DISTANCE MAPS FOR COLLISION DETECTION OF
DEFORMABLE MODELS
Athanasios Vogiannou, Michael G. Strintzis
Electrical & Computer Engineering Department,
Aristotle University of Thessaloniki, Greece
Konstantinos Moustakas, Dimitrios Tzovaras
Informatics and Telematics Institute, CERTH
P.O. Box 361, 57001, Thermi-Thessaloniki, Greece
ABSTRACT
This paper presents a method for detecting collisions between a deformable and a rigid model. The proposed approach
builds on the remarkable results achieved by distance fields and SQ-maps by introducing distance maps with multiple
entries. The computational cost of the presented extension is of the same level as the previous methods but the memory
requirements are much lower because distance maps with multiple entries can be used directly with general non-convex
3D models. The proposed method can be used in various applications at interactive performance rates.
KEYWORDS
Collision Detection, Deformable Models
1. INTRODUCTION
Collision detection for physically based modeling is a topic receiving much interest during the last years.
Various approaches have been introduced to efficiently calculate collision data and enhance perceptually
realistic interaction with deformable objects, such as clothes and human organs, in various virtual
environments (Teschner, 2004). Usually these approaches aim at improving the performance in specific
applications, for example surgery (Raghupathi, 2004) or cloth (Mezger, 2003) simulations, therefore their
requirements are closely related with the requirements of each application.
Most of the early approaches extend the relevant work on rigid bodies by using Bounding Volume
Hierarchies (Bergen, 1997; Gottschalk, 1996) with deformable models. The problem with this approach is
that a change in the shape of the object affects the algorithm and the surrounding hierarchy should be rebuild
in order to fit to the new shape. Rebuilding or refitting the new hierarchy comes at heavy computational cost.
For this reason, more sophisticated methods have been introduced and better results were produced
(Teschner, 2003; Baciu, 2004; Govindaraju, 2005).
One of the most remarkable recent approaches is the use of distance fields (Fisher, 2001; Fuhrmann,
2003; Funfzig, 2006). Distance fields have been used in the past in various applications (Jung 1997, Frisken
2000). A distance field of a 3D model represents the closest distance between the model and each point in
space. Although collision queries are dramatically simplified with this approach, calculating and storing a
distance field requires large amounts of memory. A method which compensates the memory requirements
while keeping the computational cost at low levels is the SQ-Map (Moustakas, 2007). Instead of the
excessive 3D field, SQ-Maps store the distance between the 3D model and a Super-Quadric (SQ)
surrounding the model. The only drawback of this method is that for non-convex objects it is necessary to use
decomposition in order to bound the object tightly and consequently more than one SQ-Map.
In this paper we propose an extension to SQ-Maps which can be used directly with any complex object.
Instead of the linear mapping from the 2D space of the SQ to a single distance value, we use multiple
distance values. Therefore any topological complexity of the object can be represented by a single map. The
memory requirements are significantly reduced while at the same time the performance remains at interactive
rates.
2. METHOD DESCRIPTION
The proposed approach includes a preprocessing stage where the distance map of each rigid object on
the scene is calculated. Rigid objects are represented as triangular meshes and deformable objects are
modeled using particle systems. Collision detection is performed in a vertex based approach. Every particle
of the deformable model is tested against the map of the rigid model. If collision is detected, the appropriate
response is calculated.
In the following sub-sections we describe a method to calculate and store a distance map with multiple
entries and an algorithm for quickly determining if a particle is inside or outside a rigid body using the
respective map. Experimental results indicating the importance of the proposed method are given in the last
sub-section.
2.1 Distance Map Calculation
We use spheres instead of SQs in our method because spheres better suit for our purpose, having one
centre and direct relation with the spherical coordinate system. After the bounding sphere of the object is
calculated, sampling is performed to the spherical surface using the angular coordinates (φ , θ ) . Let C and R
denote the center and the radius of the bounding sphere S. For each sample (φi , θ j ) , all the points of
intersection between the line segment from C to (φi , θ j , R ) are calculated. For each point of intersection p,
the distance p - C is calculated and stored in the (i , j ) entry of the array containing the map. Figure 1(a)
displays the case for an arbitrary object.
