An Edge and Corner Detector with Its Application in Image Segmentation

Transcription

1 An Edge and Corner Detector with Its Application in Image Segmentation LUN-CHIA KUO, SHENG-JYH WANG Institute o Electronics Engineering National Chiao Tung Uniersit 00 Ta Hsueh Road, Hsin Chu Taiwan, R.O.C. Abstract: - In this paper, we use the Hessian tensor in dierential geometr or edge and corner detection. Most edge operators, such as Cann operator, detect edge centers and use gradient magnitude to represent edge strength. In this paper, or each edge, we detect three crucial elements, edge center, edge contrast, and edge span. Once the edge contrast and the edge span hae been identiied, the can be adjusted or dierent purposes, like contrast enhancement and sharpness enhancement. In addition, the Hessian tensor can be used to ind corners, which are useul or eature selection and edge linking. Furthermore, based on detected edge elements, we also deeloped a hbrid method or image segmentation. Ke-Words: - Edge Detection, corner detection, dierential geometr, principal curature, Hessian tensor, image Segmentation. Introduction Edge detection plas an important role in image understanding, because plentiul inormation could be reealed b merel presering these edge elements. For a gien edge, we ma want to presere all its important properties, such as location and edge strength. So ar, a lot o research works hae alread been conducted []-[]. Gien step edges or ridges, most edge operators aim at detecting the edge center and measuring the edge strength based on magnitude o deriaties (see Fig., where the edge center is indicated b the smbol x and the edge strength is indicated b an arrowed line in green. Although these operators ma work well or speciic applications, some crucial inormation ma hae been lost i we onl presere the edge center and the magnitude o deriaties. Since there are ininite sets o contrast-span pair with the same slope, it would be diicult to estimate the edge contrast and edge span b merel using the magnitude o deriaties. In [], we proposed a method to represent an image using high-curature points in image suraces. We use the Hessian tensor in dierential geometr to detect high-curature points in image suraces. In this paper, we discuss edge detection urther, and demonstrate the extraction o edge center, edge contrast, and edge span with the aid o high-curature points. In addition, the Hessian tensor is also suitable or corner detection, which is useul or eature selection and edge linking. Furthermore, based on the detected high curature points, we deeloped a hbrid method or image segmentation. The rest o this paper is organized as ollows. In Section, the concept o the proposed operator is introduced. In Section 3 and Section 4, edge detection and corner detection using Hessian operator are demonstrated. In Section 5, we present image segmentation. Finall, in Section 6, we conclude this paper. Fig.. Illustration o edge center, edge span, and edge contrast. Concept o high-curature points. Concept o the Proposed Operator We irst discuss the concept o the proposed operator rom the iewpoint o a -D intensit proile. Assume a proile expressed as = (x is gien. The curature κ o the proile can be calculated as '' ( x κ ( x =, ( ' 3 ( + ( ( x where (x and (x denote the irst and second deriaties o (x, respectiel [3]. Then we detect the local extremes o κ(x. Assume κ(x reaches its dκ ( x local extremes at x ex ; that is, = 0. Then, dx x= x ex

