nips nips2005 nips2005-20 knowledge-graph by maker-knowledge-mining

20 nips-2005-Affine Structure From Sound

Source: pdf

Author: Sebastian Thrun

Abstract: We consider the problem of localizing a set of microphones together with a set of external acoustic events (e.g., hand claps), emitted at unknown times and unknown locations. We propose a solution that approximates this problem under a far ﬁeld approximation deﬁned in the calculus of afﬁne geometry, and that relies on singular value decomposition (SVD) to recover the afﬁne structure of the problem. We then deﬁne low-dimensional optimization techniques for embedding the solution into Euclidean geometry, and further techniques for recovering the locations and emission times of the acoustic events. The approach is useful for the calibration of ad-hoc microphone arrays and sensor networks. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract We consider the problem of localizing a set of microphones together with a set of external acoustic events (e. [sent-2, score-0.898]

2 , hand claps), emitted at unknown times and unknown locations. [sent-4, score-0.127]

3 We propose a solution that approximates this problem under a far ﬁeld approximation deﬁned in the calculus of afﬁne geometry, and that relies on singular value decomposition (SVD) to recover the afﬁne structure of the problem. [sent-5, score-0.312]

4 We then deﬁne low-dimensional optimization techniques for embedding the solution into Euclidean geometry, and further techniques for recovering the locations and emission times of the acoustic events. [sent-6, score-0.826]

5 The approach is useful for the calibration of ad-hoc microphone arrays and sensor networks. [sent-7, score-0.53]

6 1 Introduction Consider a set of acoustic sensors (microphones) for detecting acoustic events in the environment (e. [sent-8, score-1.169]

7 The structure from sound (SFS) problem addresses the problem of simultaneously localizing a set of N sensors and a set of M external acoustic events, whose locations and emission times are unknown. [sent-11, score-1.206]

8 The SFS problem is relevant to the spatial calibration problem for microphone arrays. [sent-12, score-0.283]

9 Classically, microphone arrays are mounted on ﬁxed brackets of known dimensions; hence there is no spatial calibration problem. [sent-13, score-0.275]

10 Ad-hoc microphone arrays, however, involve a person placing microphones at arbitrary locations with limited knowledge as to where they are. [sent-14, score-0.457]

11 Today’s best practice requires a person to measure the distance between the microphones by hand, and to apply algorithms such as multi-dimensional scaling (MDS) [1] for recovering their locations. [sent-15, score-0.256]

12 When sensor networks are deployed from the air [4], manual calibration may not be an option. [sent-16, score-0.34]

13 Others require a capability to emit and sense wireless radio signals [5] or sounds [9, 10], which are then used to estimate relative distances between microphones (directly or indirectly, as in [9]). [sent-18, score-0.335]

14 Unfortunately, wireless signal strength is a poor estimator of range, and active acoustic and GPS localization techniques are uneconomical in that they consume energy and require additional hardware. [sent-19, score-0.484]

15 In contrast, SFS relies on environmental acoustic events such as hand claps, which are not generated by the sensor network. [sent-20, score-0.858]

16 A related paper [3] describes a technique for incrementally localizing a microphone relative to a well-calibrated microphone array through external sound events. [sent-22, score-0.687]

17 In this paper, the structure from sound (SFS) problem is deﬁned as the simultaneous localization problem of N sound sensors and M acoustic events in the environment detected by these sensors. [sent-23, score-1.289]

18 Each event occurs at an unknown time and an unknown location. [sent-24, score-0.159]

19 The sensors are able to measure the detection times of the event. [sent-25, score-0.197]

20 We assume that the clocks of the sensors are synchronized (see [6]); that events are spaced sufﬁciently far apart in time to make the association between different sensors unambiguous; and we also assume absence of sound reverberation. [sent-26, score-0.877]

21 Under the assumption of independent and identically distributed (iid) Gaussian noise, the SFS problem can be formulated as a least squares problem in a space over three types of variables: the locations of the microphones, the locations of the acoustic events, and their emission times. [sent-28, score-0.871]

22 The key transformation involves a far ﬁeld approximation, which presupposes that the sound sources are relatively far away from the sensors. [sent-31, score-0.379]

