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245 nips-2010-Space-Variant Single-Image Blind Deconvolution for Removing Camera Shake

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Author: Stefan Harmeling, Hirsch Michael, Bernhard Schölkopf

Abstract: Modelling camera shake as a space-invariant convolution simpliﬁes the problem of removing camera shake, but often insufﬁciently models actual motion blur such as those due to camera rotation and movements outside the sensor plane or when objects in the scene have different distances to the camera. In an effort to address these limitations, (i) we introduce a taxonomy of camera shakes, (ii) we build on a recently introduced framework for space-variant ﬁltering by Hirsch et al. and a fast algorithm for single image blind deconvolution for space-invariant ﬁlters by Cho and Lee to construct a method for blind deconvolution in the case of space-variant blur, and (iii), we present an experimental setup for evaluation that allows us to take images with real camera shake while at the same time recording the spacevariant point spread function corresponding to that blur. Finally, we demonstrate that our method is able to deblur images degraded by spatially-varying blur originating from real camera shake, even without using additionally motion sensor information. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 In an effort to address these limitations, (i) we introduce a taxonomy of camera shakes, (ii) we build on a recently introduced framework for space-variant ﬁltering by Hirsch et al. [sent-5, score-0.369]

2 Finally, we demonstrate that our method is able to deblur images degraded by spatially-varying blur originating from real camera shake, even without using additionally motion sensor information. [sent-7, score-0.699]

3 1 Introduction Camera shake is a common problem of handheld, longer exposed photographs occurring especially in low light situations, e. [sent-8, score-0.246]

4 With a few exceptions such as panning photography, camera shake is unwanted, since it often destroys details and blurs the image. [sent-11, score-0.682]

5 The effect of a particular camera shake can be described by a linear transformation on the sharp image, i. [sent-12, score-0.634]

6 , the image that would have been recorded using a tripod. [sent-14, score-0.183]

7 Denoting for simplicity images as column vectors, the recorded blurry image y can be written as a linear transformation of the sharp image x, i. [sent-15, score-0.534]

8 , as y = Ax, where A is an unknown matrix describing the camera shake. [sent-17, score-0.311]

9 The task of blind image deblurring is to recover x given only the blurred image y, but not A. [sent-18, score-0.712]

10 Our work combines ideas of three papers: (i) Hirsch et al’s work [1] on efﬁcient space-variant ﬁltering, (ii) Cho and Lee’s work [2] on single frame blind deconvolution, and (iii) Krishnan and Fergus’s work [3] on fast non-blind deconvolution. [sent-22, score-0.197]

11 Previous approaches to single image blind deconvolution have dealt only with space-invariant blurs. [sent-23, score-0.648]

12 [8] represent space-variant blurs as projective motion paths and propose a non-blind deconvolution method. [sent-29, score-0.557]

13 Blind deconvolution of space-variant blurs in the context 1 of star ﬁelds has been considered by Bardsley et al. [sent-32, score-0.494]

14 Their method estimates PSFs separately (and not simultaneously) on image patches using phase diversity, and deconvolves the overall image using [11]. [sent-34, score-0.364]

15 [12] recently proposed a method that estimates the motion path using inertial sensors, leading to high-quality image reconstructions. [sent-36, score-0.244]

16 [1] require multiple images to perform blind deconvolution with space-variant blur, as do Sorel ˇ and Sroubek [15]. [sent-41, score-0.552]

17 2 A taxonomy of camera shakes Camera shake can be described from two perspectives: (i) how the PSF varies across the image, i. [sent-42, score-0.648]

18 , how point sources would be recorded at different locations on the sensor, and (ii) by the trajectory of the camera and how the depth of the scene varies. [sent-44, score-0.452]

19 , only the camera moves (only ego-motion), and none of the photographed objects (no object motion). [sent-47, score-0.372]

20 In this case the linear transformation is a convolution matrix. [sent-50, score-0.129]

21 Most algorithms for blind deconvolution are restricted to this case. [sent-51, score-0.502]

22 • Smooth: The PSF is smoothly varying across the image. [sent-52, score-0.13]

23 Here, the linear transformation is no longer a convolution matrix, but a more general framework is needed such as the smoothly space-varying ﬁlters in the multi-frame method of Hirsch et al. [sent-53, score-0.256]

24 For this case, our paper proposes an algorithm for single image deblurring. [sent-55, score-0.146]

25 , the distance of the camera to objects at different locations in the scene, can be classiﬁed into three categories: • Constant: All objects have the same distance to the camera. [sent-60, score-0.355]

26 • Smooth: The distance to the camera is smoothly varying across the scene. [sent-62, score-0.441]

27 • Segmented: The scene can be segmented into different objects each having a different distance to the camera. [sent-64, score-0.127]

28 The motion of the camera can be represented by a six dimensional trajectory with three spatial and three angular coordinates. [sent-67, score-0.405]

29 Furthermore, α and β describe the camera tilting up/down and left/right, and γ the camera rotation around the optical axis. [sent-69, score-0.65]

30 Exemplarily we consider the following trajectories: • • • • • Pure shift: The camera moves inside the sensor plane without rotation; only a and b vary. [sent-71, score-0.389]

