GRAPP 2007
MultiMode Tensor Representation of Motion Data
First presented at the International Conference on Computer Graphics Theory and Applications (GRAPP) 2007,
extended and revised for JVRB
urn:nbn:de:0009614197
Abstract
In this paper, we investigate how a multilinear model can be used to represent human motion data. Based on technical modes (referring to degrees of freedom and number of frames) and natural modes that typically appear in the context of a motion capture session (referring to actor, style, and repetition), the motion data is encoded in form of a highorder tensor. This tensor is then reduced by using Nmode singular value decomposition. Our experiments show that the reduced model approximates the original motion better then previously introduced PCAbased approaches. Furthermore, we discuss how the tensor representation may be used as a valuable tool for the synthesis of new motions.
Keywords: Motion Capture, Tensor Representation, NMode SVD, Motion Synthesis
Subjects: Motion Capturing
Motion capture or mocap systems allow for tracking and recording of human motions at high spatial and temporal resolutions. The resulting 3D mocap data is used for motion analysis in fields such as sports sciences, biomechanics, or computer vision, and in particular for motion synthesis in datadriven computer animation. In the last few years, various morphing and blending techniques have been suggested to modify prerecorded motion sequences in order to create new, naturally looking motions, see, e.g., [ GP00, Tro02, KGP02, SHP04, KG04, OBHK05, MZF06, CH07, SH07 ].
In view of motion reuse in synthesis applications, questions concerning data representation, data organization, and data reduction as well as contentbased motion analysis and retrieval have become important topics in computer animation. In this context, motion representations based on linear models as well as dimensionality reduction techniques via principal component analysis (PCA) have become wellestablished methods [ BSP04, CH05, FF05, LZWM05, SHP04, GBT04, Tro02, OBHK05 ]. Using these linear methods one neglects information of the motions sequences, such as the temporal order of the frames or information about different actors whose motions are included within the database.
In the context of facial animation, Vlasic et al. [ VBPP05 ] have successfully applied multilinear models of 3D face meshes that separably parameterizes semantic aspects such as identity, expression, and visemes. The strength of this technique is that additional information can be kept within a multilinear model. For example, classes of semantically related motions can be organized by means of certain modes that naturally correspond to semantic aspects referring to an actor′s identity or a particular motion style. Even though multilinear models are a suitable tool for incorporating such aspects into a unified framework, so far only little work has been done to employ these techniques for motion data [ Vas02, RCO05, MK06 ].
In this paper, we introduce a multilinear approach for modeling classes of human motion data. Encoding the motion data as a highorder tensor, we explicitly account for the various modes (e. g., actor, style, repetition) that typically appear in the context of a motion capture session. Using standard reduction techniques based on multimode singular value decomposition (SVD), we show that the reduced model approximates the original motion better then previously used PCAreduced models. Furthermore, we sketch some applications to motion synthesis to demonstrate the usefulness of the multilinear model in the motion context.
The idea of a tensor is to represent an entire class of semantically related motions within a unified framework. Before building up a tensor, one first has to establish temporal correspondence between the various motions while bringing them to the same length. This task can be accomplished by techniques based on dynamic time warping [ BW95, GP00, KG03, HPP05 ]. Most features used in this context are based on spatial or angular coordinates, which are sensitive to data variations that may occur within a motion class. Furthermore, local distance measures such as the 3D point cloud distance as suggested by Kovar and Gleicher [ KG03 ] are computationally expensive. In our approach, we suggest a multiscale warping procedure based on physicsbased motion parameters such as center of mass acceleration and angular momentum. These features have a natural interpretation, they are invariant under global transforms, and show a high degree or robustness to spatial motion variation. As a further advantage, physicsbased features are still semantically meaningful even on a coarse temporal resolution. This fact allows us to employ a very efficient multiscale algorithm for the warping step. Despite of these advantages, only few works have considered the physicsbased layer in the warping context, see [ MZF06, SH05 ].
The remainder of this paper is organized as follows. In Section 2, we introduce the tensorbased motion representation and summarize the data reduction procedure based on singular value decomposition (SVD). The multiscale approach to motion warping using physicsbased parameters is then described in Section 3. We have conducted experiments on systematically recorded motion capture data. As representative examples, we discuss three motion classes including walking, grabbing, and cartwheel motions, see Section 4. We conclude with Section 5, where we indicate future research directions. In particular, we discuss possible strategies for the automatic generation of suitable motion classes from a scattered set of motion data, which can then be used in our tensor representation.
Our tensor representation is based on multilinear algebra, which is a natural extension of linear algebra. A tensor Δ of order N ∈ and type (d_{1}, d_{2}, ... ,d_{N}) ∈ ^{N} over the real number is defined to be an element in ^{d1 x d2 x ... x dN } . The number d ≔ d_{1} ⋅ d_{2} ⋅ ... ⋅ d_{N} is referred to as the total dimension of Δ. Intuitively, the tensor Δ represents d real numbers in a multidimensional array based on N indices. These indices are also referred to as the modes of the tensor Δ. As an example, a vector v ∈ ^{d} is a tensor of order N = 1 having only one mode. Similarly, a matrix M ∈ ^{d1 x d 2 } is a tensor of order N = 2 having two modes which correspond to the columns and rows. A more detailed description of multilinear algebra is given in [ VBPP05 ].
In our context, we deal with 3D human motion data as recorded by motion capture systems. A (sampled) motionsequence can be modelled as a matrix M ∈ ^{n x f} , where the integer n ∈ refers to the degrees of freedom (DOFs) needed to represent a pose of an underlying skeleton (e. g. encoded by Euler angles or quaternions) and the integer f ∈ refers to the number of frames (poses) of the motion sequence. In other words, the ith colum of M, in the following also denoted by M(i), contains the DOFs of the ith pose, 1 ≤ i ≤ f. In the following examples, we will work either with an Euler angle representation of a human pose having n = 62 DOFs or with a quaternion representation having n = 119 DOFs (with n = 4∙m+3 where m = 29 is the number of quaternions representing the various joint orientations). In both representations 3 DOFs are used to describe the global 3D position of the root node of the skeleton.
We now describe how to construct a tensor from a given class of semantically related motion sequences. After a warping step, as will be explained in Section 3, all motion sequences are assumed to have the same number of frames. We will introduce two types of modes referred to as technical modes and natural modes. We consider two technical modes that correspond to the degrees of freedom and number of frames, respectively:

