Building hierarchical structures for 3D scenes with repeated elements

Abstract

We propose a novel hierarchy construction algorithm for 3D scenes with repeated elements, such as classrooms with multiple desk–chair pairs. Most existing algorithms focus on scenes such as bedrooms or living rooms, which rarely contain repeated patterns. Consequently, such methods may not recognize repeated patterns, which are vital for understanding the structure and context of scenes such as classrooms. Therefore, we propose a new global optimization algorithm for recognizing repeated patterns and building hierarchical structures based on repeated patterns. First, we find a repeated template by calculating the coverage ratios and frequencies of many substructures in a scene. Once the repeated template has been determined, a minimum cost maximum flow problem can be solved to find all instances (repetitions) of it in the scene and then group objects accordingly. Second, we group objects in the region outside the repeated elements according to their adjacency. Finally, based on these two sets of results, we build the hierarchy of the entire scene. We test this hierarchy construction algorithm on the Princeton and SceneNN databases and show that our algorithm can correctly find repeated patterns and construct a hierarchy that is more similar to the ground truth than the results of previous methods.

Proof that the flow network contains no negative-cost cycles

The Floyd–Warshall method works only when the network contains no negative-cost cycles. Therefore, we need to prove here that the directed cost graph from which we find the minimum path satisfies this condition. According to the Ford–Fulkerson method, we iteratively repeat two steps: (1) searching for an augmenting path in the cost network and (2) updating the cost network (corresponding to the matrix w in the algorithm) after adding the new augmenting path. The cost network may have two types of edges: positive edges, which have the same directions and cost values C(i, j) as those of the corresponding original edges in the flow network, and negative edges, which have the opposite directions and negative-cost values $-C(i, j)$. Consider a cycle in a cost network, such as that shown in Fig. 11. Note that the edges connected to the source or sink in a cycle are always paired. Because we set the costs of all edges connected to the source or sink to 1, the costs of such an edge pair, such as (e(s, u1), e(u2, s)) or (e(v1, t), e(t, v2)), always cancel to zero. Thus, the total cost of the cycle is $C_i - C_j$, where $C_i$ is the cost of edge e(u1, v1) and $-C_j$ is the cost of edge e(v2, u2). There is a negative-cost edge from v2 to u2, meaning that the flow of edge e(u2, v2) has been used as part of the flow solution. On the other hand, the positive-cost edge e(u1, v1) indicates that the flow of this edge has not been used. The edge e(u2, v2) is chosen earlier than e(u1, v1) indicates that e(u2, v2) has a smaller weight, so we can infer that $C_j \le C_i$. Therefore, the cost of the cycle cannot be negative.

Fig. 11

An arbitrary loop in the cost network always has a positive-cost value