nips nips2009 nips2009-137 knowledge-graph by maker-knowledge-mining

137 nips-2009-Learning transport operators for image manifolds

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Author: Benjamin Culpepper, Bruno A. Olshausen

Abstract: We describe an unsupervised manifold learning algorithm that represents a surface through a compact description of operators that traverse it. The operators are based on matrix exponentials, which are the solution to a system of ﬁrst-order linear differential equations. The matrix exponents are represented by a basis that is adapted to the statistics of the data so that the inﬁnitesimal generator for a trajectory along the underlying manifold can be produced by linearly composing a few elements. The method is applied to recover topological structure from low dimensional synthetic data, and to model local structure in how natural images change over time and scale. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Learning transport operators for image manifolds Bruno A. [sent-1, score-0.734]

2 edu Abstract We describe an unsupervised manifold learning algorithm that represents a surface through a compact description of operators that traverse it. [sent-6, score-0.631]

3 The operators are based on matrix exponentials, which are the solution to a system of ﬁrst-order linear differential equations. [sent-7, score-0.328]

4 The matrix exponents are represented by a basis that is adapted to the statistics of the data so that the inﬁnitesimal generator for a trajectory along the underlying manifold can be produced by linearly composing a few elements. [sent-8, score-0.343]

5 The method is applied to recover topological structure from low dimensional synthetic data, and to model local structure in how natural images change over time and scale. [sent-9, score-0.194]

6 1 Introduction It is well known that natural images occupy a small fraction of the space of all possible images. [sent-10, score-0.194]

7 Moreover, as images change over time in response to observer motion or changes in the environment they trace out particular trajectories along manifolds in this space. [sent-11, score-0.478]

8 In this paper, we derive methods for learning these transport operators from data. [sent-13, score-0.488]

9 Rather than simply learning a mapping of individual data points to a low-dimensional space, we seek a compact representation of the entire manifold via the operators that traverse it. [sent-14, score-0.598]

10 We investigate a direct application of the Lie approach to invariance [1] utilizing a matrix exponential generative model for transforming images. [sent-15, score-0.162]

11 This is in contrast to previous methods that rely mainly upon a ﬁrstorder Taylor series approximation of the matrix exponential [2,3], and bilinear models, in which the transformation variables interact multiplicatively with the input [4,5,6]. [sent-16, score-0.182]

12 It is also distinct from the class of methods that learn embeddings of manifolds from point cloud data [7,8,9,10]. [sent-17, score-0.205]

13 We share the goal of [12] of learning a model of the manifold which can then be generalized to new data. [sent-19, score-0.138]

14 Here we show how a particular class of transport operators for moving along manifolds may be learned from data. [sent-20, score-0.831]

15 Subsequently, we apply it to time-varying natural images and extrapolate along inferred trajectories to demonstrate super-resolution and temporal ﬁlling-in of missing video frames. [sent-22, score-0.496]

16 1 2 Problem formulation Let us consider an image of the visual world at time t as a point x ∈ RN , where the elements of x correspond to image pixels. [sent-23, score-0.221]

17 We describe the evolution of x as x = Ax, ˙ (1) where the matrix A is a linear operator capturing some action in the environment that transforms the image. [sent-24, score-0.209]

18 Such an action belongs to a family that occupies a subspace of RN ×N given by M A= Ψ m cm (2) m=1 for some M ≤ N 2 (usually M << N 2 ), with Ψm ∈ RN ×N . [sent-25, score-0.197]

19 The amount of a particular action from the dictionary Ψm that occurs is controlled by the corresponding cm . [sent-26, score-0.291]

20 At t = 0, a vision system takes an image x0 , and then makes repeated observations at intervals ∆t. [sent-27, score-0.146]

21 Given x0 , the solution to (1) traces out a continuously differentiable manifold of images given by xt = exp(At) x0 , which we observe periodically. [sent-28, score-0.405]

22 Our goal is to learn an appropriate set of bases, Ψ, that allow for a compact description of this set of transformations by training on many pairs of related observations. [sent-29, score-0.301]

23 This generative model for transformed images has a number of attractive properties. [sent-30, score-0.128]

24 First, it factors apart the time-varying image into an invariant part (the initial image, x0 ) and variant part (the transformation, parameterized by the coefﬁcient vector c), thus making explicit the underlying causes. [sent-31, score-0.139]

25 Second, the learned exponential operators are quite powerful in terms of modeling capacity, compared to their linear counterparts. [sent-32, score-0.413]

