nips nips2009 nips2009-172 knowledge-graph by maker-knowledge-mining

172 nips-2009-Nonparametric Bayesian Texture Learning and Synthesis

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Author: Long Zhu, Yuanahao Chen, Bill Freeman, Antonio Torralba

Abstract: We present a nonparametric Bayesian method for texture learning and synthesis. A texture image is represented by a 2D Hidden Markov Model (2DHMM) where the hidden states correspond to the cluster labeling of textons and the transition matrix encodes their spatial layout (the compatibility between adjacent textons). The 2DHMM is coupled with the Hierarchical Dirichlet process (HDP) which allows the number of textons and the complexity of transition matrix grow as the input texture becomes irregular. The HDP makes use of Dirichlet process prior which favors regular textures by penalizing the model complexity. This framework (HDP-2DHMM) learns the texton vocabulary and their spatial layout jointly and automatically. The HDP-2DHMM results in a compact representation of textures which allows fast texture synthesis with comparable rendering quality over the state-of-the-art patch-based rendering methods. We also show that the HDP2DHMM can be applied to perform image segmentation and synthesis. The preliminary results suggest that HDP-2DHMM is generally useful for further applications in low-level vision problems. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract We present a nonparametric Bayesian method for texture learning and synthesis. [sent-5, score-0.42]

2 A texture image is represented by a 2D Hidden Markov Model (2DHMM) where the hidden states correspond to the cluster labeling of textons and the transition matrix encodes their spatial layout (the compatibility between adjacent textons). [sent-6, score-1.398]

3 The 2DHMM is coupled with the Hierarchical Dirichlet process (HDP) which allows the number of textons and the complexity of transition matrix grow as the input texture becomes irregular. [sent-7, score-0.935]

4 The HDP makes use of Dirichlet process prior which favors regular textures by penalizing the model complexity. [sent-8, score-0.251]

5 This framework (HDP-2DHMM) learns the texton vocabulary and their spatial layout jointly and automatically. [sent-9, score-0.683]

6 The HDP-2DHMM results in a compact representation of textures which allows fast texture synthesis with comparable rendering quality over the state-of-the-art patch-based rendering methods. [sent-10, score-1.097]

7 We also show that the HDP2DHMM can be applied to perform image segmentation and synthesis. [sent-11, score-0.195]

8 1 Introduction Texture learning and synthesis are important tasks in computer vision and graphics. [sent-13, score-0.273]

9 The ﬁrst style emphasizes the modeling and understanding problems and develops statistical models [1, 2] which are capable of representing texture using textons and their spatial layout. [sent-15, score-0.864]

10 The second style relies on patch-based rendering techniques [3, 4] which focus on rendering quality and speed, but forego the semantic understanding and modeling of texture. [sent-17, score-0.338]

11 This paper aims at texture understanding and modeling with fast synthesis and high rendering quality. [sent-18, score-0.776]

12 We represent a texture image by a 2D Hidden Markov Model (2D-HMM) (see ﬁgure (1)) where the hidden states correspond to the cluster labeling of textons and the transition matrix encodes the texton spatial layout (the compatibility between adjacent textons). [sent-20, score-1.839]

13 The 2D-HMM is coupled with the Hierarchical Dirichlet process (HDP) [5, 6] which allows the number of textons (i. [sent-21, score-0.446]

14 hidden states) and the complexity of the transition matrix to grow as more training data is available or the randomness of the input texture becomes large. [sent-23, score-0.551]

15 The Dirichlet process prior penalizes the model complexity to favor reusing clusters and transitions and thus regular texture which can be represented by compact models. [sent-24, score-0.536]

16 This framework (HDP-2DHMM) discovers the semantic meaning of texture in an explicit way that the texton vocabulary and their spatial layout are learnt jointly and automatically (the number of textons is fully determined by HDP-2DHMM). [sent-25, score-1.496]

17 Once the texton vocabulary and the transition matrix are learnt, the synthesis process samples the latent texton labeling map according to the probability encoded in the transition matrix. [sent-26, score-1.493]

18 The ﬁnal 1 Figure 1: The ﬂow chart of texture learning and synthesis. [sent-27, score-0.388]

19 The colored rectangles correspond to the index (labeling) of textons which are represented by image patches. [sent-28, score-0.591]

