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Generalized Hough Transform: Fast Detection for HybridMulti-Circle and Multi-Rectangle*

Zi-qiang LI1,2,3

1)Department of Computer Science & Engineer, Dalian University of Technology, Dalian 116024

2)

School of Information and Engineering, Xiangtan University, Xiangtan 4111053)Key Lab for Precision and Non-traditional Machining Technology , Ministry of Education, Dalian 116024

(E-mail: xtulzq@hotmail.com)

*

This work is supported by NSF Grant #50335040, #50575031 .

Abstract- The solution to the issue of hybrid multi-circle andmulti-rectangle detection is to firstly detect all circles from adigital image with Multi-Rectangle. When detecting multi-circle,

this approach determined each possible circle by two random

sample edge pixels and the third edge pixel searched on theirperpendicular bisector. As two points are randomly picked and

the third point of the possible circle is searched, invalid sampledecreases because of the removing of isolated noises, half-linknoises and the elimination of non-co-circle points. By affirmingthe possible circle for true circle in minimal area, invalid

computation decreases. When rectangle is detected, any possiblerectangle can be obtained by tracking close curve with rule givenand can be affirmed whether it is true rectangle or not by thedefinition of rectangle. The experimental results demonstratethat the approach can implement well hybrid multi-circle and

multi-rectangle detection and the detection speed of multi-circlesis faster an order of magnitude than Randomized CircleDetection (RCD) proposed by Chen&Chung (2001) on conditionof multi-circle, multi-rectangle and/or random levels of noise.

Index Terms - Generalized Hough Transform. Multi-Circle.

Multi- Rectangle. Detection.

I. INTRODUCTION

One of the most important tasks of computer vision is therecognition of geometry graphics. It is mainly used to designautomation, probe in deeper sea and construct the nationaldefense and so on. For example, in the solution to the layoutdesign problem of satellite [1] (see Fig.1), its 2-D problem isactually a kind of complicated layout design one with constraintof performance caused by the no coincidence between thecentre of mass of the bearing and the axes, that is, many circularand rectangular objects are placed in a circular container withsome conditions (see Fig.2). It is difficult to resolution. Paper[2] primarily explores a layout design method, calledIntegration of Human, Algorithm and Knowledge. The layoutmethod is based on the information fusion of existentknowledge layout drawing (or called knowledge resolution),

numeric resolution given by human on the spot (calledartificial resolution) and the one calculated by machine(called calculated resolution). The fusion method used isevaluation algorithm (such as genetic algorithm). Before thefusion, it is required to implement transform from priorknowledge layout drawing to graphic parameters. The methodused frequently is Hough transform. Quick, reliable and

accurate transform from layout drawing into numeric resolution(i.e. detects circular and rectangular objects) is a premise toimplement the fusion of the three mentioned above with thegenetic algorithm. It is obvious that quickly and accuratelydetecting multi-circle and multi-rectangle in an image is animportant component for this layout design method. BecauseStandard Hough transform (SHT) and Generalized Houghtransform (GHT) are good at single-circle detection[3-5]but no atmulti-circle. Therefore, a lot of scholars have studied many

algorithms for multi-circle detection. For example, Xu[7]

proposed Randomized Hough Transform (RHT) in 1993. In theRHT, 3 non-collinear edge pixels taken by sampling randomlyare mapped into a point in the parameter space. The map ofmany-to-one avoids the problem of tremendous workloadcaused by map of one-to-many in SHT. But blind randomsample brings invalid sample and invalid accumulation thatlowers the detection efficiency. So, Shih-Hsuan Chiu et al[8]

brought forward an efficient voting method for circle detectionin 2005. It improved efficiency of RHT. Chen Tehchuan et al[7]

presented an non HT series algorithm called Randomized CircleDetection (RCD). Its detection speed is faster than RHT for thenoise level between the light level and the moderate level. In

addition, other numeric methods for multi-circle detection canbe found in paper[9]. The method mainly focused on therobustness and accuracy in multi-circle detection. Detectingrectangle generally is implemented by detecting line segment.By far, the algorithm of hybrid multi-circle and multi-rectangledetection cant be found. This paper gives a detection approach

based on solution to the layout design problem of satellite. Theapproach has following advantages: (1) it can eliminate isolatedand half-link noises; (2) when detecting circle, it can furtherdecrease invalid sampling; (3) it has real-time detection speed.