Frame Mode: This mode refers to the number of frames a motion sequence is composed of. The dimension of the Frame Mode is denoted by f.

DOF Mode: This mode refers to the degrees of freedom, which depends on the respective representation of the motion data. The dimension of the DOF Mode is denoted by n.
Sometimes the two technical modes are combined to form a single mode, which is referred to as data mode:
Additionally, we introduce natural modes that typically appear in the context of a motion capture session:

Actor Mode: This mode corresponds to the different actors performing the motion sequences. The dimension of the actor mode (number of actors) is denoted by a.

Style Mode: This mode corresponds to the different styles occurring in the considered motion class. The meaning of style differs for the various motion classes. The dimension of the style mode (number of styles) is denoted by s.

Repetition Mode: This mode corresponds to the different repetitions or interpretations, which are available for a specific actor and a specific style. The dimension of the repetition mode (number of repetitions) is denoted by r.
The natural modes correspond to semantically meaningful aspects that refer to the entire motion sequence. These aspects are often given by some rough textual description or instruction. The meaning of the modes may depend on the respective motion class. Furthermore, depending on the availability of motion data and suitable metadata, the various modes may be combined or even further subdivided. For example, the style mode may refer to emotional aspects (e. g., furious walking, cheerful walking), motion speed (e. g., fast walking, slow walking), motion direction (e. g., walking straight, walking to the left, walking to the right), or other stylistic aspects (e. g., limping, tiptoeing, marching). Further example will be discussed in Section 4. Finally, we note that in [ MK06 ] the authors focus on correlations with respect to joints and time only, which, in our terminology, refer to the technical modes. Furthermore, in [ Vas02 ], the authors discuss only a restricted scenario considering leg movements in walking motions.
In our experiments, we constructed several data tensors with different numbers of modes from the data base described in Section 4.1. The tensor with the smallest number of modes was created by using the three natural modes (Actors, Style, and Repetition) and the Data Mode. With this arrangement we obtain a tensor in the size of f ∙ n x a s x r. It is also possible to use the Frame and the DOF Mode, instead of the Data Mode, to arrange the same motions sequences within the tensor. The natural modes are not changed when using this strategy. Therefore a tensor of this type has a size of f x n x a s x r.
Similar to [ VBPP05 ], a data tensor Δ can be transformed by an Nmode singular value decomposition (Nmode SVD). Recall that Δ is an element in ^{d1 x d2 x ...x dN } . The result of the decomposition is a core tensor of the same type and associated orthonormal matrices U_{1}, U_{2},...,U_{N} . The matrices U_{k} are elements in ^{dk x dk } where k ∈ {1,2,...,N}. The tensor decomposition in our experiments was done by using the Nway Toolbox [ BA00 ]. Mathematically this decomposition can be expressed in the following way: Δ = x_{1} U_{1} x_{2} U_{2} x_{3}...x_{N}U_{N}. This product is defined recursively, where the modekmultiplication x_{k} with U_{k} replaces each modekvector v of x_{1}U_{1}x_{2}U_{2}x_{3}...x_{k1}U_{k1} for k > 1 (and for k = 1) by the vector U_{k}v.
One important property of is that the elements are sorted in a way, that the variance decreases from the first to the last element in each mode [ VBPP05 ]. A reduced model ' can be obtained by truncation of insignificant components of and of the matrices U_{k} , respectively. In the special case of a 2mode tensor this procedure is equivalent to principal component analysis (PCA) [ Vas02 ].
Once we have obtained the reduced model ' and its associated matrices U'_{k} , we are able to reconstruct an approximation of any original motion sequence. This is done by first modemultiplying the core tensor ' with all matrices U'_{k} , belonging to a technical mode. In a second step the resulting tensor is modemultiplied with one row of all matrices belonging to a natural mode. Furthermore, with this model at hand, we can generate an arbitrary interpolation of original motions by using linear combinations of rows of the matrices U'_{k} with respect to the natural modes.
During the last few years, several methods for motion alignment have been proposed which rely on some variant of dynamic time warping (DTW), see, e. g., [ BW95, GP00, KG03, MR06 ]. The alignment or warping result depends on many parameters including the motion features as well as the local cost measure used to compare the features. In this section, we sketch an efficient warping procedure using physicsbased motion features (Section 3.1) and applying an iterative multiscale DTW algorithm (Section 3.2).
In our approach, we use physicsbased motion features to compare different motion sequences. Physicsbased motion features are invariant under global transforms and show a high degree of robustness to spatial variations, which are often present in semantically related motions that belong to the same motion class. Furthermore, our features are still semantically meaningful even on a coarse temporal resolution, which allows us employing them in our multiscale DTW approach.
In our experiments, we used two different types of motion features: the center of mass (COM) acceleration and angular momenta for all skeletal segments.
The 3D position of the COM is calculated for all segments of the skeleton by using the anthropometric tables described in [ RW03 ]. From these positions and the mass of the segments one can calculate the COM position of the whole body by summing up the products of the 3D centers of mass of each segment and their corresponding mass and dividing this vector afterwards by the mass of the whole body. The second derivative of the resulting 3D positional data stream is the COM acceleration.
Our second feature, the angular momentum, is computed for each segment describing its rotational properties. More precisely the angular momentum how the segments rotation would continue if no external torque acts on it. It is calculated by the cross product between the vector from the point the segment rotates around to the segment′s COM and the vector expressing the linear momentum.
