Proceedings of the 2012 International Conference on Machine Learning and Cybernetics, Xian, 15-17 July, 2012

SKELETONIZATION BASED ON THE MEDIAL-AXIS AND SYMMETRY

INFORMATION

YUNG-SHENG CHEN, MING-TE CHAO

Department of Electrical Engineering, Yuan Ze University, Chung-Li 32003, Taoyuan, Taiwan, ROC E-MA IL: eeyschen@saturn.yzu.edu.tw

Abstract: The classical and potential thinning issues such as bias

effect, boundary noise immunity, and even rotation invariant,

are quite worthy of studying in image processing field. In this

paper, based on the fundamental medial axis transformation (MAT) concept, we developed an approach including extraction

of medial axis with symmetry information, clustering, linking,

and post rule-based thinning to investigate the possibility of

reducing the bias effect and increasing the boundary noise

immunity. A rotation invariant rule-based thinning algorithm

was adopted for experimental comparisons. The primary result

confirms that the proposed approach is one of the feasible

directions to overcome these issues.

Keywords: Medial axis transformation; Skeletonization; Symmetry

information; Thinning

1. Introduction

During past half century, thinning has become a quite fundamental processing used in many scientific and commercial image processing software. To preserve the connectedness and I-pixel wide thin line, a thinning algorithm is usually designed in a rule-based manner [1-6]. Based on the rule-based merit, Chen and Hsu [1] proposed a look up table; Grana et al [4] proposed a decision tree to speed up the thinning process. Chen and Hsu [2] designed a systematic approach for designing a variety of so-called 2-subcycle and pseudo I-subcycle parallel thinning algorithms. Chen [3] used the vector analysis to detect the hidden deletable pixel for obtaining the bias-reduced skeletons. Further, Ahmed and Ward [5] considered the rotation invariant for designing a rule-based algorithm, and Rockett presented its improved version.

Along the development trend based on the rule-based manner, some potential issues, e.g. bias effect and boundary noise immunity are still remained. This means that the thinning result should promise the preservation of original geometry even the boundary is influenced by something noise. Unfortunately, due to the "limited view" of small

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widow size, e.g. 3 x 3 and 4 x 4 , adopted in a thinning

method, the progress of overcoming the mentioned problems is limited. Therefore, in this paper, we return to the original for thinking this problem, i.e. skeletonization using medial axis transformation (MAT). A MAT point is defined that within a window W it possesses the local maximum of distances measuring the nearest distance from the point in W

to boundary. By the fundamental MAT algorithm, a MAT point must belong to the set of skeleton; but the skeleton may possibly not "look at " all the MAT points since some "potential " MAT points are possibly ignored in the local maximum decision. This results from some reasons like a variety of line shapes, the square-grid arrangement of digital image influencing the distance computation, and so on. Therefore, it is unavoidable that the MAT points are not guaranteed to be well-connected to form a perfect skeleton. In this paper, we will present an MAT-based skeletonization approach trying to solve this problem and pursue the goal of reducing bias effect and increasing the boundary noise immunity.

The remainder sections are organized as follows. Section 2 presents the proposed approach. Experimental results and evaluations are discussed in section 3. The conclusion is finally given in section 4.

2. Proposed Approach

Because the core of this study is based on MAT, it is apparent that the skeletonization result will be a set of disjointed MAT points. Further, due to the local maximal property in MAT, some skeleton points in tails (or ends) of a stroke pattern may not appear. In order to maintain the connectedness, prevent the serious shrinking, and preserve a I-pxiel wide thinning result, our approach proposed includes the following four parts: extraction of mediall axis with symmetry information, clustering, linking, and rule-based post thinning. The corresponding algorithms will be detailed in the following subsections.

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Proceedings of the 2012 International Conference on Machine Learning and Cybernetics, Xian, 15-17 July, 2012

2.1. Extraction of Mediall Axis with Symmetry Information

In general, MAT can be done with d4, d8, and Euclidean distance calculations to identify a pixel as a primary member of MAT point set. The Euclidean distance is adopted in this study. We define S as a binary image with dimension X x Y , background point set B c S , and foreground point set

F c S. Here B u F = S. Let Me F be the found MAT

point set, the short distance from a point p x,y E M to B may

be expressed as

ED(p,B) = min d(p, b) 'tbEE

(1)

Where

(2)

background

b2

Figure 1. Illustration for the symmetrical point decision.

Due to the transition property of local maxima, except for the disconnected phenomenon a good property in MAT may be the boundary noise immunity under enough resolution provided. However, contrarily this may result in a shrinking phenomenon, in particular on the end of a line. To overcome this problem, the symmetry information for a foreground point p is further inspected. As illustrated in Fig. 1, the p satisfies the symmetry property if there exist two

background points bp b2 E B satisfying the following two

conditions:

Condition 1:

Condition 2:

or

<J..b1pb2 ? ()2 and ED(p,B)?' 52 In this study, we select (()p51) = (180°,3.5) and

(()2' O2) = (150°,4.5) , used in our current experiments. Figure

2 shows a MAT with symmetry information as the considered skeletonization result.