2 P e (x ex, (x ex is chosen as a high-curature point o this proile [] (indicated as smbol o in Fig.. We can see rom Fig., edge contrast and edge span can be easil obtained using high-curature pair with a positie(conex sign and a negatie(concae sign. Assume the locations and curature magnitudes o the let and right high-curature point are xl, xr and κ L, κ R, respectiel (see Fig.. The location o edge center x C is deined as r κ L x C = x L + ( x R xl = xl + ( xr xl ( r + r + κ L κ R I an edge is smmetric ( κ = κ, x C is located at the midpoint between x L and xr. Thereore, i high-curature pairs are detected, edge center, edge contrast, and edge span can be easil obtained. With edge span and edge contrast, edge strength can be Contrast expressed either b edge contrast or b (. Span As indicated in [4][4], in man cases, edge contrast ma well represent the perception o edge strength. Fig.. Edge center estimation. Fig. 3. Illustration o high-curature points. The signs o high-curature points are useul in man was. For example, in Fig. 3, the edge ormed b P6 and P7 could be remoed because the contrast between the two points is relatiel small, although the slope in that section is high. This ma be helpul in combating noise. In the section ormed b P9, P0, P, and P, P9-P are actuall in the same concae trend. In this case, both P0 and P can be remoed to orm a single edge with larger contrast. 3 Edge Detection To detect edges in -D images, we irst detect high-curature pairs irst. We consider a two-d image as an image surace (, (. In L R dierential geometr, these high-curature points happen at the positions where at least one o the two principal curatures has a large enough magnitude [3]. For an image surace in the orm o (, (, the directions o these two principal curatures at a point P can be deduced b calculating the eigenectors o the ollowing matrix xx xx( + x x( + A =. (3 3/ ( + + ( + ( + x xxx x x xx x Howeer, the calculation o the matrix A is sensitie to noise intererence. Thereore, the irst order terms in A are neglected, and the Hessian tensor H expressed in (4 is used instead [9]. ( ( xx x. (4 H = [ ] = x ( ( As indicated in [], Hessian H has a much better SNR perormance than matrix A when the image is interered b noise. With this simpliication, the eigenalues and eigenectors o Hessian H are utilized to estimate the principal curatures. Let the alues o k (, and, with x k ( k( k (, represent the principal r curatures at (; while the eigenectors Λ ( and Λ r ( represent the directions o principal curatures. To obtain the high-curature points o an image, we check each pixel the magnitude o k ( and pick up these places whose k ( are local extremes and are larger than a predeined threshold T k. Then, or each high-curature point p, we tr to ind the corresponding high-curature pair r p. The direction o Λ ( proides a good clue or inding p, since Λ r ( points to the direction where a drastic surace change occurs. We search along Λ r ( to ind the nearest high-curature point p n. I p n satisies Sign ( p Sign( p' =, (5 and r Λ p Λ ( ( p', (6 p n is considered as a qualiied high-curature pair o p. Due to noise intererence, some eatures ma disappear. I p n does not satis (5 and (6, we search around the neighborhood o p, and ind the pixel which possesses the highest intensit dierence against p as p. Fig. 4 shows image peppers. Fig. 4 shows the extracted high-curature pairs o Fig. 4. In Fig. 4, negatie high-curature points are indicated in green while positie high-curature points are indicated in red.

3 Once high-curature pairs are extracted, edge center, edge contrast, and edge span can be obtained based on these high-curature pairs. Here, edge center is deined as r κ ( p r r ( p + ( ( p' ( p ; (7 + κ ( p κ ( p' edge span is deined as r r ( ( p' ( p ; (8 and edge contrast is deined as intensit(p - intensit(p. (9 Fig. 4 shows the simulation result o this edge detector, with T k =. As a comparison, edges detected b the well-known Cann Operator are shown in Fig. 4(d. Fig. 4(d is obtained using the Cann detector tool oered in Matlab 6.0. In this simulation, the low and high hsteresis thresholds o the Cann Operator are automaticall set as and 0.087, respectiel. It can be seen that the perormance o edge detection using Hessian H is comparable to that o Cann Operator. G( = πσ σ x exp x ( + σ x σ and the smbol * denotes the conolution operation. In Fig. 5, we show a step edge image. The edge center locates at x=0. In Fig. 5, we show the positions o the detected high-curature pair or the step edge image shown in Fig. 5. The ertical axis represents the alue o a scale parameter σ m o the Gaussian smooth unction. The horizontal axis represents the x axis. The reconstructed edge images using detected edge center, edge contrast, and edge span are shown in Fig. 5. It can be seen that the Hessian operator can detect the edge contrast and edge span correctl as long as σ m doesn t grow too large. The detection o edge center is less inluenced b the selection oσ m, since we use the midpoint o the high-curature pair to detect edge center. (d Fig. 4. Original image. Detected high-curature pairs. Detected edges using Hessian H. (d Detected edges using Cann Operator. We urther inestigate the perormance o Hessian H in edge detection under dierent scales. The deriaties in (4 are calculated as ( ( G( G( x = (, ( ( G( G( xx = (, and so on. Here, G( denotes the Gaussian smoothing unction Fig. 5. Original edge image. Positions o the detected high-curature pair under dierent scales. Reconstructed edge images using the detected edge center, edge contrast, and edge span or dierent scales.

