nips nips2010 nips2010-133 knowledge-graph by maker-knowledge-mining

133 nips-2010-Kernel Descriptors for Visual Recognition

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Author: Liefeng Bo, Xiaofeng Ren, Dieter Fox

Abstract: The design of low-level image features is critical for computer vision algorithms. Orientation histograms, such as those in SIFT [16] and HOG [3], are the most successful and popular features for visual object and scene recognition. We highlight the kernel view of orientation histograms, and show that they are equivalent to a certain type of match kernels over image patches. This novel view allows us to design a family of kernel descriptors which provide a uniﬁed and principled framework to turn pixel attributes (gradient, color, local binary pattern, etc.) into compact patch-level features. In particular, we introduce three types of match kernels to measure similarities between image patches, and construct compact low-dimensional kernel descriptors from these match kernels using kernel principal component analysis (KPCA) [23]. Kernel descriptors are easy to design and can turn any type of pixel attribute into patch-level features. They outperform carefully tuned and sophisticated features including SIFT and deep belief networks. We report superior performance on standard image classiﬁcation benchmarks: Scene-15, Caltech-101, CIFAR10 and CIFAR10-ImageNet.

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 We highlight the kernel view of orientation histograms, and show that they are equivalent to a certain type of match kernels over image patches. [sent-3, score-1.085]

2 This novel view allows us to design a family of kernel descriptors which provide a uniﬁed and principled framework to turn pixel attributes (gradient, color, local binary pattern, etc. [sent-4, score-1.068]

3 In particular, we introduce three types of match kernels to measure similarities between image patches, and construct compact low-dimensional kernel descriptors from these match kernels using kernel principal component analysis (KPCA) [23]. [sent-6, score-2.044]

4 Kernel descriptors are easy to design and can turn any type of pixel attribute into patch-level features. [sent-7, score-0.49]

5 In this work, we highlight the kernel view of orientation histograms and provide a uniﬁed way to low-level image feature design and learning. [sent-15, score-0.904]

6 We show how our framework is applied to gradient, color, and shape pixel attributes, leading to three effective kernel descriptors. [sent-17, score-0.681]

7 1 The most relevant work to this paper is that of efﬁcient match kernels (EMK) [1], which provides a kernel view to the frequently used Bag-of-Words representation and forms image-level features by learning compact low dimensional projections or using random Fourier transformations. [sent-20, score-0.939]

8 Another related work is based on mathematics of the neural response, which shows that the hierarchical architectures motivated by the neuroscience of the visual cortex is associated to the derived kernel [24]. [sent-22, score-0.462]

9 Instead, the goal of this paper is to provide a deep understanding of how orientation histograms (SIFT and HOG) work, and we can generalize them and design novel low-level image features based on the kernel insight. [sent-23, score-0.985]

10 Our kernel descriptors are general and provide a principled way to convert pixel attributes to patch-level features. [sent-24, score-0.973]

11 To the best of our knowledge, this is the ﬁrst time that low-level image features are designed and learned from scratch using kernel methods; they can serve as the foundation of many computer vision tasks including object recognition. [sent-25, score-0.714]

12 Our novel kernel descriptors are presented in Section 3, followed by an extensive experimental evaluation in Section 4. [sent-28, score-0.733]

13 Here we describe the kernel view of such orientation histograms features, and show how this kernel view can help overcome issues such as orientation binning. [sent-31, score-1.317]

14 Let θ(z) and m(z) be the orientation and magnitude of the image gradient at a pixel z. [sent-32, score-0.546]

15 In HOG and SIFT, the gradient orientation of each pixel is discretized into a d−dimensional indicator vector δ(z) = [δ1 (z), · · · , δd (z)] with δi (z) = dθ(z) 1, =i−1 2π 0, otherwise (1) where x takes the largest integer less than or equal to x (we will describe soft binning further below). [sent-33, score-0.564]