Figure 2. Illustration of primary skeletonization result.

2.2. Clustering

Based on the mediall axis and symmetry information extracted, they are the candidates of the final thinned pixels. However, they are distributed onto different places, clustered

or not, as well as disjointed usually. Therefore, a labeling process is necessary for information clustering before performing the linking process. This can be easily done by a traditional labeling scheme. The clustering result with labels can be illustrated in Fig. 3.

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- "'--'--'--'--'--'--'--'--'--'--'--'--'--T I I - -I- - - - - - - -I-I I · ·1· cluster 1 .. ·1· I- I

.l lilil - - --I __ I_ L-,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--,--I cluster 2 · ·1· .... . "cluster 3

• .1- �. - - - - - - f$liI$I$I$I$I$I$liI$I$I$I$I$13 -I- -- -I-: : I_ dust« 4 - c-'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--'--I :_ !I�

-

_ 1 - -I- - - - - -I-I I - -I- - - - - - - -I-I- I - -I- - - - - - - -I-I- I-- -I- I-

- -1 ______________ 1-

Figure 3. Illustration of clustering with labels.

Proceedings of the 2012 International Conference on Machine Learning and Cybernetics, Xian, 15-17 July, 2012

2.3. Linking

To maintain the connectedness of the skeleton of a line pattern, based on the line direction property the disjoined clusters should be connected. The connection points should be on the local pixel having maximum Euclidean distance and only a unique link could be established between clusters. However due to the uncertainty of searching links, the

multi-path searching may be happened and should be overcome. Therefore, the following rules are presented to deal with this problem.

For any two disjoined clusters, let N' be the total

number of link members Lm, m = 1, ... , N'. Three parameters,

namely, step count (sc), Euclidean distance summation (eds), and straight-line distance summation between link members (slds), are defined as follows for picking up the reasonable link.

N'

After performing the linking process on the clustering result in Fig. 3, the connected results may be obtained as shown in Fig. 4.

2.4. Post Rule-Based Thinning

Due to the uncertain even-number effect for a pattern width, the skeletonization results may exist a two-pixel width as shown in Fig. 4, by performing the connection process. Fortunately, a rule-based thinning algorithm may be suitably applied to eliminate the two-pixel wide phenomenon and still preserve the perfect 8-connectivity. In this approach, the pseudo I-subcycle thinning algorithm [2] is used for this purpose. Table 1 shows the used thinning windows for the rule-based decision. After this post rule-based thinning process, the result can be obtained as Fig. 5 shows.

eds = LED(Lm,B) m=l

N'-l slds = L ED(Lm,Lm+1)

(3) Table I. List of thinning windows performed by the pseudo l-subcyc1e algorithm. Here pixel p can be eliminated if (r OR s) = 1, where x denotes

don't care entry. Note in the first row, each case has other three ones with 90° rotations.

m=l Based on these definitions, the rules are given below:

1. The link lies on the pixels having local maximum of EDs.

2. No links exist between two pixels in the same cluster. 3. Close loop is not allowed. 4. The foreground boundary pixels cannot be a member of

link. 5. If many links are found, the best link is selected based

on the minima of defined, sc, eds, and elds in (3). 6. The nearest neighbor is selected for next step if more

than one neighbor have the same maximum ED. 7. For the sake of efficiency, a stop condition is set for

multi-path searching .

. . rc-,--,--,--,--,--,--,--,------�· I· 1 1 1 1

I L�����������_,_��---��������������I · . . . . . . . . . . . . . 1 · . . . . . . . . . . . . . 1 · . . . . . . . . . . . . . 1 vvvvvvvvvvvvvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvvvvvvvvvvvvv 1 1 1 1 1 ,-----,--,--,-----------------,--,--,--,-_____ 1 I I· I· I· I· 1 1 1

--------______ 1

Figure 4. Connection result.

r 1 1

0 P 1

0 0 S

1 1 X

1 P 0

1 1 x

x 1 1

0 P 1

x 1 1

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0 r 1 0 1 0 0 0 1

0 P s 0 p 1 0 P 1

0 0 0 0 0 0 0 0 1

1 1 1 0 0 S r 1 0

1 P 1 0 P 1 0 P 1

x 0 x r 1 0 0 0 s

x 0 x 0 1 s r 0 0

1 P 1 1 P 0 1 P 0

1 1 1 r 0 0 0 1 s

Figure 5. The final thinning result.

Proceedings of the 2012 International Conference on Machine Learning and Cybernetics, Xian, 15-17 July, 2012

3. Results and Evaluation

So far we have briefly presented the idea of our skeletonization approach for thinning a line pattern. To evaluate the proposed method, the rotation invariant rule-based thinning algorithms [5,6], denoted respectively as R IRB-l and RIRB-2, were used for comparisons. Further, since the merit of medial axis with symmetry information is emphasized in this study, the bias effect and boundary noise effect are worthy of investigating and will be given in the following subsections.

