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158 nips-2005-Products of ``Edge-perts

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Author: Max Welling, Peter V. Gehler

Abstract: Images represent an important and abundant source of data. Understanding their statistical structure has important applications such as image compression and restoration. In this paper we propose a particular kind of probabilistic model, dubbed the “products of edge-perts model” to describe the structure of wavelet transformed images. We develop a practical denoising algorithm based on a single edge-pert and show state-ofthe-art denoising performance on benchmark images. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Understanding their statistical structure has important applications such as image compression and restoration. [sent-6, score-0.099]

2 In this paper we propose a particular kind of probabilistic model, dubbed the “products of edge-perts model” to describe the structure of wavelet transformed images. [sent-7, score-0.594]

3 We develop a practical denoising algorithm based on a single edge-pert and show state-ofthe-art denoising performance on benchmark images. [sent-8, score-0.412]

4 Wavelet transforms, which capture most of the second order dependencies, form the basis of many successful image processing applications such as image compression (e. [sent-10, score-0.206]

5 However, the higher order dependencies can not be ﬁltered out by these linear transforms. [sent-15, score-0.114]

6 In particular, the absolute values of neighboring wavelet coefﬁcients (but not their signs) are mutually dependent. [sent-16, score-0.594]

7 This kind of dependency is caused by the presence of edges that induce clustering of wavelet activity. [sent-17, score-0.637]

8 Our philosophy is that by modelling this clustering effect we can potentially improve the performance of some important image processing tasks. [sent-18, score-0.131]

9 Our model builds on earlier work in the image processing literature. [sent-19, score-0.113]

10 The state-of-art in this area is the joint model discussed in [3] based on the “Gaussian scale mixture” model (GSM). [sent-21, score-0.103]

11 While the GSM falls in the category of directed graphical models and has a top-down structure, the PoEdges model is best classiﬁed as an (undirected) Markov random ﬁeld model and follows bottom-up semantics. [sent-22, score-0.064]

12 1) and 3) a new “iterated Wiener denoising algorithm” (section 3. [sent-24, score-0.206]

13 The statistics were collected from the vertical subband at the lowest level of a Haar ﬁlter wavelet decomposition of the ”Lena” image. [sent-31, score-0.68]

14 Note that the “bow-tie” dependencies are captured by the PoEdges model. [sent-32, score-0.095]

15 2 “Product of Edge-perts” It has long been recognized in the image processing community that wavelet transforms form an excellent basis for representation of images. [sent-35, score-0.722]

16 Within the class of linear transforms, it represents a compromise between many conﬂicting but desirable properties of image representation such as multi-scale and multi-orientation representation, locality both in space and frequency, and orthogonality resulting in decorrelation. [sent-36, score-0.081]

17 A particularly suitable wavelet transform which forms the basis of the best denoising algorithms today is the over-complete steerable wavelet pyramid [4] freely downloadable from http://www. [sent-37, score-1.535]

18 In our experiments we have conﬁrmed that the best results were obtained using this wavelet pyramid. [sent-42, score-0.594]

19 In the following we will describe a model for the statistical dependencies between wavelet coefﬁcients. [sent-43, score-0.721]

20 This model was inspired by recent studies of these dependencies (see e. [sent-44, score-0.127]

21 It also represents a generalization of the bivariate Laplacian model proposed in [2]. [sent-47, score-0.081]

22 The parameters {βj } control the shape of the contours: for β = 2 we have elliptical contours, for β = 1 the contours are straight lines while for β < 1 the contours curve inwards. [sent-53, score-0.084]

23 reﬂections of any subset of the {zi } in the origin, which implies that the wavelet coefﬁcients are necessarily decorrelated (although higher order dependencies may still remain). [sent-61, score-0.837]

24 Finally, the weights {Wij } model the scale (inverse variance) of the wavelet coefﬁcients. [sent-62, score-0.646]

25 We mention that it is possible to entertain a larger number of bases vectors than wavelet coefﬁcients (a so-called “over-complete basis”), which seems appropriate for some of the empirical joint histograms shown in [1]. [sent-63, score-0.613]