16 Aggregating feature vectors of pixels over an image patch P , we obtain the histogram of oriented gradients: Fh (P ) = m(z)δ(z) (2) z∈P m(z)2 where m(z) = m(z)/ + g is the normalized gradient magnitude, with g a small z∈P constant. [sent-35, score-0.539]

17 This is equivalent to measuring the similarity of image patches using a linear kernel in the feature map Fh (P ) in kernel space: Kh (P, Q) = Fh (P ) Fh (Q) = m(z)m(z )δ(z) δ(z ) (3) z∈P z ∈Q where P and Q are patches usually from two different images. [sent-39, score-1.277]

18 Therefore, Kh (P, Q) is a match kernel over sets (here the sets are image patches) as in [8, 1, 11, 17, 7]. [sent-42, score-0.712]

19 3 provides a kernel view of HOG features over image patches. [sent-44, score-0.69]

20 To get a kernel view of soft binning [13], we only need to replace the delta function in Eq. [sent-48, score-0.619]

21 The L2 distance between P and Q can be expressed as D(P, Q) = 2 − 2F (P ) F (Q) as we know F (P ) F (P ) = 1, and the kernel view can be provided in the same manner. [sent-51, score-0.465]

22 To measure similarity between two pixel orientation gradients θ and θ , we use the L2 norm between the normalized gradient vectors θ = [sin(θ) cos(θ)] and θ = [sin(θ ) cos(θ )]. [sent-54, score-0.559]

23 Note that the kernel km (z, z ) measuring the similarity of gradient magnitudes of two pixels is linear in gradient magnitude. [sent-60, score-0.764]

24 As can be seen, this kernel introduces quantization errors and could lead to suboptimal performance in subsequent stages of processing. [sent-63, score-0.425]

25 While soft binning results in a smoother kernel function, it still suffers from discretization. [sent-64, score-0.579]

26 This motivates us to search for alternative match kernels which can measure the similarity of image patches more accurately. [sent-65, score-0.615]

27 To estimate the difference between orientations at pixels z and z , we use the following normalized gradient vectors in the kernel function ko : θ(z) = [sin(θ(z)) cos(θ(z))] . [sent-68, score-1.099]

28 The kernel view of orientation histograms provides a simple, uniﬁed way to turn pixel attributes into patch-level features. [sent-75, score-0.91]

29 One immediate extension is to construct color match kernels over pixel values: Kcol (P, Q) = kc (c(z), c(z ))kp (z, z ) (7) z∈P z ∈Q where c(z) is the pixel color at position z (intensity for gray images and RGB values for color images). [sent-76, score-1.094]

30 The normalized linear kernel s(z)s(z ) weighs the contribution of each local binary pattern, and the Gaussian kernel kb (b(z), b(z )) = exp(−γb b(z) − b(z ) 2 ) measures shape similarity through local binary patterns. [sent-80, score-1.138]

31 Match kernels deﬁned over various pixel attributes provide a uniﬁed way to generate a rich, diverse visual feature set, which has been shown to be very successful to boost recognition accuracy [6]. [sent-81, score-0.589]

32 As validated by our own experiments, gradient, color and shape match kernels are strong in their own right and complement one another. [sent-82, score-0.515]

33 2 Learning Compact Features Match kernels provide a principled way to measure the similarity of image patches, but evaluating kernels can be computationally expensive when image patches are large [1]. [sent-85, score-0.811]

34 An important advantage of our approach is that no local minima are involved, unlike constrained kernel singular value decomposition [1]. [sent-87, score-0.453]

35 We now describe how our compact low-dimensional features are extracted from the gradient kernel Kgrad ; features for the other kernels can be generated the same way. [sent-88, score-0.94]

36 5 as inner products ko (θ(z), θ(z )) = φo (θ(z)) φo (θ(z )), kp (z, z ) = φp (z) φp (z ), we can derive the following feature over image patches: Fgrad (P ) = m(z)φo (θ(z)) ⊗ φp (z) (9) z∈P where ⊗ is the tensor product. [sent-90, score-0.722]