20 The texton vocabulary shows the correspondence between the color (states) and the examples of image patches. [sent-29, score-0.697]

21 The transition matrices show the probability (indicated by the intensity) of generating a new state (coded by the color of the top left corner rectangle), given the states of the left and upper neighbor nodes (coded by the top and left-most rectangles). [sent-30, score-0.251]

22 The inferred texton map shows the state assignments of the input texture. [sent-31, score-0.569]

23 The top-right panel shows the sampled texton map according to the transition matrices. [sent-32, score-0.542]

24 The last panel shows the synthesized texture using image quilting according to the correspondence between the sampled texton map and the texton vocabulary. [sent-33, score-1.705]

25 image is then generated by selecting the image patches based on the sampled texton labeling map. [sent-34, score-0.897]

26 Here, image quilting [3] is applied to search and stitch together all the patches so that the boundary inconsistency is minimized. [sent-35, score-0.428]

27 By contrast to [3], our method is only required to search a much smaller set of candidate patches within a local texton cluster. [sent-36, score-0.555]

28 We show that the HDP-2DHMM is able to synthesize texture in one second (25 times faster than image quilting) with comparable quality. [sent-38, score-0.593]

29 We also show that the HDP-2DHMM can be applied to perform image segmentation and synthesis. [sent-40, score-0.195]

30 2 Previous Work Our primary interest is texture understanding and modeling. [sent-42, score-0.412]

31 This model is very successful in capturing stochastic textures, but may fail for more structured textures due to lack of spatial modeling. [sent-44, score-0.228]

32 [1, 2] extend it to explicitly learn the textons and their spatial relations which are represented by extra hidden layers. [sent-46, score-0.537]

33 This new model is parametric (the number of texton clusters has to be tuned by hand for different texture images) and model selection which might be unstable in practice, is needed to avoid overﬁtting. [sent-47, score-0.873]

34 Inspired by recent progress in machine learning, we extend the nonparametric Bayesian framework of coupling 1D HMM and HDP [6] to deal with 2D texture image. [sent-49, score-0.42]

35 A new model (HDP-2DHMM) is developed to learn texton vocabulary and spatial layouts jointly and automatically. [sent-50, score-0.658]

36 Since the HDP-2DHMM is designed to generate appropriate image statistics, but not pixel intensity, a patch-based texture synthesis technique, called image quilting [3], is integrated into our system to sample image patches. [sent-51, score-1.162]

37 The texture synthesis algorithm has also been applied to image inpainting [8]. [sent-52, score-0.779]

38 xi are observations (features) of the image patch at position i. [sent-69, score-0.239]

39 [9, 10] and Varma and Zisserman [11] study the ﬁlter representations of textons which are related to our implementations of visual features. [sent-74, score-0.436]

40 But the interactions between textons are not explicitly considered. [sent-75, score-0.395]

41 [12, 13] address texture understanding by discovering regularity without explicit statistical texture modeling. [sent-77, score-0.827]

42 Our method is closely related to [16] which is not designed for texture learning. [sent-80, score-0.388]

43 They use hierarchical Dirichlet process, but the models and the image feature representations, including both the image ﬁlters and the data likelihood model, are different. [sent-81, score-0.307]

44 1 Image Patches and Features A texture image I is represented by a grid of image patches {xi } with size of 24 × 24 in this paper where i denotes the location. [sent-88, score-0.778]

45 {xi } will be grouped into different textons by the HDP-2DHMM. [sent-89, score-0.395]

46 We begin with a simpliﬁed model where the positions of textons represented by image patches are pre-determined by the image grid, and not allowed to shift. [sent-90, score-0.785]

47 l,h,b Each patch xi is characterized by a set of ﬁlter responses {wi } which correspond to values b of image ﬁlter response h at location l. [sent-92, score-0.266]

48 By contrast, we skip the clustering step and leave the learning of texton vocabulary together with spatial layout learning into the HDP2DHMM which takes over the role of k-means. [sent-107, score-0.695]

49 2 HDP-2DHMM: Coupling Hidden Markov Model with Hierarchical Dirichlet Process A texture is modeled by a 2D Hidden Markov Model (2DHMM) where the nodes correspond to the image patches xi and the compatibility is encoded by the edges connecting 4 neighboring nodes. [sent-109, score-0.773]