23 This approximation reformulates the problem as one of recovering the incident angle of the acoustic signal, which is the same for all sensors for any ﬁxed acoustic event. [sent-32, score-1.309]

24 The resulting optimization problem is still non-linear; however, by relaxing the laws of Euclidean geometry into the more general calculus of afﬁne geometry, the optimization problem can be solved by singular value decomposition (SVD). [sent-33, score-0.264]

25 The resulting solution is mapped back into Euclidean space by optimizing a matrix of size 2 × 2, which is easily carried out using gradient descent. [sent-34, score-0.192]

26 A subsequent non-linear optimization step overcomes the far ﬁeld approximation and enables the algorithm to recover locations and emission times of the deﬁning acoustic events. [sent-35, score-0.893]

27 SFM is of course deﬁned for cameras, not for microphone arrays. [sent-39, score-0.15]

28 This paper establishes an afﬁne solution to the structure from sound problem that tends to work well in practice. [sent-41, score-0.292]

29 1 Setup We are given N sensors (microphones) located in a 2D plane. [sent-43, score-0.164]

30 We shall denote the location of the i-th sensor by (xi yi ), which deﬁned the following sensor location matrix of size N × 2:   X =    x1 x2 . [sent-44, score-0.781]

31 yN (1) We assume that the sensor array detects M acoustic events. [sent-50, score-0.737]

32 Each event has as unknown coordinate and an unknown emission time. [sent-51, score-0.338]

33 The coordinate of the j-th event shall be denoted (aj bj ), providing us with the event location matrix A of size M × 2. [sent-52, score-0.592]

34 The emission time of the j-th acoustic event is denoted tj , resulting in the vector T of length M :     a1 A =  a2  . [sent-53, score-0.703]

35 In problems such as sensor calibration, only X is of interest. [sent-63, score-0.255]

36 The data establishes the detection times of the acoustic events by the individual sensors. [sent-67, score-0.636]

37 dN,M (3) aj bj (4) Here each di,j denotes the detection time of acoustical event j by sensor i. [sent-80, score-0.75]

38 Even if all acoustic events sound alike, the correspondence between different detections is easily established as long as there exists sufﬁciently long time gaps between any two sound events. [sent-82, score-1.007]

39 The matrix D is a random ﬁeld induced by the laws of sound propagation (without reverberation). [sent-83, score-0.273]

40 In the absence of measurement noise, each di,j is the sum of the corresponding emission time tj , plus the time it takes for sound to travel from (aj bj ) to (xi yi ): di,j = xi yi tj + c−1 − Here | · | denotes the L2 norm (Euclidean distance), and c denoted the speed of sound. [sent-84, score-0.831]

41 This gives us the relative location matrix for the sensors:   x2 − x 1 ¯ X y2 − y 1 y3 − y 1   . [sent-88, score-0.131]

42 = xN − x 1 (5) This relative sensor location matrix is of dimension (N − 1) × 2. [sent-94, score-0.386]

43 It shall prove convenient to subtract from the arrival time di,j the arrival time d1,j measured by the ﬁrst sensor i = 1. [sent-95, score-0.479]

44 In the relative arrival time, the absolute emission times tj cancel out: = c−1 xi yi tj + c−1 = ∆i,j aj bj − xi yi aj bj − − tj − c−1 − We now deﬁne the matrix of relative arrival times:  ∆ = d2,1 − d1,1  d3,1 − d1,1  . [sent-97, score-1.577]

45 aj bj aj bj (6)  d2,M − d1,M d3,M − d1,M   . [sent-107, score-0.804]

46 4 Least Squares Formulation The relative sensor locations X and the corresponding locations of the acoustic events A can now be recovered through the following least squares problem. [sent-111, score-1.197]

47 N A∗ , X ∗ = M argmin X,A i=2 j=1 xi yi − aj bj − aj bj 2 − ∆i,j (8) The minimum of this expression is a maximum likelihood solution for the SFS problem under the assumption of iid Gaussian measurement noise. [sent-113, score-1.051]

48 If emission times are of interest, they are now easily recovered by the following weighted mean: T∗ = 1 N N xi yi di,j − c aj bj − (9) i=1 The minimum of Eq. [sent-114, score-0.707]

