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133 nips-2007-Modelling motion primitives and their timing in biologically executed movements

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Author: Ben Williams, Marc Toussaint, Amos J. Storkey

Abstract: Biological movement is built up of sub-blocks or motion primitives. Such primitives provide a compact representation of movement which is also desirable in robotic control applications. We analyse handwriting data to gain a better understanding of primitives and their timings in biological movements. Inference of the shape and the timing of primitives can be done using a factorial HMM based model, allowing the handwriting to be represented in primitive timing space. This representation provides a distribution of spikes corresponding to the primitive activations, which can also be modelled using HMM architectures. We show how the coupling of the low level primitive model, and the higher level timing model during inference can produce good reconstructions of handwriting, with shared primitives for all characters modelled. This coupled model also captures the variance proﬁle of the dataset which is accounted for by spike timing jitter. The timing code provides a compact representation of the movement while generating a movement without an explicit timing model produces a scribbling style of output. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Modelling motion primitives and their timing in biologically executed movements Ben H Williams School of Informatics University of Edinburgh 5 Forrest Hill, EH1 2QL, UK ben. [sent-1, score-0.922]

2 uk Abstract Biological movement is built up of sub-blocks or motion primitives. [sent-10, score-0.238]

3 Such primitives provide a compact representation of movement which is also desirable in robotic control applications. [sent-11, score-0.86]

4 We analyse handwriting data to gain a better understanding of primitives and their timings in biological movements. [sent-12, score-0.899]

5 Inference of the shape and the timing of primitives can be done using a factorial HMM based model, allowing the handwriting to be represented in primitive timing space. [sent-13, score-1.76]

6 This representation provides a distribution of spikes corresponding to the primitive activations, which can also be modelled using HMM architectures. [sent-14, score-0.517]

7 We show how the coupling of the low level primitive model, and the higher level timing model during inference can produce good reconstructions of handwriting, with shared primitives for all characters modelled. [sent-15, score-1.475]

8 This coupled model also captures the variance proﬁle of the dataset which is accounted for by spike timing jitter. [sent-16, score-0.429]

9 The timing code provides a compact representation of the movement while generating a movement without an explicit timing model produces a scribbling style of output. [sent-17, score-1.0]

10 There is much evidence to suggest that biological movement generation is based upon motor primitives, with discrete muscle synergies found in frog spines, (Bizzi et al. [sent-21, score-0.462]

11 , 2002), evidence of primitives being locally ﬁxed (Kargo & Giszter, 2000), and modularity in human motor learning and adaption (Wolpert et al. [sent-24, score-0.702]

12 Compact forms of representation for any biologically produced data should therefore also be based upon primitive sub-blocks. [sent-26, score-0.445]

13 1 (A) (B) ¯t Figure 1: (A) A factorial HMM of a handwriting trajectory Yt . [sent-27, score-0.271]

14 The parameters λm indicate the probability of triggering a primitive in the mth factor at time t and are learnt for one speciﬁc character. [sent-28, score-0.589]

15 (B) A hierarchical generative model of handwriting where the random variable c indicates the currently written character and deﬁnes a distribution over random variables λm via a Markov t model over Gm . [sent-29, score-0.53]

16 There are several approaches to use this idea of motion primitives for more eﬃcient robotic movement control. [sent-30, score-0.877]

17 , 2004) use non-linear attractor dynamics as a motion primitive and train them to generate motion that solves a speciﬁc task. [sent-33, score-0.626]

18 (Amit & Matari´, 2002) use a single attractor system and generate non-linear motion by c modulating the attractor point. [sent-34, score-0.211]

19 These approaches deﬁne a primitive as a segment of movement rather than understanding movement as a superposition of concurrent primitives. [sent-35, score-0.837]

20 The goal of analysing and better understanding biological data is to extract a generative model of complex movement based on concurrent primitives which may serve as an eﬃcient representation for robotic movement control. [sent-36, score-1.182]

21 This is in contrast to previous studies of handwriting which usually focus on the problem of character classiﬁcation rather than generation (Singer & Tishby, 1994; Hinton & Nair, 2005). [sent-37, score-0.394]