50 We use zi to index the states 3 • β ∼ GEM (α) • For each state z ∈ {1, 2, 3, . [sent-112, score-0.283]

51 The likelihood model p(xi |zi ) which speciﬁes the probability of visual fetures is deﬁned by multinomial distribution parameterized by θzi speciﬁc to its corresponding hidden state zi : xi ∼ M ultinomial(θzi ) (1) where θzi specify the weights of visual features. [sent-116, score-0.408]

52 L(i) = ∅ and T (i) = ∅), the probability p(zi |zL(i) , zT (i) ) of its state zi is only determined by the states (zL(i) , zT (i) ) of the connected nodes. [sent-119, score-0.283]

53 The distribution has a form of multinomial distribution parameterized by πzL(i) ,zT (i) : zi ∼ M ultinomial(πzL(i) ,zT (i) ) (2) where πzL(i) ,zT (i) encodes the transition matrix and thus the spatial layout of textons. [sent-120, score-0.389]

54 Without loss of generality, we will skip the details of the boundary cases, but only focus on the nodes whose states should be determined by their top and left nodes jointly. [sent-125, score-0.209]

55 To make a nonparametric Bayesian representation, we need to allow the number of states zi countably inﬁnite and put prior distributions over the parameters θzi and πzL(i) ,zT (i) . [sent-126, score-0.286]

56 Therefore, the HDP makes use of a Dirichlet process prior to place a soft bias towards simpler models (in terms of the number of states and the regularity of state transitions) which explain the texture. [sent-136, score-0.195]

57 But this simpliﬁed model does not allow the textons (image patches) to be shifted. [sent-139, score-0.395]

58 We remove this constraint by introducing two hidden variables (ui , vi ) which indicate the displacements of textons associated with node i. [sent-140, score-0.52]

59 We only need to adjust the correspondence between image features xi and hidden states zi . [sent-141, score-0.508]

60 xi is modiﬁed to be xui ,vi which refers to image features located at the position with displacement of (ui , vi ) to the position i. [sent-142, score-0.211]

61 We ﬁrst instantiate a random hidden state labeling and then iteratively repeat the following two steps. [sent-153, score-0.178]

62 The next term P (zi = t|zL(i) = r, zT (i) = s) is the probability of state of t, given the states of the nodes on the left and above, i. [sent-159, score-0.173]

63 The probability of generating state t is given by: P (zi = t|zL(i) = r, zT (i) = s) = nrst + α βt t nrst + α (8) where βt refers to the weight of state t. [sent-163, score-0.204]

64 The last two terms P (zR(i) |zi = t, zT (R(i)) ) and P (zD(i) |zL(D(i)) , zi = t) are the probability of the states of the right and lower neighbor nodes (R(i), D(i)) given zi . [sent-165, score-0.449]

65 5 Figure 4: The color of rectangles in columns 2 and 3 correspond to the index (labeling) of textons which are represented by 24*24 image patches. [sent-169, score-0.591]

66 The synthesized images are all 384*384 (16*16 textons /patches). [sent-170, score-0.521]

67 Our method captures both stochastic textures (the last two rows) and more structured textures (the ﬁrst three rows, see the horizontal and grided layouts). [sent-171, score-0.355]

68 The inferred texton maps for structured textures are simpler (less states/textons) and more regular (less cluttered texton maps) than stochastic textures. [sent-172, score-1.176]

69 5 Texture Synthesis Once the texton vocabulary and the transition matrix are learnt, the synthesis process ﬁrst samples the latent texton labeling map according to the probability encoded in the transition matrix. [sent-173, score-1.493]

70 But the HDP-2DHMM is generative only for image features, but not image intensity. [sent-174, score-0.276]

71 To make it practical for image synthesis, image quilting [3] is integrated with the HDP-2DHMM. [sent-175, score-0.413]

72 The ﬁnal image is then generated by selecting image patches according to the texton labeling map. [sent-176, score-0.897]

73 Image quilting is applied to select and stitch together all the patches in a top-left-to-bottom-right order so that the boundary inconsistency is minimized . [sent-177, score-0.29]

74 By contrast to [3] which need to search over all image patches to ensure high rendering quality, our method is only required to search the candidate patches within a local cluster. [sent-179, score-0.461]