49 This shall not concern us, as we shall be content with any solution of Eq. [sent-117, score-0.188]

50 As we shall show experimentally, such gradient algorithms work poorly in practice because of the large number of local minima. [sent-122, score-0.174]

51 3 The Far Field Approximation The essence of our approximation pertains to the fact that for far range acoustic events— i. [sent-123, score-0.524]

52 , events that are (inﬁnitely) far away from the sensor array—the incoming sound wave hits each sensor at the same incident angle. [sent-125, score-1.217]

53 Put differently, the rays connecting the location of an acoustic event (aj bj ) with each of the perceiving sensors (xi yi ) are approximately parallel for all i (but not for all j! [sent-126, score-0.968]

54 Under the far ﬁeld approximation, these incident angles are entirely parallel. [sent-128, score-0.311]

55 Thus, all that matters are the incident angle of the acoustic events. [sent-129, score-0.637]

56 To derive an equation for this case, it shall prove convenient to write the Euclidean distance between a sensor and an acoustic event as a function of the incident angle α. [sent-130, score-1.057]

57 6, this gives us the following expression for ∆i,j : xi yi aj bj = c−1 ≈ c−1 = ∆i,j c−1 (cos αj sin αj ) − − aj − x i bj − y i (cos αj sin αj ) aj bj − aj bj xi yi (13) This leads to the following non-linear least squares problem for the desired sensor locations: ∗ ∗ X ∗ , α1 , . [sent-137, score-2.273]

58 ,αM X cos α1 sin α1 cos α2 sin α2 ··· ··· cos αM sin αM 2 −∆ (14) The reader many notice that in this formulation of the SFS problem, the locations of the sound events (aj , bj ) have been replaced by αj , the incident angles of the sound waves. [sent-143, score-1.685]

59 One might think of this as the “ortho-acoustic” model of sound propagation (in analogy to the orthographic camera model in computer vision). [sent-144, score-0.223]

60 Specifically, we know that the matrix ∆ can be decomposed as into three other matrices, U , V , and W : UV W T = svd(∆) (18) where U is a matrix of size (N − 1) × 2, V a diagonal matrix of eigenvalues of size 2 × 2, and W a matrix of size M × 2. [sent-153, score-0.184]

61 The key insight is that for any invertible 2 × 2 matrix C, X U V C −1 = and Γ = CW T (20) is equally a solution to the factorization problem in Eq. [sent-163, score-0.136]

62 c 8 10 N, M (here N=M) 12 14 log−error (95% confidence intervals) error (95% confidence intervals) 3. [sent-177, score-0.143]

63 −4 −5 4 6 8 10 N, M (here N=M) 12 14 Figure 1: (a) Error and (b) log error for three different algorithms: gradient descent (red), SVD (blue), and SVD followed by gradient descent (green). [sent-182, score-0.479]

64 (a) ground truth (b) gradient descent (c) SVD (d) SVD + grad. [sent-185, score-0.225]

65 sensors acoustic events Figure 2: Typical SFS results for a simulated array of nine microphones spaced in a regular grid, surrounded by 9 sounds arranged on a circle. [sent-187, score-1.149]

66 (a) Ground truth; (b) Result of plain gradient descent after convergence; the dashed lines visualize the residual error; (c) Result of the SVD with sound directions as indicated; and (d) Result of gradient descent initialized with our SVD result. [sent-188, score-0.749]

67 In (tens of thousands of) experiments with synthetic noise-free data, we ﬁnd empirically that gradient descent reliably converges to the globally optimal solution. [sent-193, score-0.251]

68 5 Recovering the Acoustic Event Locations and Emission Times With regards to the acoustic events, the optimization for the far ﬁeld case only yields the incident angles. [sent-194, score-0.686]

69 In the near ﬁeld setting, in which the incident angles tend to differ for different sensors, it may be desirable to recover the locations A of the acoustic event and the corresponding emission times T . [sent-195, score-1.086]

70 To determine these variables, we use the vector X ∗ from the far ﬁeld case as mere starting points in a subsequent gradient search. [sent-196, score-0.167]