Physicsbased features provide a lot of information about the underlying motion sequence. For example, considering the COM acceleration it is easy to detect flight phases. More precisely, in case the body has no ground contact, the COM acceleration is equivalent to the acceleration of gravity:
Figure 1. COM acceleration for a dancing motion containing three different jumps. The acceleration is spliced into its x (dotted), y (solid) and z (dashed) component, where the y component refers to the vertical direction. Note that the y component reveals two long flight phases (frames 190 to 220 and frames 320 to 350, respectively) and one short flight phase (around frame 275).
This situation is illustrated by Figure 1, which shows the COM acceleration for a dancing motion. Note that there are three flight phases, which are revealed by the vertical component (yaxis) of the COM acceleration. Further examples are discussed in Section 3.3.
Dynamic time warping (DTW) is a wellknown technique to find an optimal alignment (encoded by a socalled warping path) between two given sequences. Based on the alignment, the sequences can be warped in a nonlinear fashion to match each other. In our context, each motion sequence is converted into a sequence of physicsbased motion features at a temporal resolution of 120 Hz. We denote by V ≔ (v_{1},v_{2},...,v_{n}) and W ≔ (w_{1},w_{2},...,w_{m}) the feature sequences of the two motions to be aligned. Since one of the motions might be slower than the other, n and m do not have to be equal.
In a second step, one computes an n x m cost matrix C with respect to some local cost measure c, which is used to compare two feature vectors. In our case, we use a simple cost measure, which is based on the inner product:
for two nonzero feature vectors v and w (otherwise c(v,w) is set to zero). Note that c(v,w) is zero in case v and w coincide and assumes values in the real interval [0,1] ⊂ . Then, the cost matrix C with respect to the sequences V and W is defined by
C(i,j) ≔ c(v_{i},w_{j}
for 1 ≤ i ≤ n and 1 ≤ j ≤ m. Figure 2 shows such cost matrices with respect to different features.
Finally, an optimal alignment is determined from the cost matrix C via dynamic programming. Such an alignment is represented by a socalled (costminimizing) warping path, which, under certain constraints, optimally allocates the frame indices of the first motion with the frame indices of the second motion. In Figure 2, such optimal warping paths are indicated in red. Note that the information given by an optimal warping path can be used to timewarp the second motion (by suitably omitting or replicating frames) to match the first motion. Further details and references on DTW may be found in [ ZM06 ].
Note that the time and memory complexity of the DTW algorithm is quadratic in the number of frames of the motions to be aligned. To speed up the process, we employ an iterative multiscale DTW algorithm as described in [ ZM06 ]. Here, the idea is to proceed iteratively using multiple resolution levels going from coarse to fine. In each step, the warping path computed at a coarse resolution level is projected to the next higher level, where the projected path is refined to yield a warping path at the higher level. To obtain features at the coarse levels, we use simple windowing and averaging procedures. In this context, the physicsbased features have turned out to yield semantically meaningful features even at a low temporal resolution. In our implementation, we used six different resolution levels starting with a feature resolution of 4 Hz at the lowest level. The overall speedup of this approach (in comparison to classical DTW) depends on the length of the motion sequences. For example, the speedup amounts to a factor of roughly 10 for motions having 300 frames and a factor of roughly 100 for motions having 3000 frames.
Figure 2 shows two cost matrices, where we compared two walking motions both consisting of 6 steps forward. In dark areas the compared poses are similar with respect to the given features, whereas in lighter areas the poses are dissimilar. The red line is the optimal warping path found by the DTW algorithm. The cost matrix on the left side is based only on the COM acceleration of the entire body. Using this single feature, the checkerboardlike pattern indicates that one cannot differentiate between steps that were done with the left or the right foot. Adding the features that measures the angular momenta of the feet, the result obviously improves. The resulting cost matrix is shown on the right hand side of Figure 2. The five dark diagonals indicate that in this case only the steps made with the same foot are regarded as similar.
Figure 2. DTW cost matrices calculated on the whole body COM acceleration (left) as well as on the basis of the COM acceleration and the angular momenta of the hands and feet (right). The costminimizing warping paths are drawn red.
Depending on the motions to be timewarped, one can select specific features. For walking motions, the movement of the legs contains the most important information in case the steps are to be synchronized.
For timewarping grabbing motions as used in our experiments, aspects concerning the right hand were most important as the motions were performed by this hand. For our cartwheel motions, good correspondences were achieved when using features that concern the two hands and the two feet. For an example of warped walking motions, we refer to the accompanying video.
For our experiments, we systematically recorded several hours of motion capture data containing a number of wellspecified motion sequences, which were executed several times and performed by five different actors. The five actors all have been healthy young adult male persons. Using this data, we built up a database consisting of roughly 210 minutes of motion data. Then we manually cut out suitable motion clips and arranged them into 64 different classes and styles. Each such motion class contains 10 to 50 different realizations of the same type of motion, covering a broad spectrum of semantically meaningful variations. The resulting motion class database contains 1,457 motion clips of a total length corresponding to roughly 50 minutes of motion data [ MRC07 ]. For our experiments, we considered three motion classes. The first class contains walking motions executed in the following styles:

