連結運算在影像切割上之研究

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行政院國家科學委員會專題研究計畫成果報告

連結運算在影像切割上之研究

計畫類別：個別型計畫計畫編號： NSC91-2213-E-009-098- 執行期間： 91 年 08 月 01 日至 92 年 07 月 31 日執行單位：國立交通大學資訊科學學系計畫主持人：薛元澤計畫參與人員：張尹彬、石永靖報告類型：精簡報告處理方式：本計畫可公開查詢

中華民國 93 年 3 月 1 日

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行政院國家科學委員會專題研究計畫成果報告

連結運算在影像切割上之研究

A Study on Connected Operators on Image Segmentation

計畫編號：NSC 91-2213-E-009-098

執行期限：2002 年 8 月 1 日至 2003 年 7 月 31 日

主持人：薛元澤國立交通大學資訊科學系

計畫參與人員：張尹彬、石永靖國立交通大學資訊科學系

中文摘要在分析一連串影像時，影像切割是非常重要的影像處理方法。分水嶺切割法在數學形態學中是相當有效的影像切割法。它觀念簡單、實作容易，但卻易造成過度切割。對過度切割問題，很多學者提出各種解決之道，其中以連結運算之含意最廣，幾乎涵蓋所有解決過度切割問題的方法。本計劃探討連結運算之定義、性質與應用，並將之用於實際的影像切割上。我們獲致相當滿意的實驗結果。關鍵詞：影像切割；分水嶺切割法；連結運算 Abstract

Image segmentation is essential in sequential image analysis. In mathematical morphology, segmentation by watersheds is a well-known morphological segmentation. However, segmentation by watersheds will yield the so-called over-segmentation problem. Many operations have been proposed to solve the over-segmentation problem. Among them, the connected operators have become more and more popular. In this report, we investigate the theoretical background of connected operators and study their properties. When applying to motion segmentation, satisfactory results are obtained.

Keywords: Image Segmentation,

Segmentation by Watersheds, Connected Operators

1. INTRODUCTION

In mathematical morphology, watershed algorithm is a well-known approach to image segmentation. An image is segmented by applying the watershed algorithm to its gradient image. Due to the presence of noise, the gradient image may contain many small regional minima. Which will result in the problem of watershed oversegmentation.

Flooding from markers is a very effective way to reduce oversegmentation. However, it needs experience to choose suitable markers. Connected operators have a great potential to extract markers automatically. They form a large class of operators including opening by reconstruction, area opening, etc.

In this report, we will investigate the properties and possible applications of connected operators. Experimental results by employing connected operators are exhibited and discussed. Then, some conclusions are made and future works are suggested.

2. Connected Operators

Let E denote the 2D Euclidean space. A

partition P on E is a collection of sets in E

which are exhaustive and disjoint. For each element x in E, denote [x]_P the set in P such that x∈[x]P, called the class of x in P.

Then partition P′ is said to be finer than partition P if each class in P′ is a subset of some class in P, and P is said to be coarser than P′. A partition is called connected if all classes in it are connected.

Given a binary image A. Then A induces a unique partition, denoted P , consisting of A

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the connected components of A and its complement. An operation Ψ on binary images is called a connected operator [18] if the induced partition P_Ψ_{( A}₎ is coarser than

A

P for every binary image A.

Example. (Area opening [20])

Let A be a binary image with connected components C₁,C₂,...,C_k. For each r ≥0, the area opening

α

_r of A is the union of

those C that has area greater than or equal i

to r.

Example. (Reconstruction by markers [21]) Let A be a binary image. For any two points x

and y, the geodesic distance between them in

A, denoted d_A(x,y), is the length of the shortest path (if any) included in A linking x

and y. Then, geodesic discs are of the form

B_A_,_λ(x)={y∈E:d_A(x,y)≤λ} Given a marker M, a subimage of A, we can find the union of all connected components of A that containing M by ) (M RA ∞ < ∈ ={x A:d_A(x,y) for some y∈M} =

Υ

0 , , ( ) ≥ ∈Mλ λ y A y B

The resulted image RA(M) is called the

reconstruction of A by M.