75 The HDP-2DHMM is capable of producing high rendering quality because the patches have been grouped based on visual features. [sent-180, score-0.305]

76 We show that the HDP-2DHMM is able to synthesize a 6 Figure 5: More synthesized texture images (for each pair, left is input texture, right is synthesized). [sent-182, score-0.559]

77 texture image with size of 384*384 and with comparable quality in one second (25 times faster than image quilting). [sent-183, score-0.718]

78 1 Texture Learning and Synthesis We use the texture images in [3]. [sent-185, score-0.424]

79 For each image, it takes 100 seconds for learning and 1 second for synthesis (almost 25 times faster than [3]). [sent-193, score-0.223]

80 Figure (4) shows the inferred texton labeling maps, the sampled texton maps and the synthesized texture images. [sent-194, score-1.517]

81 The rendering quality is visually comparable with [3] (not shown) for both structured textures and stochastic textures. [sent-196, score-0.366]

82 It is interesting to see that the HMM-HDP captures different types of texture patterns, such as vertical, horizontal and grided layouts. [sent-197, score-0.422]

83 It suggests that our method is able to discover the semantic texture meaning by learning texton vocabulary and their spatial relations. [sent-198, score-0.98]

84 The ﬁrst three rows show the HDP-2DHMM is able to segment images with mixture of textures and synthesize new textures. [sent-200, score-0.231]

85 The last row shows a failure example where the texton is not well aligned. [sent-201, score-0.464]

86 2 Image Segmentation and Synthesis We also apply the HDP-2DHMM to perform image segmentation and synthesis. [sent-203, score-0.195]

87 The segmentation results are represented by the inferred state assignments (the texton map). [sent-205, score-0.649]

88 The last row in ﬁgure (6) shows a failure example where the texton is not well aligned. [sent-208, score-0.464]

89 The 2D Hidden Markov Model (HMM) is coupled with the hierarchical Dirichlet process (HDP) which allows the number of textons and the complexity of transition matrix grows as the input texture becomes irregular. [sent-210, score-0.943]

90 The HDP makes use of Dirichlet process prior which favors regular textures by penalizing the model complexity. [sent-211, score-0.251]

91 This framework (HDP-2DHMM) learns the texton vocabulary and their spatial layout jointly and automatically. [sent-212, score-0.683]

92 We demonstrated that the resulting compact representation obtained by the HDP-2DHMM allows fast texture synthesis (under one second) with comparable rendering quality to the state-of-the-art image-based rendering methods. [sent-213, score-0.947]

93 Our results on image segmentation and synthesis suggest that the HDP-2DHMM is generally useful for further applications in low-level vision problems. [sent-214, score-0.468]

94 Guo, “Statistical modeling of texture sketch,” in ECCV ’02: Proceedings of the 7th European Conference on Computer Vision-Part III, 2002, pp. [sent-224, score-0.388]

95 Freeman, “Image quilting for texture synthesis and transfer,” in Siggraph, 2001. [sent-241, score-0.748]

96 Leung, “Texture synthesis by non-parametric sampling,” in International Conference on Computer Vision, 1999, pp. [sent-244, score-0.223]

97 Mumford, “Filters, random ﬁelds and maximum entropy (frame): Towards a uniﬁed theory for texture modeling,” International Journal of Computer Vision, vol. [sent-265, score-0.388]

98 Hays, “Near regular texture analysis and manipulation,” ACM Transactions on Graphics (SIGGRAPH 2004), vol. [sent-294, score-0.42]

99 Liu, “Discovering texture regularity as a higherorder correspondence problem,” in 9th European Conference on Computer Vision, May 2006. [sent-303, score-0.439]

