Interpretation of Histogram in Statistical Image Processing
Does it assume that each pixel in images obey the same probability distribution for the histograms of images? 


Images of different scenes will definitely not obey the same probability distribution of the pixel values. 

Histograms are one way that people use to do dimensionality reduction: move from a 2D image to a 1D signal.


  Does the histogram of any image gradient obey the same probability distribution?


What you are seeing in the image gradient is the "diffs" in the image. Because images are generally low-pass in nature,  this means you are picking out the places where they change.  There will be (at least) two components to this change: how the scene being imaged changes and how the sensor capturing the image perturbs the "true" pixel values.

For the same camera taking the images, this second component should be very similar across all images.


  image statistics are spatially homogeneous  What does it mean? Does image statistics means the histogram?


means that the image statistics are very similar regardless of where in the image you look. One way the statistic show up would be in the histogram, yes.


  an assumption of Gaussianity What is Gaussian?


Gaussian means that the noise (random fluctuations in the image) follows a normal or Gaussian distribution.


  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.


If the images are not random, then they will follow some well-defined deterministic rule.

Certainly, histograms can be used.