31 Rotated shift: The camera moves inside the sensor plane with rotation; a, b, and γ vary. [sent-72, score-0.389]

32 Back and forth: The distance between camera and scene is changing; only c varies. [sent-73, score-0.351]

33 Pure tilt: The camera is tilted up and down and left and right; only α and β vary. [sent-74, score-0.311]

34 Note that only “pure shifts” in combination with “constant depths” lead to a constant PSF across the image, which is the case most methods for camera unshaking are proposed for. [sent-77, score-0.311]

35 Thus, extending blind deconvolution to smoothly space-varying PSFs can increases the range of possible applications. [sent-78, score-0.597]

36 Furthermore, we see that for segmented scenes, camera shake usually leads to blurs that are non-smoothly changing across the image. [sent-79, score-0.747]

37 in this case the model of smoothly varying PSFs is incorrect, it might still lead to better results than constant PSFs. [sent-81, score-0.13]

38 3 Smoothly varying PSF as Efﬁcient Filter Flow To obtain a generalized image deblurring method we represent the linear transformation y = Ax by the recently proposed efﬁcient ﬁlter ﬂow (EFF) method of Hirsch et al. [sent-82, score-0.447]

39 This transformation is linear in x, and thus an instance of the general linear transformation y = Ax, where the column vector a parametrizes the transformation matrix A. [sent-89, score-0.231]

40 Although being efﬁcient, the (space-invariant) convolution applies only to camera shakes which are pure shifts of ﬂat scenes. [sent-93, score-0.462]

41 The idea is (i) to cover the image with overlapping patches, (ii) to apply to each patch a different PSF, and (iii) to add the patches to obtain a single large image. [sent-95, score-0.27]

42 (1) r=0 Here, w(r) ≥ 0 smoothly fades the r-th patch in and masks out the others. [sent-97, score-0.147]

43 Note that this method does not simply apply a different PSF to different image regions, but instead yields a different PSF for each pixel. [sent-99, score-0.146]

44 The reason is that usually, the patches are chosen to overlap at least 50%, so that the PSF at a pixel is a certain linear combination of several ﬁlters, where the weights are chosen to smoothly blend ﬁlters in and out, and thus the PSF tends to be different at each pixel. [sent-100, score-0.193]

45 1 shows that a PSF array as small as 3 × 3, corresponding to p = 9 and nine overlapping patches (right panel of the bottom row), can parametrize smoothly varying blurs (middle column) that closely mimic real camera shake (left column). [sent-102, score-0.961]

46 Note that each of the MVMs with A, AT , X, and X T is needed for blind deconvolution: A and AT for the estimation of x given a, and X and X T for the estimation of a. [sent-117, score-0.213]

47 • PSF estimation step: update the PSFs given the blurry image y and the current estimate of the predicted x, using only the gradient images of x (resulting in a preconditioning effect) and enforcing smoothness between neighboring PSFs. [sent-121, score-0.361]

48 • Image estimation step: update the current deblurred image x by minimizing a leastsquares cost function using a smoothness prior on the gradient image. [sent-123, score-0.266]

49 (ii) Non-blind deblurring: given the linear transformation we estimate the ﬁnal deblurred image x by alternating between the following two steps: • Latent variable estimation: estimate latent variables regularized with a sparsity prior that approximate the gradient of x. [sent-124, score-0.288]

50 • Image estimation step: update the current deblurred image x by minimizing a leastsquares cost function while penalizing the Euclidean norm of the gradient image to the latent variables of the previous step, see “x sub-problem” of [3] for details. [sent-126, score-0.412]

51 The steps of (i) are repeated seven times on each scale of a multi-scale image pyramid. [sent-127, score-0.146]

52 The resulting PSFs in a 4 and the resulting image x at each scale are upsampled and initialize the next scale. [sent-130, score-0.146]

53 Having obtained an estimate for the linear transformation in form of an array of PSFs, the alternating steps of (ii) perform space variant non-blind deconvolution of the recorded image y using a natural image statistics prior (as in [13]). [sent-132, score-0.786]

54 While our procedure is based on Cho and Lee’s [2] and Krishnan and Fergus’ [3] methods for space-invariant single blind deconvolution, it differs in several important aspects which we presently explain. [sent-134, score-0.186]

55 Given the thresholded gradient images of the nonlinear ﬁltered image x as the output of the prediction step, the PSF estimation minimizes a regularized leastsquares cost function, ∂z y − A∂z x 2 +λ a 2 + νg(a), (4) z where z ranges over the set {h, v, hh, vv, hv}, i. [sent-143, score-0.251]

56 image details are required for PSF estimation, for images with less structured areas (such as sky) we can not estimate reasonable PSFs everywhere. [sent-154, score-0.196]

57 In both Cho and Lee’s and also in Krishnan and Fergus’ work, the image estimation step involves direct deconvolution which corresponds to a simple pixelwise divison of the blurry image by the zero-padded PSF in Fourier domain. [sent-160, score-0.731]

58 Unfortunately, a direct deconvolution does not exist in general for linear transformations represented as EFF, since it involves summations over patches. [sent-161, score-0.337]