100 Levoy, “Fast texture synthesis using tree-structured vector quantization,” in SIGGRAPH ’00: Proceedings of the 27th annual conference on Computer graphics and interactive techniques, 2000, pp. [sent-337, score-0.611]

similar papers computed by tfidf model

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In the following sections we will provide details on the speciﬁcs of each part of the model. 2 β=0.0 β=0.5 β=1.0 β=0.0 β=0.5 β=1.0 2 1 1 1 1 1 1 0 −1 0 −1 −2 −1 −2 −3 0 x1 2 0 x1 2 0 x1 2 −2 −3 −2 0 x1 2 0 −1 −2 −3 −2 0 −1 −2 −3 −2 0 −1 −2 −3 −2 0 x2 3 2 x2 3 2 x2 3 2 x2 3 2 x2 3 2 x2 3 −3 −2 0 x1 2 −2 0 x1 2 Figure 2: Shape of the conditional (Left three plots) and joint (Right three plots) density model in log scale for several values of β, from dependence to independence. 2.1 Learning tree structure In their seminal paper, Chow and Liu showed how to learn the optimal tree structure approximation for a multidimensional probability density function [12]. This algorithm is easy to apply to this scenario, and requires just a few simple steps. First, given the current estimate for the ﬁlter matrix W, we calculate the response of each of the ﬁlters with all the patches in the data set. Using these responses, we calculate the mutual information between each pair of ﬁlters (nodes) to obtain a fully connected weighted graph. The ﬁnal step is to ﬁnd a maximal spanning tree over this graph. The resulting unrooted tree is the optimal tree approximation of the joint distribution function over all nodes. We will note that the tree is unrooted, and the root can be chosen arbitrarily - this means that there is no node, or ﬁlter, that is more important than the others - the direction in the tree graph is arbitrary as long as it is chosen in a consistent way. 2.2 Joint probability density functions Gabor ﬁlter responses on natural images exhibit highly kurtotic marginal distributions, with heavy tails and sharp peaks [13, 3, 14]. Joint pair wise distributions also exhibit this same shape with varying degrees of dependency between the components [13, 2]. The density model we use allows us to capture both the highly kurtotic nature of the distributions, while still allowing varying degrees of dependence using a mixing variable. We use a mix of two forms of ﬁnite, zero mean Gaussian Scale Mixtures (GSM). In one, the components are assumed to be independent of each other and in the other, they are assumed to be spherically distributed. The mixing variable linearly interpolates between the two, allowing us to capture the whole range of dependencies: p(x1 , x2 ; β) = βpdep (x1 , x2 ) + (1 − β)pind (x1 , x2 ) (4) When β = 1 the two components are dependent (unless p is Gaussian), whereas when β = 0 the two components are independent. For the density functions themselves, we use a ﬁnite GSM. The dependent case is a scale mixture of bivariate Gaussians: 2 πk N (x1 , x2 ; σk I) pdep (x1 , x2 ) = (5) k While the independent case is a product of two independent univariate Gaussians: 2 πk N (x1 ; σk ) pind (x1 , x2 ) = k 2 πk N (x2 ; σk ) (6) k 2 Estimating parameters πk and σk for the GSM is done directly from the data using Expectation Maximization. These parameters are the same for all edges and are estimated only once on the ﬁrst iteration. See Figure 2 for a visualization of the conditional distribution functions for varying values of β. We will note that the marginal distributions for the two types of joint distributions above are the same. The mixing parameter β is also estimated using EM, but this is done for each edge in the tree separately, thus allowing our model to theoretically capture the fully independent case (ICA) and other degenerate models such as ISA. 2.3 Learning tree dependent components Given the current tree structure and density model, we can now learn the matrix W via gradient ascent on the log likelihood of the model. All learning is performed on whitened, dimensionally 3 reduced patches. This means that W is a N × N rotation (orthonormal) matrix, where N is the number of dimensions after dimensionality reduction (see details below). Given an image patch z we multiply it by W to get the response vector y: y = Wz (7) Now we can calculate the log likelihood of the given patch using the tree model (which we assume is constant at the moment): N log p(yi |ypai ) log p(y) = log p(yroot ) + (8) i=1 Where pai denotes the parent of node i. Now, taking the derivative w.r.t the r-th row of W: ∂ log p(y) ∂ log p(y) T = z ∂Wr ∂yr (9) Where z is the whitened natural image patch. Finally, we can calculate the derivative of the log likelihood with respect to the r-th element in y: ∂ log p(ypar , yr ) ∂ log p(y) = + ∂yr ∂yr c∈C(r) ∂ log p(yr , yc ) ∂ log p(yr ) − ∂yr ∂yr (10) Where C(r) denote the children of node r. In summary, the gradient ascent rule for updating the rotation matrix W is given by: t+1 t Wr = Wr + η ∂ log p(y) T z ∂yr (11) Where η is the learning rate constant. After update, the rows of W are orthonormalized. This gradient ascent rule is applied for several hundreds of patches (see details below), after which the tree structure is learned again as described in Section 2.1, using the new ﬁlter matrix W, repeating this process for many iterations. 