Non-Rigid Registration Using Kernel Correlation
4.2. REGISTRATION ERRORS AND TIME STATISTICS
where Erotate and Eregular are the same as Equation 3.5 and 3.7, and the surface fitting function is
Ef it = X
(s,t(s))∈C
αpointkA(s) − t(s)k2+ αplane|nTt(s)(A(s) − t(s))|2, (4.2)
where s is a point on the source surface and A(s) is the transformed position of s, and t(s) is the closest point of s on the target surface with the corre-sponding surface normal nt(s), and C is the set of mappings that maps one source point to its closest point and in which some of them may have been removed to filter out bad correspondences. The weights used in the test are wf it = 1, wrotate= 106, wregular = 10, αpoint = 0.1, and αplane = 1.
For comparison, the registration error of two surfaces is measured by cal-culating the distance between the shapes. The Hausdorff distance can be used to calculate the distance between two shapes. But calculating the maximum of minimum distances is bad for measuring the error between two shapes if there exists a large minimum distance between the shapes. We use an error measurement that averages the minimum distances between two shapes as follows. Given the registered surface template S and the corresponding range image T , we associate each s ∈ S with a distance ds= ks−t(s)k where t(s) is the closest point of s in T . The registration error is measured by calculating the root-mean-square error (RMSE) on the template,
RM SE(S) = sP
s∈Sd2s
|S| , (4.3)
where |S| is the number of points in S.
Figure 4.10 shows the registration errors of the face data set. Colors on the template represent the distance ds of each point s and a red color means a larger distance whereas a blue color means a smaller distance. The colors
hole regions in the range image there result in large distances ds. Another similar result is shown in Figure 4.11 for the hand data set.
Registration errors of the kernel correlation method
Registration errors of the non-rigid ICP method
Figure 4.10: Registration errors of the face data set.
Registration errors of the kernel correlation method
Registration errors of the non-rigid ICP method
Figure 4.11: Registration errors of the hand data set.
The corresponding error statistics of the two data sets are shown in Table 4.1. The numbers in Table 4.1 show the average RMSE of 90 frames in the face data set and 100 frames in the hand data set using two methods. In
4.2. REGISTRATION ERRORS AND TIME STATISTICS
summary, the average RMSE between the proposed method and non-rigid ICP are quite close, but ours are smaller. It makes sense that our RMSE is smaller since our surface fitting function fits a point on the source surface to a reasonable 3D position rather than another point on the target surface.
Face Hand
Avg. RMSE per Frame (KC) 0.78597 0.383834 Avg. RMSE per Frame (N-ICP) 0.823858 0.480465 Table 4.1: The average RMSE of the face and hand data set.
The time statistics for the two data sets are listed in Table 4.2. The com-putations are performed on a 2.3GHz Intel CoreR TM i5-2410M processor with 4GB RAM. In general, our registration algorithm is more efficient com-pared to the non-rigid ICP since the minimization process does not need to recompute the resources from iteration to iteration in the non-linear system and is solved straightforwardly. The gap of computation time for the two methods are smaller when the number of graph nodes is small, and becomes larger when the number of graph nodes grows; as listed in Table 4.2.
Face Hand
] of Frames 90 100
Avg. ] of Points per Frame ∼56k ∼37k
] of Template Vertices 8032 1441
] of Graph Nodes 802 142
Avg. Registration Time per Frame (KC) 9m:23s 7s Avg. Registration Time per Frame (N-ICP) 1h:0m:4s 20s
Table 4.2: The time statistics for the face and hand data sets.
Conclusions
We give a summary for the proposed non-rigid shape registration algorithm in this chapter and also discuss the limitations and some future works.
5.1 Summary
For non-rigid registration problem, many methods based on non-rigid ICP have been proposed to formulate the surface fitting function. The non-rigid ICP needs to find the set of closest points at each iteration of the non-linear energy optimization. Instead, in the proposed kernel-correlation based sur-face fitting function is fixed during the entire optimization process. The en-ergy optimization can be solved straightforwardly and more efficiently. The non-rigid registration algorithm we have presented relies only on the knowl-edge of point samples in 3D space and does not use any color images for additional information between adjacent frames. In summary, we have pre-sented a different way of formulating the surface fitting function for non-rigid shape registration, with better result and better computational efficiency, compared to an ICP-based method [LAGP09].
5.2. LIMITATIONS
5.2 Limitations
The input range images are assumed to have high spatial and temporal co-herence between adjacent frames. If there exists two adjacent frames with large motion gap, then the process will result in incorrect registration since it can be trapped into the local minimum of the non-linear energy system.
Figure 5.1 shows an example in which the motion gap between two adjacent frames is large, especially in the mouse region. The registration result for the second frame is bad.
(a) (b) (c)
Figure 5.1: Image (a) and (b) are adjacent frames. (c) is the registration result of the latter frame.
To handle this problem, we provide a good guess for the initial point of the non-linear system. We manually select a set of graph nodes and adjust initial translation vector t for each selected graph node. Figure 5.2(a) shows the adjusted result, which is better than before.
(a) (b) (c)
Figure 5.2: (a) The surface template before doing registration. (b) A set of graph nodes are selected and each node is given a motion vector as an initial guess for the non-linear system. (c) The adjusted registration result.