Local deformations of digital curves

Pattern Recognition Letters ELSEVIER

Pattern Recognition Letters 18 (1997) 613-620

Local deformations of digital curves 1 Azriel Rosenfeld *, Akira Nakamura Centerfor Automation Research, University of Maryland, College Park, MD 20742-3275, USA Department of Computer Science, Meiji University, Kawasaki214, Japan Received 3 September 1996; revised 27 May 1997

Abstract

A new concept of a "digitally continuous" deformation of a digital curve is introduced. Such a deformation, in general, need not be topology-preserving; but a deformation under which the regions inside and outside the curve remain disjoint is digitally continuous and does preserve topology. © 1997 Elsevier Science B.V.

1. Introduction

Let S be a connected subset of the plane, and let C, D be simple closed curves that are contained in S. If S is simply connected, any two curves in S can be continuously deformed into one another in S; but if S has holes, this is no longer true: if C surrounds a hole but D does not, C and D cannot be continuously deformed into one another in S. This paper defines analogous concepts for digital plane curves. A straightforward analog, based on a "digitally continuous" mapping of one curve into another, turns out to be too general. We give here an alternative definition in terms of how the mapping treats the regions inside and outside the two curves, and show that this yields a special class of digitally continuous mappings of curves into one another which have properties analogous to continuous deformations. We

* Corresponding author. The help of Janice Perrone in preparing this paper is gratefully acknowledged. The authors also thank Angela Wu and an anonymous referee for many useful comments and suggestions.

also discuss the possibility of extending our results to digital curves and surfaces in three dimensions. Throughout this paper we will use the standard conventions about digital images. A digital image is a mapping of the lattice points of the plane into {0, 1}, such that only finitely many lattice points are mapped into 1. We use the standard definitions of such concepts as 4- and 8-neighbors, connected components, etc., and we use opposite definitions (4- and 8-, or 8- and 4-) for the l ' s and O's.

2. Digital curves and connected sets

A digital (simple closed) c u r v e C (Rosenfeld, 1973) is a finite, non-empty connected set of pixels each of which has exactly two neighbors in C. (We will omit "digital simple closed" from now on.) It is well known that there are two versions of this concept, depending on whether we use the 4- or 8-definitions for " n e i g h b o r " and "connected". For example, the sets of l ' s in Fig. l(a) are 4-curves, and the set of l ' s in Fig. l(b) is an 8-curve. It is not hard to see that no set can be both a 4-curve and an 8-curve.

0167-8655/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 1 6 7 - 8 6 5 5 ( 9 7 ) 0 0 0 3 8 - X

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Fig. 1. (a) Two 4-curves; (b) an 8-curve.

We will assume that a 4-curve has at least eight pixels (the first example in Fig. l(a)) while an 8-curve has at least four pixels (the example in Fig. l(b)); but it will also be convenient to regard a set consisting of a single 4- (8-)isolated pixel as a trivial 4- (8-)curve. We recall (Rosenfeld, 1970) that the complement of a nontrivial 4- (8-)curve has exactly two 8(4-)components, which we will denote by C/ and C o , and refer to as the " i n s i d e " and "outside" of C. Every pixel of C is 8- (4-) adjacent to both C t and C o. In fact (Rosenfeld, 1975), it can be shown that these properties characterize nontrivial curves: if has exactly two (8-, 4-) components, and every pixel of C is (8-, 4-) adjacent to both components, C must be a nontrivial (4-, 8-) curve. Moreover, C surrounds C t, and surrounds no pixel of C o, while C o surrounds C, where " S surrounds T " means that any (8-, 4-) path from a pixel of T to the border of the image must intersect S. Finally, C separates C~ from C o - - i . e . , any (8-, 4-) path from a pixel of C t to a pixel of C o must intersect C. We briefly review some facts about finite digital connected sets (see Rosenfeld, 1970) that will be used later. Let S be a (4-, 8-) connected set, and let T be an (8-, 4-) component of the complement S of S. Exactly one component T O of S surrounds S; it is called the background of S. All other components of S, if any, are surrounded by S, and are called holes in S. The set of pixels of S that are (4-, 8-) adjacent to pixels of T is called the T-border o f S. If T = T o, the T-border is called the outer border; all other borders, if any, are called hole borders. The outer border of S surrounds S, and S surrounds its hole borders. There are welt-known algorithms (e.g., (Rosenfeld, 1970)) for " f o l l o w i n g " any border B of S: starting from any pixel P of B, such an algorithm will generate a sequence of pixels P = P0, P~ . . . . . P~ = P of B, such that every pixel of B is