x

y

Fig.1. Satellite layout. Fig.2. A floor layout of satellite

o

1-4244-0332-4/06/$20.00 2006 IEEE10130

Proceedings of the 6th World Congress on Intelligent Control

and Automation, June 21 - 23, 2006, Dalian, China

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II. THE RELATED KNOWLEDGE

Definition 1. Noise pixels and pixels on circle andrectangle are called the edge pixel (pixel value:1), others arecalled the non-edge pixel (pixel value:0) in this paper.

Theorem 1. Suppose that there are digital circles in animage without any noise and rectangle. They consist of N1,

N2,,N edge pixels, respectively. The probability that 2 edgepixels (3 edge pixels) sampled randomly lie on the same circle

is approximately1 (2) times larger than that of 3 edge pixels

picked randomly (4 edge pixels) (1,2[1

i

i

N

=

/1max i

iN

,]), and

when the difference between Ni(1i)is slight,1,2 are

approximately .Theorem 2. Suppose that there are digital circles in an

image without any noise and rectangle. They consist ofN1,N2,, N edge pixels, respectively. Then the probabilitythat 2 edge pixels sampled randomly lie on the same circle is

approximately([1/,1max i

iN

/

1

i

i

N

=

]), and when the difference

betweenNi(1i) is slight, then is 1/ approximately.

Theorem 3. Suppose that the sum of number of noise

pixels and number of edge pixels on all rectangles istimesgreater than the sum of number of edge pixels on digitalcircles in an image, then probability that 2 edge pixels (4 edge

pixels) sampled randomly lie on the same circle is

approximately 1/(1+)2 times (1/(1+)

4 times) than that of no

noise and rectangle.

III. THE MULTI-CIRCLE DETECTION

As RHT and RCD detect multi-circle in an image, possiblecircles are determined by sampling randomly so that many ofthem are false. It is better solved by reducing number of pointssampled randomly and eliminating invalid sample in this paper.

A. Eliminating Two Kinds of Noises

LetA=[aij]HeighWidth denote a matrix with Heightrow andWidth column, where Width and Height denote breadth andheight of an image, respectively, aij is a pixel value corre-sponding to a pixel in image (non-edge pixel: 0 , edge pixel:1).

Definition 2 LetP(x,y) be a edge pixel in an image, there

is no one edge pixel in its 3 3 neighborhood, thenPis called

an isolated noise.

Definition 3 Let 4 4 neighborhood of a edge pixel P(x,y)

in an image satisfy: Boundary consists of 14 non- edge pixelsand except P(x,y) there are 1~3 edge pixels in it then P iscalled a half-link noise.

By the analysis of digital image, the most noises belong tothe above two kinds. According to Theorem 3, removing all

isolated noises and half-link noises before detecting circle andrectangle can reduce remarkably invalid sample.

B. The Basic IdeaAccording to theorem 1~2, 2 edge pixels P1 (x1, y1) and

P2 (x2, y2) are picked randomly in an image. IfP1P2Td(Td is a threshold), then we begin from midpoint M(xm,ym) ofline segmentP1P2, and search a edge pixel P3(,) on perpen-dicular bisector of line segment P1P2. Suppose that circlesdont intersect each other, and ifP1 and P2 lies on the same

circle with center O(a,b) and radius r, then P3 is certainly on

the O (see Fig3). According paper[7], 3 parameters of the

O can be written as:2 2 2 2 2 2 2 2

2 1 2 1 1 1 1 2 1

2 1 1 1 2 1

( )( ) ( )( )

2(( )( ) ( )( ))

x x y y y x y y ya

x x y x y y

+ + =

(1)

2 2 2 2 2 2 2 2

1 1 2 1 2 1 2 1 3 1

2 1 1 1 2 1

( ))( ) ( ))( )

2(( )( ) ( )( ))

x y x x x x y y x xb

x x y x y y

+ + =

(2)

( ) ( )2 2

1, 2,3.i ir x a y b i= + = (3)The above stated is only an ideal case. In general, the papermakes following process according to Theorem 2~3.

