I am learning statistical image processing by myself. In papers and books, it always show the histogram of original images and gradients as the following image shows. The histograms of images vary significantly while histograms of image gradients show some similarity. Does it assume that each pixel in images obey the same probability distribution for the histograms of images? Does the histogram of any image gradient obey the same probability distribution?
In the paper Image Denoising Using Scale Mixtures of Gaussians in the Wavelet Domain by Javier Portilla, Vasily Strela, Martin J. Wainwright, and Eero P. Simoncelli there is one paragraph
Contemporary models of image statistics are rooted in the television engineering of the 1950s (see  for review), which relied on a characterization of the autocovariance function for purposes of optimal signal representation and transmission. This work, and nearly all work since, assumes that image statistics are spatially homogeneous (i.e., strict-sense stationary). Another common assumption in image modeling is that the statistics are invariant, when suitably normalized, to changes in spatial scale. The translation- and scale-invariance assumptions, coupled with an assumption of Gaussianity, provides the baseline model found throughout the engineering literature: images are samples of a Gaussian random field, with variance falling as in the frequency domain. In the context of denoising, if one assumes the noise is additive and independent of the signal, and is also a Gaussian sample, then the optimal estimator is linear.
image statistics are spatially homogeneous What does it mean? Does image statistics means the histogram?
an assumption of Gaussianity What is Gaussian?
images are samples of a Gaussian random field If one image is considered as a random field, can histograms be used? The assumption that each pixel obeys the same probability distribution will not hold.