one of the Pi's and consecutive Pi's are (4-, 8-) neighbors. In general, B is not a (4-, 8-) curve; nonconsecutive Pi's may be neighbors (B may "touch itself") and may not even be distinct (B may pass through the same pixel twice). A pixel P of S is called isolated if it is (4-, 8-) adjacent only to pixels of S. A non-isolated pixel P of S is called (4-, 8-) simple if it is (4-, 8-) adjacent to exactly one component of S. It can be shown (Rosenfeld, 1970) that if S is simply connected and has more than one pixel, it has at least two simple pixels.

3. Digital

deformations

We say that two digital curves C and D differ by if every pixel of C coincides with or is a neighbor of a pixel of D, and vice versa. (If C and D are 4-(8-) curves, "neighbor" here means 8-(4-) neighbor.) (Such a deformation can be regarded as "digitally continuous" in the sense that it takes neighboring pixels into neighboring pixels; but it is not exactly the same as a digitally continuous mapping of the digital plane into itself in the sense of (Rosenfeld, 1986; see also Boxer, 1994).) It might seem reasonable to define "digital deformation" of curves in terms of this concept of local deformation; we might say that C and D differ by a local digital deformation if there exists a sequence of digital curves C = C o , C 1 , . . . , C n = D such that C i and C i_ ~ differ by a local digital deformation, 1 ~< a local digital deformation

i <.
Unfortunately, this concept appears to be too general. In the continuous case, let C be a subset of a connected set S that has a hole H. If C surrounds (does not surround) H, and C is continuously deformed while remaining a subset of S, any such continuous deformation of C still surrounds (does not surround) H. Hence if C surrounds H and D does not, they cannot be continuous deformations of one another. In the digital case, on the other hand, Fig. 2 shows two digital 4-curves (8-curves) C and D which differ by a local digital deformation, and such that C surrounds the pixel P but D does not. (In these figures, the c's are pixels of C, the d's are pixels of D, and the l ' s are pixels of both C and D.) Since P could be a one-pixel hole, the examples in

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Fig. 2. Two curves that differ by a local digital deformation, and a pixel P that is inside one curve but outside the other. (a) 4-curves; (b) 8-curves.

Fig. 2 show that if we define digital deformation in terms of the concept of a local digital deformation, then two digital curves can differ by a digital deformation even if one of them surrounds a hole and the other does not. As Fig. 3 shows (in the 4-curve case), when two curves differ by a local digital deformation, even a large hole can be inside one curve and outside the other.

pixel of C coincides with or is adjacent to a pixel of D. Let P be a pixel of C that is not on D; then P is either in D~ or D O, say the former. Let Q be a neighbor of P that lies in C o. Since D separates D I from D o, Q cannot be in Do; and since D t and C O are disjoint, Q cannot be in D~. Hence Q must be on D, so that P is adjacent to a pixel of D. A symmetric argument shows that if P is in D o, it must also be adjacent to a pixel of D. []

4. S t r o n g digital d e f o r m a t i o n s

Note that both pairs of curves C and D in Fig. 2, and the pair of curves in Fig. 3, do not differ by a strong local digital deformation; they violate the first part of the definition. W e say that C and D differ by a strong digital deformation if there exists a sequence of digital curves C = C O, C j . . . . . C n = D such that C i and C i_ ~ differ by a strong local digital deformation, l <~ i <~n.

W e say that two digital curves C and D differ by a strong local digital deformation if Cz and D O are disjoint, and C o and D r are disjoint.

P r o p o s i t i o n 1. I f C and D differ by a strong local digital deformation, they differ by a local digital deformation. Proof. Since the definition of a (strong) local digital deformation is symmetric in C and D, we need to prove only one side of the conclusion - e.g., that any d d d d d d d d d

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Fig. 3. C and D differ by a local digital deformation, but a large hole (the P ' s ) is inside C and outside D.

