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. 2008:2008:1-8.
doi: 10.1109/CVPR.2008.4587821.

Nonlinear Image Representation Using Divisive Normalization

Siwei Lyu  1 , Eero P Simoncelli  1
Affiliations

Nonlinear Image Representation Using Divisive Normalization

Siwei Lyu et al. Proc IEEE Comput Soc Conf Comput Vis Pattern Recognit. 2008.

Abstract

In this paper, we describe a nonlinear image representation based on divisive normalization that is designed to match the statistical properties of photographic images, as well as the perceptual sensitivity of biological visual systems. We decompose an image using a multi-scale oriented representation, and use Student's t as a model of the dependencies within local clusters of coefficients. We then show that normalization of each coefficient by the square root of a linear combination of the amplitudes of the coefficients in the cluster reduces statistical dependencies. We further show that the resulting divisive normalization transform is invertible and provide an efficient iterative inversion algorithm. Finally, we probe the statistical and perceptual advantages of this image representation by examining its robustness to added noise, and using it to enhance image contrast.

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Figures

Fig. 1
Fig. 1
Conditional histogram of two adjacent wavelet coefficients. Grayscale intensities are proportional to probability, with larger values corresponding to brighter pixels. Each column is normalized to fill the full range of intensities. Red solid and dashed lines indicate conditional mean and standard deviation, respectively. Blue solid and dashed lines are best-fitting linear models for the conditional mean and standard deviation.
Fig. 2
Fig. 2
(a) A subband of the steerable pyramid decomposition of a photographic image. (b) Log histogram of subband coefficients shown in (a) (blue solid line). (c) The DNT transformation of the subband in (a). (d) Log histogram of DNT subband coefficients shown in (c) (blue solid line). In both (b) and (d) a Gaussian with the same mean and variance (red dashed line) is shown for comparison.
Fig. 3
Fig. 3
(a) A photographic image. (b)–(f) Perturbations of the image shown in (a) resulting from adding white Gaussian noise to coefficients of different image representations. All perturbed images have PSNR of 25dB in the pixel domain, and are shown along with SSIM scores. (b) Raw pixel, (c) Fourier domain, (d) wavelet domain, (e) steerable pyramid, (f) DNT domain.
Fig. 4
Fig. 4
Effects of noise perturbation for different image representations. See text for details.
Fig. 5
Fig. 5
(a) A scan line of a test image of vertical edges. (b)–(d) Contrast enhancement results of the test image with different methods: (b) global high pass filtering (“unsharp masking”), (c) local gamma correction in wavelet domain, (d) global gamma correction in DNT domain. Parameters in each operation were chosen so that the smallest intensity jump in the test image (point D) would be boosted to the same value.
Fig. 6
Fig. 6
Contrast enhancement results of natural photographic images. On the left column are the original images, and on the right are the corresponding contrast enhanced images with divisive normalization representations. Images courtesy of N. Bonnier and P. Greenspun.
See this image and copyright information in PMC

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