nips nips2002 nips2002-74 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Amos J. Storkey
Abstract: The problem of super-resolution involves generating feasible higher resolution images, which are pleasing to the eye and realistic, from a given low resolution image. This might be attempted by using simple filters for smoothing out the high resolution blocks or through applications where substantial prior information is used to imply the textures and shapes which will occur in the images. In this paper we describe an approach which lies between the two extremes. It is a generic unsupervised method which is usable in all domains, but goes beyond simple smoothing methods in what it achieves. We use a dynamic tree-like architecture to model the high resolution data. Approximate conditioning on the low resolution image is achieved through a mean field approach. 1
Reference: text
sentIndex sentText sentNum sentScore
1 uk Abstract The problem of super-resolution involves generating feasible higher resolution images, which are pleasing to the eye and realistic, from a given low resolution image. [sent-4, score-1.141]
2 This might be attempted by using simple filters for smoothing out the high resolution blocks or through applications where substantial prior information is used to imply the textures and shapes which will occur in the images. [sent-5, score-0.667]
3 We use a dynamic tree-like architecture to model the high resolution data. [sent-8, score-0.657]
4 Approximate conditioning on the low resolution image is achieved through a mean field approach. [sent-9, score-0.842]
5 1 Introduction Good techniques for super-resolution are especially useful where physical limitations exist preventing higher resolution images from being obtained. [sent-10, score-0.559]
6 For example, in astronomy where public presentation of images is of significant importance, superresolution techniques have been suggested. [sent-11, score-0.105]
7 Whenever dynamic image enlargement is needed, such as on some web pages, super-resolution techniques can be utilised. [sent-12, score-0.302]
8 This paper focuses on the issue of how to increase the resolution of a single image using only prior information about images in general, and not relying on a specific training set or the use of multiple images. [sent-13, score-0.795]
9 They range from simple use of Gaussian or preferably median filtering, to supervised learning methods based on learning image patches corresponding to low resolution regions from training data, and effectively sewing these patches together in a consistent manner. [sent-15, score-1.103]
10 There is a demand for methods which are reasonably fast, which are generic in that they do not rely on having suitable training data, but which do better than standard linear filters or interpolation methods. [sent-17, score-0.154]
11 This paper describes an approach to resolution doubling which achieves this. [sent-18, score-0.545]
12 The method is structurally related to one layer of the dynamic tree model [9, 8, 1] except that it uses real valued variables. [sent-19, score-0.189]
13 2 Related work Simple approaches to resolution enhancement have been around for some time. [sent-20, score-0.549]
14 Gaussian and Wiener filters (and a host of other linear filters) have been used for smoothing the blockiness created by the low resolution image. [sent-21, score-0.74]
15 Interpolation methods such as cubicspline interpolation tend to be the most common image enhancement approach. [sent-23, score-0.382]
16 Many authors are interested in reconstruction based on multiple slightly perturbed subsamples from an image [3, 2] . [sent-25, score-0.257]
17 They follow a supervised approach, learning a low to high resolution patch model (or rather storing examples of such maps), and utilising a Markov random field for combining them and loopy propagation for inference. [sent-31, score-0.714]
18 There are two primary difficulties with smoothing (eg Gaussian, Wiener, Median filters) or interpolation (bicubic, cubic spline) methods. [sent-34, score-0.203]
19 It occurs both within the gradual change in colour of the sky, say, as well as across the horizon, producing blurring problems. [sent-36, score-0.231]
20 Second these approaches are inconsistent: subsampling the super-resolution image will not return the original low-resolution one. [sent-37, score-0.207]
21 Hence we need a model which maintains consistency but also tries to ensure that smoothing does not occur across region boundaries (except as much is as needed for anti-aliasing). [sent-38, score-0.106]
22 3 The model Here the high-resolution image is described by a series of very small patches with varying shapes. [sent-39, score-0.3]
23 Pixel values within these patches can vary, but will have a common mean value. [sent-40, score-0.123]
24 Apriori exactly where these patches should be is uncertain, and so the pixel to patch mapping is allowed to be a dynamic one. [sent-42, score-0.577]
25 The lowest layer consists of the visible low-resolution pixels. [sent-45, score-0.11]
26 The intermediate layer is a high-resolution image (4 × 4 the size of the low-resolution image). [sent-46, score-0.268]
27 The top layer is a latent layer which is a little more than 2 × 2 the size of the low resolution image. [sent-47, score-0.938]
28 The latent variables are ‘positioned’ at the corners, centres and edge centres of the pixels of the low resolution image. [sent-48, score-1.045]
29 The values of the pixel colour of the high resolution nodes are each a single sample from a Gaussian mixture (in colour space), where each mixture centre is given by the pixel colour of a particular parent latent Latent Hi Res Low Res Figure 1: The three layers of the model. [sent-49, score-2.352]
