3D Model Retrieval With Morphing-Based

1. 3D Model Retrieval With Morphing-BasedGeometric and Topological Feature Maps*

Meng Yu, Indriyati Atmosukarto,Wee Kheng Leow, Zhiyong Huang, Rong Xu Dept. of Computer Science National University of Singapore [yumeng, indri, leowwk, huangzy, xurong]@comp.nus.edu.sg *This research is supported by NUS ARF R-252-000-137-112 CVPR 2003, Madison, Wisconsin June 16-22, 2003

2. Major Idea

3D morphing given two 3D objects, generate a sequence of intermediate objects that gradually change Measure the dissimilarity the amount of effort required to morph an object into another one

3. Feature Map

Rays are shot from the center of a bounding sphere through the object to the sphere�s surface The feature map records the distance traveled di by the ray from a point pi to the sphere�s surface, the Distance Map (DM) the number of object surfaces (solid lines; 2, in this case) penetrated by the ray since it leaves the sphere�s center, the Surface Penetration Map (SM)

4. Related Work

Global features moments, aspect ratio, volume-to-surface area ratio, ... limitation: single feature values are used to characterize the overall shape of the objects. they tend to be not very discriminative about the objects Histograms discrete frequency or probability distribution of features (angles, distance, area, ...) limitation: spatial information is not considered Spatial maps arrange object�s features in a manner that preserves the relative positions of the features in the object limitation: spatial maps preserve the spatial information of the features in an object. they are generally not invariant to linear transformations, except for specially designed maps

5. Major Steps of Our Method

The 3D object are translated so that the object�s centroid coincides with the origin of the 3D coordinate system The object is scaled so that the furthest 3D point on the object is 1 unit distance away from the centroid Principal Component Analysis (PCA) is performed on the 3D points on the object to align the major axis and minor axes of the object to the first and second eigenvectors of PCA Compute the Distance Map (DM) and the Surface Penetration Map (SM) are extracted from the object Apply the Fast Fourier Transforms to DM and SM Match the objects with the transformed DM and SM

6. Compute Distance Map (DM)

An average map is computed by dividing the bounding sphere into 64x64 pyramidal sections the longitudinal angle of ? =0o to 360o is divided into 64 equal intervals and the latitudinal angle of ?=-90o to + 90o into 64 equal intervals record the mean distance averaged over all the points pi contained in each entry (?, ?) these D(?, ?) values form the Distance Map (DM), which describes the geometry of the object

7. Compute Surface Penetration Map (SM)

Counting number of object surfaces a ray (dashed arrow) is shot from the center of the sphere through an object point obtain a cone with a small angle for the other object points within the cone, check the number of times the surface normals (arrows) at the points change direction, compared to the direction of the ray, in increasing distance of the points from the sphere's center, determines the number of surfaces that the ray passes through

8. Match the Objects with DM and SM after the FFT

Fast Fourier Transform (FFT) is performed on the feature maps the amplitudes D(u,v) and S(u,v) of the FFTs of the D(?, ?) and S(?, ?) are invariant to rotation and reflection Compute the dissimilarity between two objects

9. Experiments: Object Samples

10. Experiment: Invariance of Feature Matching

The Fourier transforms of feature maps (FT(DM), FT(SM)) are very accurate and are invariant to (a) rotation and reflection, and (b) nonuniform scaling. In contrast, the raw feature maps (DM, SM) are not invariant

11. Experiments: Precision-Recall Result

DM, SM: Distance and Surface Penetration Maps FT: Fourier transform DH, SH: Distance and Surface Penetration Histograms; WE, E: weighted and unweighted normalized Euclidean distances

12. Conclusion

A method of retrieving 3D models using geometric and topological spatial feature maps called Distance Map and Surface Penetration Map based on the amount of effort required to morph a 3D object into a canonical sphere without performing explicit 3D morphing Fourier transforms of the feature maps are used for comparison and retrieval because spatial information about the objects� features is preserved in the feature maps, retrieval performance based on the Fourier transforms of these maps is very accurate and invariant to rotation, reflection, and scaling Test results show that retrieval precision remains above 0.86 even for recall rate of 1.0

13. On-going Work

We are conducting tests to compare the retrieval performance of various features random angle, random, distance, random area, Gaussian curvature represented in histograms and 2D feature maps Our preliminary results show that the map of a feature performs better than the histogram of the same feature

14. The Preliminary Result

Comparison of retrieval performance using the best of histograms and maps DM: Distance Map SM: Surface Penetration Map RAM: Random Angle Map RRH: Random Area Histogram RDM: Random Distance Map GM: Gaussian Curvature Map