Wednesday, 23 November 2011

HACD parameters

In this post, I'll to give an overview of the HACD parameters and explain their meaning and how they should be set. The text will be improved over time. My main concern is to have things written down for reference...

  1. Parameters overview
Target number of triangles in the decimated mesh. The decimation stage was added mainly to decrease the computation costs for dense meshes (refer to Section 2.1).
Maximum number of vertices in the generated convex-hulls.
Normalization factor used to ensure that the other parameters (e.g. concavity) are expressed w.r.t. a fixed size. Refer to Section 2.3 for details

Threshold on the clusters area (expressed as a percentage of the entire mesh area) under which the cluster is considered small and it is forced to be merged with other clusters at the price of a high concavity.
If enabled an additional ray located at the center of each triangle pointing toward its normal is considered when computing the concavity of a non-flat cluster. The parameter was added to handle coarse meshes (i.e. with a low number of vertices)

If enabled additional rays are considered to handle bowl-like shapes.
Minimum number of convex-hulls to be generated
Maximum allowed concavity
If the distance between two triangles, each belonging to a different connected components (CCs), is lower than the ConnectDist threshold an additional edge connecting them is added to the dual graph. This parameter was added to handle meshes with multiple CCs.
Weight controlling the contribution of the volume related cost to the global edgecollapse cost (refer to XXX for details).
(not used)
Weight controlling the contribution of the shape factor related cost to the global edgecollapse cost (refer to XXX for details).  

Threshold expressed a percentage of ScaleFactor under which a cluster is considered flat.

Weight controlling the contribution of the computation related cost to the global edgecollapse cost (refer to XXX for details).
2. Detailed description
  • NTargetTrianglesDecimatedMesh
In order to reduce the computations times for dense meshes, the HACD library makes it possible to decimate the original mesh before running the decomposition process. More details about the decimation algorithm are provided here

  • NVerticesPerCH
this parameter was introduced in order to comply with the constraints that most physics engines put on the number of vertices/triangles per convex-hull (CH). If the function HACD::Compute() is called with the parameter fullCH=false, then the generated CHs will have a number of vertices lower than NVerticesPerCH.

In order to optimally choose the best vertices to keep in the final CH, the ICHull::Process(unsigned long nPointsCH) function implements a slightly different version of the Incremental Convex Hull algorithm (cf. demo code ). Here, at each step, the point with the highest volume increment  is chosen, until all points are processed or the CH has exactly NVerticesPerCH points.

The code looks like this:

        while (!vertices.GetData().m_tag && addedPoints < nPointsCH) // not processed
            if (!FindMaxVolumePoint())
            vertices.GetData().m_tag = true;                     
            if (ProcessPoint())
        // delete remaining points
        while (!vertices.GetData().m_tag)

  • ScaleFactor
A normalization process is applied to the input mesh in order to ensure that the other parameters (e.g. concavity) are expressed w.r.t. a fixed size. This process is inverted before producing the final CHs.

The HACD::Compute() function follows the following main steps:

    bool HACD::Compute(bool fullCH, bool exportDistPoints)
        if (m_targetNTrianglesDecimatedMesh > 0)
        return true;

The HACD::NormalizeData() function centers the mesh and scale it so its coordinates are in the interval [-m_sacle, m_sclae]x[-m_sacle, m_sclae]x[-m_sacle, m_sclae]. The code proceeds as follows:

       void HACD::NormalizeData()
              const Real invDiag = static_cast<Real>(2.0 * m_scale / m_diag);
              for (size_t v = 0; v < m_nPoints ; v++)
                     m_points[v] = (m_points[v] - m_barycenter) * invDiag;

The HACD::DenormalizeData() function invert the normalization operated by HACD::NormalizeData():
       void HACD::DenormalizeData()
              const Real diag = static_cast<Real>(m_diag / (2.0 * m_scale));
              for (size_t v = 0; v < m_nPoints ; v++)
                     m_points[v] = m_points[v] * diag + m_barycenter;
  • SmallClusterThreshold
Due to numerical stability issues (or maybe some bugs I haven't spotted yet :) ) the HACD algorithm may produce small clusters. In order to detect them and make sure they will be merged, the SmallClusterThreshold was introduced. A cluster is considered to be small if its area is smaller than SmallClusterThreshold% of the entire mesh area.