37 A straightforward way to dimension reduction is to sample sufﬁcient image patches from training images and perform KPCA for match kernels. [sent-93, score-0.482]

38 A key issue in this projection process is how to choose a set of basis vectors which makes the ﬁnite-dimensional kernel approximate well the original kernel. [sent-99, score-0.573]

39 For example, consider the Gaussian kernel ko (θ(z), θ(z )) over gradient orientation. [sent-101, score-0.891]

40 Left: the orientation kernel ko (θ(z), θ(z )) and its ﬁnite-dimensional approximation. [sent-109, score-0.94]

41 All curves show kernel values as functions of θ(z). [sent-111, score-0.425]

42 The red line is the ground truth kernel function ko , and the black, green and blue lines are the ﬁnite approximation kernels with different grid sizes. [sent-112, score-1.022]

43 Right: root mean square error (RMSE) between KPCA approximation and the corresponding match kernel as a function of dimensionality. [sent-113, score-0.576]

44 In a similar manner, we can also approximate the kernels kp , kc and kb . [sent-117, score-0.498]

45 The ﬁnite-dimensional feature for the gradient match kernel is Fgrad (P ) = z∈P m(z)φo (θ(z)) ⊗ φp (z), and may be efﬁciently used as features over image patches. [sent-118, score-0.945]

46 When the grid size is larger than 16, the ﬁnite kernel and the original kernel become virtually indistinguishable. [sent-122, score-0.903]

47 For the shape kernel over local binary patterns, because the variables are binary, we simply choose the set of all 28 = 256 basis vectors and thus no approximation error is introduced. [sent-123, score-0.702]

48 The size of basis vectors can be further reduced by performing kernel principal component analysis over joint basis vectors: {φo (x1 ) ⊗ φp (y1 ), · · · , φo (xdo ) ⊗ φp (ydp )}, where φp (ys ) are basis vectors for the position kernel and dp is the number of basis vectors. [sent-128, score-1.414]

49 (2), match kernels can be approximated rather accurately using the reduced basis vectors by KPCA. [sent-131, score-0.48]

50 Under the framework of kernel principal component analysis, our gradient kernel descriptor (Eq. [sent-132, score-1.118]

51 5) has the form t do dp t αij F grad (P ) = i=1 j=1 m(z)ko (θ(z), xi )kp (z, yj ) (12) z∈P The computational bottleneck of extracting kernel descriptors are to evaluate the kernel function ko kp between pixels. [sent-133, score-1.751]

52 Fortunately, we can compute two kernel values separately at the cost do + dp , rather than do dp . [sent-134, score-0.521]

53 Our most expensive kernel descriptor, the shape kernel, takes about 4 seconds in MATLAB to compute on a typical image (300 × 300 resolution and 16 × 16 image patches over 5 8×8 grids). [sent-135, score-0.928]

54 5 seconds for the gradient kernel descriptor, compared to about 0. [sent-137, score-0.528]

55 A more efﬁcient GPU-based implementation will certainly reduce the computation time for kernel descriptors such that real time applications become feasible. [sent-139, score-0.733]

56 4 Experiments We compare gradient (KDES-G), color (KDES-C), and shape (KDES-S) kernel descriptors to SIFT and several other state of the art object recognition algorithms on four publicly available datasets: Scene-15, Caltech101, CIFAR10, and CIFAR10-ImageNet (a subset of ImageNet). [sent-140, score-1.121]

57 For gradient and shape kernel descriptors and SIFT, all images are transformed into grayscale ([0, 1]) and resized to be no larger than 300 × 300 pixels with preserved ratio. [sent-141, score-1.088]

58 For our gradient kernel descriptors we use the same gradient computation as used for SIFT descriptors. [sent-149, score-0.939]

59 We also evaluate the performance of the combination of the three kernel descriptors (KDES-A) by simply concatenating the image-level features vectors. [sent-150, score-0.822]

