Denoising natural images based on a modified sparse coding algorithm
Introduction
Image reconstruction is generally an inverse problem, which intends to recover the original ideal image from its given bad version, such as one that is snowed by noise, blurred by atmospheric turbulence (as in certain astronomic observations), or that has some regions damaged (like in an old black–white photo or an ancient painting) [1]. Thus, an important subject is how to develop image enhancement algorithms that can reconstruct image corrupted as close as possible to the original one. In this paper, we only consider the contaminated source, noise, of natural images. In other words, the purpose of image denoising is to restore the original image with noise-free. Classical image denoising techniques are based on filtering [2]. More recently, there are more and more new denoising techniques explored, such as wavelet-based approach [3], principal components analysis (PCA) approach [4], independent component analysis (ICA) and standard sparse coding (SC) shrinkage proposed by Alpo Hyvärinen in 1997 [5], [6], etc. These methods can successfully denoise images by using different skills and strategies. The biggest advantage of wavelet methods is that a very fast algorithm can be derived to perform the transformation. However, wavelet approaches depend on, to large extent, some certain and abstract mathematic property of data and are not of adaptive ability to image data. While the significant advantage of the PCA and sparse coding is that they depend on the statistic property of data. However, the PCA technique is usually suitable for the second order accumulation variant, whereas the sparse coding method can be used for multi-dimensional mixed data. So, in recent years, the sparse coding shrinkage technique has been used widely in image denoising field. This technique can be more efficiently than wavelet transform for images and higher dimensional data. Moreover, the Ref. [7] gave an important conclusion: when ICA is applied to natural image data, ICA is equivalent to SC. However, ICA emphasizes independence over sparsity in the output coefficients, while SC requires that the output coefficients must be sparse and as independent as possible. Because of the sparse structures of natural images, SC is more suitable to process natural images than ICA. Hence, SC method has been widely used in natural image processing [7], [8], [9], [10].
However, the standard SC algorithm described in [8] can not simultaneously guarantee the sparsity and independence of coefficient components. And the objective function itself can not balance between the sparseness constraint and the reconstructed precision. To remove or avoid the disadvantages mentioned above, in this paper, we propose a modified SC algorithm, which exploits the maximum Kurtosis as the maximizing sparse measure criterion, so the natural image structure captured by the Kurtosis not only is surely sparse, but also is surely independent. At the same time, a fixed variance term of coefficients is used to yield a fixed information capacity. This term can well balance the reconstructed error and sparsity. On the other hand, we use a determinative basis function, which is obtained by a fast fixed-point independent component analysis (FastICA) algorithm, as the initialization feature basis function of our sparse coding algorithm instead of using a random initialization matrix, so that the convergent speed of SC is further speeded up. The experimental results also showed that utilizing our SC algorithm, the edge features of natural images can be extracted successfully. Further, applied the features extracted, considered the shrinkage rule similar to the wavelet-based shrinkage one, the images contaminated by additive Gaussian white-noise can be reconstructed clearly.
This paper is organized as follows. Section 2 gives a brief introduction about the early SC algorithm; Section 3 describes the novel sparse coding model in detail; Section 4 mainly introduces the sparse coding shrinkage; Section 5 is the experimental results for reconstructing natural images; Section 6 gives several conclusions.
Section snippets
Modeling natural images
In 1996, B.A. Olshausen and D.J. Field conjectured that any given natural image can be represented with linear sparse coding (SC) method [8]. In fact, the evidence for sparse structures of nature images can be seen by filtering natural images with wavelets [5]. Assume that I = (I1,I2, … ,In)T denotes the observed n-dimensional random vectors, and S = (s1,s2, … , sm)T denotes the hidden components. Further, let A = (a1, a2, … ,am) denote the feature basis vectors. Thus, the input data I can be modeled as a
The objective function
Referring to the classical SC algorithm [8], and combining the minimum image reconstruction error with Kurtosis and fixed variance, we construct the following cost function of the minimization problem:where the symbol 〈·〉 denotes the mean, X = (x1,x2, … ,xn)T denotes the n-dimensional image data, A = (a1,a2, … ,am) denotes the feature basis vectors, S = (s1,s2, … ,sm)T denotes the m- dimensional sparse coefficients (Usually, the dimension of S is less than
Sparse coding shrinkage
Denote by s the original nongaussian random variable, and by n Gaussian white noise of zero mean and variance σ2. Assume that we only observe the random variable y = s + n, then, we want to estimate the originals. According to the reference[14], [15], we applied the maximum likelihood (ML) method gives the following estimator for s:where f′(·) is the derivative of the sparse punitive function f(·), moreover, f(·) is the negative log-density (i.e., f(·) = −log [p(·)]).
Feature extraction
All test images used in our experiments are available on the following Internet web: <http://www.cns.nyu.edu/lcv/denoise>. Firstly, selecting randomly 10 noise-free natural images with 512 × 512 pixels. Then, we sampled patches of 8 × 8 pixels 5000 times from each original image, and converted every patch into one column. Thus, the input data set X with the size of 64 × 50,000 is acquired. Consequently, each image patch is represented by a 8 × 8 dimensional vector. Further, the data set X was centered
Conclusions
In this paper, a novel natural image denoising method based on a modified sparse coding (SC) algorithm developed by us is proposed. This modified SC algorithm exploited the maximum Kurtosis as the maximizing sparse measure criterion, so the natural image structure captured by the Kurtosis not only is surely sparse, but also is surely independent. At the same time, a fixed variance term of coefficients is used to yield a fixed information capacity. Edge features of natural images can be
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