Distribution of Anomalous Exponents of Natural Images

Recent studies of correlations of intensity in databases of natural images revealed a remarkable property. The two point correlations are described in terms of power law behavior, with an exponent which seems to be robust. In the present Letter we consider the statistical meaning of that result. We study many individual images of one of the databases considered. We find that the same law characterizing the correlations in the whole database governs also images randomly chosen from that database, with one essential difference. The exponent characterizing each image is specific and differs from the exponent characterizing the whole database. The distribution of single image exponents has been measured and found to exhibit a rather heavy tail. The database exponent cannot, thus, be considered as a statistical representative of a single image exponent. Possible reasons for the diversity in image exponents are discussed.

Figure 1

Averaged difference function. The average is taken over 1500 images, $\eta =0.19$.

Figure 2

Left panel: Image 1401, $\eta =-0.2$. Right panel: Image 262, $\eta =-0.05$.

Figure 3

Left panel: Image 59, $\eta =0.04$. Right panel: Image 712, $\eta =1.01$.

Figure 4

Distribution of $\eta$ for 1500 images in the archive. Fit function: $P(\eta )=\exp \{[C_1+C_2(\eta -⟨\eta ⟩)+C_3(\eta -⟨\eta ⟩{)}^2{]}^{C_4}\}$. $C_4\approx 0.5$. $\mathrm{\text{Mean}}=0.155$; $\mathrm{\text{variance}}=0.21$; $\mathrm{\text{skewness}}=-0.28$; $\mathrm{\text{kurtosis}}=2.9$.