26 These points are related through an exponentiated matrix that itself is composed of a few basis elements, plus zero-mean white i. [sent-37, score-0.189]

27 Gaussian noise, n: x1 = T(c) x0 + n T(c) = exp( Ψm cm ) . [sent-40, score-0.158]

28 Then ∂ exp(A)ij = Fαβ Uiα Vkα Ulβ Vjβ , (6) ∂Akl αβ 2 choose M ≤ N 2 initialize Ψ while stopping criteria is not met, pick x0 , x1 initialize c to zeros c ← arg minc E ∂E ∆Ψ = −η ∂Ψ sort Ψm by ||Ψm ||F M ← max m s. [sent-60, score-0.115]

29 Steps 8-9 shrink the subspace spanned by the dictionary if one or more of the elements have shrunk sufﬁciently in norm. [sent-67, score-0.182]

30 After computing two intermediate terms P and Q, P = UT (x1 x0 T + x0 x0 T TT )V Qkl = Vkα Ulβ Fαβ Pαβ , (8) (9) αβ the two partial derivatives for inference and learning are: ∂E ∂cm ∂E ∂Ψklm = Qkl Ψklm + ζ sgn(cm ) (10) kl = Qkl cm + γ Ψklm . [sent-72, score-0.251]

31 4 Experiments on point sets We ﬁrst test the model by applying it to simple datasets where the solutions are known: learning the topology of a sphere and a torus. [sent-74, score-0.255]

32 Second, we apply the model to learn the manifold of timevarying responses to a natural movie from complex oriented ﬁlters. [sent-75, score-0.431]

33 Related pairs of points on a torus are generated by choosing two angles θ0 , φ0 uniformly at random from [0, 2π]; two related angles θ1 , φ1 are produced by sampling from two von Mises distributions with means θ0 and φ0 , and concentration κ = 5 using the circular statistics toolbox of [15]. [sent-79, score-0.591]

34 Though parameterized by two angles, the coordinates of points on these surfaces are 3- and 4-dimensional; pairs of points xt for t = 0, 1 on the unit sphere are given by xt = (sin θt cos φt , sin θt sin φt , cos θt ), and points on a torus by xt = (cos θt , sin θt , cos φt , sin φt ). [sent-81, score-1.511]

35 5 −1 −1 (a) −1 (b) Figure 2: Orbits of learned sphere operators. [sent-94, score-0.203]

36 (a) Three Ψm basis elements applied to points at the six poles of the sphere, (1, 0, 0), (0, 1, 0), (0, 0, 1), (−1, 0, 0), (0, −1, 0), and (0, 0, −1). [sent-95, score-0.187]

37 The orbits are generated by setting x0 to a pole, then plotting xt = exp(Ψm t) x0 for t = [−100, 100]. [sent-96, score-0.383]

38 (b) When superimposed on top of each other, the three sets of orbits clearly deﬁne the surface of a sphere. [sent-97, score-0.326]

39 5 x3 −1 −1 x2 Figure 3: Orbits of learned torus operators. [sent-122, score-0.252]

40 Each row shows three projections of a Ψm basis element applied to a point on the surface of the torus. [sent-123, score-0.181]

41 The orbits shown are generated by setting x0 = (0, 1, 0, 1) then plotting xt = exp(Ψm t) x0 for t = [−1000, 1000] in projections constructed from each triplet of the four coordinates. [sent-124, score-0.383]

42 4 Figure 4: Learning transformations of oriented ﬁlter pairs across time. [sent-126, score-0.258]

43 The orbits of three three complex ﬁlter outputs in response to a natural movie. [sent-127, score-0.338]

44 The blue points denote the complex output for each frame in the movie sequence and are linked to their neighbors via the blue line. [sent-128, score-0.35]

45 The points circled in red were observed by the model, and the red curve shows an extrapolation along the estimated trajectory. [sent-129, score-0.126]

46 For cases where topology can be recovered, the solution is robust to the settings of γ and ζ – changing either variable by an order of magnitude does not change the solution, though it may increase the number of learning steps required to get to it. [sent-139, score-0.197]

47 In cases where the topology can not be recovered, the inﬂuence on the solution of the settings of γ and ζ is more subtle, as their relative values effectively trade-off the importance of data reconstruction and the sparsity of the vector c. [sent-140, score-0.211]