Walk four steps in a straight line.

Walk four steps in a half circle to the left side.

Walk four steps in a half circle to the right side.

Walk four steps on the place.
All motions within each of these styles had to start with the right foot and were aligned over time to the length of the first motion of actor one.
The second class of motions we considered in our experiments consists of various grabbing motions, where the actor had to pick an object with the right hand from a storage rack. In this example the style mode corresponds to three different heights(low, middle, and high) the object was located in the rack.
The third motion class consist of various cartwheels. Cartwheel motions were just available for four different actors and for one style. All cartwheels within the class start with the left foot and the left hand.
For all motion classes described in the previous section, we constructed data tensors with motion representations based on Euler angles and based on quaternions. Initially some preprocessing was required, consisting mainly of the following steps. All motions were

filtered in the quaternion domain with a smoothing filter described as in [ LS02 ],

normalized by moving the root nodes to the origin and by orienting the root nodes to the same direction,
In this section, we discuss various truncation experiments for our three representative example motion classes. In these experiments, we systematically truncated a growing number of components of the coretensors, then reconstructed the motions, and compared them with the original motions.
Based on the walking motions (using quaternions to represent the orientations), we constructed two data tensors. The first tensor was constructed by using the Data Mode as technical mode. This is indicated by the upper index, which shows the dimension of the tensor. The motions were timewarped and sampled down to a length of 60 frames. The resulting size of is 7140 x 5 x 4 x 3. Using the Frame Mode and DOF Mode, we obtained a second tensor of size 60 x 119 x 5 4 x 3. Table 1 shows the results of our truncation experiments. The first column shows the size of the core tensors '_{Walk} after truncation, where the truncated modes are colored red. The second column shows the number of entries of the core tensors, and the third one shows its size in percent compared to Δ_{Walk} . In the fourth column, the total usage of memory is shown. Note that the total memory requirements may be higher than for the original data, since besides the core tensor ' one also has to store the matrices U'_{k} . The memory requirements are particulary high in case one mode has a high dimension. The last two columns give the results of the reconstruction. E_{total} is an error measurement which is defined as the sum over the reconstruction error E_{mot} over all motions:
The reconstruction error E_{mot} of a motion is defined as normalized sum over all frames and over all joints:
where f denotes the number of frames and m the number of quaternions. Here, for each joint, the original and reconstructed quaternions q^{org} and q^{rec} are compared by means of their included angle. We performed a visual rating for some of the reconstructed motions in order to obtain an idea of the quality of our error measurement. Here, a reconstructed motion was classified as good (or better) in case one could hardly differentiate it from the original motion when both of the motions were put on top of each other. The results of our ratings are given in the last column.
Table 1. Results for truncating technical and natural modes from our tensors filled with walking motions (using quaternions).
Dimension core tensor 
Entries core tensor 
Size core tensor in percent 
Memory usage in percent 
E_{total} 
visual rating 
Truncation of Data Mode of 