59 However, we can replace the direct deconvolution by an optimization of some regularized least-squares cost function y − Ax 2 + α x p . [sent-162, score-0.337]

60 As the estimated x is subsequently processed in the prediction step, one might consider regularization redundant in the image estimation step of (i). [sent-166, score-0.17]

61 In (ii) during the ﬁnal non-blind deblurring procedure we employ a sparsity prior for x by choosing p = 1/2. [sent-168, score-0.157]

62 The main difference in the image estimation steps to [2] and [3] is that the linear transformation A is no longer a convolution but instead a space-variant ﬁlter implemented by the EFF framework. [sent-169, score-0.299]

63 5 Experiments We present results on several example images with space-variant blur, for which we are able to recover a deblurred image, while a state-of-the-art method for single image blind deconvolution does not. [sent-171, score-0.763]

64 Capturing a gray scale image blurred with real camera shake along with the set of spatially varying PSFs. [sent-173, score-0.877]

65 The idea is to create a color image where the gray scale image is shown in the red channel, a grid of dots (for recording the PSFs) is shown in the blue channel, and the green channel is set to zero. [sent-174, score-0.407]

66 We display the resulting RBG image on a computer screen and take a photo with real hand shake. [sent-175, score-0.228]

67 We split the recorded raw image into the red and blue part. [sent-176, score-0.205]

68 The red part only shows the image blurred with camera shake and the blue part shows the spatially varying PSFs that depict the effect of the camera shake. [sent-177, score-1.21]

69 To avoid a Moir´ effect the distance between the camera e and the computer screen must be chosen carefully such that the discrete structure of the computer screen can not be resolved by the (discrete) image sensor of the camera. [sent-178, score-0.616]

70 For all experiments, photos were taken with a hand-held Canon EOS 1000D digital single lens reﬂex camera with a zoom lens (Canon zoom lens EF 24-70 mm 1:2. [sent-186, score-0.413]

71 The input to the deblurring algorithm was only the red channel of the RAW ﬁle which we treat as if it were a captured gray-scale image. [sent-189, score-0.2]

72 The image sizes are: vintage car 455 × 635, butcher shop 615 × 415, elephant 625 × 455. [sent-190, score-0.453]

73 4), we compare our estimated PSFs evaluated on a regular grid of dots to the true PSFs recorded in the blue channel during the camera shake. [sent-194, score-0.463]

74 This comparison has been made for the vintage car example and is included in the supplementary material. [sent-195, score-0.139]

75 We compare with Cho and Lee’s [2] method which we consider currently the state-of-art method for single image blind deconvolution. [sent-196, score-0.311]

76 This method assumes space-invariant blurs, and thus we also compare to a modiﬁed version of this algorithm that is applied to the patches of our method and that ﬁnally blends the individually deblurred patches carefully to one ﬁnal output image. [sent-197, score-0.209]

77 The space-variant blur was modelled for the 6 Hand-shaked photo Our result Cho and Lee [14] Patchwise Cho and Lee Butcher Shop Vintage Car Elephant Figure 3: Deblurring results and comparison. [sent-205, score-0.188]

78 vintage car example by an array of 6 × 7 PSF kernels, for butcher shop by an array of 4 × 6 PSF kernels, and for the elephant by an array of 5 × 6 PSF kernels. [sent-206, score-0.436]

79 (1) we used a BartlettHanning window with 75% overlap in the vintage car example and 50% in the butcher shop and elephant example. [sent-209, score-0.333]

80 We choose for the vintage car a larger overlap to keep the patch size reasonably large. [sent-210, score-0.217]

81 For the ﬁnal non-blind deconvolution (step (ii) in Sec. [sent-211, score-0.337]

82 On the three example images our algorithm took about 30 minutes for space-variant image restoration. [sent-214, score-0.196]

83 In summary, our experiments show that our method is able to deblur space-variant blurs that are too difﬁcult for Cho and Lee’s method. [sent-215, score-0.151]

84 Interesting is the comparison with the patch-wise version of Cho and Lee: looking at the details (such as the house number 117 at the butcher shop, the licence plate of the vintage car, or the trunk of the elephant) our method is better. [sent-217, score-0.156]

85 At the door frame in the vintage car image, we see that the patch-wise version of Cho and Lee has alignment problems. [sent-218, score-0.139]

86 Our experience was that this gets more severe for larger blur kernels. [sent-219, score-0.157]

87 [18] Figure 4: Our blind method achieves results comparable to Joshi et al. [sent-223, score-0.197]

88 [12] who additionally require motion sensor information which we do not use. [sent-224, score-0.129]

89 Even though our method does not exploit the motion sensor data utilized by Joshi et al. [sent-231, score-0.161]

90 (iii) Cho and Lee are able to use direct deconvolution (division in Fourier space) for the image estimation step, while we have to solve an optimization problem, because we currently do not know how to perform direct deconvolution for the space-variant ﬁlters. [sent-242, score-0.844]

91 6 Discussion Blind deconvolution of images degraded by space-variant blur is a much harder problem than simply assuming space-invariant blurs. [sent-243, score-0.57]

