An integrated edge detection method using mathematical morphology

ISSN 1054-6618, Pattern Recognition and Image Analysis, 2006, Vol. 16, No. 3, pp. 406–412. © Pleiades Publishing, Inc., 2006.

An Integrated Edge Detection MethodUsing Mathematical Morphology

¶

C.-P. Huang and R.-Z. Wang

Department of Computer and Communication Engineering, Ming Chuan University, 5 Der-Ming Rd. Kwei-shan, Tau-yuan 333, Taiwan

e-mail: [email protected]; [email protected]

Abstract

—This paper presents an edge detection method based on mathematical morphology. The proposedscheme consists of four steps: preprocessing, edge extraction, edge decision, and postprocessing. In the prepro-cessing step, a morphological central transformation is applied to remove noise. In the edge extraction and deci-sion steps, a morphological edge extractor is designed to estimate the edge information of an image, and anedge decision criterion is followed to determine whether a pixel is an edge or not. In the postprocessing step,the morphological hit-or-miss transformation is utilized to improve the correctness of the detected edges. It isproved theoretically for the correctness and effectiveness for detecting ideal edges. Experimental results showthat the proposed method works well on both artificial and real images.

DOI:

10.1134/S1054661806030102

Keywords

: edge detection, morphological transformation, mathematical morphology, morphological filters.

Received August 5, 2005

1. INTRODUCTION

Edge detection is one of the key technologies in thefield of signal, image, and video processing. Severaltechniques based on linear filtering [6, 15] have beenproposed to detect edges. Spatial edge operators [16]are conventional edge detection methods. Statisticalapproaches such as the hypothesis test method [1] havealso been proposed. Multiscale edge detection proce-dures, such as the Canny estimation approach [7] orMallat wavelet approach [13], have gained popularityin recent years. The traditional edge detection approachuses local operators as well as a threshold scheme. Thisis done by using local operators to enhance thestrengths of edges and to smooth small variations at thesame time, and a threshold is applied to the processedimage to determine whether the edge pixel is present ornot. Sobel edge operator [11] is one of the most popularedge detection operators. This operator is a weighted-average operator that gives greater weight to pointslying close to the point of interest. These techniquesare shown to be effective in a clean image, but theywork poorly in the presence of noise [7]. There aresome limitations of using linear-based instead of non-linear-based edge detection operators. A linear filtermay either blur edges or be sensitive to noise, while anonlinear filter retains the sharpness of the edges and isinsensitive to noise. Statistical and multiscaleapproaches are based on convolution calculations andconsider multiple sizes of the windowed signal that canbe computationally expensive.

¶

The text was submitted by the authors in English.

Mathematical morphology is an approach based onset theory for extracting geometrical features out of sig-nals. The field was pioneered by Serra [4, 5], Sternberg[12], and Maragos [9, 10]. Morphology has been shownto be an effective method [2, 3, 10] to extract geometri-cal features. Morphological edge detection has beengenerally performed by using the difference of the twoprocessed images, for instance, between eroded anddilated images or between the original and erodedimage [2, 14]. These results indicate that morphologicaloperators have potential in edge detection. The resultsalso show that their efforts emphasize the operator foredge extraction.

The simplest morphological edge detectors are dila-tion and erosion residues [5]. In Serra’s book [4], heproposes the gradient method as an edge detector. Agradient is defined by averaging the difference betweendilation and erosion of the signals and corresponds tojumps in the image. This method is very simple andeffective when noise is absent. In Marago’s paper [8],he proposes the erosion residue method. He defined thestructure element

nB

=

B

⊕

B

⊕

…

B

, where

B

is a unitsize structure element,

n

is an element’s size, and

⊕

isthe morphological dilation operator. The residue

f

–(

f

Θ

nB

) will enhance the edge of a gray-scale image

f

,where

Θ

is the morphological erosion operator. Thesize

n

of

nB

will control the thickness of the edge mark-ers. The results of this method will be position-biasedto the border of the brighter side. Haralick et al. [2] pro-pose the blur-minimum morphological edge method.This method has been shown to be effective in edgeextraction, but the edges are fuzzier.

In this paper, we propose an integrated mathemati-cal morphology procedure to detect edges in images.