The condition2 in the HACD::Simplify() function forces small clusters to be merged (m_area is the area of the entire mesh):
void HACD::Simplify()
              double areaThreshold = m_area * m_smallClusterThreshold / 100.0;
              while ( !m_pqueue.empty() )
               currentEdge =;
               v1 = m_graph.m_edges[currentEdge.m_name].m_v1;
               v2 = m_graph.m_edges[currentEdge.m_name].m_v2;
               bool condition1 = (m_graph.m_edges[currentEdge.m_name].m_concavity <  
                                  m_concavity) && (globalConcavity < m_concavity) && 
                                  (m_graph.GetNVertices() > m_nMinClusters) && 
                                  (m_graph.GetNEdges() > 0);
              bool condition2 = (m_graph.m_vertices[v1].m_surf < areaThreshold || 
                                 m_graph.m_vertices[v2].m_surf < areaThreshold);
              if (condition1 || condition2)
                   m_graph.EdgeCollapse(v1, v2);
                   long idEdge;
                   for(size_t itE = 0; itE < m_graph.m_vertices[v1].m_edges.Size(); ++itE)
                      idEdge = m_graph.m_vertices[v1].m_edges[itE];


  • AddFacesPoints and AddExtraDistPoitns
The parameter AddFacesPoints was introduced to improve the precision of the concavity computation for meshes with a low number of vertices. The idea is to add additional rays located each at the center  of a triangle and pointing to the same direction as its normal.

The Parameter AddExtraDistPoints was added to handle bowl-like shapes. As illustrated below, if only the rays located on the current cluster are considered when computing its concavity, you may end up with a big cluster corresponding to the external surface (which is convex) of the bowl containing a lot of small clusters located on the internal part (which is concave). 

Bad decomposition for bowl-like shapes if AddExtraDistPoints is not enabled
In order to avoid such a bad decomposition, the idea consists in introducing new rays that would constrain the propagation of the cluster corresponding to the external surface of the bowl by taking into account rays located on the concave part. More precisely, during the initialization stage, an additional ray (the yellow ray in the figure below) is associated with each triangle (colored in red). 

The additional ray, denoted R (the yellow arrow), is defined as follows. Let N be the normal (the blue arrow) to the current triangle T (colored in red) and X be the ray starting at the center of T and with direction (-N) (the dotted green arrow). The starting point of R, denoted P0 (the yellow point), is defined a the nearest intersection point of X and the mesh. Moreover, the normal of the surface at P0 should point to the same direction as X. R has the direction of the normal of the surface at the P0.

Additional ray (yellow) is associated with the red triangle when AddExtraDistPoints is activated

The code looks like this:

    void HACD::InitializeDualGraph()
         for(unsigned long f = 0; f < m_nTriangles; f++)
            i = m_triangles[f].X();
            j = m_triangles[f].Y();
            k = m_triangles[f].Z();
            m_graph.m_vertices[f].m_distPoints.PushBack(DPoint(i, 0, false, false));
            m_graph.m_vertices[f].m_distPoints.PushBack(DPoint(j, 0, false, false));
            m_graph.m_vertices[f].m_distPoints.PushBack(DPoint(k, 0, false, false));
            u = m_points[j] - m_points[i];
            v = m_points[k] - m_points[i];
            w = m_points[k] - m_points[j];
            normal = u ^ v;

            m_normals[i] += normal;
            m_normals[j] += normal;
            m_normals[k] += normal;

