A Convex Decomposition Methodology for Collision Detection

Ashirwad Chowriappa, Raul Wirz, Aditya R. Ashammagari and Thenkurussi Kesavadas

Virtual Reality Laboratory. The State University of New York at Buffalo, NY. USA

ABSTRACT

In this paper, a shape decomposition methodology to decompose the cervical spine using an approximate convex methodology is proposed. The proposed methodology identifies the most concave L-ring neighborhood in a decomposition of a surface (manifold-2) and partitions it in order to reduce its concavity. A close approximation of the original surface by a set of convex surfaces hulls is then produced.

Keywords: Convex decomposition, Haptic, Cervical spine.

1 INTRODUCTION

Surgical simulators incorporating haptic devices are currently being used for a variety of applications in surgical skill training [1]. These systems typically include a graphical simulation of the clinical environment and a haptic rendering interface. The user can interact with the virtual environment using a haptic tool that reflect contact forces due to collisions between object in the scene and the virtual tool, e.g. biopsy needle and the cervical spine. For that reason, one of the most challenging tasks in surgical simulation and haptic rendering is the collision detection accurate.

We look at shape decomposition as a fundamental approach to effectively represent complex vertebral structures and propose a combination of different techniques for vertebral decomposition. When considering the cervical spine, although the mesh triangle count can be quite high, there are typically large regions that are planar or convex. Combining the triangles in such regions and replacing them by convex hulls can significantly reduce the computational time complexity of the colliding object pairs.

Our approach for cervical vertebral decomposition is based on identifying the most concave -ring neighbourhood in the decomposition of a surface (manifold - 2) and partition it in order to reduce its concavity. Convex decomposition and shape reduction are topics that has been significantly researched [2][3]. However, it is not well suited for vertebral structures since an exact decomposition of a complex shape can lead to a large number of components which may increase computation time for collision detection and response. To overcome this problem, we propose an approximate convex decomposition approach that is well suited for vertebral decomposition. Following, the methodology and the application will be explained.

2 DATA PROCESSING AND MESH GENERATION

The clinical data set, consisting of contrast enhanced CT data (slick thickness: 1.25 mm. stored using DICOM standards), is used for mesh generation. Volumes of interest (VOI) around identified vertebral structures are specified using maximum intensity projections (MIP). More specifically, the cervical spinal sections are reconstructed. Reconstruction is performed by first segmenting the spinal contours by means of level set evolution using the Marching Cubes algorithm [4]. A 3D model of the vertebrae is obtained as an iso-surface of intensity zero, resulting from the evolution of the level set [5].

3 MESH DECOMPOSITION

From the computed vertebrae surface representation, convex decomposition is employed to partition the mesh into a minimal set of convex sub-surfaces, 1 2{ , ,.. }ns s sS . The manifold–2 surface S is defined in

3R as 1 2 3{ , , ,.., }iv v v vV ,

1 2 3{ , , ,.., }jt t t tT and 1 2 3{ , , , , }kE e e e e (where Vis the vertex set, T the set of triangles and E is the edge set). We define the dual graph *D associated with the surface mesh S . Edge weights for *D are computed from shape indices to favor certain features over others. In mesh decimation, vertices of *Dare iteratively clustered by applying a decimation operator that minimizes a weighted cost function.

3.1 Computing Shape Index and Edge Weights

The curvature of a point on the surface can be defined by its maximum and minimum curvatures ( 1 2,k k ). Using these curvature measures we determine whether the given point lies on a concave, convex, ridge or saddle region. Saddle regions are characterized by being concave on one plane and convex from another. For each node in the graph *D ,we define a set of rings around the node as follows: the thi ring around node jv is defined as the set of vertices

*v V for which there exists a shortest path from jv to

*v containing i edges. The L ring neighborhood of node jv is defined as the set of rings i L about node jv . To capture the shape of the L ring neighborhood (in our implementation a 3-ring neighborhood, figure 3), we use the shape index introduced by [6], where the shape index SI derived from the principal curvatures is given as:

1 1

2min max

min max

k kSI arctan

k k

(1)

The values of SI vary in the closed interval [0.0, 1.0] and every distinct surface shape (table 1) corresponds to a unique value of SI (except for planar surfaces, which will be mapped to the value 0.5, together with saddle shapes). Edge weights are then computed from the SI ranges as follows: Two neighboring vertices connected by an edge in the dual graph *D are assigned an edge weight determined by (2)

1

2

3

, 0 0.5 ( )

, 0.5 0.75( )

, 0.75 1.0

SI umbilic points

SI hyperbolic points

SI

(2)

One advantage of using shape indices as edge weights is that transition from one shape type to another is continuous, hence they can be used to describe subtle shape variations.