4 Fig. 6. Concept o corner detection. Detected corners using Hessian H. Detected corners using Harris operator. mark these local extremes o k ( x, k ( as the corner pairs. Then, the mid-point o a corner pair is chosen to be a corner point. A simulation result or corner detection is shown in Fig. 6. As a comparison, the corner detection perormed b the popular Harris Operator (indicated as R [8] is shown in Fig. 6. The Harris operator R is deined as ( ( ( ( R =. (0 ( ( ( ( We can see that the perormance o corner detection using Hessian H is comparable to that o Harris Operator. In addition, since the intensit alues o the two sides o the corner are identiied, the proposed operator ma distinguish two corners with the same gradient but with dierent intensit alues or dierent properties. In Fig. 7, we use the signs o k ( x, k ( around the two sides o corners and the alues o skin color to distinguish ingertips rom the corners between two ingers. Moreoer, the capabilit o corner and junction detection beneits the process o edge linking, since the corner and junction proides clues to determine whether two edge segments should be linked together. In Fig. 8, we demonstrate the capabilit o using Hessian H to link edge elements. Since junctions can be detected, the positions o junctions can be used to link cure segments ater non-maximum suppression. Compared with the edges detected b Cann operator (see Fig. 8, the proposed method could produce a closed contour (see Fig. 8. Fig. 7. Fingertips detection. 4 Corner Detection Besides edges, the Hessian H used in edge detection can also be used to detect corners. At a corner point, both k( and k ( are expected to hae larger alues. Hence, we can detect corners b checking the product o k ( x, and k (. In Fig. 6, we show the simulation result or a snthetic image. Points with positie k( k ( are indicated in red, while points with negatie product are indicated in green. We Fig. 8. Junction image. Linked edges using Cann operator. Linked edges using Hessian H.

5 5 Image Segmentation Image segmentation methods ma be roughl diided into three major categories: region-based method, edge-based method, and hbrid method. In this paper, we propose a hbrid method or image segmentation. The high curature points extracted b Hessian H pla two dierent roles. On one hand, the indicate the erges o edges. On the other hand, high curature points indicate the erges o regions. For example, we can consider the high curature points P in Fig. 3 locate at the erge o edge PP. We can also consider P locate at the erge o region PP3. We hae demonstrated in the preious sections that high curature points with dierent properties (dierent signs can be used to detect edges. Here, we use the inormation carried b high curature points to detect smooth regions and perorm image segmentation. For an image IM, assume the sets o edge, high curature points, and regions are indicated b DE, HP, and RG, respectiel. Clearl, DE U HP U RG = IM ( The set HP is composed o high-curature points, which are obtained using Hessina tensor. The set DE contains all pixels in the detected edge span except the pixels in HP. For each element t( in RG, we ind the minimum region distance r(t( as r(t(= min ( t( t' (. ( t'( HP Then, we deine the region likelihood unction LG as r( t( LG(t(= i (t(- (t ( <T r D x + D =0 otherwise, (3 where T r indicates the intensit threshold, and D x and D represent the width and height o the test image, respectiel. Pixels with a larger region likelihood alue ma hae higher probabilities to locate at the center parts o objects, since those pixels are awa rom high curature points with similar intensit. We sort pixels in RG and label the pixel with the maximum LG alue as the irst seed point S i. The seed point S i is added into the seed point queue SQ. The corresponding region distance o S i is set as R max. We decrease the alue o R max iteratiel b one step ds. The decreasing step ds is set as in this simulation. For the k-th iteration, we ind the processing zone PZ, which deined as the pixels with region distance equal to R max -k. A pixel PZ(j will be a new seed point i it satisies PZ(j Neighborhood(SQ=Null, (4 and this pixel should be added into SQ to create a new seed point. Otherwise, PZ(j should be in the neighborhood o at least one seed point in SQ. I PZ(j is coered b onl one seed point S i, it is considered to be in the same region with S i. We tag the label o S i on PZ(j, and add PZ(j into SQ as the subset o S i. I PZ(j is coered b multiple seed points, a competitie mechanism is applied. PZ(j is classiied into the group o the seed point with minimum intensit dierence. The process continues until no pixels in RG needs to be processed. Then, the set DE and HP are cleared and onl edge centers are let. A simple region growing method is applied to ill these unlabelled pixels. To merge seed groups with similar properties, we construct a connectiit graph (see Fig. 9. Nodes in the connectiit graph represent the seed point regions in SQ, and the arc between two connected nodes indicates the aerage boundar contrast AVC o the two nodes. For two nodes with N oerlapeed boundar pixels OP, AVC is deined as N AVC = Contrast( OP( k. (5 N k= We merge two nodes with the smallest AVC irst. Ater merging, a new node is ormed. The AVC alue o each arc o the new node is recalculated to partiall update the original graph. This merging process continues until no arc in the connectiit graph is smaller than the threshold T b. Fig. 0 shows the experimental result, where the thresholds T k, T r, and T b are set as,0, and 0 empiricall. Compared with JSEG [5] (see Fig. 0, although [5] proides better segmentation results in texture regions, the proposed method ma oer superior boundar accurac. Fig. 9. Region map. Nodes and Arcs 6 Conclusions In this paper, we proposed an operator, the Hessian tensor in dierential geometr, to detect edges and corners. For an edge, we detect the edge center, edge contrast, and edge span using high-curature pairs. The proile o an edge can be approximatel reconstructed using these three elements. Meanwhile, the detected edge contrast and edge span can be adjusted or dierent purposes, such as contrast enhancement and sharpness enhancement. The Hessian tensor H can also be used to ind corners,