71 The event location matrix A is initialized by selecting points sufﬁciently far away along the estimated incident angle for the far ﬁeld approximation to be sound: A ∗ = ∗ kΓ∗ T (24) ∗ Here Γ = C W with C deﬁned in Eq. [sent-197, score-0.606]

72 22, and k is a multiple of the diameter of the locations in X. [sent-198, score-0.167]

73 With this initial guess for A, we apply gradient descent to optimize Eq. [sent-199, score-0.225]

74 1 graphs the residual error as a function of the number of sensors 2 (a) Error 3 2. [sent-205, score-0.233]

75 2 4 6 8 diameter ratio of events vs sensor array 10 log−error (95% confidence intervals) error (95% confidence intervals) 3. [sent-212, score-0.737]

76 −4 0 2 4 6 8 diameter ratio of events vs sensor array 10 Figure 3: (a) Error and (b) log-error for three different algorithms (gradient descent in red, SVD in blue, and SVD followed by gradient descent in green), graphed here for varying distances of the sound events to the sensor array. [sent-217, score-1.6]

77 hand-measured m o t e s (c) Result of gradient descent (d) SVD and GD sounds motes Figure 4: Results using our seven sensor motes as the sensor array, and a seventh mote to generate sound events. [sent-221, score-1.297]

78 (a) A mote; (b) the globally optimal solution (big circles) compared to the handmeasures locations (small circles); (c) a typical result of vanilla gradient descent; and (d) the result of our approach, all compared to the optimal solution given the (noisy) data. [sent-222, score-0.325]

79 Clearly, as N and M increase, plain gradient descent tends to diverge, whereas our approach converges. [sent-225, score-0.282]

80 Each data point in these graphs was obtained by averaging 1,000 random conﬁgurations, in which sensors were sampled uniformly within an interval of 1×1m; sounds were placed at varying ranges, from 2m to 10m. [sent-226, score-0.235]

81 This ﬁgure plots (a) a simulated sensor array consisting of 9 sensors with 9 sound sources arranged in a circle; and (b)-(d) the resulting reconstructions of our three methods. [sent-230, score-0.727]

82 An interesting question pertains to the effect of the far ﬁeld approximation in cases where it is clearly violated. [sent-232, score-0.122]

83 To examine the robustness of our approach, we ran a series of experiments in which we varied the diameter of the acoustic events relative to the diameter of the sensors. [sent-233, score-0.751]

84 If this parameter is 1, the acoustic events are emitted in the same region as the microphones; for values such as 10, the events are far away. [sent-234, score-0.897]

85 The further away the acoustic events, the better our results. [sent-237, score-0.427]

86 However, even for nearby events, for which the far ﬁeld assumption is clearly invalid, our approach generates results that are no worse than those of the plain gradient descent technique. [sent-238, score-0.347]

87 We also implemented our approach using a physical sensor array. [sent-239, score-0.255]

88 4 plots empirical results using a microphone array comprised of seven Crossbow sensor motes, one of which is shown in Panel (a). [sent-241, score-0.533]

89 Panels (b-d) compare the recovered structure with the one that globally minimizes the LMS error, which we obtain by running gradient descent using the hand-measured locations as starting point. [sent-242, score-0.421]

90 4 shows the solution of plain gradient descent applied to applied to Eq. [sent-247, score-0.326]

91 7 Discussion This paper considered the structure from sound problem and presented an algorithm for solving it. [sent-251, score-0.248]

92 Our approach makes is possible to simultaneously recover the location of a collection of microphones, the locations of external acoustic events detected by these microphones, and the emission times for these events. [sent-252, score-1.036]

93 By resorting to afﬁne geometry, our approach overcomes the problem of local minima in the structure from sound problem. [sent-253, score-0.278]

94 It shall further be interesting to investigate data association problems in the SFS framework, and to develop parallel algorithms that can be implemented on sensor networks with limited communication resources. [sent-257, score-0.35]

95 Finally, of great interest should be the incomplete data case in which individual sensors may fail to detect acoustic events—a problem studied in [2]. [sent-258, score-0.59]

96 Microphone array position calibration by basis-point classical multidimensional scaling. [sent-265, score-0.165]

97 Maximum likelihod source localization and unknown sensor location estimation for wideband signals in the near-ﬁeld. [sent-279, score-0.422]