48 Figure 2 shows orbits produced by applying each of the remaining Ψm operators to points on the sphere. [sent-145, score-0.583]

49 Similar experiments are successful for the torus; Figure 3 shows trajectories of the operators learned for the torus. [sent-146, score-0.407]

50 As an intermediate step towards modeling time varying natural images, we investigate the model’s ability to learn the response surface for a single complex oriented ﬁlter to a moving image. [sent-147, score-0.367]

51 A complex pyramid is built from each frame in a movie, and pairs of ﬁlter responses 1 to 4 frames apart are observed by the model. [sent-148, score-0.466]

52 Four 2x2 basis functions are learned in the manner described above. [sent-149, score-0.15]

53 Figure 4 shows three representative examples that illustrate how well the model is able to extrapolate from the solution estimated using the learned basis Ψ, and complex responses from the same ﬁlter within a 4 frame time interval. [sent-150, score-0.444]

54 5 Experiments on movies In the image domain, our model has potential applications in temporal interpolation/ﬁlling in of video, super-resolution, compression, and geodesic distance estimation. [sent-152, score-0.157]

55 We apply the model to moving natural images and investigate the ﬁrst three applications; the third will be the subject of future work. [sent-153, score-0.254]

56 Here we report on the ability of the model to learn transformations across time, as well as across scales of the Laplacian pyramid. [sent-154, score-0.192]

57 Our data is many short grayscale video sequences of Africa from the BBC. [sent-155, score-0.136]

58 1 Time We apply the model to natural movies by presenting it with patches of adjacent frame pairs. [sent-157, score-0.375]

59 Using an analytically generated inﬁnitesimal shift operator, we ﬁrst run a series of experiments to determine the effect of local minima on the recovery of a known displacement through the minimization of E 5 Figure 5: Shift operator learned from synthetically transformed natural images. [sent-158, score-0.417]

60 The operator Ψ1 , displayed as an array of weights that, for each output pixel, shows the strength of its connection to each input pixel. [sent-159, score-0.129]

61 Because of the 1/f 2 falloff in the power spectrum of natural images, synthetic images with a wildly different distribution of spatial frequency content, such as uncorrelated noise, will not be properly shifted by this operator. [sent-161, score-0.318]

62 00 exponential linear 1st order Taylor approximation Figure 6: Interpolating between shifted images to temporally ﬁll in missing video frames. [sent-172, score-0.45]

63 Two images x0 and x1 are generated by convolving an image of a diagonal line and a shifted diagonal line by a 3x3 Gaussian kernel with σ = 0. [sent-173, score-0.305]

64 The top row shows the sequence of images xt = exp(At) x0 for t = 0. [sent-175, score-0.229]

65 The bottom row shows the sequence of images xt = (I + At) x0 , that is, the ﬁrst-order Taylor expansion of the matrix exponential, which performs poorly for shifts greater than one pixel. [sent-181, score-0.229]

66 Doing so requires a slight alteration to our inference algorithm: now we must solve a sequence of optimization problems on frame pairs convolved with a Gaussian kernel whose variance is progressively decreased. [sent-187, score-0.281]

67 At each step in the sequence, both frames are convolved by the kernel before a patch is selected. [sent-188, score-0.242]

68 For the ﬁrst step, the c variables are initialized to zero; for subsequent steps they are initialized to the solution of the previous step. [sent-189, score-0.227]

69 For control purposes, the video for this experiment comes from a camera ﬂy over; thus, most of the motion in the scene is due to camera motion. [sent-191, score-0.292]

70 We apply the model pairs of 11x11 patches, selected from random locations in the video, but discarding patches near the horizon where there is little or no motion. [sent-192, score-0.138]

71 We initialize M = 16; after learning, the basis function with the longest norm has the structure of a shift operator in the primary direction of motion taking place in the video sequence. [sent-193, score-0.517]

72 Using these 16 operators, we run inference on 1,000 randomly selected pairs of patches from a second video, not used during learning, and measure the quality of the reconstruction as the trajectory is used to predict into the future. [sent-194, score-0.25]

73 At 5 frames into the future, our model is able to maintain an average SNR of 7, compared to SNR 5 when a ﬁrst-order Taylor approximation is used in place of the matrix exponential; for comparison, the average SNR for the identity transformation model on this data is 1. [sent-195, score-0.116]

74 Since the primary form of motion going on in these small patches is translation, we also train a single operator using artiﬁcially translated natural data to make clear that the model can learn this case completely. [sent-196, score-0.4]

