Earthenware Reconstruction Based on the Shape Similarity among Potsherds

Transcription

1 Original Paer Forma, 16, 77 90, 2001 Earthenware Reconstruction Based on the Shae Similarity among Potsherds Masayoshi KANOH 1, Shohei KATO 2 and Hidenori ITOH 1 1 Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya , Jaan 2 Toyota National College of Technology, 2-1 Eisei-cho, Toyota , Jaan (Received August 13, 2000; Acceted January 29, 2001) Keywords: Earthenware Reconstruction, Pattern Analysis, Comuter Grahics, Grahical Visualization Abstract. This aer rooses an earthenware reconstruction system, which can automatically reconstruct some earthenware from numerous given otsherds in two dimensional grayscale images. The system suoses that given otsherds are thin and moderately flat and small. Some earthenware having three dimensional shae, such as crocks, can be reconstructed by the system within the suosition. The system erforms earthenware reconstruction through two hases. At the first hase, otsherds are joining automatically in two dimensions. An efficient joint detection algorithm is roosed at the hase. At the second hase, three dimensional shae is recovered by adequate three dimensional transformation. In this aer, some exerimental results of reconstruction from numerous otsherds are also reorted. 1. Introduction In the field of archaeology, the shae of earthenware offers significant knowledge of not only the era when it is used, but also ancestor s life style and cultural interchange. Thus, it is very imortant to reconstruct earthenware from excavated otsherds. Reconstruction of earthenware, however, is attended with difficulties, due to a numerous excavation of otsherds, weathering and lack of arts. The reconstruction, therefore, imoses a heavy task on archaeologists, and a numerous earthenware has not been reconstructed yet. In order to reduce the task of the reconstruction, we have develoed a system, which can automatically reconstruct earthenware from given otsherds by KANOH (2000). In general, earthenware should be reconstructed in three dimensions, because otsherds have three dimensional shae. We are taking our stand on that some earthenware shaed in three dimensions can be reconstructed from otsherds in two dimensional images, if the earthenware consists of numerous otsherds, and if each of the otsherds is adequately small. In this aer, at the beginning of this research, we suose that otsherds, whose character in shae are restricted to thin and moderately flat and small (e.g., see Fig. 1), are given, and we consider reconstructing earthenware from the otsherds in two dimensional 77

2 78 M. KANOH et al. Fig. 1. Images of otsherds. grayscale. In this aer, firstly as techniques for reconstruction in two dimensions, we roose a method for detection of salient feature and calculation of salient value from contours of otsherds, and then roose an efficient joint detection algorithm considering comlexity of contours. We then move on to a method for recovering three dimensional shae. We have imlemented an earthenware reconstruction system with the roosed algorithm and method, and shown good erformance in reconstruction of a cylindrical clay figurine (called ento-haniwa in Jaanese), in addition to flat objects, such as lates, otlids, flat arts of earthenware, and so on. 2. System Overview The section gives a brief descrition of our earthenware reconstruction system executable on Windows PC. A full detail of the system is given in the later sections. Figure 2 shows an overview of the system. The system has two hases: ieces-joining hase and shae-recovery hase. In our system, given numerous otsherds are reconstructed into some earthenware through the two hase with the following rocedures. Phase 1: ieces-joining 1) Detection of contours The system detects contours from inut images of otsherds. In the rocedure, we adot the g-snake model roosed by LAI and CHIN (1993, 1995), which can extract a contour of any object from an observational image.

3 Earthenware Reconstruction Based on the Shae Similarity among Potsherds 79 Fig. 2. An overview of the earthenware reconstruction system. 2) Detection of salient oints Each of contours is divided into some exanded curves called sub-contour by the salient oints in the rocedure. These sub-contours are candidates for joints. 3) Joint detection For each sub-contour, salient value is calculated, and then, for all airs of subcontours, similarities are evaluated by the salient values. The most similar air is selected after that. 4) Joint If the similarity of selected air satisfies some condition, two otsherds having these sub-contours are joined. The system executes the iterations of series of rocedures 3) and 4), and finally, oututs some lane images of reconstructed earthenware. Phase 2: shae-recovery 5) Checking the ossibility In this rocedure, the lane images generated at the above hase are checked about the ossibility of three dimensional shae recovery. Once some images ass the check, the system roceeds to the next rocedure with the images; otherwise the system terminates with the lane images of reconstructed earthenware. 6) Recovery of the three dimensional shae The system recovers three dimensional shae of the earthenware from the lane images with adequate three dimensional transformation. After this hase, the system oututs three dimensional images of reconstructed earthenware. 3. Detection of Salient Points Contours detected from given otsherds should be divided into sub-contours by salient oints. In our system, the sub-contours are considered candidates for joint, and the shae of sub-contour is utilized as a criterion for joint detection. Accordingly, dividing contours is imortant in terms of erformance of the system.