Figure 1. Distance Map with Multiple Entries. In (a) the line L intersects with the object in 3 different points. The
distance from each point is stored in the data structure of the map. In (b) we see possible discretization errors in the area
around the vertex V caused by the sampling scheme.
The sampling may cause discretization errors, especially in the areas around the vertices of the model.
Figure 1(b) illustrates a 2D representation of the case. The linear interpolation between the distance values S1
and S2 of the samples D1 and D2 yields lower distance than the actual distance of the vertex. This will lead to
falsely disregarding colliding particles in the range (D1, D2). In order to correct this error, the distance values
S1 and S2 are augmented by the error difference between the interpolated value and the distance of the vertex.
The final result of the previous algorithm is a data structure describing the distance map. This map can be
used as an implicit representation of the object. Figure 2 illustrates the 3D model and the map of a horse with
6000 faces and Figure 3 the respective 3D model and map of a torus with 2880 faces. The distance maps
have been generated using 200 samples along each coordinate. Note how the uniform sampling affects the
mapping in the areas near and far of the centre of each object. Even so, prominent model parts such as the
legs and the head of the horse are still sufficiently described by the map.
Figure 2. Distance map of the horse 3D model.
Figure 3. Distance map of the torus 3D model.
2.2 Collision Detection
Consider a particle p in space together with a rigid body A. Particle p is initially tested against the
bounding sphere of A. If the particle does not penetrate the sphere then no collision is reported. On the other
r
case, let d denote the vector from the centre of the sphere to p. We then obtain the angular coordinates
r
(φd , θ d ) of d . The distance value (or values) for (φd , θ d ) are calculated using bilinear interpolation from
the four samples closer to it.
There are two error situations that can occur due to the interpolation scheme. The first happens when at
least two of the four used samples refer to different faces of the object. This case has been already treated in
the preprocessing stage by augmenting the entries of the map as this situation appears only in samples around
the vertices of the object. The second situation takes place when there are samples that have different length,
i.e. different number of distance values. In this case, we use the sample which is closer to (φd , θ d ) as the
reference sample and ignore (in the interpolation) all the other samples with different length than the
reference one. Even though this approach seems naive, it works well in practice as rarely happens to use less
than 3 samples in the interpolation.
In any case, in the end of the above procedure we will have one sample with the interpolated distance
values. These values separate space ℜ in discrete intervals. It is then enough to enumerate the intervals
r
r
(starting from 0), find the interval where | d | lies to and perform a simple parity test. If | d | is on an even
interval then p penetrates A. Note that this applies when the centre of A lies on the interior of the object
r
(there are complex objects like the torus that the centre is out of the object). In the other case, | d | should be
on an odd interval in order for p to penetrate A.
After determining collision, an estimation of the penetration depth can be retrieved directly from the
r
r
map. In order for p to exit from A, it should translate S - | d | distance along the direction of d where S
r
denotes the interpolated distance value closer to | d | .
2.3 Experimental Results
The proposed method has been tested in two simple simulations involving the horse, the torus and a cloth.
The cloth is modeled using the method described by Jakobsen (2001). During the first simulation the cloth
was falling down towards the horse, until it stopped completely. In the second simulation the torus was used
in the place of the horse. Tests were performed on a Core2 6600 2,4GHz CPU PC with 2GB of RAM and a
GeForce 7600 GS Graphics Card. Figures 3 and 4 illustrate different screenshots of each simulation
The frame rate of each simulation ranged from 15 to 60 fps, which is sufficiently enough for interactive
applications. The size of the distance map with multiple entries for the horse model is 0.63 Mbytes and for
the torus is 0.7 Mbytes. The SQ-Maps require 2.9 Mbytes and 4.1 Mbytes respectively. That happens because
the two models need more than one SQ segment to be represented sufficiently (17 for the horse and 12 for the
torus). To complete the comparison we note that the requirements of the respective distance field with the
same sampling frequency are 230 Mbytes and 180 Mbytes respectively.