30 The small boxes in the left figure (64 of them) give the position of the high resolution pixels relative to the low resolution pixels (the 4 boxes with a thick outline). [sent-50, score-1.527]
31 The positions of the latent variable nodes are given by the black circles. [sent-51, score-0.3]
32 The colour of each high resolution pixel is generated from a mixture of Gaussians (right figure), each Gaussian centred at its latent parent pixel value. [sent-52, score-1.908]
33 The closer the parent is, the higher the prior probability of being generated by that mixture is. [sent-53, score-0.222]
34 The prior mixing coefficients decay with distance in image space between the high-resolution node and the corresponding latent node. [sent-55, score-0.532]
35 Another way of viewing this is that a further indicator variable can be introduced which selects which mixture is responsible for a given high-resolution node. [sent-56, score-0.202]
36 We say a high resolution node ‘chooses’ to connect to the parent that is responsible for it, with a connection probability given by the corresponding mixing coefficient. [sent-57, score-0.909]
37 The high-resolution pixels corresponding to a visible node can be separated into two (or more) independent regions, corresponding to pixels on different sides of an edge (or edges). [sent-62, score-0.357]
38 A different latent variable is responsible for each region. [sent-63, score-0.315]
39 In other words each mixture component effectively corresponds to a small image patch which can vary in size depending on what pixels it is responsible for. [sent-64, score-0.537]
40 Let vj ∈ L denote a latent variable at site j in the latent space L. [sent-65, score-0.564]
41 Let xi ∈ S denote the value of pixel i in high resolution image space S, and let yk denote the value of the visible pixel k. [sent-66, score-1.559]
42 In other words the data is a linear transformation on the RGB colour values using the matrix 0. [sent-71, score-0.198]
43 the indicator for the responsibility) between the high-resolution nodes and the nodes in the latent layer. [sent-79, score-0.379]
44 Let zij denote this connectivity with zij an indicator variable taking value 1 when vj is a parent of xi in the belief network. [sent-80, score-0.599]
45 Every high resolution pixel has one and only one parent in the latent layer. [sent-81, score-1.257]
46 1 Distributions A uniform distribution over the range of pixel values is presumed for the latent variables. [sent-84, score-0.631]
47 The high resolution pixels are given by Gaussian distributions centred on the pixel values of the parental latent variable. [sent-85, score-1.301]
48 This Gaussian is presumed independent in each pixel component. [sent-86, score-0.42]
49 Finally the low resolution pixels are given by the average of the sixteen high resolution pixels covering the site of the low resolution pixel. [sent-87, score-2.051]
50 This pixel value can also be subject to some additional Gaussian noise if necessary (zero noise is assumed in this paper). [sent-88, score-0.341]
51 It is presumed that each high resolution pixel is allowed to ‘choose’ its parent from the set of latent variables in an independent manner. [sent-89, score-1.336]
52 A pixel has a higher probability of choosing a nearby parent than a far away one. [sent-90, score-0.484]
53 The integral is over Bi defined as the region in image space corresponding to pixel xi . [sent-92, score-0.548]
54 The connection probabilities can be illustrated by the picture in figure 2. [sent-95, score-0.082]
55 First we have P (X|Z, V ) = ijm m (xm − vj )2 1 i exp −zij 2Ωm (2πΩm )1/2 . [sent-97, score-0.134]
56 (2) where Ωm is a variance which determines how much each pixel must be like its latent parent. [sent-98, score-0.588]
57 Here the indicator zij ensures the only contribution for each i comes from the parent j of i. [sent-99, score-0.32]
58 Second P (Y |X) = km m (yk − 1 exp − (2πΛ)1/2 1 d i∈P a(k) 2Λ xm ) 2 i (3) Figure 2: An illustration of the connection probabilities from a high resolution pixel in the position of the smaller checkered square to the latent variables centred at each of the larger squares. [sent-100, score-1.439]
59 with P a(k) denoting the set of all the d = 16 high resolution pixels which go to make up the low resolution pixel yk . [sent-102, score-1.692]
60 Λ determines the additive Gaussian noise which is in the low resolution image. [sent-104, score-0.605]
61 2 Inference The belief network defined above is not tree structured (rather it is a mixture of tree structures) and so we have to resort to approximation methods for inference. [sent-108, score-0.183]
62 The posterior distribution is approximated using a factorised distribution over the latent space and over the connectivity. [sent-110, score-0.211]
63 Only in the high resolution space X do we consider joint distributions: we use a joint Gaussian for all the nodes corresponding to one low resolution pixel. [sent-111, score-1.227]
64 Here qij , µm , νj , Φm and Ψm are variational parameters to be optimised. [sent-113, score-0.307]
65 This is equivalent to maximising the negative variational free energy (or variational log likelihood) Q(Z, V, X) (6) L(Q||P ) = log P (Z, V, X, Y ) Q(Z,V,X) where Y is given by the low resolution image. [sent-115, score-0.921]
66 In this case we obtain L(Q||P ) = log Q(Z) − log P (Z) Q(Z) + log Q(V ) − log p(V ) Q(V ) + log Q(X) Q(X) − log P (X|Z, V ) Q(X,Z,V ) − log P (Y |X) Q(Y,X) . [sent-116, score-0.406]
