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Shape classification techniques for discrete 3D porous media. Application to trabecular bone. G. Aufort, R. Jennane, R. Harba Laboratoire d’Electronique, Signaux, Images Universite d’Orleans 12 rue de Blois, 45067 0rleans, France

C. L. Benhamou Equipe INSERM U658 Centre hospitalier Regional d’Orleans 1, rue Porte-Madeleine, 45032 Orleans, France

[email protected]

[email protected]

Abstract— The clinical process used to screen osteoporosis is the Bone Mineral Density (BMD). Since this density measurement does not cover the entire diagnosis range, work is beeing carried out on the segmentation of the bone and other complex porous media to provide quantitative information about their microarchitecture. Two shape classification techniques have been recently proposed in the literature. In this paper we compare these different methods and propose a new original rod/plate classification technique. The efficiency of the 3 processes is then studied on test vectors composed of both rods and plates, then applied on real trabecular bone samples. Results of this study emphasize the pros and cons of the 2 published techniques, and discuss the improvements of the new region-growth-based method. Finally, the interest of such a tool in osteoporosis screening is discussed.

shape classification of discrete samples. Then, Stauber et al. [6] published another segmentation technique for the same aim called Multicolour Dilation Algorithm. Both were based on skeletonization processes. In a first section, these 2 published rod/plate classification techniques are presented. Then an original and improved method based on surface skeletonization methodology and using shape classifiers from previous work is explained. The 3 methods are then compared on 2 synthesis test vectors composed of intersecting rods and plates. Algorithm’s accuracy and computing time are studied for each case. Finally, an application of these methods on real bone samples is presented. Results are discussed in the last section.

I. INTRODUCTION

II. M ATERIALS AND METHODS In this section, we first present the 2 test vectors used to validate and compare the different shape classification techniques. Then the 2 classification methods found in the litterature are briefly recalled. Finally, an original method combining surface thinning methodology, local topology analysis and region growth is presented.

Screening osteoporosis and other bone related alterations using the clinical routine BMD measurement is not sufficient since it does not cover the entire diagnosis. Authors have shown that associating mechanical and structural information with density would allow rising the diagnosis up to a 90% characterization [1]. Features such as Trabecular Thickness (Tb.Th) or Trabecular Spacing (Tb.Sp) are starting to become famous in microarchitecture characterization, as they complete the diagnosis of the physician. But they are computed using physical model-based algorithms. In fact, large scale porous media are often composed of a complex mix of different shapes. Plus, in the case of trabecular bone, it is clearly established that the structure is composed of both rod and plate parts. Segmentation of the trabecular bone structure into shape primitives can be useful, but is still a challenge in the field of image processing. A lot of work has been carried out on the characterization of this porous medium, especially using skeletons. However, segmentation techniques that take into account the local shape of the structure are rare. The well established SMI [2] has been designed to compute a global index featuring the rod/shape proportion of trabecular bone. But global indices do not extract precise information about the structure. In an attempt to extract more local microarchitectural information, Saha et al. [3] and Pothuaud et al. [4] managed to classify skeleton voxels according to their role into the 3D structure. But they didn’t aim to segmentate the whole sample into shape classes. Bonnassie et al. [5] were the first to develop a full algorithm for rod/plate

A. Test vectors 2 test vectors have been generated using a combination of mathematically-driven shapes in a 3D space. Perfect cylinders/torus and parallelepipeds stood respectively for rod and plate shape primitives of our object. Then a discrete volume was built, by assigning to each voxel one of the following labels : rod, plate, or background. The vectors 1 and 2 are respectively composed of : 3 parallelepipeds and 9 cylinders for vector1 (figure 1.a), and 2 parallelepipeds, 1 torus, and 1 cylinder for vector 2 (figure 2.a). B. Trabecular bone samples The 2 trabecular bone samples that were used in this study have been acquired post-mortem using a µCT scan at an isotropic resolution of 20µm. Dimensions of the 2 samples were about 7x7x10mm3 . The first sample has been extracted from a coxarthric (OA) patient, which meant that the structure of the bone was overdensified, and mostly composed of plates. The second sample has been extracted from an osteoporotic (OP) patient, which on the contrary suffered from a strong loss of bone density, leading to the alteration of plates into rods. Figures 3.a and 4.a present 2 extracts from the 2 bone samples.