5. T o p o l o g y

preservation

From now on we assume that all our digital curves (C . . . . . D) are subsets of a connected set S, and we speak of (strong) (local) digital deformations in S. We will assume, for concreteness, that S is 4-connected and the curves are 4-curves; but similar results hold in the 8-case.

P r o p o s i t i o n 2. Let H be a hole in S. I f C surrounds H but D does not, C and D cannot differ by a strong local digital deformation in S. Proof. Since C and D consist of l ' s , if C surrounds H, H must be contained in Ct, and if D does not

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modified in the neighborhood of a simple P to create a D such that (a) D passes through P; (b) D~ is contained in C1; (c) C and D differ by a strong local digital deformation. Several examples of this are shown in Fig. 4. (E.g., in the upper example in Fig. 4(a), the pixel marked d is a northeast comer pixel of Ct; in the lower left example, d is a north end pixel of C/.) From (a) and (b) it follows that D / has strictly fewer pixels than C~; hence repeating this process eventually results in a C such that Ct consists of only one pixel. Such a C, shown as the first example in Fig. l(a) and the example in Fig. l(b), is the smallest possible nontrivial curve. Evidently, such a C differs from the trivial curve D = C t by a strong local digital deformation. Thus we can construct sequences of strong local digital deformations that shrink C and D into trivial curves; and we can reverse the latter sequence to expand a trivial curve into D. []

surround H, H must be contained in Do; but if C and D differ by a strong local digital deformation, C t and D O must be disjoint. [] Corollary 3. If C surrounds H but D does not, C and D cannot differ by a strong digital deformation in S. Proof. Let Cj, 1 ~ j ~< n, be the first C i that does not surround H; then Cj_ 1 and C~ violate Proposition 2. [] Proposition 4. If neither C nor D surrounds a hole in S, they differ by a strong digital deformation in S. Proof. Clearly, any two trivial (one-pixel) " c u r v e s " in C, D ~ S differ by a strong digital deformation. In fact, since S is connected, we can define a sequence of one-pixel translations (which are evidently strong local digital deformations) which take C into D, moving only through l's. To complete the proof, we will show that there exists a sequence of strong local digital deformations that "shrinks" C (or D) into a trivial curve; evidently, the reversal of such a sequence "expands" a trivial curve into D (or C). We can construct a sequence of strong local digital deformations that shrinks C into a trivial curve by taking advantage of the fact, mentioned at the end of Section 2, that if C t is simply connected and has more than one pixel, it has at least two simple pixels P. It can be verified that C can be

From Corollary 3 and the proof of Proposition 4 we have the following theorem.

T h e o r e m 5, C differs from a trivial curve by a strong digital deformation in S iff it does not surround a hole in S. Our main goal in the remainder of this section is to prove that if C and D both surround a single hole H in S, they differ by a strong digital deformation in S. To do this we first need the following proposition. C

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Fig. 4. Examples of strong local digital deformations in the neighborhoodsof simple pixels. (a) 4-curves; (b) 8-curves.

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Proposition 6. Let S be any finite 4-connected set, and let S* be the one-step 8-neighbor dilation o f S. Then the borders o f S* are 4-curves.

(We recall that S * consists of all pixels that are in S or have 8-neighbors in S.)