22 We investigate handwriting data and analyse whether it can be modelled as a superposition of sparsely activated motion primitives. [sent-38, score-0.376]

23 Just as piano music can (approximately) be modelled as a superposition of the sounds emitted by each key we follow the idea that biological movement is a superposition of pre-learnt motion primitives. [sent-41, score-0.518]

24 This implies that the whole movement can be compactly represented by the timing of each primitive in analogy to a score of music. [sent-42, score-0.846]

25 On the lower level a factorial Hidden Markov Model (fHMM, Ghahramani & Jordan, 1997) is used to model the output as a combination of signals emitted from independent primitives (each primitives corresponds to a factor in the fHMM). [sent-44, score-1.297]

26 On the higher level we formulate a model for the primitive timing dependent upon character class. [sent-45, score-0.966]

27 The same motion primitives are shared across characters, only their timings diﬀer. [sent-46, score-0.71]

28 We train this model on handwriting data using an EM-algorithm and thereby infer the primitives and the primitive timings inherent in this data. [sent-47, score-1.252]

29 We ﬁnd that the inferred timing posterior for a speciﬁc character is indeed a compact representation for the speciﬁc character which allows for a good reproduction of this character using the learnt primitives. [sent-48, score-1.17]

30 Further, using the timing model learnt on the higher level we can generate new movement – new samples of characters (in the same writing style as the data), and also scribblings that exhibit local similarity to written characters when the higher level timing control is omitted. [sent-49, score-1.181]

31 Finally in section 4 we present results on handwriting data recorded with a digitisation tablet, show the primitives and timing code we extract, and demonstrate how the learnt model can be used to generate new samples of characters. [sent-52, score-1.3]

32 2 2 Model Our analysis of primitives and primitive timings in handwriting is based on formulating a corresponding probabilistic generative model. [sent-53, score-1.295]

33 On the lower level (Figure 1(A)) we consider a factorial Hidden Markov Model (fHMM) where each factor produces the signal of a single primitive and the linear combination of factors generates the observed movement Yt . [sent-55, score-0.642]

34 It allows the learning and identiﬁcation of primitives in the data but does not include a model of their timing. [sent-59, score-0.606]

35 In this paper we introduce the full generative model (Figure 1(B)) which includes a generative model for the primitive timing conditioned on the current character. [sent-60, score-0.92]

36 1 Modelling primitives in data Let M be the number of primitives we allow for. [sent-62, score-1.15]

37 We describe a primitive as a strongly constrained Markov process which remains in a zero state most of the time but with some ¯ probability λ ∈ [0, 1] enters the 1 state and then rigorously runs through all states 2, . [sent-63, score-0.403]

38 (1) P (St = b | St−1 = a, λm ) = t 1 for a = 0 and b = (a + 1) mod Km   0 otherwise ¯ This process is parameterised by the onset probability λm of the mth primitive at time t. [sent-71, score-0.535]

39 , WKm ) is what we call a primitive and – to stay in the analogy ¯ – can be compared to the sound of a piano key. [sent-76, score-0.462]

40 We will describe below how we learn the primitives m Ws and also adapt the primitive lengths Km using an EM-algorithm. [sent-78, score-0.978]

41 2 A timing model ¯ Considering the λ’s to be ﬁxed parameters is not a suitable model of biological movement. [sent-80, score-0.378]

42 The usage and timing of primitives depends on the character that is written and the timing ¯ varies from character to character. [sent-81, score-1.557]

43 Our model takes a diﬀerent approach to parameterise the primitive activations. [sent-83, score-0.463]

44 For instance, if a primitive is activated twice in the course of the movement we assume that there have been two signals (“spikes”) emitted from a higher level process which encode the activation times. [sent-84, score-0.682]

45 More formally, let c be a discrete random variable indicating the character to be written, see Figure 1(B). [sent-85, score-0.215]

46 We assume that for each primitive we have another Markovian process which generates a length-L sequence of states Gm ∈ {1, . [sent-86, score-0.403]