            m_graph.m_vertices[f].m_surf = normal.GetNorm();
            m_area += m_graph.m_vertices[f].m_surf;
                m_faceNormals[f] = normal;
                m_facePoints[f] = (m_points[i] + m_points[j] + m_points[k]) / 3.0;
            if (m_addExtraDistPoints)   
                Vec3<Real> seedPoint((m_points[i] + m_points[j] + m_points[k]) / 3.0);
                Vec3<Real> hitPoint;
                Vec3<Real> hitNormal;
                normal = -normal;
                if (rm.Raycast(seedPoint,normal,hitTriangle,dist, hitPoint, hitNormal))
                     faceIndex = hitTriangle;
                if (faceIndex < m_nTriangles )
                     m_extraDistPoints[f] = hitPoint;
                     m_extraDistNormals[f] = hitNormal;                
                     m_graph.m_vertices[f].m_distPoints.PushBack(DPoint(m_nPoints+f, 0, false, true));
        for (size_t v = 0; v < m_nPoints; v++)
  • Concavity
This parameter specifies the maximum allowed concavity for each cluster. As discussed in different concavity measures are considered for flat (i.e. 2D) surfaces and for non-flat surfaces.
  • ConnectDist
In order to handle meshes with different connected components. The idea consists  in adding "virtual edges" between triangles belonging to different CCs. More precisely, if the distance between two triangles T1 and T2 belonging each to a different CC is lower than a threshold distConnect then an edge connecting T1 to T2 is added to the dual graph.

The code looks like this:
       void HACD::CreateGraph()
        if (m_ccConnectDist >= 0.0)
            if (m_graph.m_nCCs > 1)
                std::vector< std::set<long> > cc2V;
                long cc;
                for(size_t t = 0; t < m_nTriangles; ++t)
                    cc = m_graph.m_vertices[t].m_cc;
                for(size_t cc1 = 0; cc1 < m_graph.m_nCCs; ++cc1)
                    for(size_t cc2 = cc1+1; cc2 < m_graph.m_nCCs; ++cc2)
                        std::set<long>::const_iterator itV1(cc2V[cc1].begin()), itVEnd1(cc2V[cc1].end());
                        for(; itV1 != itVEnd1; ++itV1)
                                                double distC1C2 = std::numeric_limits<double>::max();
                            double dist;
                            t1 = -1;
                            t2 = -1;
                            std::set<long>::const_iterator itV2(cc2V[cc2].begin()), itVEnd2(cc2V[cc2].end());
                            for(; itV2 != itVEnd2; ++itV2)
                                dist = (m_points[*itV1] - m_points[*itV2]).GetNorm();
                                if (dist < distC1C2)
                                    distC1C2 = dist;
                                    t1 = *vertexToTriangles[*itV1].begin();
                                                              std::set<long>::const_iterator it2(vertexToTriangles[*itV2].begin()),
                                                              t2 = -1;
                                                              for(; it2 != it2End; ++it2)
                                                                     if (*it2 != t1)
                                                                           t2 = *it2;
                            if (distC1C2 <= m_ccConnectDist && t1 >= 0 && t2 >= 0)
                                m_graph.AddEdge(t1, t2);                   
  •  FlatRegionThreshold
When computing concavity we need to distinguish between flat surfaces and non-float surfaces. The measure of flatness considered in HACD is related to the ratio between the convex-hull volume and the area of its boundary. If this later ration is small compared to m_scale then the mesh is considered flat. Otherwise it is considered non-flat. The parameter FlatRegionThreshold is the threshold which separate flat region from non flat region. It is expressed as percentage of m_scale.

In practice, the final concavity is computed as weighted sum of the flat region concavity and the 3D concavity. The weight is a function of the flatness of the cluster.

The code is as follows:

void HACD::ComputeEdgeCost(size_t e)
double surfCH = ch->ComputeArea() / 2.0;
double volumeCH = ch->ComputeVolume();
double vol2Surf = volumeCH / surfCH;
double concavity_flat = sqrt(fabs(surfCH-surf));
double weightFlat = std::max(0.0, 1.0 - pow(- vol2Surf * 100.0 / (m_scale * m_flatRegionThreshold), 2.0));
concavity_flat *= weightFlat;
   concavity = Concavity(*ch, distPoints);
concavity += concavity_flat;

Wednesday, 5 October 2011

HACD optimization

Today, I have updated the HACD library in order to reduce both memory usage and computation complexity (cf. 