Proof. Let B be a border of S*, say the T-border, and let Q be any pixel of B. We will show that Q can have only two 4-neighbors that are on B; hence B is a 4-curve. Let P and R be the predecessor and successor of Q when we apply border following to B. Then P and R are 4-neighbors of Q, so that the neighborhood of Q looks like one of the cases in Fig. 5 (or their rotations by multiples of 90°), where the O's are in T. We must show that no pixel of B other than P and R can be a 4-neighbor of Q. Case (a) is immediate, since the other two 4neighbors of Q are O's. In case (b), suppose the south neighbor Q' of Q is a border pixel of S* ; then Q' is not in S (since the pixels of S are interior pixels of S* ), and some neighbor Z of Q' (other than P, Q, R) is 0. Now P and R are in S* but not in S; hence they have 8-neighbors P', R' that are in S. Let 7r be a shortest 4-path in S from any such P' to any such R'. Since 7r is shortest, it must be a simple arc (Rosenfeld, 1973), so that it cannot intersect or be 4-adjacent to itself (i.e., nonconsecutive pixels of 7r cannot be the same and cannot be 4-neighbors), and it is not 4-adjacent to P or R except possibly at its endpoints; also, 7r cannot be 4-adjacent to Q since the 4-neighbors of Q are not in S. Hence we can complete 7r into a simple closed 4-curve s: in S* by adjoining to it P, Q and R (and common 4-neighbors of P and P', R and R' in S*, if necessary). The column containing Q crosses s: (at Q); hence (as shown in the proof of the digital Jordan curve theorem in (Rosenfeld, 1970)) K separates Q' from Q ' s north neighbor; and since K is a 0 PQO R

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Fig. 5. Neighborhoodsof a border pixel.

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4-curve, Z mad Q' are in the same 8-component of the complement of s:. Hence Z and Q ' s north neighbor are in different 8-components of the complement of S *, so that Q' is on a different border of S" than Q is. The proof in case (c) is similar; there are several subcases, depending on whether the east or north neighbor Q' of Q is a border pixel of S*. Here the row or column containing Q may not cross K; but a nearly horizontal or nearly vertical digital straight line must cross K. For example, if Q' is the east neighbor of Q, we can use the nearly horizontal line that consists of the part of Q ' s row to the right of Q (including Q itself) and part of R ' s row to the left of R (including R itself). Evidently P and R' are on opposite sides of this line, so the line crosses K. The details of the case (c) proof are left to the reader. []

Proposition 7. Let B and C be curves such that C surrounds B, and there are no O's " b e t w e e n " B and C (i.e., no holes in B o ( 3 C l ) ; then B and C differ by a strong digital deformation in S. Proof. If every pixel of C is on B, we evidently have B = C and we are done. Let P be a pixel of C that is not on B, hence is in B o. Since B separates B o from B z, the neighbors of P cannot be in B r Thus C can be locally diverted around P, through neighbors of P that are in CI but not in Bt (so that these neighbors cannot be O's). If this diversion C' is still a simple closed curve (see below), it is evidently a strong local digital deformation of C. Moreover, C' still surrounds B, and the intersection of C ' U C'~ with B o has smaller area than the intersection of C U Cz with B o. Thus repeating this process eventually gives us a C (n) such that the intersection of C (") U C~") with B o is empty, which implies that C (n) = B. It remains to show that there is always some pixel P on C that is in B o, and around which C can be diverted while still remaining a simple closed curve. Suppose P is a pixel of C ~ B o such that C cannot be diverted around P, into C 1, while remaining a simple closed curve. Then P must have an 8-neighbor R which is on C, or which is 4-adjacent to a pixel Q of C, such that R or Q (respectively) is not

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close to P (specifically, is at least three steps away from P ) along C. Suppose first that R is on C; then it cannot be a 4-neighbor of P, and since R is not close to P along C, the common 4-neighbors of P and R cannot be on C. Let R' be one of these common 4-neighbors. R' must be inside C, and must be on or outside B since B cannot pass between P and R; hence R' cannot be 0. It is not hard to see that at most one of the arcs PR and RP of C, say PR, can be 4-adjacent to R' at a pixel of C that is not close to P or R along C, and if there are two such pixels, U and V, they must be two steps apart along PR. In any case, the arc PR (or the arcs PU and UR, or PV and VR) together with R', and the arc RP together with R', must be simple closed curves. Evidently B cannot pass between P and R; hence one of these simple closed curves must still surround B, while there are neither O's nor pixels of B in the interior(s) of the other(s). By induction on the length of C, the curve that surrounds B can be diverted as in the previous paragraph, while the other curve(s) can be eliminated entirely by a sequence of strong local digital deformations using the methods described earlier in this section for curves that do not surround holes. The argument is similar if R is not on C but is 4-adjacent to a pixel Q of C. Here R may be a 4-neighbor of P; in this case each of the arcs PQ and QP of C can be 4-adjacent to R at pixels other than their endpoints P and Q, but this can happen at at most two pixels, say U and V, and the subarcs of C between P , Q, and these pixels (e.g., PU, UV, VQ), together with R, must be simple closed curves one of which surrounds B. Finally, if R is an 8-neighbor of P, one or both of their common 4-neighbors (R 1 and R 2) may be consecutive to P on C; in any case, since R has only one other 4-neighbor (in addition to R~, R 2 and Q), only one of the arcs PQ (or R1Q) and QP (or QR 2) can be 4-adjacent to R at a non-endpoint, say U, and the subarcs between the pixels at which they are adjacent together with R, must be simple closed curves one of which surrounds B. [] W e can now prove Theorem 8. T h e o r e m 8. If C and D both surround a single hole H in S, they differ by a strong digital deformation in S.