7140 x 5 x 4 x 3 
428 400 
100 % 
12 000 % 
0.0000 
excellent 
60 x 5 x 4 x 3 
3 600 
0.8403 % 
100.8520 % 
0.0000 
excellent 
53 x 5 x 4 x 3 
3 180 
0.7423 % 
89.0873 % 
0.0000 
excellent 
52 x 5 x 4 x 3 
3 120 
0.7283 % 
87.4066 % 
0.0634 
excellent 
50 x 5 x 4 x 3 
3 000 
0.7003 % 
84.0453 % 
0.2538 
very good 
40 x 5 x 4 x 3 
2 400 
0.5602 % 
67.2386 % 
1.1998 
very good 
30 x 5 x 4 x 3 
1 800 
0.4202 % 
50.4318 % 
2.1221 
very good 
20 x 5 x 4 x 3 
1 200 
0.2801 % 
33.6251 % 
3.6258 
good 
10 x 5 x 4 x 3 
600 
0.1401 % 
16.8184 % 
6.3961 
good 
5 x 5 x 4 x 3 
300 
0.0700 % 
8.4150 % 
9.3932 
satisfying 
4 x 5 x 4 x 3 
240 
0.0560 % 
6.7344 % 
10.4260 
satisfying 
3 x 5 x 4 x 3 
180 
0.0420 % 
5.0537 % 
10.9443 
sufficient 
2 x 5 x 4 x 3 
120 
0.0280 % 
3.3730 % 
11.5397 
poor 
1 x 5 x 4 x 3 
60 
0.0140 % 
1.6923 % 
11.8353 
poor 
Truncation of Actor Mode of 

60 x 4 x 4 x 3 
2 880 
0.6723 % 
100.6828 % 
4.3863 
satisfying 
60 x 3 x 4 x 3 
2 160 
0.5042 % 
100.5135 % 
6.4469 
satisfying 
60 x 2 x 4 x 3 
1 440 
0.3361 % 
100.3443 % 
8.2369 
satisfying 
60 x 1 x 4 x 3 
720 
0.1681 % 
100.1751 % 
10.7773 
sufficient 
Truncation of Style Mode of 

60 x 5 x 3 x 3 
2 700 
0.6303 % 
100.6410 % 
3.5868 
good 
60 x 5 x 2 x 3 
1 800 
0.4202 % 
100.4300 % 
5.8414 
sufficient 
60 x 5 x 1 x 3 
900 
0.2101 % 
100.2190 % 
8.5770 
poor 
Truncation of Repetition Mode of 

60 x 5 x 4 x 2 
285 600 
66,667 % 
100.5712 % 
2.7639 
good 
60 x 5 x 4 x 1 
142 800 
33,333 % 
100.2904 % 
5.0000 
good 
Truncation of Frame and DOF Mode of 