92 Our experiments show that even state-of-the-art algorithms such as Cho and Lee’s [2] are not able to recover image details for such blurs without unpleasant artifacts. [sent-244, score-0.271]

93 We have proposed an algorithm that is able to tackle space-variant blurs with encouraging results. [sent-245, score-0.125]

94 Presently, the main limitation of our approach is that it can fail if the blurs are too large or if they vary too quickly across the image. [sent-246, score-0.125]

95 We believe there are two main reasons for this: (i) on the one hand, if the blurs are large, the patches need to be large as well to obtain enough statistics for estimating the blur. [sent-247, score-0.197]

96 For instance, a PSF that looks like a thick horizontal line is challenging, because the resulting image feature might be misunderstood by the prediction step to be horizontal lines in the image. [sent-251, score-0.146]

97 Another limitation of our method are image areas with little structure. [sent-253, score-0.146]

98 On such patches it is difﬁcult to infer a reasonable blur kernel, and our method propagates the results from the neighboring patches to these cases. [sent-254, score-0.338]

99 Rotational motion deblurring of a rigid object from a single image. [sent-314, score-0.229]

100 Space-variant deblurring using one blurred and one underexposed image. [sent-356, score-0.255]

similar papers computed by tfidf model

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The success of these methods in applications such as denoising is a poor measure of the quality of the generative model that has been learned: Setting the parameters to random values works almost as well for eliminating independent Gaussian noise [17], because this can be done quite well by just using a penalty for high-frequency variation. In this work, we show that the generative quality of these models can be drastically improved by jointly modelling both pixel mean intensities and pixel covariances. This can be achieved by using two sets of latent variables, one that gates pair-wise interactions between pixels and another one that sets the mean intensities of pixels, as we already proposed in some earlier work [4]. Here, we show that this modelling choice is crucial to make the gated MRF work well on high-resolution images. Finally, we show that the most widely used method of sharing weights in MRF’s for high-resolution images is overly constrained. Earlier work considered homogeneous MRF’s in which each potential is replicated at all image locations. This has the subtle effect of making learning very difﬁcult because of strong correlations at nearby sites. Following Gregor and LeCun [18] and also Tang and Eliasmith [19], we keep the number of parameters under control by using local potentials, but unlike Roth and Black [6] we only share weights between potentials that do not overlap. 2 Augmenting Gated MRF’s with Mean Hidden Units A Product of Student’s t (PoT) model [15] is a gated MRF deﬁned on small image patches that can be viewed as modelling image-speciﬁc, pair-wise relationships between pixel values by using the states of its latent variables. It is very good at representing the fact that two-pixel have very similar intensities and no good at all at modelling what these intensities are. Failure to model the mean also leads to impoverished modelling of the covariances when the input images have nonzero mean intensity. The covariance RBM (cRBM) [20] is another model that shares the same limitation since it only differs from PoT in the distribution of its latent variables: The posterior over the latent variables is a product of Bernoulli distributions instead of Gamma distributions as in PoT. We explain the fundamental limitation of these models by using a simple toy example: Modelling two-pixel images using a cRBM with only one binary hidden unit, see ﬁg. 1. This cRBM assumes that the conditional distribution over the input is a zero-mean Gaussian with a covariance that is determined by the state of the latent variable. Since the latent variable is binary, the cRBM can be viewed as a mixture of two zero-mean full covariance Gaussians. The latent variable uses the pairwise relationship between pixels to decide which of the two covariance matrices should be used to model each image. When the input data is pre-proessed by making each image have zero mean intensity (the empirical histogram is shown in the ﬁrst row and ﬁrst column), most images lie near the origin because most of the times nearby pixels are strongly correlated. Less frequently we encounter edge images that exhibit strong anti-correlation between the pixels, as shown by the long tails along the anti-diagonal line. A cRBM could model this data by using two Gaussians (ﬁrst row and second column): one that is spherical and tight at the origin for smooth images and another one that has a covariance elongated along the anti-diagonal for structured images. If, however, the whole set of images is normalized by subtracting from every pixel the mean value of all pixels over all images (second row and ﬁrst column), the cRBM fails at modelling structured images (second row and second column). It can ﬁt a Gaussian to the smooth images by discovering 2 Figure 1: In the ﬁrst row, each image is zero mean. In the second row, the whole set of data points is centered but each image can have non-zero mean. The ﬁrst column shows 8x8 images picked at random from natural images. The images in the second column are generated by a model that does not account for mean intensity. The images in the third column are generated by a model that has both “mean” and “covariance” hidden units. The contours in the ﬁrst column show the negative log of the empirical distribution of (tiny) natural two-pixel images (x-axis being the ﬁrst pixel and the y-axis the second pixel). The plots in