IMAGE PROCESSING, ANALYSIS, RECOGNITION, AND UNDERSTANDING

PATTERN RECOGNITION AND IMAGE ANALYSIS

Vol. 16

No. 3

2006

AN INTEGRATED EDGE DETECTION METHOD USING MATHEMATICAL MORPHOLOGY 407

Morphological operators are nonlinear; they are notonly very effective for detecting edges but also wellsuited for efficient implementation using parallel orsequential computing [5]. The proposed method isshown to be theoretically effective for ideal edges.Experimental results are provided to show the validityand performance of our method in practice for artificialand real images.

2. THE PROPOSED METHOD

The proposed edge detection method consists of thefollowing four steps: (1) preprocessing, (2) edge extrac-tion, (3) edge decision, and (4) postprocessing. Thepurpose of the preprocessing step is to remove noisefrom an image in order to enhance the validity of thesubsequent edge detection process. A morphologicaledge extractor is proposed to evaluate the edge infor-mation of an image, and the morphological decisionstep is applied to decide whether or not edges arepresent. In the postprocessing step, a morphologicalhit-or-miss transformation is designed to remove extra-neous noisy edges. Details of each step are presented inthe following subsections.

2.1. Preprocessing

A good edge detection algorithm should not only becapable of detecting the edge information but also beinsensitive to noise. A morphological transformation isproposed here to remove noise while keeping the edgeinformation. In order to deal with both positive andnegative noise, we combine the positive and negativemorphological smoothing filters to form the morpho-logical central transformation (CT). The central trans-formation is derived from the combination of morpho-logical opening-closing and morphological closing-opening filters to obtain the central area of the signal,that is,

(1)

where

co

(

f

(

n

)) is the morphological opening operationfollowed by the morphological closing operationapplied to signal

f

(

n

) and the

oc

(

f

(

n

)) is the morpholog-ical closing operation followed by the morphologicalopening operation applied to signal

f

(

n

). The centraltransformation has the following important properties:.

Property 1.

The CT

(

f

(

n

))

is bounded to

co

(

f

(

n

))

and

oc

(

f

(

n

)),

that is

,

(2)

Property 2.

The CT filter is an idempotent filter, thatis

,

(3)

These properties can be proved using the antiextensive,extensive, and idempotent properties of

co

and

oc

filters

CT f n( )( )= Min oc f n( )( ) Max co f n( )( ) f n( ),{ },{ },

co f n( )( ) CT f n( )( ) oc f n( )( ).≤ ≤

CT CT f n( )( )( ) CT f n( )( ).=

[3]. These properties show the validity of the CT for thesignal smoothing.

To compare with the linear smoothing (LS) operatorshown in Eq. (4) used in many edge detection algo-rithms [2, 16],

(4)

where N (odd, positive integer) is the number of pixelsof the mask and the median smoothing (MD) operatoris constructed by replacing the value of each point withthe median value in the masked neighborhood of thatpoint. The CT method is a better choice for noiseremoval in edge detection applications due to the fol-lowing observations: (1) Using LS, sharp edges arewidened. However, using the CT and MD, sharp edgesare preserved. (2) The CT method smoothes out morethan LS or MD. (3) In the noisy case, the CT and MDremove all the noise, while the LS cannot. These obser-vations indicate that the morphological filter is a betterchoice for signal smoothing than the linear or mediansmoothing.

2.2. Edge Extraction

In this section, an effective edge extraction schemebased on the morphological transformation is proposedand its validity is investigated. The edge extractionoperator is defined as follows:

(5)

where

f

ed

is the extracted edge information,

f

is an inputsignal such as an image, and

LS

(.) is the linear operatordefined in Eq. (4). The size of the linear smoothingmask is the same as the size of the structuring functionused in the erosion and dilation residues. The erosionresidue

e

r

is defined as

(6)

where

k

is a structuring function with a flat top and zeroheight,

f

is the input signal, and

Θ

is the morphologicalerosion operator. The dilation residue

d

r

is defined as

(7)

where

f

and

k

are the same as in the erosion residue and

⊕

is the morphological dilation operator. We list someproperties of the proposed edge extraction operator tosupport the validity of the operator.

Property 3.

If input signal

f

(

n

)

is constant over thesize of the mask

(

or support K of the structuring func-tion

)

and the structuring function has a flat top withzero height

,

then the extracted edge informationf

ed

(

n

) = 0.