60 Instead of spatial pyramid kernels, we compute image-level features using efﬁcient match kernels (EMK), which has been shown to produce more accurate quantization. [sent-151, score-0.525]

61 We consider 1 × 1, 2 × 2 and 4 × 4 pyramid sub-regions (see [1]), and perform constrained kernel singular value decomposition (CKSVD) to form image-level features, using 1,000 visual words (basis vectors in CKSVD) learned by K-means from about 100,000 image patch features. [sent-152, score-0.818]

62 We have experimented both with linear SVMs and Laplacian kernel SVMs and found that Laplacian kernel SVMs over efﬁcient match kernel features are always better than linear SVMs (see (§4. [sent-154, score-1.515]

63 We use Laplacian kernel SVMs in our experiments (except for the tiny image dataset CIFAR10). [sent-156, score-0.614]

64 1 Hyperparameter Selection We select kernel parameters using a subset of ImageNet. [sent-158, score-0.425]

65 We choose basis vectors for ko , kc , and kp from 25, 5 × 5 × 5 and 5 × 5 uniform grids, respectively, which give sufﬁcient approximations to the original kernels (see also Fig. [sent-160, score-0.957]

66 We optimize the dimensionality of KPCA and match kernel parameters jointly using exhaustive grid search. [sent-162, score-0.629]

67 Our experiments suggest that the optimal parameter settings are r = 200 (dimensionality of kernel descriptors), γo = 5, γc = 4, γb = 2, γp = 3, g = 0. [sent-163, score-0.425]

68 As we see, both gradient and shape kernel descriptors outperform SIFT with a margin. [sent-174, score-0.937]

69 Gradient kernel descriptors and shape kernel descriptors have similar performance. [sent-175, score-1.567]

70 It is not surprising that the intensity kernel descriptor has a lower accuracy, as all the images are grayscale. [sent-176, score-0.673]

71 The combination of the three kernel descriptors further boosts the performance by about 2 percent. [sent-177, score-0.733]

72 Another interesting ﬁnding is that Laplacian kernel SVMs are signiﬁcantly better than linear SVMs, 86. [sent-178, score-0.425]

73 We also tried to replace SIFT features with our gradient and shape kernel descriptors in SPM, and both obtained 83. [sent-183, score-1.026]

74 To our best knowledge, our gradient kernel descriptor alone outperforms the best published result 84. [sent-185, score-0.685]

75 left: Accuracy as functions of feature dimensionality for orientation kernel (KDES-G) and shape kernel (KDES-S), respectively. [sent-206, score-1.144]

76 Methods Linear SVM Laplacian kernel SVM SIFT 76. [sent-209, score-0.425]

77 4 Table 1: Comparisons of recognition accuracy on Scene-15: kernel descriptors and their combination vs SIFT. [sent-229, score-0.82]

78 We compare our kernel descriptors with recently published results obtained both by low-level feature learning algorithms, convolutional deep belief networks (CDBN), and sparse coding methods: invariant predictive sparse decomposition (IPSD) and locality-constrained linear coding. [sent-236, score-0.955]

79 Both our gradient kernel descriptor and shape kernel descriptor are superior to CDBN by a large margin. [sent-239, score-1.318]

80 Notice that both naive Bayesian nearest neighbor (NBNN) and locality-constrained linear coding should be compared to our kernel descriptors over multiple patch sizes because both of them used multiple scales to boost the performance. [sent-241, score-0.861]

81 Another ﬁnding is that the combination of three kernel descriptors outperforms any single kernel descriptor. [sent-244, score-1.158]

82 Our goal in this paper is to evaluate the strengths of kernel descriptors. [sent-246, score-0.425]

83 To improve accuracy further, kernel descriptors can be combined with other types of image features. [sent-247, score-0.918]

84 This dataset consists of 60,000 32x32 color images in 10 categories, with 5,000 images per category as training set and 1,000 images per category as test set. [sent-250, score-0.502]

85 We extract kernel descriptors over 8×8 image patches per pixel. [sent-252, score-0.996]

86 The resulting feature vectors have a length of (1+4+9)∗1000(visual words)= 14000 per kernel descriptor. [sent-254, score-0.56]

87 We compare our kernel descriptors to deep networks [14, 9] and several baselines in table 3. [sent-316, score-0.83]

88 5%, respectively, over 30 percent lower than deep belief networks and our kernel descriptors. [sent-319, score-0.6]

89 To validate our intuitions, we also evaluated the color kernel descriptor without spatial information (kernel features are extracted on 1 × 1 spatial grid), and only obtained 38. [sent-326, score-0.852]

90 5% accuracy, 18 percent lower than the color kernel descriptor over pyramid spatial grids. [sent-327, score-0.788]

91 The shape kernel feature, KDES-S, has accuracy of 68. [sent-329, score-0.575]

92 Combing the three kernel descriptors, we obtain the best performance of 76%, 5 percent higher than the most sophisticated deep network mcRBM-DBN, which model pixel mean and covariance jointly using factorized third-order Boltzmann machines. [sent-331, score-0.725]

93 Both gradient and shape kernel descriptors achieve higher accuracy than SIFT features, which again conﬁrms that our gradient kernel descriptor and shape kernel descriptor outperform SIFT features on high resolution images with the same category as CIFAR-10. [sent-339, score-2.56]

94 5 Conclusion We have proposed a general framework, kernel descriptors, to extract low-level features from image patches. [sent-343, score-0.65]

95 Kernel descriptors are based on the insight that the inner product of orientation histograms is a particular match kernel over image patches. [sent-345, score-1.255]

96 We have performed extensive comparisons and conﬁrmed that kernel descriptors outperform both SIFT features and hierarchical feature learning, where the former is the default choice for object recognition and the latter is the most popular low-level feature learning technique. [sent-346, score-1.042]

97 To our best knowledge, we are the ﬁrst to show how kernel methods can be applied for extracting low-level image features and show superior performance. [sent-347, score-0.65]

98 This opens up many possibilities for learning low-level features with other kernel methods. [sent-348, score-0.514]

99 Considering the huge success of kernel methods in the last twenty years, we believe that this direction is worth being pursued. [sent-349, score-0.425]

100 In the future, we plan to investigate alternative kernels for low-level feature learning and learn pixel attributes from large image data collections such as ImageNet. [sent-350, score-0.568]