26 x 91 x 5 x 4 x 3 
141960 
20.9288 % 
22.8968 % 
0.5492 
very good 
21 x 91 x 5 x 4 x 3 
114660 
16.9040 % 
18.8020 % 
0.7418 
very good 
21 x 46 x 5 x 4 x 3 
57960 
8.5449 % 
9.6534 % 
0.9771 
very good 
15 x 34 x 5 x 4 x 3 
30 600 
4.5113 % 
5.3252 % 
1.9478 
good 
14 x 35 x 5 x 4 x 3 
29 400 
4.3344 % 
5.1519 % 
1.9817 
good 
13 x 37 x 5 x 4 x 3 
28 860 
4.2548 % 
5.0933 % 
1.9724 
good 
12 x 39 x 5 x 4 x 3 
28 800 
4.1398 % 
4.9994 % 
1.9907 
good 
If the Data Mode is split up into the Frame and DOF Mode, as in , one can truncate the two modes separately. The results are shown in the lower part of Table 1 and Figure 3.
For example, reducing the DOF Mode from 60 to 26, the error E_{total} is still less than one degree. A similar result is reached by reducing the Frame Mode down to a size of 20. This shows that there is a high redundancy in the data with respect to the technical modes.
We also conducted experiments, where we reduced the dimensions of the natural modes. As the experiments suggest, the dimensions of the natural modes seem to be more important than the ones of the technical modes. The smallest errors (when truncating natural modes) resulted by truncating the Repetition Mode. This is not surprising since the actors were asked to perform the same motion several times in the same fashion. Note that different interpretations by one and the same actor reveal a smaller variance than motions performed by different actors or motions performed in different styles. Some results of our experiments are illustrated by Figure 4.
Figure 4. Error E_{total} , of reconstructed motions where two natural Modes were truncated. Actor and Style Mode are truncated (left). Actor and Repetition Mode are truncated (middle). Style and Repetition Mode are truncated (right).
The displacement grows with the size of truncated values from Style and Personal Mode.
For building the data tensors and , all motions were warped to the length of one reference motion. has a size of 8449 x 5 x 3 x 3, while has a size of 71 x 119 x 5 x 3 x 3. The exact values for truncating the Data Mode and the DOF Mode can be found in Table 2.
Table 2. Truncation results for grabbing motions (using quaternions).
Dimension core tensor 
Entries core tensor 
Size core tensor in percent 
Memory usage 
E_{total} 
visual rating in percent 
Truncation of Data Mode of


8449 x 5 x 3 x 3 
380 205 
100 % 
18 775 % 
0.0000 
excellent 
60 x 5 x 3 x 3 
2 700 
0.7101 % 
134.0548 % 
0.0000 
excellent 
55 x 5 x 3 x 3 
2 475 
0.6510 % 
122.8845 % 
0.0000 
excellent 
50 x 5 x 3 x 3 
2 250 
0.5918 % 
111.7142 % 
0.0000 
excellent 
45 x 5 x 3 x 3 
2 025 
0.5326 % 
100.5439 % 
0.0000 
excellent 
40 x 5 x 3 x 3 
1 800 
0.4734 % 
89.3736 % 
1.2632 
very good 
35 x 5 x 3 x 3 
1 575 
0.4143 % 
78.2033 % 
2.1265 
very good 
30 x 5 x 3 x 3 
1 350 
0.3551 % 
67.0330 % 
2.9843 
very good 
25 x 5 x 3 x 3 
1 125 
0.2959 % 
55.8628 % 
3.9548 
good 
20 x 5 x 3 x 3 
900 
0.2367 % 
44.6925 % 
5.1628 
good 
15 x 5 x 3 x 3 
675 
0.1775 % 
33.5222 % 
6.6799 
satisfying 
10 x 5 x 3 x 3 
450 
0.1184 % 
22.3519 % 
8.8702 
sufficient 
5 x 5 x 3 x 3 
225 
0.0592 % 
11.1816 % 
11.5604 
sufficient 
4 x 5 x 3 x 3 
180 
0.0473 % 
8.9475 % 
12.4463 
poor 
3 x 5 x 3 x 3 
135 
0.0355 % 
6.7135 % 
12.7304 
poor 
2 x 5 x 3 x 3 
90 
0.0237 % 
4.4794 % 
13.4234 
poor 
1 x 5 x 3 x 3 
45 
0.0118 % 
2.2454 % 
13.7150 
poor 
Truncation of DOF Mode of 

71 x 91 x 5 x 3 x 3 
290 745 
76.4706 % 
80.6560 % 
0.0000 
excellent 
71 x 86 x 5 x 3 x 3 
274 770 
72.2689 % 
76.2978 % 
0.0001 
excellent 
71 x 61 x 5 x 3 x 3 
194 895 
51.2605 % 
54.5069 % 
0.1311 
excellent 
71 x 51 x 5 x 3 x 3 
162 945 
42.8571 % 
45.7906 % 
0.4332 
good 
71 x 41 x 5 x 3 x 3 
130 995 
34.4538 % 
37.0742 % 
1.0450 
satisfying 
71 x 31 x 5 x 3 x 3 
99 045 
26.0504 % 
28.3579 % 
2.2182 
sufficient 
71 x 21 x 5 x 3 x 3 
67 095 
17.6471 % 
19.6415 % 
3.9491 
sufficient 
71 x 11 x 5 x 3 x 3 
35 145 
9.2437 % 
10.9252 % 
7.1531 
poor 
71 x 6 x 5 x 3 x 3 
19 170 
5.0420 % 
6.5670 % 
10.1546 
poor 
71 x 1 x 5 x 3 x 3 
3 195 
0.8403 % 
2.2088 % 
14.3765 
poor 
In our third example, we consider a motion class consisting of cartwheel motions. The core tensor has a size of 7497 x 4 x 1 x 3. Here, all motions could be reconstructed without any visible error for a size of no more than 12 dimensions for the Data Mode. Further results are shown in Table 3.
Table 3. Truncation results for cartwheel motions (using quaternions).
Dimension core tensor 
Entries core tensor 
Size core tensor in percent 
Memory usage 
E_{total} 
visual rating in percent 
Truncation of Data Mode of


