It is easy to note that R is in the range [−1, 1] with high values when the spin images are similar and low values when they are not similar. However, there is a problem when we compare two spin images with this measure. Due to occlusions and cluttered scenes, many times a spin image contains more information than others so, to limit this effect, it is necessary to take only those ‘pixels’ where data exists. Since the cross-correlation depends of the number of pixels to compute it, the amount of overlap will have an effect on the comparison. Obviously, the confidence in the comparison is better when more pixels are used. In addition, the confidence in the match can be measured by the variance associated with the cross-correlation, so by combining both the cross-correlation R and its variance, we get a new similarity measure C, defined as:

where N is the number of overlapping pixels (pixels different from zero), λ is a constant and R is calculated using the N overlapping pixels. Note that the hyperbolic arctangent function transforms the correlation coefficient, R, into a distribution that has better statistical properties and the variance of this distribution is N1−3 . The measure, C, has a high value when the spin images are highly correlated and a large number of pixels overlap. In experiments, λ was configured to 3.

7.3.2.1 Matching

Given a scene, a random set of points is selected for matching. For each point, a set of correspondences is established using spin images from the scene and those calculated from models. Given a point from the scene, we calculate its spin image as previously described, thus this is compared with all the spin images in the huge collection using Eq. (7.8). From the comparison with the stored spin images, a histogram is built quantizing the similarity measure. This histogram maintains the information about occurrences of similarity measures when a comparison is being performed and it can be seen as a distribution of similarities between the input spin image and the stored ones. As we are interested in high similarity values, these can be found as outliers in the histogram.

In practice, outliers are found by automatically evaluating the histogram. A standard way to localize outliers is to determine the fourth spread of the histogram defined as the difference between the median of the largest N/2 measurements and the median of the smallest N/2 measurements. Let fs be the fourth spread, extreme outliers are 3fs units above the median of the largest N/2 measurements. Note that

with this method, the number of outliers can be greater than or equal to zero, so many correspondences per point can be found.

Once we have the set of correspondences for each point in the scene, we need to organize them in order to recognize the correct model object in the scene. As a large number of correspondences have been detected, it is necessary to filter them. Firstly, correspondences with a similarity measure of less than half of the maximum similarity are discarded. Secondly, by using geometric consistency, it is possible to eliminate bad correspondences. Given two correspondences C1 = (s1, m1) and C2 = (s2, m2), the geometric consistency is defined as follows

where SO(p) denotes the spin map function of point p using the local basis of point O, as defined in Eq. (7.2).

This geometric consistency measures the consistency in position and normals. Dgc is small when C1 and C2 are geometrically consistent. By using geometric consistency, correspondences which are not geometrically consistent with at least a quarter of the complete list of correspondences are eliminated. The final set of correspondences has a high probability of being correct, but it is still necessary to group and verify them.

Now, we group correspondences in order to calculate a good transformation and hence do the matching. A grouping measure is used which prioritizes correspon-

The same measure can also be defined between a correspondence C and a group

Therefore, given a set of possible correspondences L = {C1, C2, . . . , Cn}, the following algorithm has to be used for generating groups:

•For each correspondence Ci L, initialize a group Gi = {Ci }

–Find a correspondence Cj L − Gi , such that Wgc(Cj , Gi ) is minimum. If Wgc(Cj , Gi ) < Tgc then update Gi = Gi {Cj }. Tgc is set between zero and one. If Tgc is small, only geometrically consistent correspondences remains. A commonly used value is 0.25.

–Continue until no more correspondences can be added.

As a result, we have n groups, which are used as starting point for final matching. For each group of correspondences {(mi , si )}, a rigid transformation T is calculated by minimizing the following error using the least squares method:

where T (mi ) = R(mi ) + t, R and t are the rotation matrix and the translation vector, representing the rotation and position of the viewpoint si in the coordinate system of mi .

As a last step, each transformation needs to be verified in order to be validated as a matching. The model points in the correspondences are transformed using T . For each point in the scene and the correspondences set, we extend correspondences for neighboring points in both points of a correspondence under a distance constraint. A threshold distance equal to twice the mesh resolution is used. If the final set of correspondences is greater than a quarter or a third of the number of vertices of the model, the transformation is considered valid and the matching is accepted. Finally, with the final correspondences set, the transformation is refined by using an iterative closest point algorithm.

7.3.2.2 Evaluation

Unfortunately, in the original work by Johnson [56], the models used in experiments were obtained using a scanner and they are not available to date. Johnson and Hebert [57] presented results trying to measure the robustness of their method against clutter and occlusion. They built 100 scenes involving four shapes, using a range scanner. The experiments were based on querying an object in a scene and determining if the object was present or not. In addition, the levels of occlusion and clutter were also determined.

The four shapes were used in each scene, so the number of runs was 400. Interestingly, there were no errors at levels of occlusion under 70 % and the rate of recognition was above 90 % at 80 % of occlusion. In addition, the recognition rate was greater than 80 % at levels of clutter under 60 %.

On the other hand, spin images have recently been evaluated in the Robust Feature Detection and Description Benchmark [21] (SHREC 2010 track). In this track, the goal was to evaluate the robustness of the descriptors against mesh transformations such as isometry, topology, holes, micro holes, scale, local scale, sampling, noise, and shot noise. The dataset consisted of three shapes taken from the TOSCA dataset. Subsequently, several transformations, at different levels, were applied to each shape. The resulting dataset contained 138 shapes. In addition, a set of correspondences was available in order to measure the distance between descriptors over corresponding points.

The evaluation was performed using the normalized Euclidean distance, Q(X, Y ), between the descriptors of corresponding points of two shapes X and Y ,

where (xj , yk ) are corresponding points, f (·) and g(·) are the descriptors of a point, and F (X) is the set of vertices to be considered. Here, we present the results obtained using F (X) = X.

The best results were obtained for isometry and topology transformations with 0.10 and 0.11 average distance respectively. This is because spin images were extracted locally, and these transformations do not modify the local structure of the mesh. On the other hand, the noise and shot noise transformations got higher distances (up to 0.20 and 0.18, respectively). It is clear that stronger levels of noise modify considerably the distribution of points on the surface, so spin images are not constructed robustly. See Tables 7.4 and 7.5 for the complete results. Table 7.5 shows the performance for dense heat kernel signatures calculated on 3D meshes. Clearly, regarding robustness, spin images show some drawbacks. However, an important aspect of this approach is its support to occlusion. In that sense, its application in recognition has been proved.

7.3.2.3 Complexity Analysis

Let S be a 3D object with n vertices. In addition, let W be the number of rows and columns of a resulting spin image (we assume square spin images for the analysis). The complexity of each stage is given as follows: