nips nips2008 nips2008-118 knowledge-graph by maker-knowledge-mining

118 nips-2008-Learning Transformational Invariants from Natural Movies

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Author: Charles Cadieu, Bruno A. Olshausen

Abstract: We describe a hierarchical, probabilistic model that learns to extract complex motion from movies of the natural environment. The model consists of two hidden layers: the ﬁrst layer produces a sparse representation of the image that is expressed in terms of local amplitude and phase variables. The second layer learns the higher-order structure among the time-varying phase variables. After training on natural movies, the top layer units discover the structure of phase-shifts within the ﬁrst layer. We show that the top layer units encode transformational invariants: they are selective for the speed and direction of a moving pattern, but are invariant to its spatial structure (orientation/spatial-frequency). The diversity of units in both the intermediate and top layers of the model provides a set of testable predictions for representations that might be found in V1 and MT. In addition, the model demonstrates how feedback from higher levels can inﬂuence representations at lower levels as a by-product of inference in a graphical model. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract We describe a hierarchical, probabilistic model that learns to extract complex motion from movies of the natural environment. [sent-4, score-0.671]

2 The model consists of two hidden layers: the ﬁrst layer produces a sparse representation of the image that is expressed in terms of local amplitude and phase variables. [sent-5, score-1.219]

3 The second layer learns the higher-order structure among the time-varying phase variables. [sent-6, score-0.698]

4 After training on natural movies, the top layer units discover the structure of phase-shifts within the ﬁrst layer. [sent-7, score-0.567]

5 We show that the top layer units encode transformational invariants: they are selective for the speed and direction of a moving pattern, but are invariant to its spatial structure (orientation/spatial-frequency). [sent-8, score-1.005]

6 The diversity of units in both the intermediate and top layers of the model provides a set of testable predictions for representations that might be found in V1 and MT. [sent-9, score-0.318]

7 1 Introduction A key attribute of visual perception is the ability to extract invariances from visual input. [sent-11, score-0.312]

8 In the realm of object recognition, the goal of invariant representation is quite clear: a successful object recognition system must be invariant to image variations resulting from different views of the same object. [sent-12, score-0.436]

9 While spatial invariants are essential for forming a useful representation of the natural environment, there is another, equally important form of visual invariance, namely transformational invariance. [sent-13, score-0.688]

10 A transformational invariant refers to the dynamic visual structure that remains the same when the spatial structure changes. [sent-14, score-0.61]

11 For example, the property that a soccer ball moving through the air shares with a football moving through the air is a transformational invariant; it is speciﬁc to how the ball moves but invariant to the shape or form of the object. [sent-15, score-0.463]

12 Here we seek to learn such invariants from the statistics of natural movies. [sent-16, score-0.244]

13 There have been numerous efforts to learn spatial invariants [1, 2, 3] from the statistics of natural images, especially with the goal of producing representations useful for object recognition [4, 5, 6]. [sent-17, score-0.471]

14 However, there have been few attempts to learn transformational invariants from natural sensory data. [sent-18, score-0.468]

15 Furthermore, it is unclear to what extent these models have captured the diversity of transformations in natural visual scenes or to what level of abstraction their representations produce transformational invariants. [sent-20, score-0.544]

16 However, this type of model does not capture the abstract property of motion because each unit is bound to a speciﬁc orientation, spatial-frequency and location within the image—i. [sent-22, score-0.318]

17 1 Here we describe a hierarchical probabilistic generative model that learns transformational invariants from unsupervised exposure to natural movies. [sent-25, score-0.625]

18 The latter approach characterizes most models of form and motion processing in the visual cortical hierarchy [6, 12], but suffers from the fact that information about these properties is not bound together—i. [sent-27, score-0.397]

19 , it is not possible to reconstruct an image sequence from a representation in which form and motion have been extracted by separate and independent mechanisms. [sent-29, score-0.439]

20 In the model we propose here, form and motion are factorized, meaning that extracting one property depends upon the other. [sent-31, score-0.318]

21 We show that when such a model is adapted to natural movies, the top layer units learn to extract transformational invariants. [sent-33, score-0.815]

22 The diversity of units in both the intermediate layer and top layer provides a set of testable predictions for representations that might be found in V1 and MT. [sent-34, score-0.949]