The following is an interest property of connected operators.

Proposition 1. [18]

An operator Ψ on binary images is a

connected operator if and only if the symmetric difference of a binary image A and

) ( A

Ψ consists of connected components of

A and the complement of A.

The notion of connected operators can be easily extended to grayscale images by using the concept of flat zones. Given a grayscale image f. We consider the level set

of f at level t:

L_t(f)={x∈E: f(x)=t}

Then a flat zone [16] of f is a maximal

connected component of a level set. Note that flat zones of f constitute a partition of E,

which will be written as P . Then an f

operation Ψ on grayscale images is called a

connected operator [16] if partition P_Ψ_{( f}₎

is coarser than P for every grayscale image f

f.

Area openings and reconstruction by markers can also be extended to grayscale

images. We denote X_t( f) the set of all

points x with f(x)≥t and Xt( f)

∧

the set

of all points x with f(x)≤t .

Example. (Area opening)

Let f be a grayscale image. For each r ≥0,

the area opening

α

r of f is given by

α

_r(f)=sup{t:x∈

α

_r

(

X_t(f)

)

}

Example. (Reconstruction of f by g)

Let f and g be two grayscale images. If g

f ≥ , the reconstruction of f by g is given

by

Rf(g)=sup{t:x∈RX_t(f)

(

Xt(g)

)

}

If f ≤ g, the dual reconstruction of f by g

is given by ( ) sup{ : ( ) } ) (      ∈ = ∧ ∗ ∧ X g R x t g R t f X f t Proposition 2. [7]

An operator Ψ on grayscale images is a

connected operator if for any grayscale image f and for any two neighbors x and y,

) )( ( ) )( (f x ≠Ψ f y Ψ implies f(x)≠ f(y). 3. APPLICATIONS

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Connected operators can be used to reduce the oversegmentation problem when images are segmented by watershed algorithm [1, 19, 20]. For instance, the area

opening

α

r can be used to eliminate those

regions with areas less than r. The

reconstruction of f by f −h will remove

any local maximum with relative height less than h, while the dual reconstruction of f by

h

f + will remove any local minimum with

relative depth less than h.

In our experiment, we apply reconstruction operators to motion segmentation. Consider extracting a frame from the sequence Hall-Monitor using the method proposed by Chang [3] with two different threshold values. The followings are the extracted results after smoothing.

Figure 1 Figure 2

Figure 1 is the extracted image with smaller threshold value. Now, we let f to be the image in Figure 1 and g the image in Figure 2. Then the dual reconstruction of f by g will produce the image shown in Figure 3.

Figure 3

Observe that the image in Figure 3 preserves more object information than that in Figure 2.

4. CONCLUSIONS

In this study we investigate the theoretical background and properties of connected operators. These operators do not process images by pixels, instead, they

process images by zones. This property enables them to become a very powerful approach to reduce oversegmentation.

The property listed in proposition 2 is very useful to derive the notion of levelings [7, 8]. They are the fundamental operations to build segmentation pyramids. We will work on this subject in the future.

5. REFERENCES

[1] A. Bieniek and A. Moga, A Connected

Component Approach to the Watershed Segmentation, In H. Heijmans and J. Roerdink, Editors, Mathematical

Morphology and It’s Application to Image and Signal Processing, pages

215-222, Kluwer Academic, 1998.

[2] U. Braga-Neto and J. Goutsias,

Multiresolution Connectivity: An Axiomatic Approach, In J. Goutsias, L. Vincent, and D. Bloomberg, Editors,

Mathematical Morphology and It’s Application to Image and Signal Processing, pages 159-168, Kluwer

Academic, 2000.

[3] Y.P. Chang, Fast and automatic video

object segmentation using background information and morphological operations, Master thesis, National Chiao Tung University, Taiwan, 2003.