26 the wavelet coefﬁcients) undergo a nonlinear transformation zi → |zi |βi → u = W |z|β → uα . [sent-77, score-0.667]

27 What is the reason that the PoEdges model captures the clustering of wavelet activities? [sent-80, score-0.647]

28 Consider a local model describing the statistical structure of a patch of wavelet coefﬁcients and recall that the weighted sum of these activities is penalized. [sent-81, score-0.653]

29 This “sparse” pattern of activity incurs less penalty than for instance the same amount1 of activity distributed equally over all images because of the concave shape of the penalty function, i. [sent-84, score-0.143]

30 1 Related Work Early wavelet denoising techniques were based on the observation that the marginal distribution of a wavelet coefﬁcient is highly kurtotic (peaked and heavy tails). [sent-88, score-1.418]

31 (1) p(z) = 1 exp [−(w|z|) ] , 2Γ( α ) This has lead to the successful wavelet coring and shrinkage methods. [sent-90, score-0.656]

32 A bivariate generalization of that model describing a wavelet coefﬁcient zc and its “parent” zp at a higher level in the pyramid jointly, was proposed in [2]. [sent-91, score-0.843]

33 The probability density, p(zc , zp ) = w 2 2 exp − w(zc + zp ) 2π (2) is easily seen to be a special case of the PoEdges model proposed here. [sent-92, score-0.11]

34 This model, unlike the univariate model, captures the bow-tie dependencies described above resulting a signiﬁcant gain in denoising performance. [sent-93, score-0.301]

35 “Gaussian scale mixtures” (GSM) have been proposed to model even larger neighborhoods of wavelet coefﬁcients. [sent-94, score-0.669]

36 In particular, very good denoising results have been obtained by including within subband neighborhoods of size 3 × 3 in addition to the parent of a wavelet coefﬁcient [3]. [sent-95, score-0.956]

37 A GSM is deﬁned in terms of a precision √ variable u, the squareroot of which multiplies a multivariate Gaussian variable: z = u y, y ∼ N [0, Σ], resulting in the following expression for the distribution over the wavelet coefﬁcients: p(z) = du Nz [0, uΣ] p(u). [sent-96, score-0.594]

38 1 We assume the total amount of variance in wavelet activity is ﬁxed in this comparison. [sent-99, score-0.614]

39 3 Edge-pert Denoising Based on the PoEdges model discussed in the previous sections we now introduce a simpliﬁed model that forms the basis for a practical denoising algorithm. [sent-102, score-0.296]

40 Recent progress in the ﬁeld has indicated that it is important to model the higher order dependencies which exist between wavelet coefﬁcients [2, 3]. [sent-103, score-0.74]

41 This can be realized through the estimation of a joint model on a small cluster of wavelet coefﬁcients around each coefﬁcient. [sent-104, score-0.645]

42 Therefore, in order to keep computations tractable, we proceed with a simpliﬁed model, ˆ wj aj T z p(z) ∝ exp − 2 α . [sent-106, score-0.09]

43 1 Model Estimation Our next task is to estimate the parameters of this model efﬁciently. [sent-109, score-0.051]

44 We will learn separate models for each wavelet coefﬁcient jointly with a small neighborhood of dependent coefﬁcients. [sent-110, score-0.649]

45 Each such model is estimated in three steps: I) determine the coefﬁcients that participate in each model, II) transform each model into a decorrelated domain (this implicitly estimates the {ˆj }) and III) estimate the remaining parameters w, α in the decorrelated a domain using moment matching. [sent-111, score-0.519]

46 By zi , zi we will denote the clean and noisy wavelet coefﬁcients respectively. [sent-113, score-0.789]

47 With yi , yi ˜ ˜ we denote the decorrelated clean and noisy wavelet coefﬁcients while ni denotes the Gaussian noise random variable in the wavelet domain, i. [sent-114, score-1.452]

48 Both due to ˜ the details of the wavelet decomposition and due to the properties of the noise itself we assume the noise to be correlated and zero mean: E[ni ] = 0, E[ni nj ] = Σij . [sent-117, score-0.694]

