We also capture dependencies that exist between spatially neighboring luminance subband responses by modeling the bivariate distributions of horizontally adjacent subband responses sampled from all locations,
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and
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, of each orientation subband at different scales on each image patch. Since we have observed similar statistics from both horizontally and vertically neighboring responses (Su et al.,
2014b,
2015a), and used subband orientations covering 0 to
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(rad), we exploit only horizontal adjacency to achieve the same efficacy with reduced computational complexity. This also applies to the correlation NSS feature, which will be detailed in the next section. To model these empirical joint histograms, we utilize a multivariate generalized Gaussian distribution (MGGD), which includes both the multivariate Gaussian and Laplacian distributions as special cases. The probability density function of an MGGD is defined as:
where
x ∈ ℝ
N,
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is an
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symmetric scatter matrix,
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and
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are scale and shape parameters, respectively, and
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is the density generator:
where
y ∈ ℝ
+. Note that when
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,
Equation 10 becomes the multivariate Laplacian distribution, and when
Equation 10 corresponds to the multivariate Gaussian distribution. When
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,
Equation 10 corresponds to the multivariate Gaussian distribution. When
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, the MGGD converges to a multivariate uniform distribution, and when
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, it becomes a 2D heavy-tailed “sparsity” density. The scatter matrix
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is a sample statistic that can be used to estimate the covariance matrix of
x ∈ ℝ
N, which may embed dependencies in
x ∈ ℝ
N (i.e., the spatially neighboring bandpass image responses). In order to capture these second-order statistics, we adopt a closed-form correlation model, which is described in detail in the next subsection, to extract the corresponding NSS features. In our implementation, we model the bivariate empirical histograms of horizontally adjacent subband responses of each image patch using a bivariate generalized Gaussian distribution (BGGD) with
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in
Equation 10. Specifically, from each subband of an image patch, we collect all pairs of horizontally adjacent subband responses to form
x ∈ ℝ
2, and estimate the BGGD model parameters using the maximum likelihood estimator (MLE) algorithm described in (Su, Cormack, & Bovik,
2014a). In our case, the scatter matrix
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is a
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matrix, which can be written as:
and
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is the number of horizontally adjacent pairs in an image patch of size
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. Both the BGGD scale and shape parameters,
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and
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, from all eight subbands are included in each image patch's feature set.