the other columns are toy examples showing how each model could represent the empirical distribution using a mixture of Gaussians with components that have one of two possible covariances (corresponding to the state of a binary “covariance” latent variable). Models that can change the means of the Gaussians (mPoT and mcRBM) can represent better structured images (edge images lie along the anti-diagonal and are ﬁtted by the Gaussians shown in red) while the other models (PoT and cRBM) fail, overall when each image can have non-zero mean. the direction of strong correlation along the main diagonal, but it is very likely to fail to discover the direction of anti-correlation, which is crucial to represent discontinuities, because structured images with different mean intensity appear to be evenly spread over the whole input space. If the model has another set of latent variables that can change the means of the Gaussian distributions in the mixture (as explained more formally below and yielding the mPoT and mcRBM models), then the model can represent both changes of mean intensity and the correlational structure of pixels (see last column). The mean latent variables effectively subtract off the relevant mean from each data-point, letting the covariance latent variable capture the covariance structure of the data. As before, the covariance latent variable needs only to select between two covariance matrices. In fact, experiments on real 8x8 image patches conﬁrm these conjectures. Fig. 1 shows samples drawn from PoT and mPoT. mPoT (and similarly mcRBM [4]) is not only better at modelling zero mean images but it can also represent images that have non zero mean intensity well. We now describe mPoT, referring the reader to [4] for a detailed description of mcRBM. In PoT [9] the energy function is: E PoT (x, hc ) = i 1 [hc (1 + (Ci T x)2 ) + (1 − γ) log hc ] i i 2 (1) where x is a vectorized image patch, hc is a vector of Gamma “covariance” latent variables, C is a ﬁlter bank matrix and γ is a scalar parameter. The joint probability over input pixels and latent variables is proportional to exp(−E PoT (x, hc )). Therefore, the conditional distribution over the input pixels is a zero-mean Gaussian with covariance equal to: Σc = (Cdiag(hc )C T )−1 . (2) In order to make the mean of the conditional distribution non-zero, we deﬁne mPoT as the normalized product of the above zero-mean Gaussian that models the covariance and a spherical covariance Gaussian that models the mean. The overall energy function becomes: E mPoT (x, hc , hm ) = E PoT (x, hc ) + E m (x, hm ) 3 (3) Figure 2: Illustration of different choices of weight-sharing scheme for a RBM. Links converging to one latent variable are ﬁlters. Filters with the same color share the same parameters. Kinds of weight-sharing scheme: A) Global, B) Local, C) TConv and D) Conv. E) TConv applied to an image. Cells correspond to neighborhoods to which ﬁlters are applied. Cells with the same color share the same parameters. F) 256 ﬁlters learned by a Gaussian RBM with TConv weight-sharing scheme on high-resolution natural images. Each ﬁlter has size 16x16 pixels and it is applied every 16 pixels in both the horizontal and vertical directions. Filters in position (i, j) and (1, 1) are applied to neighborhoods that are (i, j) pixels away form each other. Best viewed in color. where hm is another set of latent variables that are assumed to be Bernoulli distributed (but other distributions could be used). The new energy term is: E m (x, hm ) = 1 T x x− 2 hm Wj T x j (4) j yielding the following conditional distribution over the input pixels: p(x|hc , hm ) = N (Σ(W hm ), Σ), Σ = (Σc + I)−1 (5) with Σc deﬁned in eq. 2. As desired, the conditional distribution has non-zero mean2 . Patch-based models like PoT have been extended to high-resolution images by using spatially localized ﬁlters [6]. While we can subtract off the mean intensity from independent image patches to successfully train PoT, we cannot do that on a high-resolution image because overlapping patches might have different mean. Unfortunately, replicating potentials over the image ignoring variations of mean intensity has been the leading strategy to date [6]3 . This is the major reason why generation of high-resolution images is so poor. Sec. 4 shows that generation can be drastically improved by explicitly accounting for variations of mean intensity, as performed by mPoT and mcRBM. 3 Weight-Sharing Schemes By integrating out the latent variables, we can write the density function of any gated MRF as a normalized product of potential functions (for mPoT refer to eq. 6). In this section we investigate different ways of constraining the parameters of the potentials of a generic MRF. Global: The obvious way to extend a patch-based model like PoT to high-resolution images is to deﬁne potentials over the whole image; we call this scheme global. This is not practical because 1) the number of parameters grows about quadratically with the size of the image making training too slow, 2) we do not need to model interactions between very distant pairs of pixels since their dependence is negligible, and 3) we would not be able to use the model on images of different size. Conv: The most popular way to handle big images is to deﬁne potentials on small subsets of variables (e.g., neighborhoods of size 5x5 pixels) and to replicate these potentials across space while 2 The need to model the means was clearly recognized in [21] but they used conjunctive latent features that simultaneously represented a contribution to the “precision matrix” in a speciﬁc direction and the mean along that same direction. 3 The success of PoT-like models in Bayesian denoising is not surprising since the noisy image effectively replaces the reconstruction term from the mean hidden units (see eq. 5), providing a set of noisy mean intensities that are cleaned up by the patterns of correlation enforced by the covariance latent variables. 