Space-time Radial Basis Function

If we have any two joints Jointi and Jointj whose coordinates are (xi, yi, zi, ti) and (xj, yj, zj, t_j) in hyper-space. Typically, the distance term should be defined as:

( ) (

) (

) (

)

However, we reorganized the joint sequence due to the processing mentioned previously.

In our thesis, we redefine the basis with distance as:

⎟⎟

s jo index jo index

r = − (10)

( )

2 _ _i _ _j

t frame index frame index

r = − (11)

where ct and cs are constants which control the shape of basis.

6.3 Approximation

Different composition of centers may have great influence on the compression or approximation results. Since our goal is to approximate the motion variations by radial basis function, we utilize an iterative greedy-algorithm propose by Carr [8]. For each cluster, we choose initial centers to train the radial basis functions network at beginning. If current cluster forms a surface, we choose the corners of surface as our initial centers. If current cluster is a curve, we choose the initial centers at first and last frame. Then we use this function to reconstruct the animation segment. Samples with larger residuals will be chosen as new centers and we re-train the radial basis functions. The iterative step will continue until stop criterions are satisfied. (i.e. approximation error smaller than the predefined threshold.)

Figure 14 is the iteration flowchart which illustrates how we use radial basis functions to approximate motion data gradually. Figure 15 is an example of approximating a curve. The green curve is the real data. The red curve represents the approximated curve. And blue dots mean centers.

Setup initial centers

Train RBFs

Evaluate error & Pick up new centers

Criterion Satisfied?

Finish Yes No

Figure 14: An illustration of the approximation steps.

Figure 15: An example of approximating a curve. Green curve and red curve are real data and approximated function respectively. Blue dots are centers of radial basis functions.

Figure 16: An example of real data surface (green one) and approximated surface (red one).

Experiment and Result

In order to prove our thesis utilizing spatial and temporal coherence, we prepared 16 testing data and design several experiments. The original file format is the commonly used BVH. However, as we mentioned previously, we converted the motion data from hierarchical angular domain to Euclidean system before starting compression. We stored such converted data in “BIN” file format and the file extension is “bin”. The BIN file contains the global positions of joints frame by frame in binary. The compressed motion data were encoded in

“R” file format. Therefore, our compression ratio is defined as:

Compression_ratio = ( size of BIN file )/( size of R file) (12) Since our thesis is a lossy compression method. We defined the error threshold is 5 cm if the height of the subject is 1.8 m. Such error threshold is sensible hardly for human eyes.

The first experiment compressed these testing data directly. It is worth noting that different motion data have different coherence that we can utilize. Therefore we have to adjust the basis shape (i.e. Cs & Ct) to produce better results. Generally speaking, when the target surfaces or curves are smoother, C_t and C_s should be larger such that far fewer centers are needed.

Motion Cs Ct .bin .r Ratio

Ballet05 64 256 200,687 5,187 1:38.7

Ballet23 4 256 117,059 3,869 1:30.3

Cowboy3 16 32 191,027 10,127 1:18.9

Cowboy4 16 32 165,083 12,246 1:13.4

Drunk5 16 64 269,135 26,962 1:9.9

Faint5 4 32 95,255 9,571 1:9.9

ShotShoulder03 4 32 52,475 6,146 1:8.5

Sit21 4 256 72,899 5,540 1:13.2

Sneak01 32 128 148,523 8,359 1:17.8

Stand03 32 128 62,963 5,390 1:11.7

Tired05 32 128 203,999 16,622 1:12.3

Walk25* 32 64 248,711 12,774 1:19.5

Walk34* 32 64 241,811 10,987 1:22

Second, we compared two compression approaches: a space-time method and a traditional approximation method in temporal domain only. In this experiment, the temporal constant must be same value between these two methods.

Table 1: Compression Results

Motion .bin s-t ratio

The third experiment shows how the segment length affects our compression results.

Since IPCA will produce uncontrollable segment length, we propose use several fixed length segment to test.

Table 2: The Comparison time domain only and space-time approach.

Motion Frame Seg. 100 Seg. 200 Seg. 300

Obviously, the compression ratios are better in the longer segments. This is because longer segment may propose more coherence in temporal domain. However, if segments were too long, we may find fewer joints with similar trajectories. In other words, we miss mush spatial coherence. Therefore, the length of each segment is an important tradeoff we have to concern.

The final experiment discussed the performance of our approach. The platform is P4 Table 3: A discussion of segment length.

motion

Obviously, the compression time is much longer than decompression time. This is because iteration process. Solving the coefficients of radial basis functions is inverting a large matrix essentially. Fortunately, decompression speed is more important than compression ones in practical. Therefore this thesis is still practicable and feasible.

Table 4: Algorithm performance

8. Conclusion

In this thesis, we propose a compression method for human motion data. Unlike previous studies, our method utilizes radial basis functions to approximate motion in both spatial and temporal domain simultaneously. In order to find out more coherence, we analyze motion data in temporal and spatial domain and reorder the joint sequence such that we have a smooth curve or surface.

However, if we can not find out sufficient spatial coherence in motion data, the compression ratio will be close temporal-domain compression only. This is because most clusters form curves. Besides, solving the coefficients of radial basis functions is inversing a matrix essentially. This means the matrix size can not be too large. Although we may find a cluster with much useful coherence, we still need to care about the amount of available samples.

9. Future Work

Important future research directions are listed as follows: First, since our method is a flexible compression component. It can be combined with other compression component to achieve better results. For example, there may be repetitive motion behaviors in some motion data. We can retrieve such motion and represent it more efficiency.

Second, in the typical motion data, the length of bone is fixed and this data is usually known in advance. Thus, we can use it as an additional constraint. In details, we can approximate motion data roughly. During decompression, we may utilize this constraint to enhance the reconstructed joint position more correctly.

Although we proposed a efficient framework to exploit coherence, more sophisticated methods, e.q. PPCA segmentation may improve our analysis.

Hierarchical Radial Basis Functions

Automatically deciding the width of each center is very difficult. Different human motions are quite different in behavior. Therefore, approximating surfaces with same ct and cs can not have a good compression result. Future researches may adopt the hierarchical radial basis functions to overcome this problem. For example, we use wider basis to capture the rough shapes of surfaces and use thicker basis to capture the high-variation part. In other words, we may approximate a surface in several levels. Each level represents the different details or frequencies.

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