A number of algorithms have been implemented for use in the SCULPTEUR project (Ad- dis et al., 2005b) and are used within the work presented here. This section gives some more details of the algorithms implemented although Chapter 2 gives a more compre- hensive overview. The majority of 3-D objects used in this work do not have any colour information associated with them, so only shape based descriptors have been used. Four- teen algorithms have been implemented. These are the Area Volume Ratio (Tung and Schmitt, 2004), the Cord Histograms (six versions) (Paquet and Rioux, 1999b), Extended Gaussian Image (EGI) (Horn, 1984) and 3-D Hough Transform (Zaharia and Prêteux, 2001b) (both are implemented using spherical and octagon decomposition methods as described below), the D2 Shape Distribution (Osada et al., 2001) and the modied D2 Shape Distribution (Ohbuchi et al., 2003a) and nally the augmented Multi-resolution Reeb Graph (MRG) (Tung and Schmitt, 2004).
Chapter 5 3-D Content-Based Retrieval 75
Short Name Descriptor Name
Area Volume Area Volume
Cord Hist 1 Cord Histogram (Lengths)
Cord Hist 2 Cord Histogram (First Principal Axis) Cord Hist 3 Cord Histogram (Second Principal Axis) Cord Hist 4 Cord Histogram (Joint First/Second Principal Axes) Cord Hist 5 Cord Histogram (Combined First/Second Principal Axes)
Cord Histogram Combined Cord Histogram
EGI Oct Extended Gaussian Image (Oct Method)
EGI Sphere Extended Gaussian Image (Sphere Method)
Hough Oct 3-D Hough Transform (Oct Method)
Hough Sphere 3-D Hough Transform (Sphere Method)
MD2 Modied Shape D2
Shape D2 Shape Distributions (D2 variant)
MRG Augmented Multi-resolution Reeb Graph
Table 5.1: Short names of descriptors
There are several desirable characteristics for a 3-D descriptor. It should be invariant to changes in rotation and position. Scale invariance can also be desirable. However, in some cases the original scale may be desirable if the objects are all captured in the same manner, or encode the scale somehow. Scaling methods can also scale the model such that the largest axis is in the range [0.0:1.0] and the other axes are scaled by the same factor, or each axis can be separately scaled such that they are all in the range [0.0:1.0]. It is also desirable for the descriptor to ignore how the model is composed; that is it should work on the surface of the model and not directly on the polygons composing it. Table 5.1 lists the short names for the descriptors used in this work.
5.2.1 Area Volume Ratio
The Area to Volume ratio descriptor (Tung and Schmitt, 2004) calculates the ratio between the surface area and the volume of a 3-D object. See Section 2.6.1 in Chapter 2 for more details. The principal drawback of this method is that the triangles need to be orientated consistently for a correct volume calculation and a closed mesh is required. The area volume ratio descriptor is likely to perform badly against the PSB models as they are more likely to have inconsistently orientated triangles and have holes in the mesh. Both of these conditions will result in an incorrect volume calculation. Typically the museum objects will have consistently orientated triangles due to the acquisition process, however some objects may contain holes in the mesh. Therefore varying, but typically low performance is expected, however it is fast to compute which may be advantageous in some circumstances.
Chapter 5 3-D Content-Based Retrieval 76 5.2.2 Cord Histograms
The Cord Histograms were introduced by Paquet and Rioux (1999b) and are described in Section 2.6.2
The histograms are rotation and translation invariant but again normalisation for scale is required. Principal Components Analysis (PCA) is used as the normalisation step. This also adjusts the rst principal axis to the x-axis, and the second to the y-axis making
the angle based Cord histograms easier to calculate. Histograms with 16 bins have been used for the Cord Hist 1, 2 and 3 descriptors. The Cord Hist 4 descriptor has 32 (16 + 16) bins and the Cord Hist 5 descriptor has 256 (16 * 16) bins. The Combined Cord Histogram has a histogram size of 48 (16 + 16 + 16) bins.