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Instead of the typical KFT we use the Extended Kalman Filter (EKFT) to implement a nonlinear trajectory model by linearizing the dynamics around the best estimate at each time-step [17]. With this approach we add only small computational overhead to the KFT recursions. 4.2 Tr a i n i n g t h e t i m e w a r p i n g mo d e l The filter parameters were trained using a variant of the Expectation Maximization (EM) algorithm [18]. For extended Kalman filter learning the initialization for the variables may matter. S was initialized with the ground truth average reach speeds for each movement relative to the average speed across all movements. The state transition parameters were estimated using nonlinear least squares regression, while C, Q and R were estimated linearly for the new system, using the maximum likelihood solution [18] (M-step). For the E-step we used a standard extended Kalman smoother. We thus found the expected values for t he states given the current filter parameters. For this computation, and later when testing the algorithm, xs was initialized to its average value across all reaches while the remaining states were initialized to their true values. The smoothed estimate fo r xs was then used, along with the true values for the other states, to re-estimate the filter parameters in the M-step as before. We alternated between the E and M steps until the log likelihood converged (which it did in all cases). Following the training procedure, the diagonal of the state covariance matrix Q corresponding to xs was set to the variance of the smoothed xs over all reaches, according to how much this state should be allowed to change during prediction. This allowed the estimate of xs to develop over the course of the reach due to the evidence provided by the observations, better capturing the dynamics of reaches at different speeds. 4.3 P e r f o r ma n c e o f t h e t i m e - w a r p e d E K F T Incorporating time warping explicitly into the trajectory model pro duced a noticeable increase in decoding performance over the KFT. As the speed state xs is estimated throughout the course of the reach, based on the evidence provided by the observations, the trajectory model has the flexibility to follow the dynamics of the reach more accurately (Fig. 3). While at the normal self-selected speed the difference between the algorithms is small, for the slow and fast speeds, where the dynamics deviate from average, there i s a clear advantage to the time warping model. 5 Figure 3: Hand positions and predictions of the KFT and EKFT for sample reaches at A slow, B normal and C fast speeds. Note the different time scales between reaches. The models were first trained using data from all speeds (Fig. 4A). The EKFT was 1.8% more accurate on average (p<0.01), and the effect was significant at the slow (1.9%, p<0.05) and the fast (2.8%, p<0.01), but not at the normal (p=0.3) speed. We also trained the models from data using only reaches at the self-selected normal speed, as we wanted to see if there was enough variation to effectively train the EKFT (Fig. 4B). Interestingly, the performance of the EKFT was reduced by only 0.6%, and the KFT by 1.1%. The difference in performance between the EKFT and KFT was even more pronounced on aver age (2.3%, p<0.001), and for the slow and fast speeds (3.6 and 4.1%, both p< 0.0001). At the normal speed, the algorithms again were not statistically different (p=0.6). This result demonstrates that the EKFT is a practical option for a real BMI system, as it is not necessary to greatly vary the speeds while collecting training data for the model to be effective over a wide range of intended speeds. Explicitly incorporating speed information into the trajectory model helps decoding, by modeling the natural variation in volitional speed. Figure 4: Mean and standard error of EKFT and KFT accuracy at the different subjectselected speeds. Models were trained on reaches at A all speeds and B just normal speed reaches. Asterisks indicate statistically significant differences between the algorithms. 5 Mi xtu res of Target s So far, we have assumed that the targets of our reaches are perfectly known. In a real-world system, there will be uncertainty about the intended target of the reach. However, in typical applications there are a small number of possible objectives. Here we address this situation. Drawing on the recent literature, we use a mixture model to consider each of the possible targets [11, 13]. We condition the posterior probability for the state on the N possible targets, T: (6) 6 Using Bayes' Rule, this equation becomes: (7) As we are dealing with a mixture model, we perform the Kalman filter recursion for each possible target, xT, and our solution is a weighted sum of the outputs. The weights are proportional to the prior for that target, , and the likelihood of the model given that target . is independent of the target and does not need to be calculated. We tested mixtures of both algorithms, the mKFT and mEKFT, with real uncert ain priors obtained from eye-tracking in the one-second period preceding movement. As the targets were situated on two planes, the three-dimensional location of the eye gaze was found by projecting its direction onto those planes. The first, middle and last eye samples were selected, and all other samples were assigned to a group according to which of the three was closest. The mean and variance of these three groups were used to initialize three Kalman filters in the mixture model. The priors of the three groups were assigned proportional to the number of samples in them. If the subject looks at multiple positions prior to reaching, this method ensures with a high probability that the correct target was accounted for in one of the filters in the mixture. We also compared the MTM approach of Yu et al. [13], where a different KF model was generated for each target, and a mixture is performed over these models. This approach explicitly captures the dynamics of stereotypical reaches to specific targets. Given perfect target information, it would reduce to the STM described above. 