30 x 4 x 3 
360 
0.4002 % 
250.4279 % 
0.0000 
excellent 
12 x 4 x 3 
144 
0.1601 % 
100.1879 % 
0.0000 
excellent 
11 x 4 x 3 
132 
0.1467 % 
91.8412 % 
1.6780 
good 
10 x 4 x 3 
120 
0.1334 % 
83.4945 % 
3.2163 
satisfying 
9 x 4 x 3 
108 
0.1200 % 
75.1478 % 
5.7641 
sufficient 
8 x 4 x 3 
96 
0.1067 % 
66.8012 % 
8.1549 
poor 
5 x 4 x 3 
60 
0.0667 % 
41.7611 % 
13.2847 
poor 
1 x 4 x 3 
12 
0.0133 % 
8.3745 % 
22.4095 
poor 
The number of necessary components of the data mode varied a lot in the different motion classes. One would expect that a cartwheel motion is more complex than a grabbing or walking motion. The results of previous experiments do not support this prospect. But the results of this truncation experiments are not comparable as they depend on all dimensions of the constructed tensors. To get comparable results for the three motion classes, we constructed a tensor including three motions from three actors for each motion class. The style mode is limited to a size of one, since we have no different styles for cartwheel motions. Therefor the resulting tenors have a size of f ⋅ n x 3 x 3 x 1. When truncating the data mode of these tensors, one gets the result that is shown in Figure 5. All motions are reconstructed perfectly until the size of the data mode gets smaller than 12. At this size the core tensor ' and the matrices U_{k} have as many entries as the original data tensor Δ. Then the error E_{total} grows different for the three motion classes. The smallest error is observed for the walking motions (solid). This could be expected for a cyclic motion that contains a lot of redundant frames. The used grabbing motions (dotted) are more complex. The reason may be that the motions sequences disparate since some sequences include a step to the storage rack while others do not. The cartwheel motions (dashed) are the most complex class in this experiment, as we expected.
Figure 5. Reconstruction error E_{total} for walking (solid), grabbing (dotted) and cartwheel (dashed) motions, depending on the size of the Data Mode of the core tensor.
To compare our multilinear model with linear models, as they are used for principal component analysis (PCA), we constructed two tensors for our model and two matrices for the PCA. The first tensor and the first matrix were filled with walking motions, the second tensor and the second matrix were filled with grabbing motions. The orientations were represented by Euler angles. The resulting tensors had a size of 81 x 62 x 3 x 3 x 3 (walking) and 64 x 62 x 3 x 3 x 3 (grabbing), respectively.
Figure 6. Mean error of reconstructed motions with reconstructions based on our model (blue) and based on a PCA (green). The result is shown for walking motions (left) and grabbing motions (right).
After some data reduction step, we compared the reconstructed motions with the original motions by measuring the differences between all orientations of the original and the reconstructed motions. Averaging over all motions and differences, we obtained a mean error as is also used in [ SHP04 ] (we used this measure to keep the results comparable to the literature). Figure 6 shows a comparison of the mean errors in the reconstructed motions for the walking (left) and grabbing (right) examples. The mean errors depend on the size of the DOF Mode and the number of principal components, respectively. Note that the errors for motions reconstructed from the multimodemodel are smaller than the errors from the motions reconstructed from principal components. For example, a walking motion can be reconstructed with a mean error of less than one degree (in the average) from a core tensor when the DOF Mode is truncated to just three components (see left part of Figure 6). Therefore, in cases where a motion should be approximated by rather few components the reduction based on the multilinear model may be considerably better than the one achieved by PCA.
As it was described in Sect. 2.3, it is possible to synthesize motions with our multilinear model. For every mode k there is an appropriate matrix U_{k} , where every row u_{k,j} with j ∈ {1,2,...,d_{k}} represents one of the dimensions, of mode k. Therefore an inter or extrapolation between the d_{k} dimensions e.g. between the styles, actors and repetitions, can be done by inter or extrapolation between any rows of U_{k} before they are multiplied with the core tensor to synthesize a motion. To prevent our results from artifacts such as turns and unexpected flips resulting from a representation based on Euler angles we used our quaternion based representation to synthesize motions.
For the following walking example, we constructed a motion that was interpolated between two different styles. The first style was walking four steps straight forward and the second one was walking four steps on a left circle. We made a linear interpolation by multiplying the corresponding rows with the factor 0.5. The result is a four step walking motion that describes a left round with a larger radius. One sample frame of this experiment can be seen in Figure 7. Another synthetic motion was made by an interpolation of grabbing styles.
Figure 7. Screenshot from the original motions that are from the styles walking forward (left) and walking a left circle (right), the synthetic motion (middle) is produced by a linear combination of these styles.