Proof:

By definition,

f

ed

=

Max

{

|

LS

(

f

(

n

)) –

f

(

n

)

|

,

Min

{

e

r

(

n

),

d

r

(

n

)}}. If

f

(

n

) is constant over the mask

LS f n( )( ) 1N---- f n j–( ),

j N 1–( )/2–=

N 1–( )/2

∑=

f ed

= Max LS f n( )( ) f n( )– Min er n( ) dr n( ),{ },{ },

er f fΘk( ),–=

dr f k⊕( ) f ,–=

408

PATTERN RECOGNITION AND IMAGE ANALYSIS Vol. 16 No. 3 2006

HUANG, WANG

area, then LS(f(n)) = 0, and |LS(f(n)) – f(n)| = 0. If f(n)is constant over the support K of the structuring func-tion, then mini ∈ Kf(n + i) = f(n) and maxi ∈ Kf(n – i) =f(n) hold, and thus er = f – (fΘk) = f(n) – mini ∈ Kf(n +i) = 0. Similarly, it can be derived that dr = (f ⊕ k) – f = 0.

Thus, we have fed = Max{|LS( f(n)) – f(n)|,Min{er(n), dr(n)}} = 0.

This property shows that our edge extractor has theproperty that the edge detected is 0 if there is no edge.

Property 4. If the input signal f(n) is not constantover the size of the mask (or support K of the structur-ing function) and the structuring function has a flat topwith zero height, then, for at least one point n' ∈ K, wehave fed(n') > 0.

Proof: By definition, fed = Max[|LS(f(n)) – f(n)|,Min[er(n), dr(n)]]. If f(n) is not constant over the maskarea, then we can find at least two points n1, n2 ∈ K,f(n1) ≠ f(n2) in the mask area such that LS(f(n1)) – f(n1)≠ 0 or LS(f(n2)) – f(n2) ≠ 0. Let us assume the point n'∈ K to be n1 or n2. Then, |LS(f(n')) – f(n')| > 0.

We have fed(n') = Max[|LS(f(n')) – f(n')|, Min[er(n'),dr(n')]] > 0.

This property shows our edge extractor has theproperty that, if there is an edge (not flat a surface), thenthe edge detected is greater than 0.

Property 5. If f(n) is a step edge u(n) defined as

(8)

where C is a positive constant value and the size of themask is odd (≥3), then fed(n0 – 1) = fed(n0) > 0 and thosetwo points have the maximum value.

Proof: By definition, we have

(9)

where k is a structuring function with a flat top withzero height and K is the support of the structuring func-tion. Let Nk be the size of the mask or structuring func-tion. If f(n) = u(n), then we have

(10)

(11)

From Eqs. (10) and (11), we get Min[er(n), dr(n)] = 0.

u n( )C if n n0,≥0 otherwise,⎩

⎨⎧

=

f ed n( ) = Max LS f n( )( ) f n( )– Min er n( ) dr n( ),[ ],[ ],

er n( ) = f n( ) f n( )Θk( )–( ) = f n( ) f n i+( ),i K∈min–

dr n( ) = f n( ) k⊕( ) f n( )– = f n i–( )i K∈max f n( ),–⎩

⎪⎨⎪⎧

er n( )C if n0 n n0 Nk 3–( )/2[ ],+≤ ≤0 otherwise.⎩

⎨⎧

=

dr n( )C if n0 Nk 3–( )/2[ ]– n n0 1,–≤ ≤0 otherwise.⎩

⎨⎧

=

By definition,

(12)

(13)

Let LSr(f(n)) = |LS(u(n)) – u(n)|. Then,

(14)

and

(15)

We have

(16)

This implies

(17)

Thus, fed(n0) = fed(n0 – 1) > 0, and these two points havethe maximum value.

This property shows that our edge extractor extractstwo pixels with the maximum value in the masked areafor the step edge and that these two edge pixels are at ahigher and lower gray-scales of that edge. This propertyconfirms that our edge extractor is an unbiased two-pixel edge detector for an ideal step edge.

Property 6. If f(n) is a ramp edge r(n) defined as

(18)

where C is a positive constant value and the size of themask is odd (≥3), then the edge detected at the middlepoint, nm, is greater than zero, i.e., fed(nm) > 0, wherem = ⎣(i + 1)/2⎦ and ⎣x⎦ denotes the largest integer partof x.