4 80 M. KANOH et al. Fig. 3. Contours of otsherds. Fig. 4. Divided contours by salient oints. Paying attention to reconstructed earthenware, a contour of a otsherd is almost divided into joints by salient oints such that the oints become vertexes constructing a olygon which aroximates the contour with as few vertexes as ossible. We, therefore, adot ROSENFELD and JOHNSTON (1973) s method for detecting salient oints having above roerty. Figure 4 shows the result of detecting salient oints from contours shown in Fig Joint Detection 4.1. Salient value We give the P-tye Fourier descritors roosed by UESAKA (1985, 1986) to salient values for shae of sub-contours. Using the descritors, we evaluate similarity between two

5 Earthenware Reconstruction Based on the Shae Similarity among Potsherds 81 sub-contours. The descritors are reresented by the Fourier transform of curves and characterized as follows: It is alicable to exanded curves. Two endoints of a reroduced curve always corresond to two endoints of the original one, resectively. Information about a certain shae of an original curve is concentrated in low frequency range. A Curve is reresented by only the angular function. In earthenware reconstruction, the shae of a sub-contour is not comletely equal to the shae of its joining air because of weathering and so on. The P-tye Fourier descritors are, therefore, alicable to joint detection. A salient value c N (k) of a sub-contour c is calculated by low-ass sectrum with cut frequency N of Fourier transformation as follows: c N n 1 1 ( k)= ω() i ex ( j2πik / n) k = N,..., N, 1 n i= 0 ( ) () where ω(i) is the P-tye exression of c and n is an enough large natural number. For more recise, lease refer to UESAKA (1985, 1986). Using the salient value, similarity ε a,b (φ) between sub-contour a and b can be evaluated by the following equation: N 2 ( )= ( ) ( ) ( ) ( = ) ( ) εab, φ an k ex jφ bn k φ 0,..., 2π, 2 k= N where a N (k) and b N (k) denote the low-ass sectrums of P-tye Fourier descritors of a and b, resectively Shae similarity The P-tye Fourier descritors are invariant under translations, and changes of the scale. Rotations of an original curve are reresented by the descritors multilied by ex(jφ). It follows from Eq. (2) that b becomes more similar to a when ε a,b (φ) gets nearer to zero. It also follows that ε a,b (φ) = 0 when the shae of b is comletely equivalent to the shae of a. In our system, we, therefore, define the shae similarity between a and b by the following equation: ss( a, b)= min 1 φ εab,. φ ( ) () 3 The above salient value and similarity are also adoted by the existent work in HORI (1999).

6 82 M. KANOH et al JDID In our system, an efficient joint detection algorithm considering comlexity of contours, JDID roosed by KANOH (2000), is imlemented. The algorithm reduces comutational cost of joint detection drastically, therefore numerous otsherds can be reconstructed at once. For a contour, the algorithm considers some contours aroximating it by different recision, shae-recision, and suitably selects one aroximation for calculating shae similarity by comlexity of the contour. For each sub-contour c, the algorithm considers sequential lines s(c)s with case that shae-recision s S = 0,..., s S (shown in Fig. 5), where s S is uer bound of shaerecision. It should be noticed that for all c, s(c) can aroximate c more accurately with case that s S is higher. Here, s i (c) denotes the sequential line of sub-contour c with its shae-recision s S = i. Figure 6 shows an outline of search trees constructed by JDID. The search begins at the lowest shae-recision. In the figure, the leftmost tree, for examle, shows joint detection from two otsherds A and B. The leftmost branch in the tree, for examle, shows Fig. 5. Sequential lines with different shae-recisions. Fig. 6. An outline of search trees constructed by JDID.