75 For this last experiment we take a 360x360 pixel frame of our natural movie, and continuously translate the entire frame in the Fourier domain by a displacement chosen uniformly at random from [0, 3] pixels. [sent-197, score-0.549]

76 We then randomly select a 15x15 region on the interior of the pair of frames and use the two 225 pixel vectors as our x0 and x1 . [sent-198, score-0.157]

77 We modify the objective function to be 1 γ E = ||W [x1 − T(c) x0 ]||2 + ||Ψm ||2 + ζ||c||1 , (12) 2 F 2 2 m where W is a binary windowing function that selects the central 9x9 region from a 15x15 patch; thus, the residual errors that come from new content translating into the patch are ignored. [sent-199, score-0.196]

78 After learning, the basis function Ψ1 (shown in ﬁgure 5) is capable of translating natural images up to 3 pixels while maintaining an average SNR of 16 in the 9x9 center region. [sent-200, score-0.324]

79 Figure 6 shows how this operator is able to correctly interpolate between two measurements of a shifted image in order to temporally up-sample a movie. [sent-201, score-0.394]

80 2 Scale The model can also learn to transform between successive scales in the Laplacian pyramid built from a single frame of a video sequence. [sent-203, score-0.423]

81 Figure 7 depicts the system transforming an image patch from scale 2 to 3 of a 256x256 pixel image. [sent-204, score-0.413]

82 We initialize M = 100, but many basis elements shrink during learning; we use only the 16 Ψm with non-negligible norm to encode a scale change. [sent-205, score-0.284]

83 The basis Ψ is initialized to mean-zero white Gaussian noise with variance 0. [sent-206, score-0.2]

84 01; the same inference and learning procedures as described for the point sets are then run on pairs x0 , x1 selected at random from the corpus of image sequences in the following way. [sent-207, score-0.152]

85 First, we choose a random frame from a random sequence, then up-sample and blur scale 3 of its Laplacian pyramid. [sent-208, score-0.237]

86 Second, we select an 8x8 patch from scale 2 (x0 ) of the corresponding up-blurred image patch (x1 ). [sent-209, score-0.38]

87 Were it not for the highly structured manifold on which natural images live, the proposition of ﬁnding an operator that maps a blurred, subsampled image to its high-resolution original state would seem untenable. [sent-210, score-0.552]

88 6 Discussion and conclusions We have shown that it is possible to learn low-dimensional parameterizations of operators that transport along non-linear manifolds formed by natural images, both across time and scale. [sent-212, score-0.823]

89 Our focus thus far has been primarily on understanding the model and how to properly optimize its parameters, 7 Scale 3 Estimated Scale 2 Actual Scale 2 Error (a) (b) (c) (d) Figure 7: Learning transformations across scale. [sent-213, score-0.18]

90 (a) Scale 3 of the Laplacian pyramid for a natural scene we wish to code, by describing how it transforms across scale, in terms of our learned dictionary. [sent-214, score-0.242]

91 (b) The estimated scale 2, computed by transforming 8x8 regions of the up-sampled and blurred scale 3. [sent-215, score-0.255]

92 Early attempts to model the manifold structure of images train on densely sampled point clouds and ﬁnd an embedding into a small number of coordinates along the manifold. [sent-221, score-0.378]

93 However such an approach does not actually constitute a model, since there is no function for mapping arbitrary points, or moving along the manifold. [sent-222, score-0.124]

94 Here, by learning operators that transport along the manifold we have been able to learn a compact description of its structure. [sent-226, score-0.788]

95 Previous attempts to learn Lie group operators have focused on linear approximations. [sent-229, score-0.34]

96 Here we show that utilizing the full Lie operator/matrix exponential in learning, while computationally intensive, is tractable, even in the extremely high dimensional cases required by models of natural movies. [sent-230, score-0.164]

97 Our spectral decomposition is the key component that enables this, and, in combination with careful mitigation of local minima in the objective function using a coarse-to-ﬁne technique, gives us the power to factor out large transformations from data. [sent-231, score-0.222]

98 One shortcoming of the approach described here is that transformations are modeled in the original pixel domain. [sent-232, score-0.22]

99 Potentially these transformations may be described more economically by working in a feature space, such as a sparse decomposition of the image. [sent-233, score-0.142]

100 (2004) Unsupervised learning of image manifolds by semideﬁnite programming. [sent-294, score-0.246]

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