5.2.3 Extended Gaussian Image
The Extended Gaussian Image (EGI) method is a way of indexing features. It is described in Section 2.6.14
Two methods are used to perform the indexing to the histogram. These are the Oct method and the Sphere method (Zaharia and Prêteux, 2001b, 2002). The Oct method subdivides an octahedron twice such that there are 128 faces. Each face has a surface normal and each quantised surface normal represents a bin for the histogram. The Sphere method uses spherical co-ordinates as the index into a bi-dimensional histogram. Each axis is quantised into ve sections. The problem with the sphere method is that each bounded region can be a dierent size, with larger regions at the equator and smaller regions near the poles. Misalignment during PCA can cause this to be a problem. The EGI Oct method has a histogram of 386 (128 * 3) bins and the EGI Sphere method has a histogram of 50 (5 * 5 * 2) bins.
5.2.4 Hough Transform
The 3-D Hough Transform developed by Zaharia and Prêteux (2002) takes its roots from the 2-D generalised Hough Transform (Ballard, 1981). See Section 2.6.15 for more details. Like with the EGI technique, the Oct and Sphere methods are employed in indexing surface normals in this implementation. The Hough Transform creates a table indexed by surface normal orientation (represented as spherical co-ordinates in the Sphere method or as a face index in the Oct method) and distance from centre of mass storing the surface area for each polygon in the mesh. Similarity matching is performed by comparing the tables treated as histograms. The true Hough Transform method creates an accumulator that gathers evidence for the existence of an object based on parameters
Chapter 5 3-D Content-Based Retrieval 77 calculated from a reference. Peaks in the accumulator space identify possible matches. However, this is quite slow compared to matching just the histograms calculated here. The Hough Sphere method has a histogram of 250 (5 * 5 * 10) bins and the Hough Oct method has a histogram of 1280 (10 * 128) bins.
5.2.5 Shape Distributions
The Shape Distributions (Osada et al., 2001) are a collection of descriptors that capture distributions of various features of the shape of an object. See Section 2.6.4. The work done by Osada et al. (2001) determined that the D2 variant performed best overall and hence this variant is used here. The Shape D2 descriptor captures the distribution of distances between random pairs of points on the shape surface. It is rotation and translation invariant and robust to changes in mesh resolution. However, it is not scale invariant and so requires some pre-processing. It is created using a large number of sample points (10242) recorded into 64 histogram bins as used in Shilane et al. (2004);
Osada et al. (2001).
5.2.6 Modied Shape Distributions
Based upon the Shape D2 algorithm, the modied Shape D2 (MD2) (Ohbuchi et al., 2003a) has several modications that aim to improve it and these are described in Sec- tion 2.6.5. The MD2 uses a 64 bin histogram and 1024 sample points.
5.2.7 Augmented Multi-resolution Reeb Graph
The augmented Multi-resolution Reeb Graph (MRG) (Tung and Schmitt, 2004) stores geometric attributes associated with nodes of the Reeb Graph (see Section 2.6.21). These geometric attributes are typically various 3-D descriptors applied to the section of mesh the node represents. In this implementation, the attributes are the value of mu (the function of the graph), surface area, volume, Cord Histograms (Cord Hist 1, 2 & 3), sur- face curvature (as in the 3-D Shape Spectrum Descriptor (Zaharia and Prêteux, 2001a)) and the 3-D Hough Transform.
Unlike the other methods, simple histogram matching will not suce. At the lower level, histogram matching between geometric attributes is still used, however similarity between nodes is much more complicated and is based on graph matching.
Chapter 5 3-D Content-Based Retrieval 78
Short Name Distance Metric
City-block City-block Distance, L1 Norm
Euclidean Euclidean Distance, L2 Norm
Intersect Histogram Intersection
Chi χ2 Distance
Bhattacharyya Bhattacharyya Distance
Kullback Kullback-Leibler (symmetric) Distance Kullback-ns Kullback-Leibler (non-symmetric) Distance
Quadratic Quadratic Distance
Table 5.2: Short names of distance metrics