7 Earthenware Reconstruction Based on the Shae Similarity among Potsherds 83 a comarison of shae between two sub-contours a 1 and b 1 in A and B resectively. In the branch, a node at the deth of i (i > 0) has two sequential lines s i 1 (a 1 ) and s i 1 (b 2 ). A node at the deeer deth has sequential lines with their higher shae-recision. The search roceeds in the breadth-first manner with iterative deeening (KORF, 1985a, b). Node exansion at the deth of i behaves intuitively as follows: i) calculate shae similarity ss(s i 1 (a), s i 1 (b)), and then judge whether the air is a joint or not, ii) from the above judgment, delete nodes containing inconsistent airs (see nodes marked with in Fig. 6), iii) go on in either of following three ways according to the number of remaining nodes at the same deth: zero: one: terminate the search in failure, terminate the search in success, and detect the air of sub-contours in the node as joint, more than one: generate a descendant node, which contains air (s i (a), s i (b)) of subcontours, with their shae-recision s S = i. The search iterates the above node exansion in breadth-first manner with successively heightening shae-recision. In the last of this section, let us discuss the of JDID for our earthenware reconstruction system. Although there generally exists more than one joint air in given otsherds, JDID detects only one joint air at once. In other words, our system erforms reconstruction of earthenware by executing JDID successively for each ste of joining. At one ste, JDID dose not consider global evaluation (i.e., considering not all combination of sub-contours for all stes of joining) for selection of the most similar air, but considers local evaluation (i.e., considering all combination of sub-contours at the one ste). Earthenware reconstructed by our system, thus, cannot always have an otimal combination of joint airs, while the most similar air selected by JDID is the best in each ste, that is, our system finds quasiotimal solution. Considering global evaluation for selection causes that earthenware reconstruction becomes a NP-hard roblem, and it is almost imossible to roose an otimal algorithm which can solve the roblem in adequate execution time. JDID, therefore, aims to detect a air efficiently, which can be adequate joint air for earthenware reconstruction. For more recise about JDID algorithm, lease refer to KANOH (2000). 5. Three Dimensional Shae Recovery In this section, we describe a technique for shae-recovery hase in our system: recovery of three dimensional shae from lane images of earthenware reconstructed at ieces-joining hase (see Sec. 2). In general, earthenware having solid shae should be reconstructed from otsherds reresented in three dimensions. The technique is an attemt for our system to reconstruct earthenware, having solid shae, from two dimensional images of its otsherds. In our system, it should be noticed that all inut otsherds have character restricted in shae: thin and moderately flat and small. In shae-recovery hase, by two rocedures, a lane image of reconstructed earthenware is transformed into a three dimensional image reresenting solid shae of the earthenware. The rocedures are described briefly in the following sections.

8 84 M. KANOH et al Checking the ossibility This rocedure checks the lane image about the ossibility of three dimensional shae recovery. The check is whether a contour of the lane image contains any joint air of sub-contours and the air is configurationally consistent, or not. The rocedure can be done efficiently by utilizing our roosed JDID algorithm. If some joint air is found, the lane image is cognizable as a figure to which a solid object is develoed Recovery of the three dimensional shae In this rocedure, the system recovers three dimensional shae of the earthenware from the develoment figure with adequate three dimensional transformation. By the transformation, a develoment figure behaves intuitively as follows. Firstly, the airs in the figure detected by the above rocedure are joined, and the figure has one closed-sace constructed in three dimensions. And then, the figure is gradually modified so that the volume of its closed-sace becomes the largest. In case that the figure is to grow into cylinder shae or cone shae (such like a aer cu), the transformation can achieve drastic efficiency by means of some geometrical characteristics. The secialized transformation is shown in Aendix. 6. Exerimental Results We have imlemented earthenware reconstruction system in C++ on a PentiumIII 500 MHz running Windows PC. We have also made some exeriments of earthenware reconstruction from excavated otsherds. This section shows two reresentative exeriments verifying effectiveness of our system; one is for efficiency in ieces-joining hase, and another is for erformance of shae-recovery hase. In the exeriments, arameters for our system, N = 5 and n = 64 2 i, where i is a value of shae-recision Efficiency in ieces joining In order to verify efficiency of ieces-joining hase, we have given 35 otsherds shown in Fig. 1 to our system. Figure 7 shows earthenware reconstructed by our system. In this articular examle, the result indicates that our system can correctly and comletely reconstruct 8 ieces of earthenware from given otsherds. We have, moreover for effectiveness of JDID, comared execution time of our system with a naive system, by changing the number of otsherds. In the naive system, joint detection is executed with shae-recision fixed on the uer bound. Figure 8 shows the exerimental results. The solid line in the figure shows the runtime by our system, while the broken line is those by the naive system. The results indicates that our system embedded with JDID becomes faster than the naive system as the number of otsherds increases. Our system, for examle of 35 otsherds shown in Fig. 1, takes (sec.) to erform all reconstruction, and the execution time is about 7.6 times faster than naive system Performance of three dimensional shae recovery We have also made an exeriment for recovery of three dimensional shae of cylindrical clay figurines. Our aim of this exeriment is to ascertain that our system can correctly recover the three dimensional shae from given otsherds. This exeriment,