23 The model consists of an input layer and two hidden layers as shown in Figure 1. [sent-37, score-0.519]

24 The input layer represents the time-varying image pixel intensities. [sent-38, score-0.518]

25 The ﬁrst hidden layer is a sparse coding model utilizing complex basis functions, and shares many properties with subspace-ICA [13] and the standard energy model of complex cells [14]. [sent-39, score-0.946]

26 The second hidden layer models the dynamics of the complex basis function phase variables. [sent-40, score-0.947]

27 The sparse coding model imposes a kurtotic, independent prior over the coefﬁcients, and when adapted to natural image patches the Ai (x) converge to a set of localized, oriented, multiscale functions similar to a Gabor wavelet decomposition of images. [sent-43, score-0.546]

28 We propose here a generalization of the sparse coding model to complex variables that is primarily motivated from two observations of natural image statistics. [sent-44, score-0.451]

29 These are most clearly seen as circularly symmetric, yet kurtotic distributions among pairs of coefﬁcients corresponding to neighboring basis functions, as ﬁrst described by Zetzsche [17]. [sent-46, score-0.312]

30 As pointed out by Hyvarinen [3], the temporal evolution of a coefﬁcient in response to a movie, ui (t), can be well described in terms of the product of a smooth amplitude envelope multiplied by a quickly changing variable. [sent-51, score-0.304]

31 A similar result from Kording [1] indicates that temporal continuity in amplitude provides a strong cue for learning local invariances. [sent-52, score-0.301]

32 Note that the basis functions are only functions of space. [sent-58, score-0.232]

33 Therefore, the temporal dynamics within image sequences will be expressed in the temporal dynamics of the amplitude and phase. [sent-59, score-0.572]

34 The second term in the exponential imposes temporal stability on the time rate of change of the amplitudes and is given by: Sla (ai (t), ai (t−1)) = (ai (t) − ai (t−1))2 . [sent-63, score-0.82]

35 2 Phase Transformations Given the decomposition into amplitude and phase variables, we now have a non-linear representation of image content that enables us to learn its structure in another linear generative model. [sent-67, score-0.709]

36 In particular, the dynamics of objects moving in continuous trajectories through the world over short epochs will be encoded in the population activity of the phase variables φi . [sent-68, score-0.421]

37 Furthermore, because we have encoded these trajectories with an angular variable, many transformations in the image domain that would otherwise be nonlinear in the coefﬁcients ui will now be linearized. [sent-69, score-0.342]

38 This linear relationship allows us to model the time-rate of change of the phase variables with a simple linear generative model. [sent-70, score-0.329]

39 We thus model the ﬁrst-order time derivative of the phase variables as follows: ˙ φi (t) = Dik wk (t) + νi (t) (6) k ˙ where φi = φi (t) − φi (t−1), and D is the basis function matrix specifying how the high-level ˙ variables wk inﬂuence the phase shifts φi . [sent-71, score-0.925]

40 The additive noise term, νi , represents uncertainty or noise in the estimate of the phase time-rate of change. [sent-72, score-0.259]

41 As before, we impose a sparse, independent distribution on the coefﬁcients wk , in this case with a sparse cost function given as: Sw (wk (t)) = β log 1 + wk (t) σ 2 (7) The uncertainty over the phase shifts is given by a von Mises distribution: p(νi ) ∝ exp(κ cos(νi )). [sent-73, score-0.554]

42 Thus, the log-posterior over the second layer units is given by ˙ κ cos(φi − [Dw(t)]i ) + E2 = − t Sw (wk (t)) k i∈{ai (t)>0} 3 (8) Figure 1: Graph of the hierarchical model showing the relationship among hidden variables. [sent-74, score-0.606]

43 Because the angle of a variable with 0 amplitude is undeﬁned, we exclude angles where the corresponding amplitude is 0 from our cost function. [sent-75, score-0.436]

44 Note that in the ﬁrst layer we did not introduce any prior on the phase variables. [sent-76, score-0.625]

45 With our second hidden layer, E2 can be viewed as a log-prior on the time rate of change of the phase variables: ˙ ˙ φi (t). [sent-77, score-0.311]

46 ˙ Activating the w variables moves the prior away from φi (t) = 0, encouraging certain patterns of phase shifting that will in turn produce patterns of motion in the image domain. [sent-79, score-0.788]