98 Deployment and connectivity repair of a sensor net with a ﬂying robot. [sent-289, score-0.255]

99 Wireless sensor networks: A new regime for time synchronization. [sent-300, score-0.255]

100 Position calibration of microphones and loudspeakers in distributed computing platforms. [sent-318, score-0.283]

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The current splitter takes an input current (Ibias, figure 4) and divides it into branches which recursively halve the current, i.e., the first branch gives ½ Ibias, the second branch ¼ Ibias, the third branch 1/8 Ibias and so on. These currents can be used together with digitally controlled switches as a Digital-to-Analogue converter. By holding default low and setting C5:C0 appropriately, any gain – from 4 to 0.125 – can be set. To save on output pins the program bits (C5:C0) for each of the three gain cells are set via a single 18-bit shift register in bit-serial fashion. Summing the output of the three gain circuits in the current domain simply involves connecting three wires together. Therefore, a natural option for the filters that follow is to use current domain filters. In our case we have chosen to implement log-domain filters using MOS transistors operating in weak inversion. Figure 5 shows the basic building blocks for the filters – the Tau Cell [8] and the multiplier cell – and block diagrams showing how these blocks were connected to create the necessary filtering blocks. The Tau Cell is a log-domain filter which has the firstorder response: I out 1 , = I in sτ + 1 where τ = nC aVT Ia and n = the slope factor, VT = thermal voltage, Ca = capacitance, and Ia = bias current. In figure 5, the input currents to the Tau Cell, Imult and A*Ia, are only used when building a second-order filter. The multiplier cell is simply a translinear loop where: I out1 ∗ I mult = I out 2 ∗ AI a or Imult = AIaIout2/Iout1. The configurations of the Tau Cell to get particular responses are covered in [8] along with the corresponding equations. The high frequency filter of Figure 2 is implemented by the high-pass filter in Figure 5 with a corner frequency of 17kHz. The low frequency filter, however, is divided into two parts since the biological filter’s response (see for example Figure 3A in [9]) separates well into a narrow second-order band-pass filter with a 10kHz resonant frequency and a wide band-pass filter made from a first-order high-pass filter with a 3kHz corner frequency followed by a first-order low-pass filter with a 12kHz corner frequency. These filters are then added together to reproduce the biological filter. The filters’ responses can be adjusted post fabrication via their bias currents. This allows for compensation due to processing and matching errors. Figure 4: The Gain Cell above is used to convert the differential voltage input from the delay cells into a single-ended current output. The gain of each cell is controllable via a programmable current cell (p_splitter). An on-chip bias generator [7] was used to create all the necessary current biases on the chip. All the main blocks (delays, gain cells and filters), however, can have their on-chip bias currents overridden through external pins on the chip. The chip was fabricated using the MOSIS AMI 1.6µm technology and designed using the Cadence Custom IC Design Tools (5.0.33). 3 Methods The chip was tested using sound generated on a computer and played through a soundcard to the chip. Responses from the chip were recorded by an oscilloscope, and uploaded back to the computer on completion. Given that the output from the chip and the gain circuits is a current, an external current-sense circuit built with discrete components was used to enable the output to be probed by the oscilloscope. Figure 5: The circuit diagrams for the log-domain filter building blocks – The Tau Cell and The Multiplier – along with the block diagrams for the three filters used in the aVLSI model. Initial experiments were performed to tune the delays and gains. After that, recordings were taken of the directional frequency responses. Sounds were generated by computer for each chip input to simulate moving the forelegs by delaying the sound by the appropriate amount of time; this was a much simpler solution than using microphones and moving them using motors. 