C. Deceasing Invalid Sample FarthestTheorem 2 shows that even ifP1 and P2 are not noise

pixels, the invalid sample rate is much larger for multi-circledetection. For example, the invalid sample rate goes up 9/10

possibly when =10. So, it is quite necessary to reduce invalidsample most. The judge method is as follows: If there isntany edge pixel on the perpendicular bisector of line segment

P1P2, thenP1 andP2 are invalid sample; or else, the slope a1 offitting line y=a0+a1x can be figured out by the equation set(4) obtained by least- squares line fitting with 9 edge pixels in

3 3 neighborhood ofP3 and element value corresponding tothem in matrixA. According to character of circle, ifka1(kdenotes slope of line segment P1P2), then P1,P2 and P3 arentcertainly on the same circle and belong to invalid sample(seeFig.4). In programming the condition k=a1 is replaced by |k-a1|

0, on line lwhen s(Pi,M)=

0, on 2 sides of line lwhen s(Pi,M)

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Fig.7. Tracking process of rectangle

(a)

(b)

(c)

(d)

(e)

AF

C

D

B2B1

Q1Q2

Q1

Q1 Q2

Q2

Q3

C E

F

DE

After working out centre O(a,b) and radius r of the

possible circleonly when statistic of the edge pixels on Ois conducted in the smaller area contained the circle, can thedetection speed be improved. The area D in the paper is thedifference of 2 square areas. Side-lengths of 2 square are2(r+1) and 1.1rrespectively. Their sides run paralleling withthe coordinate axes respectively, and their symmetry centre

both are the circle centre O(a,b).

( ) ( )2 2a b r + < T (6)

Let T=0.5, ifP(,)D and satisfies inequality(6), Then Pisaffirmed on the possible circle. Because the number of edge

pixels on a digital circle is different from radius. Whether thepossible circle in area D is true circle is affirmed by the factwhether the edge pixels Mc satisfying inequality (6) in areaD

is larger than the least edge pixels Tm. Set Tm= 2r, where

is proportional coefficient. When Mc>Tm the possible circle

is true circle marked with C(a,b,r).

IV. THE MULTI-RECTANGLE DETECTION

In this paper, we prescribe that any side of a rectangle

can't overlap and intersect with any side of others in an image.A. Warm-up workAfter taking out the circles detected, all rectangles

intersected with the circle are divided into several segmentsand there are some new noises in the image. In order to carryout contour track of rectangles, we connect all broken pointsaccording to rule 1 after removing isolated noise and half-linknoise by Definition 2,3 in section III( B).

Rule 1 IfP1and P2 are end-points of two curves and

|P1P2|3, thenP1,P2 are connected.B. Contour track

After all broken points are connected we begin scanning theimage from left to right and from up to down. Once an edge

pixel is met we pause scanning and regard it as the first trackpixel, marking with Pb. After all edge pixels in its adjacentpixels (on eight directions) are found out, one with the largestdirection value (see Fig.6) is regarded as the next track pixel.After the track pixel is renewed twice, we search for the next

track pixel by the rule 2 until we meet an edge pixel with Nu=0. In this way, we can obtain all the points on the possiblerectangle. After taking out all the edge pixels of the possiblerectangle we begin at Pb and repeat the process above until asequence of points of all possible rectangles can be found out.

Rule 2 Suppose there are Nu edge pixels untracked

among edge pixels adjacent to current track pixel, then 0Nu7. IfNu=0 then we finish tracking; ifNu=1 then we regardthe unique untracked edge pixel adjacent to current track pixel

as the next track pixel; if 2Nu7 and they are marked by

Qj(j=1, 2,,Nu) according to direction value from small to

large. WhenNu =2 andN(Q1)N(Q2), the edge pixel Qj withthe smallestN(Qj) is regarded as the next track pixel. WhenNu=2 and N(Q1)=N(Q2) or Nu3, the edge pixel Qj with thesmallest direction value is regarded as the next track pixel.

