Single-source shortest path [5] describes the problem how to find the shortest distance to other vertices from the certain source vertex in a graph. The connections between the vertex and all other vertices on the path from the starting vertex to the target vertex will affect the distance between these two vertices. When there are multiple pathways between the source vertex and target vertex, the shortest distance between them is regarded as the shortest path.

This thinking can be used in similarity matching of 3D model retrieval. The shortest path method can be used to expand the breadth of geometric distance metric in the 3D similarity matching method based on geometric distance. Here, the similarity matching no longer depends on one geometric distance measure result, but on the minimum value among the multi geometric distance metric results. It will improve the retrieval rate to some extent.

Assuming that there are model in the collections of 3D models and the distance metric function d in the similarity matching method based on geometric distance, then there is an inequality: . Introducing the thinking of single-source shortest path in the similarity matching method, we can find a new distance metric function d'. There will be an inequality: . So it will improve the previous matching results.

In the webpage search method, a kind of correlation matching method is often used. The retrieved webpage depends on correlation between the keywords in the webpage and the given keywords. When a webpage has a strong correlation with the webpage that matches the retrieval condition best, the webpage will be regarded as retrieval result. So a lot of matching webpage will be found. In view of this idea, we use the label propagation to the similarity matching of 3D model.

Label propagation [6-8] is a kind of semi-supervised learning method. It is assumed that the classification label is known for small quantities in a data set, then a propagation algorithm is used to push out the labels to the unlabeled data in the rest of the data set. In the similarity matching of 3D model retrieval, it is assumed that only the label of query model is known. The label propagation method is based on graph structure. Whether the label of a node can propagate to the neighbor node or not is determined by the similarity between them.

Here, we assume that the labeled data set is , in which is a classification label. The C is known and is defined as the total number of categories. The x is the data in the data set. is unlabeled data set, in which. The whole data set is. In label propagation method, the unlabeled data will be mapped to set through label propagation algorithm. Usually, the l is much smaller than u.

Each data in the data set X is taken as a node and weighted graph is constructed. The weight value connecting and can be calculated in several methods. Here, we use the simple Euclidean distance to calculate the weight. The is inversely proportional to its Euclidean distance between them. The greater the distance, the less the similarity. It will also be affected by the parameters. The formula of is

. The labels are propagated through the edges connecting with the nodes. The greater the weight on the edge connecting with two nodes is, the easier the label is propagated. The total number of nodes is known. Constructing a matrix, in which

represents the probability of propagation from node to node.

Constructing a matrix, in which represents the probability that node is belong to the classification of. The classification label is propagated as below:

1) The node classification is propagated to neighboring nodes according to a certain probability;

2) The data in the matrix Y from the line 1 to line l will be reverted to the initial value and the data in the rest lines will be normalized;

3) Repeating the above steps until the Y is converged.

Case-based Reasoning (CBR) has become a very popular AI technique [9,10]. It is based on the assumption that problems can be solved efficiently by reusing knowledge about similar, already solved problems stored in the collections of cases. The case-based learning is also called passive learning. Because the training cases are simply stored in memory without any evaluation or classification, they will be treated only when a new case comes. An important advance in this passive learning method is that it is not one-time estimated objective function on the whole instances space, but on local space and using different estimation for each new instance.

The case-based K-nearest neighbor method assumes that all cases X corresponds to the points in n-dimensional space. The distance between case and case is defined as

. The value of target function can be a discrete value or can also be real. For the distance-weighted K nearest neighbor method, the inverse of square of distance between each neighbor and the query node the is used to calculate the weight. If the query node coincides with the training case, the denominator will be 0. So under this situation, we define

.

Here, is the classification for the training cases. The target function is f and the query case is V. The learning process is below:

1) Firstly, each training case will be added into training_examples;

2) Input a query case, select k cases that are nearest to in the training_examples and represent them with;

3) Calculate and return.

In 3D model retrieval system, only a good feature extraction method is not enough. It there is no good method for similarity matching, the retrieval results will not be good. Here, the learning-based dissimilarity calculation is proposed to reduce the error similarity matching. It will improve the search accuracy.

In general model retrieval method, when the query model is compared to model with similarity, the neighbor models of will not be considered. In fact, in the learning-based dissimilarity calculation method, when model a is more similar with model b, the model a is more similar with the neighbor models of model b. This is consistent with the facts and it can improve the retrieval accuracy and recall rate. The existing dissimilarity calculation does not reflect this characteristic [11].

In this paper, the new learning-based dissimilarity calculation method not only takes into the individual characteristics of compared model, but also considers the characteristics of its nearest neighbor model. Here, the single-source shortest path, label propagation, and casebased K-nearest neighbor learning method are combined together in order to improve the retrieval accuracy and recall rate.

Assume that the original target function of dissimilarity calculation is, the value of this function is the value of Gaussian density function of distance

between the node. Here, weight value is defined as. The smaller the value is, the more similar the nodes are. In the learningbased dissimilarity calculation, a new function is introduced as the dissimilarity between the model and query model. The subscript t represents the number of iterations.

When the iteration is terminated, the t is up to T and will be the value of dissimilarity between the model and query model. The function is the improved function of the existing similarity metric function. It is combined the learning-based dissimilarity calculation, shortest-path, and label propagation.

In the existing dissimilarity calculation method,

, in which

is the mean distance of k nearest neighbor between model and model. The k and are determined by experiments [11]. For matrix W, a new matrix P is introduced and improves the weight calculation. Here, , it represents the probability of similarity between model and model. This comes from the thinking of label propagation. When a model has very high similarity with many models in the collection of model, the is smaller and is bigger. That means the model has strong correlation with other models and it has a greater probability to be retrieved.

In the case-based K nearest neighbor method, the key function is introduced. It represents the value of dissimilarity between model and model. Here, assume, in which model is the query model and model is the training model in the collection of models. The solution of this iteration function is below:

1) Initialize, , i = 2, ∙∙∙, n;

2) Iteration process:, i = 2, ∙∙∙n and;

3) The terminal condition is when the tends to infinitesimal. Then setting a threshold, when t is up to T, the calculation result of is the finial result.

After finishing the calculation of matrix W, the matrix P can be calculated as above. When the satisfies the threshold condition, the iteration process stops [12].

When this iteration is over, the dissimilarity of model and model can be calculated as. Then the values of dissimilarity are sorted in ascending order. The smaller the value, the more similar the models.