The new version:

  • uses John's Micro Allocator to avoid intensive dynamic memory allocation (thanks John for the great work),
  • exploits a simplified version of  an axis-aligned-bounding-volume AABB tree to accelerate dual graph generation (the code is inspired by John's post on this subject, thanks again John :) )
  • has an integrated mesh simplification pre-processing step, which makes it possible to handle dense meshes.
In this post, I'll give more details about this last feature.

To enable mesh decimation, the user specifies the target number of triangles that should be produced before running the convex decomposition. HACD will handle automatically the simplification and the decomposition processes. For the details of the mesh decimation algorithm, have a look at Michael Garland's  webpage To turn this feature off just set the parameter targetNTrianglesDecimatedMesh=0.

I have tested the updated HACD algorithm on the 3D model "Bunny", which has 70K triangles. The HACD's parameters were set as follows:
  • minNClusters = 12
  • maxConcavity = 1000
  • addExtraDistPoints = 1
  • addFacesPoints = 1
  • ConnectedComponentsDist = 30
  • targetNTrianglesDecimatedMesh = 500, 1000, 2000, 4000, 8000 and 16000.

The snapshots of the produced convex decompositions are reported below. The computation times on my machine (Mac Intel Core 2 Duo, 4 GB RAM DDR3) ranged between 3 sec. for 500 triangles and 200 sec. for 16000 triangles (cf. Table 1 for the details).

# triangles  Time (sec.)
500 2.8
1000 4.6
2000 10
4000 21
8000 70
16000 192
Table 1. Computation times for targetNTrianglesDecimatedMesh= 500, 1000, 2000, 4000, 8000 and 16000 
# triangles after simplification - 500

# triangles after simplification - 1000

# triangles after simplification - 2000

# triangles after simplification - 4000

# triangles after simplification - 8000

# triangles after simplification - 16000

Sunday, 2 October 2011

HACD: Hierarchical Approximate Convex Decomposition

Check out V-HACD 2.0!

I discovered the convex decomposition problem two years ago thanks to a very instructive discussion I had with Stan Melax. At that time, we needed a convex decomposition tool for the game we were working on.  Stan pointed me to John Ratcliff's approximate convex decomposition (ACD) library. Inspired by John's work I developed the HACD algorithm, which was published in a scientific paper at ICIP 2009.

The code I developed back then was  heavily relaying on John's ACD library and Stan's convex-hull implementation (thanks John and Stan :)! ). My implementation was horribly slow and the code was unreadable. When Erwin Coumans contacted me six moths ago asking for an open source implementation of the algorithm, I had no choice than re-coding the method from scratch (i.e. my code was too ugly :) !) . One month later, I published the first version of the HACD library.  Since then, I have been improving it thanks to John's, Erwin's and a lot of other peoples' comments and help. John re-factored the HACD code in order to provide support for user-defined containers (cf. John's HACD). Thanks to Sujeon Kim and Erwin, the HACD library was recently integrated into Bullet.

In this post, I'll try to briefly describe the HACD algorithm and give more details about the implementation. I'll provide also an exhaustive description of the algorithm parameters and discuss how they should be chosen according to the input meshes specificities. By doing so, I hope more people would get interested in the library and would help me improving it.

Why do we need approximate convex decomposition?

Collision detection is essential for realistic physical interactions in video games and computer animation. In order to ensure real-time interactivity with the player/user, video game and 3D modeling software developers usually approximate the 3D models composing the scene (e.g. animated characters, static objects...) by a set of simple convex shapes such as ellipsoids, capsules or convex-hulls. In practice, these simple shapes provide poor approximations for concave surfaces and generate false collision detections.
Original mesh


Approximate convex decomposition

A second approach consists in computing an exact convex decomposition of a surface S, which consists in partitioning it into a minimal set of convex sub-surfaces. Exact convex decomposition algorithms are NP-hard and non-practical since they produce a high number of clusters. To overcome these limitations, the exact convexity constraint is relaxed and an approximate convex decomposition of S is instead computed. Here, the goal is to determine a partition of the mesh triangles with a minimal number of clusters, while ensuring that each cluster has a concavity lower than a user defined threshold.
Exact Convex Decomposition produces 7611 clusters
Approximate convex decomposition generates 11 clusters

What is a convex surface?
Definition 1: A set A in a real vector space is convex if any line segment connecting two of its points is contained in it.
A convex set in IR2.