9 Earthenware Reconstruction Based on the Shae Similarity among Potsherds 85 Fig. 7. Outut images. Fig. 8. Exerimental results. therefore, requires a comlete set of otsherds, which can show the erfect shae of reconstructed earthenware in advance. We, thus, took hotograhs* of a figurine reconstructed by archaeologists, and made inut images of otsherds by croing the hotograhs along their contours. After the above manner, 39 otsherds shown in Fig. 9 are inutted to our system. *At the shooting, the figurine is settled on a turntable aligned lumb in the face of a camera, and hotograhs are taken from the whole direction.

10 86 M. KANOH et al. Fig. 9. Inut images. Fig. 10. An outut image. Figure 10 shows the result of ieces-joining hase for the otsherds. Figures 11 and 12 show the result of three dimensional shae recovery from the lane image in Fig. 10. In this articular examle, the result shows satisfactory erformance as three dimensional shae recovery. 7. Conclusion In this aer, we roosed an earthenware reconstruction system, which can automatically reconstruct earthenware from given otsherds. As methods for imroving the system erformance, we roosed an efficient joint detection algorithm (JDID) considering comlexity of contours, and imlemented a method of recovering three

11 Earthenware Reconstruction Based on the Shae Similarity among Potsherds 87 Fig. 11. The three dimensional image of a reconstructed figurine. Fig. 12. The three dimensional image of a reconstructed figurine (another angle). dimensional shae of reconstructed earthenware. We also made some exeriments on reconstructing many excavated otsherds, and reorted two reresentative results. One result shows high erformance of correct and comlete ieces joining with JDID. Another indicates satisfactory erformance as three dimensional shae recovery of a cylindrical clay figurine. In general, it is difficult to imlement an efficient, accurate and robust three dimensional transformation which can recover the solid shae from an arbitrary develoment figure. We have, resently, considered an energy minimization method as a key strategy to solve the above roblem, and we will introduce the method into our earthenware reconstruction system in forthcoming aers.

12 88 M. KANOH et al. We are grateful to Nagoya City Miharashidai Archaeological Museum for excavated otsherds. This work was funded in art by Grant-in-Aid for Scientific Research Program of Jaan Society for the Promotion of Science. Aendix: Three Dimensional Shae Recovery Using Geometrical Characteristic A develoment figure can be transformed into cylinder shae or circular cone shae (such like a aer cu) by the following three dimensional coordinate transformation. Let c and d be sub-contours included in the develoment figure, and let (c, d) be a joint air of sub-contours detected from the develoment figure. Further let (v c1, v d1 ) and (v c2, v d2 ) be airs of corresonding endoints of c and d when c and d are joined, resectively (see Fig. A1). Firstly, the develoment figure is rotated on an x-y orthogonal coordinate system, such that a straight line which connects v c1 and v d1 becomes arallel to x-axis. And then, transformed shae from the figure are refigured as either a cylinder or a circular cone by the following condition. a cylinder: a circular cone: ( vc1vd1 vc2vd2 ) < α vc1vd1 ( vc1vd1 vc2vd2 ) α, v v c1 d1 ( A1) where α is a arameter. Secondly, any oints in the develoment figure are maed onto the three dimensional coordinate system. Let be an arbitrary oint in the develoment figure laid out above, and Fig. A1. Two tyes of ceveloment figure which is alicable to the efficient transformation method.

13 Earthenware Reconstruction Based on the Shae Similarity among Potsherds 89 let the coordinates of be (x, y ). Further let be an arbitrary oint in an orthogonal three dimensional coordinate system, and let the coordinates of be (x, y, z ). Then, is maed onto by either of the following two ways. The case of a cylinder: x y z v v c1 d1 = cos( φ ) 2π = y v v c1 d1 = ( φ ) 2π sin, ( A2) where φ is determined by the following formula: φ 2πx =. ( A3) v v c1 d1 The case of a circular cone: x y z = r cos( φ ) = l = r sin ( φ ), ( A4) where l, r and φ are determined as follows: l = o, ( A5) c r φ lθ 2π, = ( A6 ) 2πθ = ( ) Θ, A7 where o c is the center of a circle including arc vc1vd1, and Θ be an center angle of vc v and θ denotes the angle between x-axis and o c. 1 d1,

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