N(Qj) denotes number of untracked edge pixels adjacent toQj(j=1, 2,,Nu).

For example, if Fig.7(a) is scanned from top to bottomand from left to right, the edge pixel met firstly is A. i.e. A

become the first track pixel. Edge pixels neighbored on pixelA areFandB1. Their direction values are 4 and 6 respectively.

Because of 6>4, the next track pixel isB1. AfterwardB2 turninto next track pixel ofB1. When C is current track pixel,untracked edge pixels adjacent to Care Q1, Q2 (see Fig.7(b)).Because N(Q1), N(Q2) are 3, 4 respectively, Q1 (orD) is thenext track pixel ofC. the untracked edge pixels adjacent toDare Q1,Q2,Q3 (see Fig.7(c)). The 3 direction valuescorresponding to Q1,Q2,Q3 are 4,5,6 respectively. Thus thenext track pixel ofD is Q1. Being similar with C, the nexttrack pixel ofEis Q2 (see Fig.7(d)), namely the vertex of the

possible rectangle. But for 3 cases as Fig.8, the next trackpixel ofEis Q1. Only when true rectangle is affirmed, can thepossible rectangles vertex Q2be found out. WhenFbecomethe current track pixel (see Fig.7(e)),Nu=0. Therefore the track

is ended.

C. Affirm true rectangleLet G={Gi(xi,yi),i=1,2,,m} be a set composed by all edge

pixels of a possible rectangle, and xminxixmax, yminyi

ymax(i=1,2,,m), the possible rectangle is determined byGr1(xmin,ymin), Gr2(xmin,ymax), Gr3(xmax,ymax), Gr4(xmax,ymin) (or Gt1(xmin,yt1), Gt2(xt2,ymin), Gr3 (xmax, yr3),Gt4 (xt4, ymax)). Where 1

ri,tim(see Fig. 9 (a),(b)). If each of the 4 sides of the possible

rectangle is a true line segment and the possible rectanglesatisfies the definition of rectangle, the possible rectangle is

Fig. 6. 8 directions of a pixelFig.5. P1,P

2is onO

1and P

3is onO

2

P1

P2

PO

1

O2

M

1 0

2

3 4 5

6

7

E E E

Q1

Q2 Q1 Q2 Q

1Q

2

Fig.8. The rectangle verte[RPLWWHGGXHWRWUDFking.

(a) (b) (c)

(a) (b)

Gr3Gr2

Gr1 Gr4Gt2

Gt3

Gt4

Gt1

Fi . 9. Two case of the ossible rectan le

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true. Left bottom vertex (x,y), length d, width w, slant angle

of rectangleR(x,y,d,w,) are finally calculated.

V. THE SIMPLE DATA STRUCTURE

In this approach, we initialize the matrixA with value 0 or1 ofHeightWidth pixels in an image, and the 2-D list V(setV) with row and column value of all edge pixels in an image.Once a possible circle or rectangle is affirmed for true, all

elements corresponded to the edge pixels on the circle orrectangle are assigned 0 in matrixA, and elements in set Varerenewed by row and column value of the nonzero elements inmatrix A. Thus, the changing set A and V plays the role ofdynamic list in the process of detection. By using it we canobtain fleetly the information of circle or rectangle in animage and speed up detection.

VI. PSEUDO CODE OF THE APPROACH

From the above description, C/C++ pseudo code in theapproach can be written as following:Initialization Given Tf,Nc,Td,Tk,T,T,.setf=N=Mc=0.