4 Results The aVLSI chip was tested to measure its gains and delays, which were successfully tuned to the appropriate values. The chip was then compared with the simulation to check that it was faithfully modelling the system. A result of this test at 4kHz (approximately the cricket calling-song frequency) is shown in Figure 6. Apart from a drop in amplitude of the signal, the response of the circuit was very similar to that of the simulator. The differences were expected because the aVLSI circuit has to deal with real-world noise, whereas the simulated version has perfect signals. Examples of the gain versus frequency response of the two log-domain band-pass filters are shown in Figure 7. Note that the narrow-band filter peaks at 6kHz, which is significantly above the mating song frequency of the cricket which is around 4.5kHz. This is not a mistake, but is observed in real crickets as well. As stated in the introduction, a range of further testing results with both the circuit and the simulator are being published in [6]. 5 D i s c u s s i on The aVLSI auditory sensor in this research models the hearing of the field cricket Gryllus bimaculatus. It is a more faithful model of the cricket auditory system than was previously built in [4], reproducing all the acoustic inputs, as well as the responses to frequencies of both the co specific calling song and bat echolocation chirps. It also generates outputs corresponding to the two sets of behaviourally relevant auditory receptor fibres. Results showed that it matched the biological data well, though there were some inconsistencies due to an error in the specification that will be addressed in a future iteration of the design. A more complete implementation across all frequencies was impractical because of complexity and size issues as well as serving no clear behavioural purpose. Figure 6: Vibration amplitude of the left (dotted) and right (solid) virtual tympana measured in decibels in response to a 4kHz tone in simulation (left) and on the aVLSI chip (right). The plot shows the amplitude of the tympanal responses as the sound source is rotated around the cricket. Figure 7: Frequency-Gain curves for the narrow-band and wide-band bandpass filters. The long-term aim of this work is to better understand simple sensorimotor control loops in crickets and other insects. The next step is to mount this circuitry on a robot to carry out behavioural experiments, which we will compare with existing and new behavioural data (such as that in [10]). This will allow us to refine our models of the neural circuitry involved. Modelling the sensory afferent neurons in hardware is necessary in order to reduce processor load on our robot, so the next revision will include these either onboard, or on a companion chip as we have done before [11]. We will also move both sides of the auditory system onto a single chip to conserve space on the robot. It is our belief and experience that, as a result of this intelligent pre-processing carried out at the sensor level, the neural circuits necessary to accurately model the behaviour will remain simple. Acknowledgments The authors thank the Institute of Neuromorphic Engineering and the UK Biotechnology and Biological Sciences Research Council for funding the research in this paper. References [1] R. Wehner. Matched ﬁlters – neural models of the external world. J Comp Physiol A, 161: 511–531, 1987. [2] A. Michelsen, A. V. Popov, and B. Lewis. Physics of directional hearing in the cricket Gryllus bimaculatus. Journal of Comparative Physiology A, 175:153–164, 1994. [3] A. Michelsen. The tuned cricket. News Physiol. Sci., 13:32–38, 1998. [4] H. H. Lund, B. Webb, and J. Hallam. A robot attracted to the cricket species Gryllus bimaculatus. In P. Husbands and I. Harvey, editors, Proceedings of 4th European Conference on Artiﬁcial Life, pages 246–255. MIT Press/Bradford Books, MA., 1997. [5] R Reeve and B. Webb. New neural circuits for robot phonotaxis. Phil. Trans. R. Soc. Lond. A, 361:2245–2266, August 2003. [6] R. Reeve, A. van Schaik, C. Jin, T. Hamilton, B. Torben-Nielsen and B. Webb Directional hearing in a silicon cricket. Biosystems, (in revision), 2005b [7] T. Delbrück and A. van Schaik, Bias Current Generators with Wide Dynamic Range, Analog Integrated Circuits and Signal Processing 42(2), 2005 [8] A. van Schaik and C. Jin, The Tau Cell: A New Method for the Implementation of Arbitrary Differential Equations, IEEE International Symposium on Circuits and Systems (ISCAS) 2003 [9] Kazuo Imaizumi and Gerald S. Pollack. Neural coding of sound frequency by cricket auditory receptors. The Journal of Neuroscience, 19(4):1508– 1516, 1999. [10] Berthold Hedwig and James F.A. Poulet. Complex auditory behaviour emerges from simple reactive steering. Nature, 430:781–785, 2004. [11] R. Reeve, B. Webb, A. Horchler, G. Indiveri, and R. Quinn. New technologies for testing a model of cricket phonotaxis on an outdoor robot platform. Robotics and Autonomous Systems, 51(1):41-54, 2005.

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