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The asterisk indicates a statistically significant effects between the two training types, where real priors are used. Here, only reaches at normal speed were used to train the models, as this is a more realistic training set for a BMI application. This accounts for the degraded performance of the MTM with perfect priors relative to the STM from above (Fig. 2). With even more stereotyped training data for each target, the MTM doesn't generalize as well to new speeds. 7 We also wanted to know if the algorithms could generalize to new targets. In a real application, the available training data will generally not span the entire useable worksp ace. We compared the algorithms where reaches to all targets except the one being tested had been used to train the models. The performance of the MTM was significantly de graded unsurprisingly, as it was designed for reaches to a set of known targets. Performance of the mKFT and mEKFT degraded by about 1%, but not significantly (both p>0.7), demonstrating that the continuous approach to target information is preferable when the target could be anywhere in space, not just at locations for which training data is available. 6 Di scu ssi on and concl u si on s The goal of this work was to design a trajectory model that would improve decoding for BMIs with an application to reaching. We incorporated two features that prominently influence the dynamics of natural reach: the movement speed and the target location. Our approach is appropriate where uncertain target information is available. The model generalizes well to new regions of the workspace for which there is no training data, and across a broad range of reaching dynamics to widely spaced targets in three dimensions. The advantages over linear models in decoding precision we report here could be equally obtained using mixtures over many targets and speeds. While mixture models [11, 13] could allow for slow versus fast movements and any number of potential targets, this strategy will generally require many mixture components. Such an approach would require a lot more training data, as we have shown that it does not generalize well. It would also be run-time intensive which is problematic for prosthetic devices that rely on low power controllers. In contrast, the algorithm introduced here only takes a small amount of additional run-time in comparison to the standard KF approach. The EKF is only marginally slower than the standard KF and the algorithm will not generally need to consider more than 3 mixture components assuming the subject fixates the target within the second pre ceding the reach. In this paper we assumed that subjects always would fixate a reach target – along with other non-targets. While this is close to the way humans usually coordinate eyes and reaches [15], there might be cases where people do not fixate a reach target. Our approach could be easily extended to deal with such situations by adding a dummy mixture component that all ows the description of movements to any target. As an alternative to mixture approaches, a system can explicitly estimate the target position in the state vector [9]. This approach, however, would not straightforwardly allow for the rich target information available; we look at the target but also at other locations, strongly suggesting mixture distributions. A combination of the two approaches could further improve decoding quality. We could both estimate speed and target position for the EKFT in a continuous manner while retaining the mixture over target priors. We believe that the issues that we have addressed here are almost universal. Virtually all types of movements are executed at varying speed. A probabilistic distribution for a small number of action candidates may also be expected in most BMI applications – after all there are usually only a small number of actions that make sense in a given environment. While this work is presented in the context of decoding human reaching, it may be applied to a wide range of BMI applications including lower limb prosthetic devices and human computer interactions, as well as different signal sources such as electrode grid recordings and electroencephalograms. The increased user control in conveying their intended movements would significantly improve the functionality of a neuroprosthetic device. A c k n o w l e d g e me n t s T h e a u t h o r s t h a n k T. H a s w e l l , E . K r e p k o v i c h , a n d V. Ravichandran for assistance with experiments. This work was funded by the NSF Program in Cyber-Physical Systems. R e f e re n c e s [1] L.R. Hochberg, M.D. Serruya, G.M. Friehs, J.A. Mukand, M. Saleh, A.H. Caplan, A. 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Nascimento, “Estimation of foot orientation with respect to ground for an above knee robotic prosthesis,” Proceedings of the 2009 IEEE/RSJ international conference on Intelligent robots and systems, St. Louis, MO, USA: IEEE Press, 2009, pp. 1112-1117. I. Cikajlo, Z. Matjačić, and T. Bajd, “Efficient FES triggering applying Kalman filter during sensory supported treadmill walking,” Journal of Medical Engineering & Technology, vol. 32, 2008, pp. 133144. S. Kim, J.D. Simeral, L.R. Hochberg, J.P. Donoghue, and M.J. Black, “Neural control of computer cursor velocity by decoding motor cortical spiking activity in humans with tetraplegia,” Journal of Neural Engineering, vol. 5, 2008, pp. 455-476. L. Srinivasan, U.T. Eden, A.S. Willsky, and E.N. Brown, “A state-space analysis for reconstruction of goal-directed movements using neural signals,” Neural computation, vol. 18, 2006, pp. 2465–2494. G.H. Mulliken, S. Musallam, and R.A. 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