[4] J. Crespo, Space Connectivity and

Translation-Invariance, In P. Maragos, R. Schafer, and M. Butt, Editors,

Academic, 1996.

[5] J. Crespo and R. Schafer, The flat zone

approach and color images, In J. Serra and P. Soille, Editors, Mathematical

[6] J. Crespo, J. Serra, and R. Schafer,

Theoretical aspects of morphological filters by reconstruction, Signal

Processing, 47(2): 201-225, 1995.

[7] F. Meyer, From Connected Operators to Levelings, In H. Heijmans and J. Roerdink, Editors, Mathematical

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[8] F. Meyer, The Levelings, In H. Heijmans and J. Roerdink, Editors, Mathematical

[9] A. Popov, Approximate Connectivity

and Mathematical Morphology, In J. Goutsias, L. Vincent, and D. Bloomberg, Editors, Mathematical Morphology and

It’s Application to Image and Signal Processing, pages 149-158, Kluwer

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[10] F. Potjer, Region adjacency graphs and

connected morphological operators, In P. Maragos, R. Schafer, and M. Butt, Editors, Mathematical Morphology and

Academic, 1996.

[11] C. Ronse, Set-theoretical algebraic

approaches to connectivity in continuous or digital spaces, Journal of

Mathematical Imaging & Vision, 8(1):

41-58, 1998

[12] C. Ronse and J. Serra, Geodesy and

connectivity in lattices, Fundamenta Informaticae, 46(4): 349-395, 2001.

[13] P. Salembier and L. Garrido, Connected

Operators Based on Region-Tree Pruning, In J. Goutsias, L. Vincent, and D. Bloomberg, Editors, Mathematical

[14] P. Salembier, L. Garrido, and D. Garcia, Auto-dual Connected Operators Based on Iterative Merging Algorithms, In H. Heijmans and J. Roerdink, Editors,

Academic, 1998.

[15] P. Salembier and A. Oliveras, Pratical Extensions of Connected Operators, In P. Maragos, R. Schafer, and M. Butt, Editors, Mathematical Morphology and

Academic, 1996.

[16] P. Salembier and J. Serra, Flat zones

filtering, connected operators and filters by reconstruction, IEEE Transactions on

Image Processing, 3(8): 1153-1160,

1995.

[17] J. Serra, Connectivity on complete

lattices, In P. Maragos, R. Schafer, and M. Butt, Editors, Mathematical

[18] J. Serra and P. Salembier, Connected

operators and pyramids, In SPIE, editor,

Image Algebra and Mathematical Morphology, Vol. 2030, pages 65-76,

1993

[19] V. Vilaplana and F. Marques, Face

Segmentation Using Connected Operators, In H. Heijmans and J. Roerdink, Editors, Mathematical

[20] L. Vincent, Grayscale area openings and closings, their efficient implementation and applications, In J. Serra and P. Salembier, editors, First Workshop on

Mathematical Morphology and its Applications to Signal Processing, pages

22-27, Barcelona, Spain, 1993.

[21] L. Vincent, Morphological gray scale

reconstruction in image analysis: Applications and efficient algorithms,

IEEE Transactions on Image Processing,

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連結運算在影像切割上之研究

行政院國家科學委員會專題研究計畫 成果報告

連結運算在影像切割上之研究

中 華 民 國 93 年 3 月 1 日

行政院國家科學委員會專題研究計畫成果報告

連結運算在影像切割上之研究

A Study on Connected Operators on Image Segmentation

計畫編號：NSC 91-2213-E-009-098

執行期限：2002 年 8 月 1 日至 2003 年 7 月 31 日

主持人：薛元澤 國立交通大學資訊科學系

計畫參與人員：張尹彬、石永靖 國立交通大學資訊科學系

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行政院國家科學委員會專題研究計畫成果報告

中華民國 93 年 3 月 1 日

主持人：薛元澤國立交通大學資訊科學系

計畫參與人員：張尹彬、石永靖國立交通大學資訊科學系