Proof. We will show in the next paragraph that for any hole H in S, there is a " s m a l l e s t " curve B H that surrounds H - i.e., if C is any curve that surrounds H, then B H is contained in C U C 1. (It should be pointed out that there need not exist a curve that surrounds only a given hole H. For example, if H has a narrow-necked concavity, there can be another hole K in the l ' s inside the concavity; and any curve that surrounds H must also surround K.) By Proposition 7, C and B H differ by a strong digital deformation, and so do D and B/4; this implies the Proposition. To see that there is a " s m a l l e s t " curve surrounding any H, note first that the set of l ' s adjacent to the outer border of H (the outer " c o b o r d e r " of H) is not necessarily a curve, since if H has a narrow concavity, this set of l ' s may intersect or touch itself. However, by Proposition 6, if we perform a one-step 8-neighbor dilation of H, the outer border of the resulting set H * is a curve. This outer border consists entirely of l ' s (of the original image, prior to the dilation). It surrounds H * , hence surrounds H. Moreover, if C is any curve that surrounds H, evidently H* is contained in C U C r Hence the outer border of H* is the smallest curve that surrounds H. [] The proof of Proposition 7 says nothing about possible O's in B/; it is valid even if B (and therefore C) surrounds many holes. In general, if C and D surround the same set of holes, and there are no holes outside either of them, they differ by a strong digital deformation in S. (Proof: If the outer border of S is a curve (call it E), it surrounds both C and D, and since there are no holes outside C and D, there are no O's between either of them and E. Thus by the proof of Proposition 7, C and D differ by strong digital deformation from E, and hence from each other. If the outer border of S is not a curve, we can proceed as follows: Temporarily change the O's in the holes of S to 1 's; call the result S (~). Perform a one-step erosion of S (1) (change l ' s to O's if there are O's in their 8-neighborhoods); call the result S (2). Perform a one-step dilation of S(2); call the result S (3). It is easy to see that S (3) is a subset of S (~), and contains all of the original hole pixels, but has none of them on its border, so we can change the hole pixels back to 0's; call this final result S*. Any

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Fig. 6. C and D both surround holes H and K, and hole L is outside both of them, but C and D do not differ by a strong digital deformation.

curve C in S must still be contained in S* (the interior of C must have survived the erosion, so C must be contained in the re-dilation). Note that S* is not necessarily connected, but except for the holes that were in S, its components must be simply connected. Since C is connected, it must be contained in some component T of S*, and if C surrounds a set of holes, these holes must also be surrounded by T. Thus if C and D both surround the same set of holes, they must be contained in the same T. But since T is the result of a dilation, its outer border is a curve, and this curve E surrounds both C and D.) By Theorems 5 and 8, if C and D surround no holes or surround the same (one) hole, they differ by a strong digital deformation in S even if there are other holes outside them. On the other hand, if C and D both surround the same two holes, and there is another hole outside them, they may not differ by a strong digital deformation, as we see from the example in Fig. 6, where H, K and L are one-pixel holes in S. (Note that in this example, any curve E that surrounds both C and D also surrounds L; thus E does not surround the same set of holes that C and D do.) To see that C and D in Fig. 6 cannot differ by a strong digital deformation, let C * = C U Cz, D* = D U D t, and note that C* U D * has a hole which contains a 0 (the pixel L). If C and D differed by a strong digital deformation, there would exist a sequence of strong local digital deformations that would make C and D coincide, thus eliminating the hole. But O's cannot cross the hole's coborder, which consists of l's belonging to C and D; hence no sequence of strong local digital deformations can eliminate the hole.