4 sharing their parameters at each image location [23, 24, 6]. This yields a convolutional weightsharing scheme, also called homogeneous ﬁeld in the statistics literature. This choice is justiﬁed by the stationarity of natural images. This weight-sharing scheme is extremely concise in terms of number of parameters, but also rather inefﬁcient in terms of latent representation. First, if there are N ﬁlters at each location and these ﬁlters are stepped by one pixel then the internal representation is about N times overcomplete. The internal representation has not only high computational cost, but it is also highly redundant. Since the input is mostly smooth and the parameters are the same across space, the latent variables are strongly correlated as well. This inefﬁciency turns out to be particularly harmful for a model like PoT causing the learned ﬁlters to become “random” looking (see ﬁg 3-iii). A simple intuition follows from the equivalence between PoT and square ICA [15]. If the ﬁlter matrix C of eq. 1 is square and invertible, we can marginalize out the latent variables and write: p(y) = i S(yi ), where yi = Ci T x and S is a Student’s t distribution. In other words, there is an underlying assumption that ﬁlter outputs are independent. However, if the ﬁlters of matrix C are shifted and overlapping versions of each other, this clearly cannot be true. Training PoT with the Conv weight-sharing scheme forces the model to ﬁnd ﬁlters that make ﬁlter outputs as independent as possible, which explains the very high-frequency patterns that are usually discovered [6]. Local: The Global and Conv weight-sharing schemes are at the two extremes of a spectrum of possibilities. For instance, we can deﬁne potentials on a small subset of input variables but, unlike Conv, each potential can have its own set of parameters, as shown in ﬁg. 2-B. This is called local, or inhomogeneous ﬁeld. Compared to Conv the number of parameters increases only slightly but the number of latent variables required and their redundancy is greatly reduced. In fact, the model learns different receptive ﬁelds at different locations as a better strategy for representing the input, overall when the number of potentials is limited (see also ﬁg. 2-F). TConv: Local would not allow the model to be trained and tested on images of different resolution, and it might seem wasteful not to exploit the translation invariant property of images. We therefore advocate the use of a weight-sharing scheme that we call tiled-convolutional (TConv) shown in ﬁg. 2-C and E [18]. Each ﬁlter tiles the image without overlaps with copies of itself (i.e. the stride equals the ﬁlter diameter). This reduces spatial redundancy of latent variables and allows the input images to have arbitrary size. At the same time, different ﬁlters do overlap with each other in order to avoid tiling artifacts. Fig. 2-F shows ﬁlters that were (jointly) learned by a Restricted Boltzmann Machine (RBM) [29] with Gaussian input variables using the TConv weight-sharing scheme. 4 Experiments We train gated MRF’s with and without mean hidden units using different weight-sharing schemes. The training procedure is very similar in all cases. We perform approximate maximum likelihood by using Fast Persistence Contrastive Divergence (FPCD) [25] and we draw samples by using Hybrid Monte Carlo (HMC) [26]. Since all latent variables can be exactly marginalized out we can use HMC on the free energy (negative logarithm of the marginal distribution over the input pixels). For mPoT this is: F mPoT (x) = − log(p(x))+const. = k,i 1 1 γ log(1+ (Cik T xk )2 )+ xT x− 2 2 T log(1+exp(Wjk xk )) (6) k,j where the index k runs over spatial locations and xk is the k-th image patch. FPCD keeps samples, called negative particles, that it uses to represent the model distribution. These particles are all updated after each weight update. For each mini-batch of data-points a) we compute the derivative of the free energy w.r.t. the training samples, b) we update the negative particles by running HMC for one HMC step consisting of 20 leapfrog steps. We start at the previous set of negative particles and use as parameters the sum of the regular parameters and a small perturbation vector, c) we compute the derivative of the free energy at the negative particles, and d) we update the regular parameters by using the difference of gradients between step a) and c) while the perturbation vector is updated using the gradient from c) only. The perturbation is also strongly decayed to zero and is subject to a larger learning rate. The aim is to encourage the negative particles to explore the space more quickly by slightly and temporarily raising the energy at their current position. Note that the use of FPCD as opposed to other estimation methods (like Persistent Contrastive Divergence [27]) turns out to be crucial to achieve good mixing of the sampler even after training. We train on mini-batches of 32 samples using gray-scale images of approximate size 160x160 pixels randomly cropped from the Berkeley segmentation dataset [28]. We perform 160,000 weight updates decreasing the learning by a factor of 4 by the end of training. The initial learning rate is set to 0.1 for the covariance 5 Figure 3: 160x160 samples drawn by A) mPoT-TConv, B) mHPoT-TConv, C) mcRBM-TConv and D) PoTTConv. On the side also i) a subset of 8x8 “covariance” ﬁlters learned by mPoT-TConv (the plot below shows how the whole set of ﬁlters tile a small patch; each bar correspond to a Gabor ﬁt of a ﬁlter and colors identify ﬁlters applied at the same 8x8 location, each group is shifted by 2 pixels down the diagonal and a high-resolution image is tiled by replicating this pattern every 8 pixels horizontally and vertically), ii) a subset of 8x8 “mean” ﬁlters learned by the same mPoT-TConv, iii) ﬁlters learned by PoT-Conv and iv) by PoT-TConv. ﬁlters (matrix C of eq. 1), 0.01 for the mean parameters (matrix W of eq. 4), and 0.001 for the other parameters (γ of eq. 1). During training we condition on the borders and initialize the negative particles at zero in order to avoid artifacts at the border of the image. We learn 8x8 ﬁlters and pre-multiply the covariance ﬁlters by a whitening transform retaining 99% of the variance; we also normalize the norm of the covariance ﬁlters to prevent some of them from decaying to zero during training4 . Whenever we use the TConv weight-sharing scheme the model learns covariance ﬁlters that mostly resemble localized and oriented Gabor functions (see ﬁg. 3-i and iv), while the Conv weight-sharing scheme learns structured but poorly localized high-frequency patterns (see ﬁg. 3-iii) [6]. The TConv models re-use the same 8x8 ﬁlters every 8 pixels and apply a diagonal offset of 2 pixels between neighboring ﬁlters with different weights in order to reduce tiling artifacts. There are 4 sets of ﬁlters, each with 64 ﬁlters for a total of 256 covariance ﬁlters (see bottom plot of ﬁg. 3). Similarly, we have 4 sets of mean ﬁlters, each with 32 ﬁlters. These ﬁlters have usually non-zero mean and exhibit on-center off-surround and off-center on-surround patterns, see ﬁg. 3-ii. In order to draw samples from the learned models, we run HMC for a long time (10,000 iterations, each composed of 20 leap-frog steps). Some samples of size 160x160 pixels are reported in ﬁg. 3 A)D). Without modelling the mean intensity, samples lack structure and do not seem much different from those that would be generated by a simple Gaussian model merely ﬁtting the second order statistics (see ﬁg. 3 in [1] and also ﬁg. 2 in [7]). By contrast, structure, sharp boundaries and some simple texture emerge only from models that have mean latent variables, namely mcRBM, mPoT and mHPoT which differs from mPoT by having a second layer pooling matrix on the squared covariance ﬁlter outputs [11]. A more quantitative comparison is reported in table 1. We ﬁrst compute marginal statistics of ﬁlter responses using the generated images, natural images from the test set, and random images. The statistics are the normalized histogram of individual ﬁlter responses to 24 Gabor ﬁlters (8 orientations and 3 scales). We then calculate the KL divergence between the histograms on random images and generated images and the KL divergence between the histograms on natural images and generated images. The table also reports the average difference of energies between random images and natural images. All results demonstrate that models that account for mean intensity generate images 4 The code used in the experiments can be found at the ﬁrst author’s web-page. 6 MODEL F (R) − F (T ) (104 ) KL(R G) KL(T G) KL(R G) − KL(T PoT - Conv 2.9 0.3 0.6 PoT - TConv 2.8 0.4 1.0 -0.6 mPoT - TConv 5.2 1.0 0.2 0.8 mHPoT - TConv 4.9 1.7 0.8 0.9 mcRBM - TConv 3.5 1.5 1.0 G) -0.3 0.5 Table 1: Comparing MRF’s by measuring: difference of energy (negative log ratio of probabilities) between random images (R) and test natural images (T), the KL divergence between statistics of random images (R) and generated images (G), KL divergence between statistics of test natural images (T) and generated images (G), and difference of these two KL divergences. Statistics are computed using 24 Gabor ﬁlters. that are closer to natural images than to random images, whereas models that do not account for the mean (like the widely used PoT-Conv) produce samples that are actually closer to random images. 4.1 Discriminative Experiments on Weight-Sharing Schemes In future work, we intend to use the features discovered by the generative model for recognition. To understand how the different weight sharing schemes affect recognition performance we have done preliminary tests using the discriminative performance of a simpler model on simpler data. We consider one of the simplest and most versatile models, namely the RBM [29]. Since we also aim to test the Global weight-sharing scheme we are constrained to using fairly low resolution datasets such as the MNIST dataset of handwritten digits [30] and the CIFAR 10 dataset of generic object categories [22]. The MNIST dataset has soft binary images of size 28x28 pixels, while the CIFAR 10 dataset has color images of size 32x32 pixels. CIFAR 10 has 10 classes, 5000 training samples per class and 1000 test samples per class. MNIST also has 10 classes with, on average, 6000 training samples per class and 1000 test samples per class. The energy function of the RBM trained on the CIFAR 10 dataset, modelling input pixels with 3 (R,G,B) Gaussian variables [31], is exactly the one shown in eq. 4; while the RBM trained on MNIST uses logistic units for the pixels and the energy function is again the same as before but without any quadratic term. All models are trained in an unsupervised way to approximately maximize the likelihood in the training set using Contrastive Divergence [32]. They are then used to represent each input image with a feature vector (mean of the posterior over the latent variables) which is fed to a multinomial logistic classiﬁer for discrimination. Models are compared in terms of: 1) recognition accuracy, 2) convergence time and 3) dimensionality of the representation. In general, assuming ﬁlters much smaller than the input image and assuming equal number of latent variables, Conv, TConv and Local models process each sample faster than Global by a factor approximately equal to the ratio between the area of the image and the area of the ﬁlters, which can be very large in practice. In the ﬁrst set of experiments reported on the left of ﬁg. 4 we study the internal representation in terms of discrimination and dimensionality using the MNIST dataset. For