49 In this paper we further assume that we know the noise covariance in the image domain from which one can easily compute the noise covariance in the wavelet domain, however only minor changes are needed to estimate it from the noisy image itself. [sent-118, score-0.995]

50 Step I: We start with a 7 × 7 neighborhood from which we will adaptively select the best candidates to include in the model. [sent-119, score-0.055]

51 In addition, we will always include the parent coefﬁcient in the subband of a coarser scale if it exists (this is done by ﬁrst up-sampling this band, see [3]). [sent-120, score-0.153]

52 The coefﬁcients that participate in a model are selected by estimating their dependencies relative to the center coefﬁcient. [sent-121, score-0.158]

53 Anticipating that (second order) correlations will be removed by sphering we are only interested in higher order dependencies, in particular dependencies between the variances. [sent-122, score-0.114]

54 The following cumulant is used to obtain these estimates, Hcj = E[˜c zj ] − 2E[˜c zj ]2 − E[˜c ]E[˜j ] z 2 ˜2 z ˜ z2 z2 (4) where c is the center coefﬁcient which will be denoised. [sent-123, score-0.16]

55 Effectively, we meaz sure the (higher order) dependencies between squared wavelet coefﬁcients after subtraction of all correlations. [sent-128, score-0.689]

56 Finally, we select the participants of a model centered at coefﬁcient zc by ˜ ranking the positive Hcj and picking all the ones which satisfy: Hci > 0. [sent-129, score-0.103]

57 Step II: For each model (with varying number of participants) we estimate the covariance, Cij = E[zi , zj ] = E[˜i zj ] − Σij z˜ (5) and correct it by setting to zero all negative eigenvalues in such a way that the sum of the eigenvalues is invariant (see [3]). [sent-131, score-0.187]

58 (6) In this new space (which is different for every wavelet coefﬁcient) we can now assume ˆ aj = ej , the axis aligned basis vector. [sent-135, score-0.66]

59 Step III: In the decorrelated space we estimate the single edge-pert model by moment matching. [sent-136, score-0.22]

60 The moments of the edge-pert model in this space are easily computed using Np 2 wj yj ) E ( =Γ j=1 Np + 2 2α / Γ Np 2α (7) where Np is the number of participating coefﬁcients in the model. [sent-137, score-0.13]

61 permutations of the variables uj = wj yj we ﬁnd wj = Γ Np +2 2α / Np (E[˜i ] − 1) Γ y2 Np 2α . [sent-145, score-0.125]

62 (9) A common strategy in the wavelet literature is to estimate the averages E[·] by collecting samples in a local neighborhood around the coefﬁcient under consideration. [sent-146, score-0.689]

63 We have adopted this strategy and used a 11 × 11 box around each coefﬁcient to collect 121 samples in the decorrelated wavelet domain. [sent-148, score-0.723]

64 The estimation of α depends on the fourth moment and is thus very sensitive to outliers, which is a commonly known problem with the moment matching method. [sent-150, score-0.08]

65 2 The Iterated Wiener Filter To infer a wavelet coefﬁcient given its noisy observation in the decorrelated wavelet domain, we maximize the a posteriori probability of our joint model. [sent-155, score-1.364]

66 z (10) z When we assume Gaussian pixel noise, this translates into, z∗ = argmin z 1 2 (z ˜ ˜ − z)T K(z − z) + 2 wj zj α (11) j ˜ where J is the (linear) wavelet transform z = Jx, K = J #T Σ−1 J # with J # = n (J T J)−1 J T the pseudo-inverse of J (i. [sent-157, score-0.77]

67 In the decorrelated wavelet domain we simply set K = I. [sent-160, score-0.761]

68 Determine coefﬁcients participating in joint model by using Eqn. [sent-212, score-0.074]

69 Estimate parameters {α, wi } on a local neighborhood using Eqn. [sent-224, score-0.055]

70 Denoise all wavelet coefﬁcients in the neighborhood using IWF from section 3. [sent-228, score-0.649]

71 Transform denoised cluster back to the wavelet domain and retain the “center coefﬁcient” only. [sent-231, score-0.671]

72 4 Experiments Denoising experiments were run on the steerable wavelet pyramid with oriented highpass residual bands (FSpyr) using 8 orientations as described in [3]. [sent-234, score-0.678]