V= V-all the isolated noise and half-link noise;

do{do { randomly sampleP1,P2in V}while (|P1P2|Td);

calculate slope kand midpoint M(xm,ym) ofP1P2;P3(,) = M(xm,ym);do {P3(,)=next pixel ofP3(,) on digital perpendicular bisector ofP1P2

with the same direction ;calculate a1 of fitting line with Eq.(4);if(|k-a1|>Tk) continue ;

else { calculate 0, 1, 2 with Eq.(5);if (0+1-2< T) continue; else break;}

} while (P3 is in the image space);if(P3 is out of a image space) continue;calculate C(a,b,r) throughP1,P2,P3 with Eqs. (1),(2),(3);if (there is C(a,b,r) in tableP) continue;

else {calculate set V1{ P(,)|P(,)DVand it can satisfy

inequality (6)};if(size Mc of set V1> 2r)

{ N++, store C(a,b,r) into tableP;if(NTf)output no circle is detected; break;}

}while ( true );all elements inA correspond to all the noise pixels =0;connect curves broken by detecting circle with rule 1;

Pf=Pb=(0,0);N=0;P=NULL;do{Pb=Pf;

scanA fromPband find the first edge pixel G1andPf=G1;

obtain edge pixel set G of the possible rectangle;calculatexmax,xmin,ymax,ymin for the G ;affirm true rectangle Gr1Gr2Gr3 Gr4, Gt1Gt2Gr3 Gt4;if (rectangle Gr1Gr2Gr3 Gr4 orGt1Gt2Gr3 Gt4 is true)

{N++; storeR (xmin,ymin,d,w,) into tableP;}

if (N=Nr) {output all rectangles inP; break; }}while(PfPb);

Where, V1 denotes set of edge pixel for a possible circle,Nc,Nr stands for the number of circles and rectanglesrespectively;Nis the number of circles or rectangles detected;

f stands for the number of failures; Tf denotes the number offailures that we can tolerate. The others are the same as whatis mentioned in the previous context.

VII. EXPERIMENTAL RESULTS AND DISCUSSION

A. ExperimentIn order to test time performance of RCD and our

approach, 3 experiments are performed on a Pentium 933

MZ computer, with the program written using visual C++ 6.0.

In our approach, is set in turn to be 0.6, 0.5, 0.7; Tf, Tk, T, T

are 500, 1.4, 0.5, 1 respectively.

Experiment 1. The original 181 243 image with 3270

pixels is shown in Fig.10(a). It consists of 5 circles, 7 rectanglesand 1638 noise points. We use algorithm proposed by paper [7]and our approach to detect Fig.10(a) 50 times individually. Theaverage time to detect 5 circles by paper [7] is 14013 (ms). 7rectangles and 5 circles (see Fig.10(b) and TABLE I) are detected

by our approach. But the average time is only 1275 (ms).

TABLE I

RESULT DETECTED FIG.10(a) BY OURAPPROACH

No. Circle Rectangle

1 C1 (24,20,17) R1 (140,8,46,28,90)

2 C2(86,67,44) R2 (17,26,51,36,0)

3 C3(42,106,28) R3 (92,38,92,51,35.5)4 C4(140,117,24) R4 (148,58, 23,14, 0)

5 C5(128,28,20) R5 (7,68, 38,25, 90)

6 R6 (57,117,25,19,0)

7 R7 (81,122,26,19,90)

Experiment 2. It is a real image shown as Fig.11 (a), the

size of which is 166 210 with 4 circles. After we extract the

edge of the image by Kirsch operator, its result is shown inFig.11(b) with 1580 edge pixels. We apply the approach

proposed by paper [7] and ours to detect Fig.11(b) 50 timesindividually. The average time to detect 4 circles by paper [7]is 1143 (ms). 4 circles (see TABLE ) and 2 rectangles

R1(9,5,153, 20,0), R2(184,29,129,22,90) are detected by our

approach. But the average time is only 211 (ms).