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Fig. 1. Result of the 3 shape classification algorithms on test vector 1. (a) test vector 1, (b) result of the MAC technique, (c) result of the MDA technique, (d) result of the SCA technique.

Fig. 2. Result of the 3 shape classification algorithms on test vector 2. (a) test vector 2, (b) result of the MAC technique, (c) result of the MDA technique, (d) result of the SCA technique.

C. Medial Axis-based shape Classification (MAC)

to its type (arc, boundary, etc...). Since the drawback of surface thinning is that it generates surfaces even in rod parts which are not perfectly cylindrical, a curve-thinning-like process is used on surface-end labeled voxels. This is when the slenderness parameter s appears, which determines how many curve thinning iterations are to be applied onto those surface border voxels. To sum up, the slenderness parameter s intrinsically determines the classification between rods and plates by defining a threshold on the rod section’s ellipsoidal aspect ratio. The advantage of this step is that it turns non significant plates into rods, but the terrible drawback is that it strongly depends on the number of iteration, i.e. the slenderness parameter s. Finally, Stauber et al. implemented an isotropic region growth segmentation of the original binary object seeded by the previously classified skeleton.

The MAC technique [5] is based on the distance preservation and reversibility properties of the medial axis skeleton [7]. After computing this skeleton, 4 types of voxels are identified, according to the local topology of the structure: boundary, branching, regular and arc. The labels are determined by studying the intersection of the maximal ball of each medial axis voxel with the object. For this purpose, the Betti Numbers [8], which fully characterize the ball’s neighborhood topology, are used. Then, the reversibility property of the medial axis is used to spread the skeleton labels back to the entire object, leading to a complete 4 classes classification. The radius of the maximal ball at any voxel is computed using the thickness map of the original object as described in [2]. Plus, it is slightly increased to add tolerance to the process, which makes the algorithm parameter dependant. This parameter called ε is set for the entire volume. It is definitely the limitation of this algorithm. However, among the papers found in the literature on discrete shape classifiers, this technique seems to be the most reliable in a mathematical point of view. In order to only consider rod and plate primitives, we completed the MAC algorithm with a simple local regiongrowth process to gather the 4 labels into 2 labels: rod and plate. The process consists in spreading isotropically the rod and plate labels over the arc and boundary labels. D. Multicolour Dilation Algorithm (MDA) Stauber et al. [6] have developed a protocol for the segmentation of rod and plate shape primitives in a porous medium. They initiated their work by using a homotopic surface thinning technique presented by Manzanera [9], which generates 2D 26-connex surface sets. Then a set of tools inspired from the Saha [3] skeleton classification rules is applied, in order to label each skeleton voxel according

E. Surface-thinning-based rod/plate shape Classification (SCA) Based on our experience in thinning and the implementation of the 2 previous techniques, we developed an original method to segmentate rods and plates in a complex porous medium. Since the MAC method is very efficient but slow, and since the MDA method computes faster but suffers from the influence of its slenderness parameter, we associated the use of a fast computed surface skeleton with the MAC’s local topological classification criterion. This way, we intended to take advantage of both the speed of surface thinning and region-growth and the precision of local classification. The MAC process that uses the maximal ball and the Betti numbers is slow and should be repeated the less times possible. In fact, thinning before classifying voxels is not mandatory. Its interest resides in the limitation of the number of voxels to be classified. Our algorithm processes as follows:

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Fig. 3. Result of the 3 shape classification algorithms on a coxarhtric trabecular bone sample. (a) bone sample, (b) result of the MAC technique, (c) result of the MDA technique, (d) result of the SCA technique.