We synthesized a motion by an interpolation of the styles grabbing low and grabbing high. The result is a motion that grabs in the middle. One sample frame of this synthetic motion is shown in Figure 8.
With this technique we are able to make interpolation between all modes simultaneous. One example is a walking motion that is an interpolation between the Style and Actors Mode. One snapshot taken from the accompanying animation video of this example is given in Figure 8.
Figure 8. Left: Screenshot from the original motions that are from the styles grabbing low (left) and grabbing high (right), the synthetic motion (middle) is produced by a linear combination of these styles. Right: Screenshot from four original walking motions and one synthetic motion, that is an result of combining two, the personal and the style mode. The original motions of the first actor are on the left side, the original motions of the second actor are on the right side and the synthetic example can be seen in the middle.
In Table 4 the computation times of our MATLAB implementations of the NMode SVD and PCA are given (on an 1.66 GHz Intel Core2 CPU T5500). For the decomposition of a data tensor consisting of 95 x 119 x 4 x 3 x 2 = 273600 entries, the NMode SVD needs 5.675 seconds, while the PCA needs 0.636 seconds for a matrix of comparable size having 119 x 2280 = 273600 entries. As Table 4 shows, the computing time for the SVD increase with the dimension of the tensor, while the computation time for the PCA is nearly constant.
The SVD decomposition can be seen as a preprocessing step, where all further calculations can be done on the core tensor and the corresponding matrices. The reconstruction of a motion from the tensor and the matrices U_{k} can be performed at interactive frame rates—even in our MATLAB implementation the reconstruction only requires 0.172 seconds. As a combination of motions of different modes is just a reconstruction with modified weights, the creation of synthetic motions is also possible with a similar computational cost.
Table 4. Computation times for PCA and NModeSVD for the data used in the examples.
Dimension Motion Matrix 
Time PCA (in sec.) 
Dimension Core Tensor 
Time NMode SVD (in sec.) 
119 x 2280 
0.636 
95 x 119 x 4 x 3 x 2 
5.675 
95 x 2280 
0.621 
95 x 95 x 4 x 3 x 2 
4.968 
80 x 2280 
0.615 
95 x 80 x 4 x 3 x 2 
4.461 
5 x 2280 
0.600 
95 x 5 x 4 x 3 x 2 
2.295 
In this paper, we have shown how multilinear models can be used for analyzing and processing human motion data. The representation is based on explicitly using various modes that correspond to technical as well as semantic aspects of some given motion class. Encoding the data as highorder tensors allows for reducing the model with respect to any combination of modes, which often yields better approximation results than previously used PCAbased methods. Furthermore, the multilinear model constitutes a unified and flexible framework for motion synthesis applications, which allows for controlling each motion aspect independently in the morphing process. As a further contribution, we described an efficient multiscale approach for motion warping using physicsbased motion features.
Multilinear motion representations constitute an interesting alternative and additional tool in basically all situations, where current PCAbased methods are used. We expect that our multimodal model is helpful in the context of reconstructing motions from lowdimensional control signals, see, e. g., [ CH05 ]. Currently, we also investigate how one can improve auditory representations of motions as described in [ RM05, EMWZ05 ] by using strongly reduced motion representations.
In order to construct a highorder tensor for a given motion class, one needs a sufficient number of example motions for each mode to be considered in the model. In practice, this is often problematic, since one may only have sparsely given data for the different modes. In suchsituations, one may employ similar techniques as have been employed in the context of face transfer, see [ VBPP05 ], to fill up the data tensor. Another important research problem concerns the automatic extraction of suitable example motions from a large database, which consists of unknown and unorganized motion material. For the future, we plan to employ efficient contentbased motion retrieval strategies as described, e. g., in [ KG04, MRC05, MR06 ] to support the automatic generation of multimodal data tensors for motion classes that have a sufficient number of instances in the unstructured dataset.
We are grateful to Bernd Eberhardt and his group from HDM Stuttgart for providing us with the mocap data. We also thank the anonymous referees for constructive and valuable comments.
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Additional Material
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Morphing of motion by interpolation of two different styles (Walking,Grabbing). Examples for motion reconstruction. Motion Preprocessing. Time Warping Example. Examples for extreme style interpolation. 

Tensor Representation 
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Björn Krüger, Jochen Tautges, Meinard Müller, and Andreas Weber, MultiMode Tensor Representation of Motion Data. JVRB  Journal of Virtual Reality and Broadcasting, 5(2008), no. 5. (urn:nbn:de:0009614197)
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