFk (i) the i-th subfield if Z3 is partitioned into k subfields (k = 2, 4, 8; i = 0, . . . , k − 1). The considered divisions are illustrated in Fig. 1b-d. In order to reduce the noise sensitivity and the count of skeletal points, N´emeth et al. introduced a new subfield-based thinning scheme [16]. The iteration level border detection strategy takes the border points into consideration at the beginning of iteration steps as potential deletable points within the entire iteration. Next, we present our six new parallel 3D subfield-based thinning algorithms. The scheme of the subfield-based parallel thinning algorithm 3D-k-SF-I (k = 2, 4, 8; I ∈ {IC , IS }) with iteration-level checking is sketched in Algorithm 3. The 3D-k-SF-i-deletable points are defined as follows (k = 2, 4, 8; i = 0, . . . , k − 1):
3D Parallel Thinning Algorithms Based on Isthmuses
Algorithm 3. Algorithm 3D-k-SF-I 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14:
331
(k = 2, 4, 8; I ∈ {IC , IS })
Input: picture (Z3 , 26, 6, X) Output: picture (Z3 , 26, 6, Y ) Y =X I=∅ repeat // one iteration step E = { p | p is a border point in Y } I = I ∪ { p | p ∈ E and p is an I-isthmus } for i = 0 to k − 1 do // subfield SFk (i) is activated D(i) = { q | q ∈ E \ I and is 3D-k-SF-i-deletable in Y } Y = Y \ D(i) end for until D(0) ∪ D(1) ∪ . . . ∪ D(k − 1) = ∅
Definition 5. A black point p is 3D-k-SF-i-deletable if p ∈ SFk (i) and all conditions of Theorem 2 hold. It is easy to see that all of the six subfield-based algorithms 3D-k-SF-I (k = 2, 4, 8; I ∈ {IC , IS }) are topology preserving. Note that conditions of Theorem 2 can be simplified if elements of a subfield are considered as potential delatable points [23].
4
Results
In experiments the proposed algorithms were tested on objects of various images. Due to the lack of space, here we present two illustrative examples below (see Figs. 2-3). The new algorithms were compared with the corresponding endpointpreserving 3D parallel thinning algorithms presented in [23]. We can state that the isthmus-based algorithms produce fewer unwanted side branches or surface patches than the thinning algorithms that preserve curve-endpoints of type CE or surface-endpoints of type SE [23]. Figure 2 presents ten kinds of centerlines produced by the five new isthmus-based curve-thinning algorithms (3D-FP-IC , 3D-6-SI-IC , 3D-2-SF-IC , 3D-4-SF-IC , and 3D-8-SF-IC ) and the corresponding five algorithms (3D-FP-CE, 3D-6-SI-CE, 3D-2-SF-CE, 3D-4-SF-CE, and 3D8-SF-CE) that use curve-endpoints as geometrical constraint [23]. Ten kinds of medial surfaces are presented in Fig. 3: five of them are produced by the proposed isthmus-based surface-thinning algorithms (3D-FP-IS , 3D-6-SI-IS , 3D-2-SF-IS , 3D-4-SF-IS , and 3D-8-SF-IS ), and these medial surfaces can be compared with the results of the five endpoint-preserving surface thinning algorithms (3D-FPSE, 3D-6-SI-SE, 3D-2-SF-SE, 3D-4-SF-SE, and 3D-8-SF-SE) [23]. Note that the reported algorithms are not time consuming and it is easy to implement them on conventional sequential computers by adapting the efficient implementation method presented in [22]. Centerlines and medial surfaces can be extracted from large 3D shapes within one second on a usual PC.
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original image (273 743)
3D-FP-IC (1 297)
3D-FP-CE (1 399)
3D-6-SI-IC (1 332)
3D-6-SI-CE (1 360)
3D-2-SF-IC (1 353)
3D-2-SF-CE (1 387)
3D-4-SF-IC (1 344)
3D-4-SF-CE (1 377)
3D-8-SF-IC (1 353)
3D-8-SF-CE (1 357)
Fig. 2. A 102 × 381 × 255 image of a helicopter and its centerlines produced by the five new isthmus-based curve-thinning algorithms (left column) and the centerlines produced by the five existing endpoint-based curve-thinning algorithms (right column). (Numbers in parentheses mean the count of black points.)
3D Parallel Thinning Algorithms Based on Isthmuses
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original image (2 150 151)
3D-FP-IS (70 386)
3D-FP-SE (94 114)
3D-6-SI-IS (58 097)
3D-6-SI-SE (107 028)
3D-2-SF-IS (68 367)
3D-2-SF-SE (82 539)
3D-4-SF-IS (67 136)
3D-4-SF-SE (81 853)
3D-8-SF-IS (66 869)
3D-8-SF-SE (82 179)
Fig. 3. A 515 × 455 × 110 image of an airplane and its medial surfaces produced by the five new isthmus-based surface-thinning algorithms (left column) and the medial surfaces produced by the five existing endpoint-based surface-thinning algorithms (right column). (Numbers in parentheses mean the count of black points.)
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Conclusions
Fast and reliable extraction of skeleton-like 3D shape features (i.e., centerlines and medial surfaces) is extremely important in numerous applications for large shapes (binary objects). This paper presents ten new variations for parallel thinning algorithms: five of them are curve-thinning algorithms (3D-FP-IC , 3D6-SI-IC , 3D-2-SF-IC , 3D-4-SF-IC , and 3D-8-SF-IC ), and the remaining five (3D-FP-IS , 3D-6-SI-IS , 3D-2-SF-IS , 3D-4-SF-IS , and 3D-8-SF-IS ) are capable of producing medial surfaces. Deletion rules of the proposed algorithms were not given by matching templates (as is usual), they were derived from some sufficient conditions for topology preserving parallel reductions. Hence their topological correctness is guaranteed. All of the reported algorithms are based on isthmuses, and they can produce fewer unwanted branches or surface patches than the corresponding conventional algorithms that preserve endpoints. Acknowledgements. This research was supported by the European Union and ´ the European Regional Development Fund under the grant agreement TAMOP4.2.1/B-09/1/KONV-2010-0005, and the grant CNK80370 of the National Office for Research and Technology (NKTH) & the Hungarian Scientific Research Fund (OTKA).
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