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First, the 26-connex surface skeleton of the object is computed using a fast method called MESPTA presented by Mukherjee et al. [10]. Then, voxels are classified using the MAC method, by computing the Betti Numbers inside the maximal ball of each medial axis voxel. The radius of the ball is still extended by a constant factor ε. Finally, an isotropic iterative region-growth method close to MDA’s is applied to spread the rod and plate labels back to the entire original object. III. C OMPARISON OF THE SHAPE CLASSIFICATION TECHNIQUES

The 3 shape classification techniques presented above have been compared on 2 test vectors, then applied on 2 trabecular bone samples. In the case of test vectors, classification efficiency and computing time were measured. In the case of real bone data, morphometric features were measured such as rod and plate densities or rod/plate proportion. A. Comparison on test vectors For each of the 2 test vectors, we processed the 3 segmentation techniques (MAC, MDA, SCA) and verified their efficiency by comparing them to the classified reference. Relative error of the rod/plate proportion was computed for each method. Computing time was also measured. For each of the 2 vectors, and each of the 3 algorithms, we investigated the whole range of the ε parameter for MAC and SCA and the s parameter for MDA. Results from the different methods for vector 1 and 2 are respectively illustrated in figures 1 and 2. Figure 5 plots the evolution of the relative classification error for the 2 test vectors. Table I exposes the results of our comparative study on the 2 test vectors,

Fig. 4. Result of the 3 shape classification algorithms on an osteoporotic trabecular bone sample. (a) bone sample, (b) result of the MAC technique, (c) result of the MDA technique, (d) result of the SCA technique.

ε, s value Err (%) Time (s)

MAC 26 0.86 541

Vector1 MDA 3 8.71 28

SCA 23 0.76 307

MAC 23 0.70 372

Vector2 MDA 3 7.38 74

SCA 28 1.58 221

TABLE I S HAPE CLASSIFICATION VECTORS

1

AND

EFFICIENCY OF THE

3

ALGORITHMS APPLIED ON TEST

2. PARAMETERS ε OR s AND TIME

ARE INDICATED FOR THE

MINIMUM RELATIVE ERROR CASE .

where the best case is considered for each algorithm (i.e. the parameter value that generates the smallest relative error value). B. Application to trabecular bone MAC, MDA and SCA were applied on the 2 trabecular bone samples of figures 3 and 4. Since there was no ground truth for our real bone data, the relative classification error was impossible to determine. However, the interest of this study was to check if the shape classification methods were able to quantify microarchitectural changes in trabecular bone, since the 2 samples were known to be different. Results of this real case study are reported in table II. IV. D ISCUSSION The aim of this comparative study was to investigate the efficiency differences between 2 shape classification techniques found in the litterature for discrete 3D objects: MAC [5] and MDA [6]. We also presented an original method (SCA) based on surface thinning, topology classification and region growth. These 3 techniques have been compared in terms of classification efficiency and computing

Fig. 5. Influence of the parameter (ε for MAC and SCA, s for MDA) on the classification relative error for test vector 1 (a) and test vector 2 (b).

MAC MDA SCA

Rod.D(%) OA OP 5,36 43,92 19,14 76,70 12,42 70,91

P late.D(%) OA OP 94,64 56,08 80,86 23,30 87,58 29,09

Rod/P late OA OP 0.056 1.783 0.237 3.292 0.142 2.438

TABLE II C OMPARISON

OF THE

BONE SAMPLES .

3 SHAPE CLASSIFICATION

ROD .D, P LATE .D

DENSITY OF ROD AND PLATES IN

AND

%

TECHNIQUES ON

ROD /P LATE

2

TRABECULAR

ARE RESPECTIVELY THE

OF SOLID VOXELS , AND THEIR RATIO .

time on 2 test vectors for which the classification solution was known, then applied on 2 trabecular bone samples. In a first part, plotting the evolution of the relative error depending on the parameter value (see figure 5) helped us acknowledge that both of the 2 published techniques (MAC and MDA) were strongly parameter-dependant. This limitation has already been pointed out in the litterature [5]. The MDA technique, besides suffering from the effect of the slenderness parameter on the algorithm efficiency, is proved to be strongly related to the object’s discrete resolution. In fact, the s parameter range is very small, about half the maximal local thickness value of the object in voxels. The comparison between the shape classification techniques has been pursued by considering the 3 algorithms in the minimal error case. Table I shows that for the 2 test vectors, the MAC technique is always under 1% of error. This result emphasizes the accuracy of the MAC method, which is consistent with its strong mathematical background. The MDA method minimizes the error at about 8%. The SCA method that we proposed, gathering the power of both MAC and MDA, gives less than 2% of relative error for both vectors. This is a good result considering its computing time