Figure 4. Different screenshots of the simulation with the horse model
Figure 5. Different screenshots of the simulation with the torus model
3. CONCLUSION
We presented an extension of SQ-Maps for collision detection by introducing distance maps with
multiple entries. Instead of mapping a single distance value of the point closer to the bounding surface, the
distances from all the intersection points are stored and used to construct an implicit representation of the
object. The proposed method requires much lower memory than SQ-Maps and distance fields while the
performance remains at interactive rates. The major limitations of the presented approach are related to the
accuracy and derive from the approximations introduced by sampling the 3D model and the vertex based
collision detection method. Even so, the method is suitable for practical graphics applications as it is simple
to implement and very efficient in computational time.
Further work may include the investigation of other bounding surfaces instead of the sphere and
improving the sampling method along the 2D surface. Also we would like perform tests in more demanding
interactive applications to highlight the advantages and the limitations of the proposed method.
REFERENCES
Baciu G. and Wong Wingo S.-K., 2004. Image-Based Collision Detection for Deformable Cloth Models. In IEEE
Transactions on Visualization and Computer Graphics, Vol. 10, No. 6, pp 649 - 663.
Bergen v. d. G., 1997. Efficient Collision Detection of Complex Deformable Models using AABB Trees. In Journal of
Graphics Tools, Vol. 2, No. 4, pp 1-14.
Fisher S. and Lin M. C., 2001. Deformed distance fields for simulation of non-penetrating flexible bodies. Proceedings
of the Eurographic workshop on Computer animation and simulation, pp. 99-111.
Frisken, S.F. and Perry, R.N. and Rockwood, A.P. and Jones, T.R., 2000. Adaptively Sampled Distance Fields: A
General Representation of Shape for Computer Graphics, ACM SIGGRAPH, pp 249-254.
Funfzig C. and Ullrich T. and Fellner D. W., 2006. Hierarchical Spherical Distance Fields for Collision Detection. In
IEEE Computer Graphics and Applications, Vol. 26, No. 1, pp 64-74.
Fuhrmann A. and Sobottka G. and Gross C., 2003. Distance Fields for Rapid Collision Detection in Physically Based
Modeling. Proceedings of Graphicon 2003. Moscow, Russia, pp. 58-65.
Gottschalk S. and Lin M. C. and Manocha D., 1996. OBBTree: A Hierarchical Structure for Rapid Interference
Detection. In Computer Graphics, Vol. 30, Annual Conference Series, pp 71-80.
Govindaraju N. K. et al., 2005. Interactive collision detection between deformable models using chromatic
decomposition. In ACM Trans. Graph., Vol. 24, No. 3, pp 991 - 999.
Jakobsen T., 2001. Advanced character physics. In Proceedings of the Game Developers Conference. pp. 139-149.
Jung D. and Gupta K., 1997. Octree-based hierarchical distance maps for collision detection. In Journal of Robotics
Systems, Vol. 14, No. 11, pp 789-806.
Mezger J. and Kimmerle S. and Etzmu O., 2003. Hierarchical Techniques in Collision Detection for Cloth Animation.
Journal of WSCG. Vol. 11, No. 2, pp 322-329.
Moustakas K. and Tzovaras D. and Strintzis M.G., 2007. SQ-Map: Efficient Layered Collision Detection and Haptic
Rendering. In IEEE Transactions on Visualization and Computer Graphics, Vol. 13, No. 1, pp 80-93.
Raghupathi, L. et al, 2004. An Intestinal Surgery Simulator: Real-Time Collision Processing and Visualization. In IEEE
Transactions on Visualization and Computer Graphics, Vol. 10, No. 6, pp 708-718.
Teschner M. et al, 2003. Optimized spatial hashing for collision detection of deformable objects. In Proceedings of
Vision, Modeling, Visualization (VMV 2003). Munich, Germany, pp. 47-54.
Teschner M. et al, 2004. Collision Detection for Deformable Objects. Eurographics State-of-the-Art Report (EG-STAR).
Grenoble, France, pp. 119-139.