3.2 Mesh decimation

Following the assignment of the edge weights to the dual graph,*D convex decomposition is initiated by iteratively applying a

half edge collapse operation on neighboring vertices. A half edge collapse operator defined as ( , )hcol u v , when applied to two vertices ( , )u v connected by an edge in * D , merges v with uand all incident edges on v are connected to u . The decimation process using the half edge collapse operator is governed by a cost function that is weighted on and minimizes for concavity. The cost associated with ( , )hcol u v is given by

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( ( , ))( , ) ( , )

Con S u vC u v AR u v

N (3)

where, the concavity of surface S(u,v) [7] and N is the normalization factor which is set as the diagonal of the bounding box of S(u,v). The parameters and control contributions of the aspect ratio and concavity Con(S(u,v)), where 2 favor the generation of compact disks in which case the cost is unity, and the aspect ratio (AR) of the surface S(u,v) is defined by

2( ( , ))( , )

2 * ( ( , ))

S u vAR u v

S u v

(4)

where,  is the perimeter and  is the area of the surface S(u,v). After each edge collapse operation is locally recomputed for surface S(u,v) and the new edge weight  new obtained from equation (5) is used in the update of

*D .

1new

(5)

An influence parameter is used in order to minimize the influence of newly formed surface features caused by the decimation in successive iterations of hcol .

3.3 Mesh partitioning

With each iteration of the hcol operator, the lowest mesh simplification cost is applied to a new partition

1 2 3 4 ( ) ( ) { , , , , , }n n n n nP nn minimizing the cost

function C .

( , )C u v , 1, , ( ) , ( )n n nk k kk P n p H p ò (6)

where, ( {1 ,) , ( )}nkp k P nò represents the dual graph *D

obtained after n edge collapse operations on P clusters. This procedure is iteratively performed until all edges of *D are in clusters with concavity lower than a determined concavity resolution value . We illustrate the decomposition of the cervical vertebrae in the figure 1.

t=12,804

k=14

Figure 1: Decomposition of the C1 Atlas. Where, t is the number of mesh triangles and k is the convex clusters.

The following algorithm computation time was measured on a PC running Windows 7, 64-bits with an I-7 (2.66 GHz, 6 GB RAM). On average the estimated decomposition time for a triangulated mesh with 3000 vertices was 250 seconds. INPUT: A mesh , , , . . , , , and

, , , . . ,

FOR EACH , { IF , < THEN /section 2.2.3 RETURN , ELSE COMPUTE , UPDATE UPDATE /section 2.2.4 RETURN }

4 EXPERIMENT AND RESULTS

The proposed vertebral decomposition methodology is tested on a virtual biopsy simulator for needle biopsy of cervical spine, where

the convex decomposed model was used to provide haptic feedback. The cervical spine is simulated using six degrees of freedom joint between each two vertebrae. The motion presented is the relative motion, i.e. of one vertebra moving with respect to the other vertebra below it.

Preliminary tests were conducted to validate the convex decomposition methodology on the needle biopsy simulator. Participants were asked to perform the needle insertion operation on the haptic simulator. The biopsy needle was inserted in the simulation till contact was made with a highlighted section of vertebrae. Preliminary trials showed that all participants were able to perform the task effectively. Participants felt the collisions between the needle and the cervical spine was accurate and, the haptic feedback was consistent with the contact of the virtual biopsy needle and the vertebrae.

With this methodology, the collision detection is improved since the collision model is very realistic as the real model. Moreover, the simulation performance is not decreased since the convex hulls are produced and saved before the simulation once and after, they are loaded in the simulation where the simulation doesn’t decrease below 100 FPS. (For K<20 using Nvidia PhysX by GPU).

5 DISCUSSION AND CONCLUSION

The methodology proposed is tested for effective collision shape representation on a haptic needle biopsy simulator of the cervical spine. An advantage of this decomposition for collision detection over triangles is that convex hulls, unlike triangles, have a clearly defined interior. This means that the convex hulls should approximate the vertebral mesh surfaces as close as possible and also be large enough to be effective for a fast realistic detection. Since the application requires high precision, even a small approximation error may cause significant visual and haptic discrepancy between the collision result and the actual geometry seen and the haptic feedback felt by the user. We have demonstrated that the proposed approach is a promising method for cervical spinal decomposition capable of producing effective convex representations.

Our main contribution in this area is a convex decomposition methodology. To our knowledge this is the first time that this methodology has been used for physics and haptic rendering of the cervical spine.

REFERENCES

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[5] J.A.Sethian, “Level Set Methods and Fast Marching Methods,” Cambridge University Press: Cambridge, 1999.

[6] W.Cees, F.R.Vincent, M.V.Frans, J.V.Lucas, “Detection and Segmentation of Colonic Polyps on Implicit Isosurfaces by Second Principal Curvature Flow,” IEEE Trans. on Medical Imaging, Vol. 29, Issue 3, March 2010, pp. 688-698.

[7] A.Chowriappa, T.Kesavadas, M.Mokin, P.Kan, S.Salunke, S.Natarajan, P.Scott. Vascular decomposition using weighted approximate convex decomposition. International Journal of Computer Assisted Radiology and Surgery. 1-13. 2012

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