which is about half the one of the MAC. Although the MAC method provides the best accuracy, it needs up to 9 minutes to compute a 2003 voxels porous sample. On the contrary, the MDA method is quick to compute (ten times less than MAC), but still generates an 8% error. SCA represents a great compromise between computing time and accuracy. Finally, in the case of real bone data, table II shows that the 3 techniques are able to measure significant differences between the 2 samples. Plate density and rod/plate proportion clearly indicate for the 3 techniques that the coxarthric sample (OA) contains much more plates than the osteoporotic one (OP), which is an expected result. However, it can be noticed that the 3 methods do not intrinsically give the exact same densities of rods and plates. To conclude, the apparent limitation of all these techniques is their parameters ε and s. In the image community, automatic or semi-automatic processed are always prefered. Although a minimum relative error value has been found for each of the 3 methods, it is still impossible to justify mathematically that a value is better than another. Furthermore, in real applications, the ground truth cannot be obtained, and the parameter’s optimal value cannot be determined precisely. However, shape classification is already used in a large field range such as sandstones, rocks, or trabecular bone analysis. The Hybrid Skeleton Graph Analysis [11] [12] is the example of a technique that uses the MAC and SCA methods and gives interesting results in trabecular bone characterization. Work is in progress to understand and improve the mechanisms induced by the ε parameter. R EFERENCES [1] R. W. Goulet, S. A. Goldstein, M. J. Ciarelli, J. L. Kuhn, M. B. Brown, and L. A. Feldkamp. The relationship between the structural and orthogonal compressive properties of trabecular bone. J. of Biomechanics, 27(4):375–377, 1994. [2] T. Hilderbrand and P. Ruegsegger. A new method for the model independent assessment of thickness in three-dimensional images. J. of Microscopy, 185:65–67, 1997. [3] P. K. Saha and B. B. Chaudhuri. 3d digital topology under binary transformation with applications. Computer Vision and Image Understanding, 63:418–429, 1996. [4] L. Pothuaud, P. Orion, E. Lespessailles, C. L. Benhamou, and P. Levitz. A new method for three-dimensional skeleton graph analysis of porous media: application to trabecular bone microarchitecture. J. of Microscopy, 199(2):149–161, 2000. [5] A. Bonnassie, F. Peyrin, and D. Attali. A new method for analyzing local shape in three-dimensional images based on medial axis transformation. IEEE. Trans. Sys. Man. Cyber., 44(4):700–705, 2003. [6] M. Stauber and R. Muller. Volumetric spatial decomposition of trabecular bone into rods and plates–a new method for local bone morphometry. Bone, 38(4):475–484, 2005. [7] G. Matheron. Examples of topological properties of skeletons., volume 2. Theoretical Advances, Academic Press, London, 1988. [8] D. G. Morgenthaler. Three-dimensional simple points: serial erosion, parallel thinning and skeletonization. Tech. Rep. TR-1005, 1981. [9] A. Manzanera, T. Bernard, F. Preteux, and B. Longuet. Medial faces from a concise 3d thinning algorithm. In IEEE International Conference on Computer Vision (ICCV’99), pages 337–343, 1999. [10] J. Mukherjee, P. P. Das, and B. N. Chatterji. On connectivity issues of espta. Pattern Recogn. Lett., 11(9):643–648, 1990. [11] G. Aufort, R. Jennane, R. Harba, and C. L. Benhamou. A new shapedependant skeletonization method. application to porous media. In Proc. EUSIPCO 2006, Florence, Italy, September 2006. [12] G. Aufort, R. Jennane, and R. Harba. Hybrid skeleton graph analysis of disordered porous media. application to trabecular bone. In Proc. IEEE ICASSP 2007, pages II 781–784, Toulouse, France, May 2006.