《statistics of image》.pdf
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Statistics of Images, the TV Algorithm of Rudin-Osher-Fatemi
for Image Denoising and an Improved Denoising Algorithm
Mark L. Green*, Dept. of Mathematics and
Institute for Pure and Applied Mathematics, UCLA
October 1, 2002
The goal of this paper is to present a new basic model for the joint density function
for a broad class of spatial and time-series data. As evidence that this model is indeed
useful in practical problems, an application to image denoising in the presence of textures
will be explained.
Section 1 discusses the joint density distribution for pixel intensities in naturally occur-
ring images. This reflects an experimental discovery made by examining pixel intensities
for a variety of naturally occurring imagesóthat the pixel intensities in many images have a
property I call Differentially Laplacian. The idea here is to consider not just differences
between measurements, but all linear combinations of measurements where the coefficients
add up to 0ówhen the data points are adjacent, these correspond to discretizations of lin-
ear differential operators, but it is fruitful to consider such differences for all data points
in k k sub-blocks of the image for k of modest size. The invertibility of the Radon
transform uniquely specifies the density function of a Differentially Laplacian collection of
random variables once one knows the autocorrelation function. Further experimental data
describes how the autocorrelation between pixel intensities in many images dies off as a
function of the distance d between themóthis is very often of the form (1+γd)−α . Although
there are many much more sophisticated statistical models for specific textures, the model
given here has a wide range of applicabilityóspecific images of plants, lava, galactic clus-
ters, bacteria,
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