47 The resulting dynamics for the amplitudes and phases in the ﬁrst layer are given by ∆ai (t) ∝ {bi (t)} − Sp (ai (t)) − Sl (ai (t), ai (t−1)) (9) ˙ i (t) − [Dw(t)]i ) + κ sin(φi (t+1) − [Dw(t+1)]i ) (10) ˙ ∆φi (t) ∝ {bi (t)} ai (t) − κ sin(φ with bi (t) = σ1 e−jφi (t) 2 N ∗ {zi (t) Ai (x)} . [sent-87, score-1.214]

48 The learning rule for the ﬁrst layer basis functions is given by the gradient of E1 with respect to Ai (x), using the values of the complex coefﬁcients inferred in eqs. [sent-92, score-0.632]

49 1 Simulation procedures The model was trained on natural image sequences obtained from Hans van Hateren’s repository at http://hlab. [sent-95, score-0.241]

50 They contain a variety of motions due to the movements of animals in the scene, camera motion, tracking (which introduces background motion), and motion borders due to occlusion. [sent-102, score-0.345]

51 We trained the ﬁrst layer of the model on 20x20 pixel image patches, using 400 complex basis functions Ai in the ﬁrst hidden layer initialized to random values. [sent-103, score-1.233]

52 During this initial phase of learning only the terms in E1 are used to infer the ai and φi . [sent-104, score-0.591]

53 Once the ﬁrst layer reaches convergence, we begin training the second layer, using 100 bases, Di , initialized to random values. [sent-105, score-0.366]

54 The second ˙ layer bases are initially trained on the MAP estimates of the ﬁrst layer φi inferred using E1 only. [sent-106, score-0.732]

55 After the second layer begins to converge we infer coefﬁcients in both the ﬁrst layer and the second layer simultaneously using all terms in E1 + E2 (we observed that this improved convergence in the second layer). [sent-107, score-1.098]

56 The bootstrapping of the second layer was used to speed convergence and we did not observe much change in the ﬁrst layer basis functions after the initial convergence. [sent-109, score-0.917]

57 2 Learned complex basis functions After learning, the ﬁrst layer complex basis functions converge to a set of localized, oriented, and bandpass functions with real and imaginary parts roughly in quadrature. [sent-113, score-1.026]

58 Figure 2(a) shows the real part, imaginary part, amplitude, and angle of two representative basis functions as a function of space. [sent-116, score-0.266]

59 Examining the amplitude of the basis function we see that it is localized and has a roughly Gaussian envelope. [sent-117, score-0.356]

60 The angle as a function of space reveals a smooth ramping of the phase in the direction perpendicular to the basis functions’ orientation. [sent-118, score-0.435]

61 Figure 2(b) displays the resulting image sequences produced by two representative basis functions as the amplitude and phase follow the indicated time courses. [sent-122, score-0.814]

62 The amplitude has the effect of controlling the presence of the feature within the image and the phase is related to the position of the edge within the image. [sent-123, score-0.668]

63 Importantly for our hierarchical model, the time derivative, or slope of the phase through time is directly related to the movement of the edge through time. [sent-124, score-0.314]

64 Figure 2(c) shows how the population of complex basis functions tiles the space of position (left) and spatial-frequency (right). [sent-125, score-0.345]

65 Each dot represents a different basis function according to its maximum amplitude in the space domain, or its maximum amplitude in the frequency domain computed via the 2D Fourier transform of each complex pair (which produces a single peak in the spatialfrequency plane). [sent-126, score-0.797]

66 This visualization will be useful for understanding what the phase shifting components D in the second layer have learned. [sent-128, score-0.677]

67 Some functions we believe arise from aliased temporal structure in the movies (row 1, column 5), and others are unknown (row 2, column 4). [sent-138, score-0.286]

68 Spatial Domain Frequency Domain Spatial Domain Frequency Domain Figure 3: Learned phase shifting components. [sent-140, score-0.311]

69 The phase shift components generate movements within the image that are invariant to aspects of the spatial structure such as orientation and spatial-frequency. [sent-141, score-0.679]

70 4 Discussion and conclusions The computational vision community has spent considerable effort on developing motion models. [sent-146, score-0.335]

71 Of particular relevance to our work is the Motion-Energy model [14], which signals motion via the amplitudes of quadrature pair ﬁlter outputs, similar to the responses of complex neurons in V1. [sent-147, score-0.469]