73 In each experiment an image was artiﬁcially contaminated with independent Gaussian pixel noise of some predetermined variance and denoised using 20 iterations of the proposed algorithm. [sent-236, score-0.197]

74 In table 1 we compare performance between the PoEdges and GSM based denoising algorithms on six test images and ten different noise levels. [sent-242, score-0.303]

75 66 Table 1: Comparison of image denoising results between PoEdges (EP above) and its closest competitor (GSM). [sent-363, score-0.322]

76 Note that PoEdges outperforms GSM for low noise levels while the GSM performs better at high noise levels. [sent-367, score-0.119]

77 Also, PoEdges performs best at all noise levels on the Barbara image, while GSM is superior on the boat image. [sent-368, score-0.156]

78 FSpyr against various methods published in the literature [3, 2, 9] on the images “Lena” and “Barbara”. [sent-369, score-0.065]

79 In comparing PoEdges with GSM the general trend seems to be that PoEdges performs superior at lower noise levels while the reverse is true for higher noise levels. [sent-371, score-0.178]

80 We observe that the PoEdges give signiﬁcantly better results on the ”Barbara” image than any other published method (by a large magin). [sent-372, score-0.099]

81 Increasing the estimation window in step 3ii of the algorithm let the denoising results drop down to the GSM solution (not reported here). [sent-374, score-0.206]

82 Comparing the quality of restored images in detail (as in ﬁgure 3) we conclude that the GSM produces slightly sharper edges at the expense of more artifacts. [sent-375, score-0.066]

83 Denoising a 512 × 512 pixel sized image on a pentium 4 2. [sent-376, score-0.108]

84 8GHz PC for our adaptive neighborhood selection model took 26 seconds for the QMF9 and 440 seconds for the FSpyr. [sent-377, score-0.087]

85 We also compared GSM and EP using a separable orthonormal pyramid (QMF9). [sent-378, score-0.058]

86 However the results are signiﬁcantly inferior because the wavelet representation plays a prominent role for denoising performance. [sent-380, score-0.8]

87 5 Discussion We have proposed a general “product of edge-perts” model to capture the dependency structure in wavelet coefﬁcients. [sent-382, score-0.648]

88 This was turned into a practical denoising algorithm by simplifying to a single edge-pert and choosing βj = 2 ∀j. [sent-383, score-0.206]

89 The parameters of this model can be adapted based on the noisy observation of the image. [sent-384, score-0.06]

90 In comparison with the closest competitor (GSM [3]) we found superior performance at low noise levels while the reverse is true for high noise levels. [sent-385, score-0.194]

91 Also, the PoEdges model performs better than any competitor on the Barbara image, but consistency less well than GSM on the boat image. [sent-386, score-0.133]

92 The GSM model aims at capturing the same statistical regularities as the PoEdges but using a very different modelling paradigm: where PoEdges is best interpreted as a bottom-up constraint satisfaction model, the GSM is a causal generative model with top-down semantics. [sent-387, score-0.093]

93 We have found that these two modelling paradigms exhibit different denoising accuracies 2 3 Personal communication http://www. [sent-388, score-0.235]

94 69) on Barbara image (cropped to 150 × 150 to enhance artifacts). [sent-393, score-0.081]

95 For example, we can lift the ˆ restriction on βj = 2, allow more basis-vectors aj than coefﬁcients or extend the neighborhood selection to subbands of different scales and/or orientations. [sent-400, score-0.095]

96 However, approximations in the estimation of these models will become necessary to keep the denoising algorithm practical. [sent-402, score-0.206]

97 Further performance gains may still be expected through the development of new wavelet pyramids and through modelling of new dependency structures such as the phenomenon of phase alignment at the edges. [sent-405, score-0.645]

98 Image denoising using scale mixtures of Gaussians in the wavelet domain. [sent-427, score-0.82]

99 Modeling the joint statistics of images in the wavelet domain. [sent-439, score-0.66]

100 Low-complexity image denoising based on statistical modeling of wavelet coefﬁcients. [sent-470, score-0.881]

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