67 Here for simplicity we only solve for qij and for the means µm i m and νj which turn out to be independent of the variational variance parameters. [sent-118, score-0.343]
68 We obtain qij xm i m m νj = i and µm = ρm + Dc(i) where ρm = qij vi (8) i i i qij i j where c(i) is the child of i, i. [sent-119, score-0.712]
69 The update for the qij is given by qij ∝ pij exp − m m (xm − vk )2 i m 2Ω (10) where the constant of proportionality is given by normalisation: j qij = 1. [sent-123, score-0.737]
70 For each Q(Z) optimisation (10), equations (8a) and (8b) are iterated a number of times. [sent-125, score-0.132]
71 Each optimisation loop is either done a preset number of times, or until a suitable convergence criterion is met. [sent-126, score-0.191]
72 The former approach is generally used, as the basic criterion is a limit on the time available for the optimisation to be done. [sent-127, score-0.164]
73 If this is not known to be zero, then it will vary from image to image, and needs to be found for each image. [sent-130, score-0.239]
74 This can be done using variational maximum likelihood, where Λ is set to maximise the variational log likelihood. [sent-131, score-0.258]
75 Σ is presumed to be independent of the images presented, and is set by hand by visualising changes on a test set. [sent-132, score-0.137]
76 To optimise automatically based on the variational log likelihood is possible but does not produce as good results due to the complicated nature of a true prior or error-measure for images. [sent-136, score-0.187]
77 For example, a highly elaborate texture offset by one pixel will give a large mean square error, but look almost identical, whereas a blurred version of the texture would give a smaller mean square error, but look much worse. [sent-137, score-0.542]
78 5 Implementation The basic implementation involves setting the parameters, running the mean field optimisation and then looking at the result. [sent-138, score-0.197]
79 The final result is a downsampled version of the 4 × 4 image to 2 × 2 size: the larger image is used to get reasonable anti-aliasing. [sent-139, score-0.476]
80 To initialise the mean field optimisation, X is set equal to the bi-cubic interpolated image with added Gaussian noise. [sent-140, score-0.237]
81 Although in the examples here we used 25 optimisations Q(Z), each of which involves 10 cycles through the mean field equations for Q(X) and Q(V ), it is possible to get reasonable results with only three Q(Z) optimisation cycles each doing 2 iterations through the mean field equations. [sent-142, score-0.299]
82 6 Demonstrations and assessment The method described in this paper is compared with a number of simple filtering and interpolation methods, and also with the methods of Freeman et al. [sent-146, score-0.195]
83 The image from Freeman’s website is used for comparison with that work (figure 3). [sent-147, score-0.254]
84 Full colour comparisons for these and other images can be found at http://www. [sent-148, score-0.256]
85 (a) gives the 70x70 low resolution image, (b) the true image, (c) a bi-cubic interpolation (d) Freeman et al result (taken from website and downsampled), (e) dynamic structure super-resolution, (f) median filter. [sent-159, score-1.049]
86 compare the results of this approach with standard filtering methods using a root mean squared pixel error on a set of 8, 128 by 96 colour images, giving 0. [sent-160, score-0.569]
87 0452 for the original low resolution image, bicubic interpolation, the median filter and dynamic structure super-resolution respectively. [sent-164, score-0.864]
88 Unfortunately the unavailability of code prevents representative calculations for the Freeman et al approach. [sent-165, score-0.137]
89 Dynamic structure resolution requires approximately 30 − 60 flops per 2 × 2 high resolution pixel per optimisation cycle, compared with, say, 16 flops for a linear filter, so it is more costly. [sent-166, score-1.536]
90 Qualitatively, the results for dynamic structure super-resolution are significantly better than most standard filtering approaches. [sent-169, score-0.095]
91 The texture is better represented because it maintains consistency, and the edges are sharper, although there is still some significant difference from the true image. [sent-170, score-0.102]
92 The method of Freeman et al is perhaps comparable at this resolution, although it should be noted that their result has been downsampled here to half the size of their enhanced image. [sent-171, score-0.132]
93 Their method can produce 4 × 4 the resolution of the original, and so this does not accurately represent the full power of their technique. [sent-172, score-0.501]
94 Furthermore this image is representative of early results from their work. [sent-173, score-0.245]
95 However their approach does require learning large numbers of patches from a training set. [sent-174, score-0.093]
96 Fundamentally the dynamic structure super-resolution approach does a good job at resolution doubling without the need for representative training data. [sent-175, score-0.678]
97 The edges are not blurred and much of the blockiness is removed. [sent-176, score-0.149]
98 Dynamic structure super-resolution provides a technique for resolution enhancement, and provides an interesting starting model which is different from the Markov random field approaches. [sent-177, score-0.501]
99 A Bayesian approach to image expansion for improved definition. [sent-224, score-0.207]
100 Dynamic trees: A structured variational method giving efficient propagation rules. [sent-229, score-0.13]
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