72 Simoncelli & Heeger have shown how it is possible to extract motion by pooling over a population of such units lying within a common plane in the 3D Fourier domain [12]. [sent-148, score-0.516]

73 Our model addresses each of these problems: it learns from the statistics of natural movies how to best tile the joint domain of position and motion, and it captures complex motion beyond uniform translation. [sent-151, score-0.799]

74 The use of phase information for computing motion is not new, and was used by Fleet and Jepson [20] to compute optic ﬂow. [sent-153, score-0.546]

75 In addition, as shown in Eero Simoncelli’s Thesis, one can establish a formal equivalence between phase-based methods and motion energy models. [sent-154, score-0.287]

76 Here we argue that phase provides a convenient representation as it linearizes trajectories in coefﬁcient space and thus allows one to capture the higher-order structure via a simple linear generative model. [sent-155, score-0.37]

77 Whether or how phase is represented in V1 is not known, 6 (a) (c) (b) (d) Figure 4: Visualization of learned transformational invariants (best viewed as animations in movie TransInv Figure4x. [sent-156, score-0.757]

78 Each phase-shift component produces a pattern of motion that is invariant to the spatial structure contained within the image. [sent-158, score-0.573]

79 Each panel displays the induced image transformations for a different basis function, Di . [sent-159, score-0.36]

80 Induced motions are shown for four different image patches with the original static patch displayed in the center position. [sent-160, score-0.384]

81 The ﬁnal image in each sequence shows the pixel-wise variance of the transformation (white values indicate where image pixels are changing through time, which may be difﬁcult to discern in this static presentation). [sent-162, score-0.304]

82 The example in (a) produces global motion in the direction of 45 deg. [sent-163, score-0.376]

83 The next example (b) produces local vertical motion in the lower portion of the image patch only. [sent-166, score-0.635]

84 Note that in the ﬁrst patch the strong edge in the lower portion of the patch moves while the edge in the upper portion remains ﬁxed. [sent-167, score-0.262]

85 Again, this component produces similar transformations irrespective of the spatial structure contained in the image. [sent-168, score-0.272]

86 The example in (c) produces horizontal motion in the left part of the image in the opposite direction of horizontal motion in the right half (the two halves of the image either converge or diverge). [sent-169, score-1.039]

87 Note that the oriented structure in the ﬁrst two patches becomes more closely spaced in the leftmost patch and is more widely spaced in the right most image. [sent-170, score-0.261]

88 This is seen clearly in the third image as the spacing between the vertical structure is most narrow in the leftmost image and widest in the rightmost image. [sent-171, score-0.389]

89 Instead of independent processing streams focused on form perception and motion perception, the two streams may represent complementary aspects of visual information: spatial invariants and transformational invariants. [sent-177, score-0.975]

90 Importantly though, in our model information about form and motion is bound together since it is computed by a process of factorization rather than by independent mechanisms in separate streams. [sent-179, score-0.318]

91 Our model also illustrates a functional role for feedback between higher visual areas and primary visual cortex, not unlike the proposed inference pathways suggested by Lee and Mumford [22]. [sent-180, score-0.251]

92 The ﬁrst layer units are responsive to visual information in a narrow spatial window and narrow spatial frequency band. [sent-181, score-0.924]

93 However, the top layer units receive input from a diverse population of ﬁrst layer units and can thus disambiguate local information by providing a bias to the time rate of change of the phase variables. [sent-182, score-1.305]

94 Because the second layer weights D are adapted to the statistics of natural movies, these biases will be consistent with the statistical distribution of motion occurring in the 7 natural environment. [sent-183, score-0.769]

95 Most obviously, the graphical model in Figure 1 begs the question of what would be gained by modeling the joint distribution over the amplitudes, ai , in addition to the phases. [sent-186, score-0.363]

96 To some degree, this line of approach has already been pursued by Karklin & Lewicki [2], and they have shown that the high level units in this case learn spatial invariants within the image. [sent-187, score-0.398]

97 We are thus eager to combine both of these models into a uniﬁed model of higher-order form and motion in images. [sent-188, score-0.318]

98 A selection model for motion processing in area MT of primates. [sent-233, score-0.318]

99 Invariant global motion recognition in the dorsal visual system: A unifying theory. [sent-248, score-0.445]

100 Computation of component image velocity from local phase information. [sent-316, score-0.444]

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