47 l l−1 (3) l=2 The states Gm encode which primitives are activated and how they are timed, as seen in l Figure 2(b). [sent-89, score-0.601]

48 (b) Scatter plot of the MAP onsets of a single t primitive for diﬀerent samples of the same character ‘p’. [sent-99, score-0.681]

49 of a primitive at time t, which we call a “spike”. [sent-101, score-0.403]

50 We can observe at l most L spikes in one primitive, the spike times between diﬀerent primitives are dependent, but we have a Markovian dependency between the presence and timing of spikes within a primitive. [sent-113, score-1.115]

51 The whole process is parameterised by the initial state distribution P (Gm | c), 1 m the transition probabilities P (Gm | Gm , c), the spike means µm and the variances σr . [sent-114, score-0.182]

52 l 3 Inference and learning In the experiments we will compare both the fHMM without the timing model (Figure 1(A)) and the full model including the timing model (Figure 1(B)). [sent-121, score-0.676]

53 5 5 Distance /mm (c) Figure 3: (a) Reconstruction of a character from a training dataset, using a subset of the primitives. [sent-168, score-0.215]

54 The posterior probability of primitive onset is shown on the left, highlighting why a spike timing representation is appropriate. [sent-170, score-0.882]

55 (c) Generative samples using a ﬂat primitive onset prior, showing scribbling behaviour of uncoupled model. [sent-172, score-0.528]

56 In the full model, inference is an iterative process of inference in the timing model and inference in the fHMM. [sent-174, score-0.416]

57 We couple this iteration to inference in the timing model in both directions: In each iteration, m the posterior over St deﬁnes observation likelihoods for inference in the Markov models Gm . [sent-176, score-0.359]

58 Standard M-steps are then used t to train all parameters of the fHMM and the timing model. [sent-178, score-0.276]

59 In addition, we use heuristics to adapt the length Km of each primitive: we increase or decrease Km depending on whether the learnt primitive is signiﬁcantly diﬀerent to zero in the last time steps. [sent-179, score-0.554]

60 By this we mean that we take a trained model, use inference to compute the MAP spikes λ for a speciﬁc data sample, then we use these λ’s and the deﬁnition of our generative model (including the learnt primitives W ) to generate a trajectory which can be compared to the original data sample. [sent-182, score-0.991]

61 1 Results Primitive and timing analysis using the fHMM-only We ﬁrst consider a data set of 300 handwritten ‘p’s recorded using an INTUOS 3 WACOM digitisation tablet http://www. [sent-185, score-0.334]

62 Our choice of parameter was M = 10 primitives and we initialised all Km = 20 and constrained them to be smaller than 100 throughout learning. [sent-192, score-0.575]

63 This clean posterior is the motivation for introducing a model of the spike timings as a compact representation 5 −15 −20 −25 −30 −35 −40 −45 −50 4 2 −20 −10 0 Distance /mm (a) 10 0 x 10 x position y position pressure 1. [sent-195, score-0.294]

64 (b) Histogram of the reconstruction error, which is 3-dimensional pen movement velocity space. [sent-204, score-0.303]

65 Equally the reconstruction (using the Viterbi aligned MAP spikes) shows the suﬃciency of the spike code to generate the character. [sent-208, score-0.216]

66 Figure 3(b) shows the primitives W m (translated back into pen-space) that were learnt and implicitly used for the reconstruction of the ‘p’. [sent-209, score-0.798]

67 These primitives can be seen to represent typical parts of the ‘p’ character; the arrows in the reconstruction indicate when they are activated. [sent-210, score-0.647]

68 To show the importance of this spike timing information, we can demonstrate the eﬀects of removing it. [sent-212, score-0.398]

69 When using the fHMM-only model as a generative model with ¯ the learnt stationary spike probabilities λm the result is a form of primitive babbling, as can be seen in Figure 3(c). [sent-213, score-0.834]

70 Since these scribblings are generated by random expression of the learnt primitives they locally resemble parts of the ‘p’ character. [sent-214, score-0.755]