A non-convex set in IR2.

Let us note that with this definition a sphere (i.e. the two-dimensional spherical surface embedded in IR3)  is a non-convex set in IR3. However, a ball (i.e. three-dimensional shape consisting of a sphere and its interior) is a convex set of IR3.

When dealing with two dimensional surfaces embedded in IR3, the convexity property is re-defined as follows.

Definition 2: A closed surface S in IR3 is convex if the volume it defines is a convex set of IR3.

The definition 2 characterizes only closed surfaces. So, what about open surfaces?

Definition 3: An oriented surface S in IR3 is convex if there is a convex set of IR3 such that is exactly on the boundary of A and the normal of each point of points toward the exterior of A.

The definition 3 do not provide any indication of how to choose the convex set A. A possible choice is to consider the convex-hull of (i.e. the minimal convex set of IR3 containing S), which lead us to this final definition.

Definition 4: An oriented surface S in IR3 is convex if it is exactly on the boundary of its convex-hull CH(S) and the normal of each point of points toward the exterior of CH(S).

With Definition 4, the surface defined by the half of a sphere is convex. The half of a torus is not convex.
Half of a sphere is convex.

Half of a torus is not convex.

How to measure concavity?
There is no consensus in the literature on a quantitative concavity measure. In this work, we define the concavity C(S) of a 3D mesh S, as follows:
C(S) = argmax ∥M − P (M )∥, M∈S

where P(M) represents the projection of the point M on the convex-hull CH(S) of S, with respect to the half-ray with origin M and direction normal to the surface S at M.

Concavity measure for 3D meshes: distance between M0 and P(M0) (Remark:  S is a non-flat surface. It is represented in 2D to simplify the illustration)

Let us note that the concavity of a convex surface is zero. Intuitively, the more concave a surface the ”farther” it is from its convex-hull. The definition extends directly to open meshes once oriented normals are provided for each vertex.

In the case of a flat mesh (i.e., 2D shape), concavity is measured by computing the square root of the area difference between the convex-hull and the surface:
C_flat(S) = sqrt(Area(CH)-Area(S)).

Concavity measure for closed 2D surfaces: the square root of the green area.

Here again, the concavity is zero for convex meshes. The higher C_flat(S) the more concave the surface is. This later definition applies only to closed 2D surfaces, which is enough for HACD decomposition of 3D meshes.

Overview of the HACD algorithm
The HACD algorithm exploits a bottom up approach in order to cluster the mesh triangles while minimizing the concavity of each patch. The algorithm proceeds as follows. First, the dual graph of the mesh is computed. Then its vertices are iteratively clustered by successively applying topological decimation operations, while minimizing a cost function related to the concavity and the aspect ratio of the produced segmentation clusters.

Dual Graph

The dual graph G associated to the mesh S is defined as follows:

• each vertex of G corresponds to a triangle of S,

• two vertices of G are neighbours (i.e., connected by an edge of the dual graph) if and only if their corresponding triangles in S share an edge.
Original Mesh

Dual Graph
Dual Graph Decimation
Once the dual graph G is computed, the algorithm starts the decimation stage which consists in successively applying halfe-edge collapse decimation operations. Each half- edge collapse operation applied to an edge (v,w), denoted hecol(v, w), merges the two vertices v and w. The vertex w is removed and all its incident edges are connected to v.

HACD implementation details
How to improve HACD?
HACD parameter tweaking
HACD decomposition results

HACD vs John Ratcliff's ACD library