nips nips2002 nips2002-126 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Phil Sallee, Bruno A. Olshausen
Abstract: We describe a method for learning sparse multiscale image representations using a sparse prior distribution over the basis function coefficients. The prior consists of a mixture of a Gaussian and a Dirac delta function, and thus encourages coefficients to have exact zero values. Coefficients for an image are computed by sampling from the resulting posterior distribution with a Gibbs sampler. The learned basis is similar to the Steerable Pyramid basis, and yields slightly higher SNR for the same number of active coefficients. Denoising using the learned image model is demonstrated for some standard test images, with results that compare favorably with other denoising methods. 1
Reference: text
sentIndex sentText sentNum sentScore
1 edu Abstract We describe a method for learning sparse multiscale image representations using a sparse prior distribution over the basis function coefficients. [sent-7, score-0.806]
2 The prior consists of a mixture of a Gaussian and a Dirac delta function, and thus encourages coefficients to have exact zero values. [sent-8, score-0.248]
3 Coefficients for an image are computed by sampling from the resulting posterior distribution with a Gibbs sampler. [sent-9, score-0.303]
4 The learned basis is similar to the Steerable Pyramid basis, and yields slightly higher SNR for the same number of active coefficients. [sent-10, score-0.2]
5 Denoising using the learned image model is demonstrated for some standard test images, with results that compare favorably with other denoising methods. [sent-11, score-0.416]
6 1 Introduction Increasing interest has been given to the use of overcomplete representations for natural scenes, where the number of basis functions exceeds the number of image pixels. [sent-12, score-0.845]
7 This may translate into gains in coding efficiency for image compression, and improved accuracy for tasks such as denoising. [sent-14, score-0.358]
8 Overcomplete representations have been shown to reduce Gibbs-like artifacts common to thresholding methods employing critically sampled wavelets [4, 3, 9]. [sent-15, score-0.56]
9 Common wavelet denoising approaches generally apply either a hard or softthresholding function to coefficients which have been obtained by filtering an image with a the basis functions. [sent-16, score-0.834]
10 One can view these thresholding methods as a means of selecting coefficients for an image based on an assumed sparse prior on the coefficients [1, 2]. [sent-17, score-0.652]
11 This statistical framework provides a principled means of selecting an appropriate thresholding function. [sent-18, score-0.275]
12 When such thresholding methods are applied to overcomplete representations, however, problems arise due to the dependencies between coefficients. [sent-19, score-0.479]
13 Choosing optimal thresholds for a non-orthogonal basis is still an unsolved problem. [sent-20, score-0.232]
14 In one approach, orthogonal subgroups of an overcomplete shift-invariant expansion are thresholded separately and then the results are combined by averaging [4, 3]. [sent-21, score-0.32]
15 Here we address two major issues regarding the use of overcomplete representations for images. [sent-23, score-0.413]
16 First, current methods make use of various overcomplete wavelet bases. [sent-24, score-0.566]
17 What is the optimal basis to use for a specific class of data? [sent-25, score-0.157]
18 To help answer this question, we describe how to adapt an overcomplete wavelet basis to the statistics of natural images. [sent-26, score-0.723]
19 Secondly, we address the problem of properly inferring the coefficients for an image when the basis is overcomplete. [sent-27, score-0.49]
20 We avoid problems associated with thresholding by using the wavelet basis as part of a generative model, rather than a simple filtering mechanism. [sent-28, score-0.718]
21 We then sample the coefficients from the resulting posterior distribution by simulating a Markov process known as a Gibbs-sampler. [sent-29, score-0.058]
22 Our previous work in this area made use of a prior distribution peaked at zero and tapering away smoothly to obtain sparse coefficients [7]. [sent-30, score-0.216]
23 However, we encountered a number of significant limitations with this method. [sent-31, score-0.032]
24 First, the smooth priors do not force inactive coefficients to have values exactly equal to zero, resulting in decreased coding efficiency. [sent-32, score-0.231]
25 Efficiency may be partially regained by thresholding the near-zero coefficients, but due to the non-orthogonality of the representation this will produce sub-optimal results as previously mentioned. [sent-33, score-0.279]
26 The maximum a posteriori (MAP) estimate also introduced biases in the learning process. [sent-34, score-0.097]
27 These effects can be partially compensated for by renormalizing the basis functions, but other parameters of the model such as those of the prior could not be learned. [sent-35, score-0.292]
28 Finally, the gradient ascent method has convergence problems due to the power spectrum of natural images and the overcompleteness of the representation. [sent-36, score-0.199]
29 Here we resolve these problems by using a prior distribution which is composed of a mixture of a Gaussian and a Dirac delta function, so that inactive coefficients are encouraged to have exact zero values. [sent-37, score-0.469]
30 Similar models employing a mixture of two Gaussians have been used for classifying wavelet coefficients into active (high variance) and inactive (low variance) states [2, 5]. [sent-38, score-0.599]
31 Such a classification should be even more advantageous if the basis is overcomplete. [sent-39, score-0.19]
32 A method for performing Gibbs-sampling for the Delta-plus-Gaussian prior in the context of an image pyramid is derived, and demonstrated to be effective at obtaining very sparse representations which match the form of the imposed prior. [sent-40, score-0.642]
33 Biases in the learning are overcome by sampling instead of using a MAP estimate. [sent-41, score-0.054]
34 Unlike typical wavelet transforms which use a single 1-D mother wavelet function to generate 2-D functions by inner product, we do not constrain the functions ψi (x, y) to be 1-D separable. [sent-44, score-0.901]
35 The functions ψi (x, y) provide an efficient way to perform computations involving W by means of convolutions. [sent-45, score-0.079]
36 Basis functions of coarser scales are produced by upsampling the ψi (x, y) functions and blurring with a low-pass filter φ(x, y), also known as the scaling function. [sent-46, score-0.251]
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