71 The primitives generalise to other characters if the training dataset contained suﬃcient variation. [sent-215, score-0.65]

72 Further investigation has shown that 20 primitives learnt from 12 character types are suﬃciently generalised to represent all remaining novel character types without further learning, by using a single E-step to ﬁt the pre-learnt parameters to a novel dataset. [sent-216, score-1.156]

73 2 Generating new characters using the full generative model Next we trained the full model on the same ‘p’-dataset. [sent-218, score-0.273]

74 To the right we see the reconstruction errors in velocity space showing at many time points a perfect reconstruction was attained. [sent-220, score-0.185]

75 Since the full model includes a timing model it can also be run autonomously as a generative model for new character samples. [sent-221, score-0.727]

76 Figure 4(c) displays such new samples of the character ‘p’ generated by the learnt model. [sent-222, score-0.403]

77 As a more challenging problem we collected a data set of over 450 character samples of the letters a, b and c. [sent-223, score-0.252]

78 The full model includes the written character class as a random variable and can thus be trained on multi-character data sets. [sent-224, score-0.277]

79 Note that we restrict the total number of primitives to M = 10 which will require a sharing of primitives across characters. [sent-225, score-1.15]

80 Coupling the timing and the primitive model during learning has the eﬀect of trying to learn primitives from data that are usually in the same place. [sent-229, score-1.285]

81 Thus, using the full model the inferred spikes are more compactly clustered at the Gaussian components due to the prior imposed from the timing model (the thick black lines correspond to Equation (4)). [sent-230, score-0.44]

82 (b) Reconstruction of dataset using 10 primitives learnt from the dataset in (a). [sent-232, score-0.726]

83 8 Time /ms (a) (b) Figure 6: (a) Scatter plot of primitive onset spikes for a single character type across all samples and primitives, showing the clustering of certain primitives in particular parts of a character. [sent-250, score-1.36]

84 The thick black lines displays the prior over λ’s imposed from the timing model via Equation (4). [sent-253, score-0.307]

85 Finally, we run the full model autonomously to generate new character samples, see Figure 5(c). [sent-254, score-0.337]

86 Here the character class, c is ﬁrst sampled uniform randomly and then all learnt parameters are used to eventually sample a trajectory Yt . [sent-255, score-0.407]

87 5 Conclusions In this paper we have shown that it is possible to represent handwriting using a primitive based model. [sent-257, score-0.582]

88 The timing of activations is crucial to the accurate reproduction of the character. [sent-260, score-0.32]

89 With a small amount of timing variation, a distorted version of the original character is reproduced, whilst large (and coordinated) diﬀerences in the timing pattern produce diﬀerent character types. [sent-261, score-0.982]

90 The spike code provides a compact representation of movement, unlike that which has previously been explored in the domain of robotic control. [sent-262, score-0.24]

91 We have proposed to use Markov processes conditioned on the character as a model for these spike emissions. [sent-263, score-0.368]

92 7 An assumption made in this work is that the primitives are learnt velocity proﬁles. [sent-267, score-0.767]

93 We have not included any feedback control systems in the primitive production, however the presence of low-level feedback, such as in a spring system (Hinton & Nair, 2005) or dynamic motor primitives (Ijspeert et al. [sent-268, score-1.156]

94 We make no assumptions about how the primitives are learnt in biology. [sent-271, score-0.726]

95 It would be interesting to study the evolution of the primitives during human learning of a new character set. [sent-272, score-0.79]

96 This could be related to a more accurate and eﬃcient use of primitives already available. [sent-274, score-0.575]

97 However, it might also be the case that new primitives are learnt, or old ones adapted. [sent-275, score-0.575]

98 Shared and speciﬁc muscle synergies in natural motor behaviors. [sent-311, score-0.179]

99 Combinations of muscle synergies in the construction of a natural motor behavior. [sent-317, score-0.179]

100 A primitive based generative model to infer timing information in unpartitioned handwriting data. [sent-374, score-0.963]

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