3.1. Bias Effect

Bias effect usually appears in thinning the patters like "T-shape", "L-shape", and the line-intersection part. In [3], Chen presented a hidden deletable pixel detection approach trying to solve this problem in order to preserve the geometry property of original line patters. However there are still many inherent properties to be worthy of studying. Figure 6 shows our result compared with those by R IRB-l and RIRB-2. A less bias-effect result is obtained by the proposed approach.

(a) (b) (c)

Figure 6. Thinning results by (a) proposed approach, (b) RIRB-l, and (c)

RIRB-2, for investigating the bias effect.

3.2. Boundary Noise Effect

To further confirm the feasibility of the proposed method, given an original "H" pattern with I-pixel wide, as the work in [7], a set of thickened digital patters with different boundary noise added were built as follows for

S SNR=lOlog­

N (4)

Where Sand N denote the number of unchanged boundary pixels and that of thickened boundary pixels, respectively. Figure 7 shows the results with SNR ranging from 1 to 30 dB. Corresponding to the original thin pattern of "H", the numbers of unmatched points of thinning results for all cases are listed in Table 2. From this experiment, we can observe that our method is less influenced by the boundary noise added and still preserves the good property of less bias effect. This further confirms the merit of better boundary nOIse immunity for the proposed approach.

HHHHHHH SIIR1dB SIIR5dB SUR 10d6 StiR HdB SIIR20dB SIIR25dB SIIR 30118

(a)

HHHHHHH SIIR1dB SIIR SdD SUR 10d8 SIIR 15dB StiR 20d8 StiR 25118 StiR 30d8

(b)

HHHHHHH SUR idB SIIR SliD SUR 10d8 SUR 15<18 SUR 20d8 StiR 25dB SUR 30d8

(e)

Figure 7. Thinning results by (a) proposed approach, (b) RIRB-l, and (c)

RIRB-2, for investigating the boundary noise effect.

TABLE 2. LIST OF THE NUMBERS OF UNMATCHED POINTS OF THINNING RESULTS FOR ALL CASES FOR INVESTIGATING THE BOUNDARY NOISE

EFFECT.

SNR, dB "H" pattern Algorithm

Pattern Size Our RIRB-1 RIRB-2

Method

1 256 20 31 32

5 256 19 31 32

10 256 24 34 37

15 256 18 36 38

20 256 19 31 33

25 256 19 29 31

30 256 16 29 31

Average 19.28 31.57 33.42

experiments. The boundary noise added to the "H " pattern are 4. Conclusion generated by a random number generator in C language. SNR is defined as Conventional rule-based thinning algorithms have been

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Proceedings of the 2012 International Conference on Machine Learning and Cybernetics, Xian, 15-17 July, 2012

developed a long time and applied widely in many image processing and pattern recognition fields. In past ten years, the rotation invariant issue for a thinning result is raised [3, 5, 6], and conducts some researches refocus on the potential topics on thinning. In this paper, we used traditional MAT method combined with symmetry information, and developed a linking process followed a post-thinning for trying to investigate this concerned issue. The primary results of reducing the bias effect and increasing boundary noise immunity by the proposed approach are encouraging. However, in this study, the time-complexity was not

considered due to the high computing power of nowadays. Further, our method is suitable on a truly line-like pattern. If a non-line-like pattern is given or a narrow-short-length noise added on a truly line pattern boundary, it may result in other complex issues, e.g. the wanted thinned results disappear and the unwanted branches appear. These issues will be our future works to continuously improve the proposed approach.

References

[1] Y.S. Chen and W. H. Hsu, "A modified fast parallel algorithm for thinning digital patterns," Pattern Recognition Letters, Vol. 7, No. 2, 99-106, 1988.

[2] Y.S. Chen and W. H. Hsu, "A systematic approach for designing 2-subcycle and pseudo l-subcycle parallel thinning algorithms," Pattern Recognition, Vol. 22, No. 3,267-282, 1989.

[3] Y.S. Chen, "Hidden deletable pixel detection using vector analysis in parallel thinning to obtain bias-reduced skeletons," Computer Vision and Image Understanding, Vol. 71, No. 3, 294-311,1998.

[4] C. Grana, D. Borghesani, and R. Cucchiara, "Decision trees for fast thinning algorithms," IEEE Proc. of International Conference on Pattern Recognition, 2836-2839,2010.

[5] M. Ahmed and R. Ward, "A rotation invariant rule-based thinning algorithm for character recognition," IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 24, No. 12, 1672-1678, 2002.

[6] P. I. Rockett, "An improved rotation-invariant thinning algorithm," IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 27, No. 10, 1671-1674, 2005.

[7] Y.S. Chen and Y. T. Yu, "Thinning approach for noisy digital patterns," Pattern Recognition, Vol. 29, No. 11, 1847-1862,1996.

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