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6. C o n c l u d i n g r e m a r k s

It would be of interest to develop analogs of the results in this note for (simple closed) surfaces in three-dimensional digital images. A simple approach would be to define a surface as a connected set S of voxels whose complement S consists of exactly two components, and such that each voxel of S is adjacent to both components; of course, the appropriate types of connectedness (6-, 18- or 26-) and adjacency must be used for S and S. (This simple definition is actually not strong enough; see (CohenOr et al., 1996).) Given a suitable definition, it should be possible to define local digital deformations and strong local digital deformations of a surface, and to prove results analogous to those in this note. Extending the results of this note to simple closed (6-, 18- or 26-) curves in three-dimensional digital space is not so easy. It is straightforward to define a local digital deformation of such a curve and to show that it does not preserve topological properties; for example, sequences of local digital deformations can be defined that link two unlinked curves or vice versa. On the other hand, it is not clear how to define a concept of strong local digital deformation for 3D curves, since the complement of a 3D curve has only one component (which surrounds the curve). The authors are currently investigating a general approach to defining strong local digital deformations of arbitrary sets S, based on deletion of simple pixels or voxels from S or S. (Note that this process does not take curves into curves or surfaces into surfaces.) Another approach to studying deformations of digital (plane) curves is to regard them as composed of unit line segments ( " c r a c k s " between pairs of edge-adjacent pixels; see (Rosenfeld, 1973)). A "crack curve" is a " 4 - c u r v e " if vertex-adjacency of nonconsecutive line segments is permitted; if not, it is an "8-curve".) For any crack curve C, we can associate a pair of edge-adjacent pixels P,Q with each line segment of C, where P is inside C and Q outside it. Let C't denote the set of P s and C O the set of Qs. (Note that C'I and C O may not be digital curves; at a narrow convexity or concavity of C, C'z (C o) can touch or intersect itself.) A weak definition of local digital deformation for crack curves C and D might be: For each pair P,Q associated with C,

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either P or Q is in D'1 U D o, and vice versa. (Equivalently: Each pair of C overlaps a pair of D, and vice versa.) Evidently this definition is too weak; it is satisfied by the examples in Fig. 2, if we regard them as crack curves (where the c ' s and d ' s are the P ' s of C and D, respectively). A strong definition might be: C t___D/UDo; C o c_D o U D z D tc_C tU t C'o; D O c_ C o U Cz. It is easy to see that this implies the weak definition. To insure " t o p o l o g y preservation", we must also require that any O's in C l ( C O , D l , D O ) must be in D t ( D o , C I, Co), and not in D O (D'~, C o, C't). It would be of interest to study analogs of the results of this paper for crack curves, and also to extend them to digital surfaces defined by sets of unit squares corresponding to pairs of face-adjacent voxels, one inside the surface and the other outside; see (Rosenfeld, et al., 1991).

References Boxer, L. (1994). Digitally continuous functions. Pattern Recognition Letters 15 (1994) 833-839. Cohen-Or, D., A.E. Kaufman and T.Y. Kong (1996). On the soundness of surface voxelizations, In: T.Y. Kong and A. Rosenfeld, eds., Topological Algorithms for Digital Image Processing. North-Holland, Amsterdam, 181-204. Rosenfeld, A. (1970). Connectivity in digital pictures. J. ACM 17, 146-160. Rosenfeld, A. (1973). Arcs and curves in digital pictures. J. ACM 20, 81-87. Rosenfeld, A. (1975). A converse to the Jordan Curve Theorem for digital curves. Inform. and Control 29, 292-293. Rosenfeld, A. (1986). "Continuous" functions on digital pictures. Pattern Recognition Letters 4, 177-184. Rosenfeld, A., T.Y. Kong and A. Wu (1991). Digital surfaces. CVG1P: Graphical Models and Image Processing 53, 305312.