Proof: By definition, fed(n) = Max{|LS(f(n)) – f(n)|,Min{er(n), dr(n)}}.

LS f n( )( ) 1/Nk f n ni–( ),ni Nk 1–( )/2–=

Nk 1–( )/2

∑=

LS u n( )( )

=

0 if n n0 n j, n j– Nk 1+( )/2,≥≤C if n n0 n j, n j+ Nk 1+( )/2,≥ ≥Nk 2n j 1+ +( )C/ 2Nk( ) if n n0 n j,+≥

0 n j Nk 1+( )/2,<≤Nk 2n j– 1+( )C/ 2Nk( ) if n n0 n j,–≤

1 n j Nk 1+( )/2.<≤⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧

LSr u n0( )( ) LSr u n0 1–( )( ) Nk 1–( )2Nk

-------------------- 0,>= =

f ed n0( ) f ed n0 1–( ) 0.>=

LSr u n0( )( ) LSr u n0 n j+( )( ), n j 0,> >LSr u n0 1–( )( ) LSr u n0 n j–( )( ), n j 1.> >⎩

⎨⎧

f ed n0( ) f ed n0 n j+( ), n j 0,> >f ed n0 1–( ) f ed n0 n j–( ), n j 1.> >⎩

⎨⎧

r n( )C if n ni≥C n n0–( )/ ni n0–( ) if n0 n ni,< <0 otherwise,⎩

⎪⎨⎪⎧

=



Let f(n) = r(n). Then, LSr(r(n)) = |LS(r(n)) – r(n)| ≥0, ∀n. If LSr(r(nm)) > 0, then fed(nm) > 0 holds. IfLSr(r(nm)) = 0, then we need to show Min[er(nm),dr(nm)]] > 0 first, then we can get fed(nm) > 0.

By definition, r(n) = C if n ≥ ni, C(n – n0)/(ni – n0) ifn0 < n < ni, 0 otherwise.

We have r(nm) = C(nm – n0)/(N – n0) > 0, er(nm) =r(nm) – Minj ∈ Kr(nm + j) > 0, and dr(nm) = Maxj ∈ Kr(nm –j) – r(nm) > 0, where K is the support of the structuringfunction. We get Min[er(nm), dr(nm)]] > 0.

This property confirms that our edge extractordetects the middle point of the ramp edge.

2.3. Edge Decision

In addition to the edge information, the image con-tains many spurious pixels, illustrating the problemassociated with thresholding the edge information. Thesimplest thresholding technique would be to select asingle value Th and assign all pixels with an edgestrength above the selected value Th to the set of edgepixel. But in real image it seems hard to select a singlethreshold value that will extract the edge without spuri-ous noise.

In order to solve this problem, we apply a morpho-logical operator Rm to the extracted edge first and then

(a) (b) (c)

Fig. 1. Artificial test images: (a) the clean image, (b) the noisy version with SNR 10dB, (c) the noisy version with SNR 7dB.

Fig. 2. Edge detection results of Fig. 1 using the proposed method.

Fig. 3. Edge detection results of Fig. 1 using the blur-minimum method.

410


HUANG, WANG

thresholding to get a binary signal. The morphologicaloperator Rm is defined as

(19)

where fed denotes the extracted edge information image,oc is the opening–closing operation, and Min[x]denotes the minimum of x.

Property 7. Rm(n) is a nonnegative value for all n.

Rm n( ) f ed n( ) Min oc f ed n( )( ) f ed n( ),[ ],–=

The property can be shown easily by using the anti-extensive property of the oc filter. The advantage of thisoperator is not only that it can be used to select thepeaks of interest but also that it is above most of the tex-ture. Thus, this operator can be used for prethreshold-ing purposes. The width of the structuring functionused in this stage should not be more than that used inthe edge extraction. Then, to determine the edge pixel

Fig. 4. Real test images: Lena, House.

Fig. 5. Edge detection results of Fig. 4 using the proposed method.

Fig. 6. Edge detection results of Fig. 4 using the blur-minimum method.



in the image, just set a single threshold value Th is nec-essary. The thresholding is

(20)

where F is a binary signal or image, C is a fixed con-stant (C > 0), for the binary image case C = 1, Rm is themorphological operation signal, and Th is set to a smallconstant.