each choice of dimensionality all models are trained using the same number of operations. This is set to the amount necessary to complete one epoch over the training set using the Global model. This experiment shows that: 1) Local outperforms all other weight-sharing schemes for a wide range of dimensionalities, 2) TConv does not perform as well as Local probably because the translation invariant assumption is clearly violated for these relatively small, centered, images, 3) Conv performs well only when the internal representation is very high dimensional (10 times overcomplete) otherwise it severely underﬁts, 4) Global performs well when the representation is compact but its performance degrades rapidly as this increases because it needs more than the allotted training time. The right hand side of ﬁg. 4 shows how the recognition performance evolves as we increase the number of operations (or training time) using models that produce a twice overcomplete internal representation. With only very few ﬁlters Conv still underﬁts and it does not improve its performance by training for longer, but Global does improve and eventually it reaches the performance of Local. If we look at the crossing of the error rate at 2% we can see that Local is about 4 times faster than Global. To summarize, Local provides more compact representations than Conv, is much faster than Global while achieving 7 6 2.4 error rate % 5 error rate % 2.6 Global Local TConv Conv 4 3 2 1 0 2.2 Global Local 2 Conv 1.8 1000 2000 3000 4000 5000 dimensionality 6000 7000 1.6 0 8000 2 4 6 8 # flops (relative to # flops per epoch of Global model) 10 Figure 4: Experiments on MNIST using RBM’s with different weight-sharing schemes. Left: Error rate as a function of the dimensionality of the latent representation. Right: Error rate as a function of the number of operations (normalized to those needed to perform one epoch in the Global model); all models have a twice overcomplete latent representation. similar performance in discrimination. Also, Local can easily scale to larger images while Global cannot. Similar experiments are performed using the CIFAR 10 dataset [22] of natural images. Using the same protocol introduced in earlier work by Krizhevsky [22], the RBM’s are trained in an unsupervised way on a subset of the 80 million tiny images dataset [33] and then “ﬁne-tuned” on the CIFAR 10 dataset by supervised back-propagation of the error through the linear classiﬁer and feature extractor. All models produce an approximately 10,000 dimensional internal representation to make a fair comparison. Models using local ﬁlters learn 16x16 ﬁlters that are stepped every pixel. Again, we do not experiment with the TConv weight-sharing scheme because the image is not large enough to allow enough replicas. Similarly to ﬁg. 3-iii the Conv weight-sharing scheme was very difﬁcult to train and did not produce Gabor-like features. Indeed, careful injection of sparsity and long training time seem necessary [31] for these RBM’s. By contrast, both Local and Global produce Gabor-like ﬁlters similar to those shown in ﬁg. 2 F). The model trained with Conv weight-sharing scheme yields an accuracy equal to 56.6%, while Local and Global yield much better performance, 63.6% and 64.8% [22], respectively. Although Local and Global have similar performance, training with the Local weight-sharing scheme took under an hour while using the Global weight-sharing scheme required more than a day. 5 Conclusions and Future Work This work is motivated by the poor generative quality of currently popular MRF models of natural images. These models generate images that are actually more similar to white noise than to natural images. Our contribution is to recognize that current models can beneﬁt from 1) the addition of a simple model of the mean intensities and from 2) the use of a less constrained weight-sharing scheme. By augmenting these models with an extra set of latent variables that model mean intensity we can generate samples that look much more realistic: they are characterized by smooth regions, sharp boundaries and some simple high frequency texture. We validate our approach by comparing the statistics of ﬁlter outputs on natural images and generated images. In the future, we plan to integrate these MRF’s into deeper hierarchical models and to use their internal representation to perform object recognition in high-resolution images. The hope is to further improve generation by capturing longer range dependencies and to exploit this to better cope with missing values and ambiguous sensory inputs. References [1] E.P. Simoncelli. Statistical modeling of photographic images. Handbook of Image and Video Processing, pages 431–441, 2005. 8 [2] A. Hyvarinen, J. Karhunen, and E. Oja. Independent Component Analysis. John Wiley & Sons, 2001. [3] G.E. Hinton and R. R Salakhutdinov. Reducing the dimensionality of data with neural networks. Science, 313(5786):504–507, 2006. [4] M. Ranzato and G.E. Hinton. Modeling pixel means and covariances using factorized third-order boltzmann machines. In CVPR, 2010. [5] M.J. Wainwright and E.P. Simoncelli. Scale mixtures of gaussians and the statistics of natural images. In NIPS, 2000. [6] S. Roth and M.J. Black. Fields of experts: A framework for learning image priors. In CVPR, 2005. [7] U. Schmidt, Q. Gao, and S. Roth. A generative perspective on mrfs in low-level vision. In CVPR, 2010. [8] S. Geman and D. Geman. Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. PAMI, 6:721–741, 1984. [9] M. Welling, G.E. Hinton, and S. Osindero. Learning sparse topographic representations with products of student-t distributions. In NIPS, 2003. [10] S.C. Zhu and D. Mumford. Prior learning and gibbs reaction diffusion. PAMI, pages 1236–1250, 1997. [11] S. Osindero, M. Welling, and G. E. Hinton. 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