6 which are useul or eature selection and edge linking. We urther demonstrate the easibilit o using high curature points to perorm image segmentation. For urther research, we aim at enhancing the noise-combating capabilit o the proposed operator, and improing the segmentation results in texture regions. Reerences: [] J.F. Cann, A Computational Approach to Edge Detection, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 8, No. 6, 986, pp [] C.K. Chow, Neural Edge Estimator, WSEAS Int. Con. on Signa Processing, Robitics and Automation, Salzburg, Austria, Feb [3] G. Costantini, D. Casali, and R. Peretti, Almost Space-Inariant Cellular Neural Network or Edge Detection, WSEAS Int. Con. on NNA-FSFS-EC, Athens, Greece, Ma 003. [4] M.A. Ruzon and C. Tomasi, Edge, Junction, and Corner Detection Using Color Distributions, IEEE Trans. Pattern Anal. Mach. Intell., 00, pp [5] Z. Zheng, H. Wang and E.K. Teoh, Analsis o Gra Leel Corner Detection, Pattern Recognition Letters, Vol. 0, 999, pp [6] F. an der Heijden, Edge and Line Feature Extraction Based on Coariance Models, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 7, No., 995, pp [7] K. R. Rao and J. Ben-Arie, Optimal Edge Detection Using Expansion Matching and Restoration, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 6, No., 994, pp [8] C. Harris and M. Stephens, A Combined Corner and Edge Detector, in Proceeding o the Fourth Ale Vision Conerence, 988, pp [9] T. Qian and L. Zhang, Radon Measure Model or Edge Detection Using Rotational Waelet, WSEAS Int. Con. on MMACTEE-WAMUS-NOLASC, Athens, Greece, Dec. 00. [0] S.M. Smith and M. Brad, SUSAN - A New Approach to Low Leel Image Processing, International Journal o Computer Vision, Vol. 3, No., 997, pp [] F. A. Pellegrino, W. Vanzella, and V. Torre, Edge Detection Reisited, IEEE Trans. Sstem, Man and Cbernetics, Vol. 34, No. 3, 004, pp [] S.J. Wang, L.C. Kuo, H.H. Jong, and Z.H. Wu, Representing Images Using Verge Points on Image Suraces, to appear in IEEE Trans. Image Processing. [3] M.P. Do Carmo, Dierential Geometr o Cures and Suraces. Prentice Hall, 976 [4] H.C. Chen, W.J. Chien and S.J. Wang, Contrast-Based Color Image Segmentation, IEEE Signal Processing Letters, Vol., No.7, 004, pp [5] Y. Deng and B.S. Manjunath, Unsuperised Segmentation o Color-Texture Regions in Images and Video, IEEE Trans Pattern Anal. Machine Intell., Vol. 3, No. 8, 00, pp Fig. 0. Original image. Segmentation result using the proposed method. Segmentation result using JSEG.