2.4. Postprocessing

Although the edge can theoretically be correctlydetected, in practice the resulting edge image will havesome broken and isolated edge pixels. If we assumededges are connected and not isolated, then some kind ofnoise removal and edge growing is required to improvethe quality. This procedure can be efficiently performedusing morphological operators. Since the image in thisstep is already a binary image, the morphological oper-ator is also binary. The hit-or-miss transformation [9] isproposed to remove the spurious noise and to grow bro-ken edges by designing proper structuring elementpairs.

The hit-or-miss transformation is defined as

(21)

where F is a binary image, S is a structuring elementpair S = (A, B) (one to probe the inside and one to probethe outside of F), Fc is the complement of F, and ∩ is alogical AND operator. It is assumed that A and B aredisjoint, or otherwise it is impossible for both fits tooccur simultaneously. Since the hit-or-miss transforma-tion is a pattern-matching process, so it can be used tolocate the desired pattern for removal or recovery.

3. EXPERIMENTAL RESULTS

In this section, two experiments are conducted totest the validity of the proposed method. A structuringfunction of dimensions 3 × 3 with a flat top and zeroheight is used in the experiments. In the first test, anartificial checkerboard image and two of its noisy ver-sions shown in Fig. 1 are selected to test the algorithm.Figures 1b and 1c show added Gaussian noise to Fig. 1awith a signal-to-noise ratio (SNR) of 10.0 and 7.0 dB,respectively. The edge detection results of the proposedalgorithm and the blur-minimum method [2] are shownin Figs. 2 and 3, respectively. These results indicate thatthe visual quality of these two methods in the “clean”artificial images is almost the same. However, the edgedetection results using the proposed method are moreinsensitive to noise.

In the second test, we apply the algorithm to popularreal test images of Lena and House, shown in Fig. 4.The edge detection results using the proposed methodand the blur-minimum method are shown in Figs. 5 and

FC if Rm Th,>0 otherwise,⎩

⎨⎧

=

F S n( )⊗ FΘA( ) FcΘB( ),∩=

6, respectively. Figure 6 indicates that some thin edgesare lost and the detected edges appear fuzzy when usingthe blur-minimum based method. Figure 5 shows thatthe proposed method successfully detects the edges ofthe test images. The results indicate that the proposedalgorithm is effective in the edge detection of compleximages. A visual evaluation gives the impression thatthe proposed algorithm has a better performance in thetest images.

4. CONCLUSIONS

In this paper, the problem of edge detection using amorphological approach has been studied. Existingmethods for edge detection emphasize the edge extrac-tion only. In this paper, we use morphologicalapproaches to solve the edge detection problem with amore complete and robust solution. We show theoreti-cally the correctness and effectiveness for the proposedmethod in manipulating ideal edges. Experimentalresults are also provided to show the validity and highperformance of our method in practice for artificial andreal images.

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Chin-Pan Huang was born in1959 in Taiwan, Republic of China.He received the B.S. and M.S. degreesin electrical engineering from ChungCheng Institute of Technology, Tai-wan, in 1981 and in 1985, respec-tively. In 1996, he received the Ph.D.degree in electrical engineering fromthe University of Pittsburgh in theUnited States. From 1996 to 2002, hewas an associate scientist of the Elec-tronic System Division in Chung Shan

Institute of Science and Technology. He then joined theDepartment of Computer and Communication Engineering atMing Chuan University in August 2002 and is currently anassistant professor there. His recent research interests includedata compression, computer vision, digital image processing,and pattern recognition.

Ran-Zan Wang was born in 1972in Fukien, Republic of China. Hereceived his B.S. degree in computerengineering and science in 1994 andM.S. degree in electrical engineeringand computer science in 1996, bothfrom Yuan-Ze University. In 2001, hereceived his Ph.D. degree in computerand information science fromNational Chiao Tung University. In2001–2002, he was an assistant pro-fessor at the Department of Computer

Engineering at the Van Nung Institute of Technology. Hejoined the Department of Computer and CommunicationEngineering at Ming Chuan University in August 2002 and iscurrently an assistant professor there. His recent researchinterests include data hiding and digital watermarking, imageprocessing, and pattern recognition. Dr. Wang is a member ofthe Phi Tau Phi Scholastic Honor Society.

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An integrated edge detection method using mathematical morphology