3 Results and analysis 3.1 Validation Before running the full algorithm on natural image data, we wanted to validate that it does produce sensible results with simple synthetic data. We generated noise from four different models, one is 1/f independent Gaussian noise with 8 Discrete Cosine Transform (DCT) ﬁlters, the second is a simple ICA model with 8 DCT ﬁlters, and highly kurtotic marginals. The third was a simple ISA model - 4 subspaces, each with two ﬁlters from the DCT ﬁlter set. Distribution within the subspace was a circular, highly kurtotic GSM, and the subspaces were sampled independently. Finally, we generated data from a simple synthetic tree of DCT ﬁlters, using the same joint distributions as for the ISA model. These four synthetic random data sets were given to the algorithm - results can be seen in Figure 3 for the ICA, ISA and tree samples. In all cases the model learned the ﬁlters and distribution correctly, reproducing both the ﬁlters (up to rotations within the subspace in ISA) and the dependency structure between the different ﬁlters. In the case of 1/f Gaussian noise, any whitening transformation is equally likely and any value of beta is equally likely. Thus in this case, the algorithm cannot ﬁnd the tree or the ﬁlters. 3.2 Learning from natural image patches We then ran experiments with a set of natural images [9]1 . These images contain natural scenes such as mountains, ﬁelds and lakes. . The data set was 50,000 patches, each 16 × 16 pixels large. The patches’ DC was removed and they were then whitened using PCA. Dimension was reduced from 256 to 128 dimensions. The GSM for the density model had 16 components. Several initial 1 available at http://www.cis.hut.ﬁ/projects/ica/imageica/ 4 Figure 3: Validation of the algorithm. Noise was generated from three models - top row is ICA, middle row is ISA and bottom row is a tree model. Samples were then given to the algorithm. On the right are the resulting learned tree models. Presented are the learned ﬁlters, tree model (with white edges meaning β = 0, black meaning β = 1 and grays intermediate values) and an example of a marginal histogram for one of the ﬁlters. It can be seen that in all cases all parts of the model were correctly learned. Filters in the ISA case were learned up to rotation within the subspace, and all ﬁlters were learned up to sign. β values for the ICA case were always below 0.1, as were the values of β between subspaces in ISA. conditions for the matrix W were tried out (random rotations, identity) but this had little effect on results. Mini-batches of 10 patches each were used for the gradient ascent - the gradient of 10 patches was summed, and then normalized to have unit norm. The learning rate constant η was set to 0.1. Tree structure learning and estimation of the mixing variable β were done every 500 mini-batches. All in all, 50 iterations were done over the data set. 3.3 Filters and tree structure Figures 4 and 5 show the learned ﬁlters (WQ where Q is the whitening matrix) and tree structure (T ) learned from natural images. Unlike the ISA toy data in ﬁgure 3, here a full tree was learned and β is approximately one for all edges. The GSM that was learned for the marginals was highly kurtotic. It can be seen that resulting ﬁlters are edge ﬁlters at varying scales, positions and orientations. This is similar to the result one gets when applying ICA to natural images [5, 15]. More interesting is Figure 4: Left: Filter set learned from 16 × 16 natural image patches. Filters are ordered by PCA eigenvalues, largest to smallest. Resulting ﬁlters are edge ﬁlters having different orientations, positions, frequencies and phases. Right: The “feature” set learned, that is, columns of the pseudo inverse of the ﬁlter set. 5 Figure 5: The learned tree graph structure and feature set. It can be seen that neighboring features on the graph have similar orientation, position and frequency. See Figure 4 for a better view of the feature details, and see text for full detail and analysis. Note that the ﬁgure is rotated CW. 6 Optimal Orientation Optimal Frequency 3.5 Optimal Phase 7 3 0.8 6 2.5 Optimal Position Y 0.9 6 5 5 0.7 0.6 3 Child 1.5 4 Child 2 Child Child 4 3 2 1 1 0.4 0.3 2 0.5 0.5 0.2 0 1 0.1 0 0 1 2 Parent 3 4 0 0 2 4 Parent 6 8 0 0 1 2 3 Parent 4 5 