Fig.10. (a) The image with 5 circles, 7 rectangles and

1638 noises (b) detection result

(a) (b)

(a) (b)

Fig.11. An example of image detection

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TABLE

RESULT DETECTED FIG.11(b) BY OURAPPROACH

No. Detect value Accurate value

1 C1 (48,71,42) C1 (48,70,42)

2 C2(40,127,33) C2(40,127,33)

3 C3(129,128,32) C3(129,128,33)

4 C4(138,73,42) C3(129,128,33)

Experiment 3. Fig.1 is a 3D optimal layout of satellite

module. Fig.2 is the planner layout of a bearing surface for thesatellite, which contain 10 circles and 7 rectangles. We applythe algorithm proposed by literature [7] and our approach toFig.2 to detect 50 times individually. The average time todetect 10 circles by literature [7] is 2713 (ms). 10 circles and 7rectangles are detected by our approach (see TABLE ). But

the average time is only 750 (ms).

TABLE

RESULT DETECTED FIG.2 BY OURAPPROACH

No. Circle Rectangle

1 C1 110,38,13) R1 (56,10,26,17,0)

2 C2(79,40,13) R2 (27,33, 17,14,0)

3 C3(32,91,10) R3 (19,47,35,21,90)

4 C4(53,44,13) R4 (41,57, 22,18, 0)

5 C5(87,84,13) R5 (44,77, 31,21, 0)

6 C6(122,59,10) R6 (98,92,15,15,0)

7 C7(115,80,13) R7 (54,98,92,120,0)

8 C8(68,68,64)

9 C9(77,60,13)

10 C10(71,63,12)

B. Discussion1) Applicable scope: The approach can suit the detection

of the image with multi-circle and multi-rectangle. But anyside of a rectangle can't overlap and intersect with any side ofothers in a digital image.

2) Precision: The precision of circles detection is sameas paper[7], controlled by T in inequality(6) and in sectionIII(D). So, the detection precision of circles is the same as

paper [7]. If the rectangles side is single pixel width thereisnt error in its detected value.

3) Workload: Experiments show that main workload ismulti-circle detection in our approach. So, our approachmerely compares circle detection with RCD.

Suppose that there are circles in an image. After a circleis detected, all pixels of the circle are removed. So then, when

there are i circles (i=,,2,1) in the image,1,2,,of

Theorem 1~3 are marked asi1,i

2,i,1i +

2i (i=,,2,1)

respectively, where 1i = 1jL

=

/1

j

j

N

=

( jL denotes number of pixels

of rectangle j (j=1,2,,)); 2i is noise level. The meaningsof

Vand D are the same as the above mentioned. In term ofTheorem 1~3 and procedure of RCD, RCD and our approach

samples ( )4

1 2 1 21 /1

i i ii ii

+ +

=and ( )

211 /

1ii

i

+

=times respectively.

Sincei1,i

2 (i=1,2,, ) are related to , andi (i=1,

2,,) is related to reciprocal of, and 1i >

1

1 ,2

i >2

1 (i=1,

2,,), the average number of the sample times of RCD

increase sharply at the same order as O(3(1+ 1i +

2

i )4)

quantitative level. Average number of the sample times of our

approach increases at the same order as O((1+1

i )2)

quantitative level. Furthermore, invalid sample times in RCDare more than those in our approach. When it affirms truecircle, RCD does more invalid computing meanwhile. Thewhole detection time increases with exponential of and

1+

1

i +

2

i . If there are more circles and rectangles or somelevel noises in an image, the detection speed of our approachis faster an order of magnitude than that of RCD averagely.

VIII. CONCLUSION

This paper proposes the fast detection approach forhybrid multi-circle and multi-rectangle. Our approach onlyrequires 0,1 pixel value of all pixels in the image, not anyother information and can eliminate isolated, half-link noiseand has very fast detection speed. So, in the condition of morecircles and more rectangles and some noises defined bydefinition 2~3, the advantages lay with our approach. Theexperimental results demonstrate our approach can realize the

graphics-parameters transform of hybrid multi-circle and non-overlapped multi-rectangle. The future work is detectingmulti-circle and hybrid multi-rectangle with overlappedrectangles from a digital image.

ACKNOWLEDGMENT

This work is supported by the National Natural ScienceFoundation of China (Grants No. 50335040, 50575031).

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