6 0 0.2 0.4 0.6 Parent 0.8 1 Figure 6: Correlation of optimal parameters in neighboring nodes in the tree graph. Orientation, frequency and position are highly correlated, while phase seems to be entirely uncorrelated. This property of correlation in frequency and orientation, while having no correlation in phase is related to the ubiquitous energy model of complex cells in V1. See text for further details. Figure 7: Left: Comparison of log likelihood values of our model with PCA, ICA and ISA. Our model gives the highest likelihood. Right: Samples taken at random from ICA, ISA and our model. Samples from our model appear to contain more long-range structure. the tree graph structure learned along with the ﬁlters which is shown in Figure 5. It can be seen that neighboring ﬁlters (nodes) in the tree tend to have similar position, frequency and orientation. Figure 6 shows the correlation of optimal frequency, orientation and position for neighboring ﬁlters in the tree - it is obvious that all three are highly correlated. Also apparent in this ﬁgure is the fact that the optimal phase for neighboring ﬁlters has no signiﬁcant correlation. It has been suggested that ﬁlters which have the same orientation, frequency and position with different phase can be related to complex cells in V1 [2, 16]. 3.4 Comparison to other models Since our model is a generalization of both ICA and ISA we use it to learn both models. In order to learn ICA we used the exact same data set, but the tree had no edges and was not learned from the data (alternatively, we could have just set β = 0). For ISA we used a forest architecture of 2 node trees, setting β = 1 for all edges (which means a spherical symmetric distribution), no tree structure was learned. Both models produce edge ﬁlters similar to what we learn (and to those in [5, 15, 6]). The ISA model produces neighboring nodes with similar frequency and orientation, but different phase, as was reported in [2]. We also compare to a simple PCA whitening transform, using the same whitening transform and marginals as in the ICA case, but setting W = I. We compare the likelihood each model gives for a test set of natural image patches, different from the one that was used in training. There were 50,000 patches in the test set, and we calculate the mean log likelihood over the entire set. The table in Figure 7 shows the result - as can be seen, our model performs better in likelihood terms than both ICA and ISA. Using a tree model, as opposed to more complex graphical models, allows for easy sampling from the model. Figure 7 shows 20 random samples taken from our tree model along with samples from the ICA and ISA models. Note the elongated structures (e.g. in the bottom left sample) in the samples from the tree model, and compare to patches sampled from the ICA and ISA models. 7 40 40 30 30 20 20 10 1 10 0.8 0.6 0.4 0 0.2 0 0 1 2 3 Orientation 4 0 0 2 4 6 Frequency 8 0 2 4 Phase Figure 8: Left: Interpretation of the model. Given a patch, the response of all edge ﬁlters is computed (“simple cells”), then at each edge, the corresponding nodes are squared and summed to produce the response of the “complex cell” this edge represents. Both the response of complex cells and simple cells is summed to produce the likelihood of the patch. Right: Response of a “complex cell” in our model to changing phase, frequency and orientation. Response in the y-axis is the sum of squares of the two ﬁlters in this “complex cell”. Note that while the cell is selective to orientation and frequency, it is rather invariant to phase. 3.5 Tree models and complex cells One way to interpret the model is looking at the likelihood of a given patch under this model. For the case of β = 1 substituting Equation 4 into Equation 3 yields: 2 2 ρ( yi + ypai ) − ρ(|ypai |) log L(z) = (12) i 2 Where ρ(x) = log k πk N (x; σk ) . This form of likelihood has an interesting similarity to models of complex cells in V1 [2, 4]. In Figure 8 we draw a simple two-layer network that computes the likelihood. The ﬁrst layer applies linear ﬁlters (“simple cells”) to the image patch, while the second layer sums the squared outputs of similarly oriented ﬁlters from the ﬁrst layer, having different phases, which are connected in the tree (“complex cells”). Output is also dependent on the actual response of the “simple cell” layer. The likelihood here is maximized when both the response of the parent ﬁlter ypai and the child yi is zero, but, given that one ﬁlter has responded with a non-zero value, the likelihood is maximized when the other ﬁlter also ﬁres (see the conditional density in Figure 2). Figure 8 also shows an example of the phase invariance which is present in the learned

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