01-03-2013, 10:43 AM
Exact Histogram Specification
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Abstract
While in the continuous case, statistical models of histogram
equalization/specification would yield exact results, their
discrete counterparts fail. This is due to the fact that the cumulative
distribution functions one deals with are not exactly invertible.
Otherwise stated, exact histogram specification for discrete images
is an ill-posed problem. Invertible cumulative distribution functions
are obtained by translating the problem in a -dimensional
space and further inducing a strict ordering among image pixels.
INTRODUCTION
HISTOGRAM specification (or modeling) refers to a
class of image transforms which aims to obtain images
the histograms of which have a desired shape [1]–[3]. Even
if specifying a meaningful histogram for a certain image is
not obvious, there are some general ones (such as uniform,
Gaussian, exponential) whose usefulness is clearly understood.
Thus, obtaining a uniform histogram image corresponds to the
well-known image enhancement technique called histogram
equalization. By means of histogram equalization, graylevels
are spread over the entire scale and an equal number of pixels
is allocated to each graylevel.
ORDERING EVALUATION
With the proposed ordering relation, a pixel turns out to be
brighter than another pixel when its local mean is greater than
the local mean of the other one. The initial ordering of the
graylevels is refined. Our aim is to achieve a strict ordering, or,
in a less restrictive setting, a strict ordering almost everywhere,
i.e., having very few equalities in (1).
Obviously, the induced ordering depends on as well as on
the image: Original graylevel distribution, graylevel range and
image size. For images with very large uniform areas (like synthetic
images), a strict ordering may not be possible. We will
assume we deal with natural images having enough graylevels
and enough details (or noise). A too large value for means an
increase in the computational complexity of the ordering procedure.
Moreover, when is increased, the rank of a pixel depends
on pixels located far apart (which is of no physical relevance).
Therefore, a moderate value for is desired.
Theoretical Analysis
In order to quantify the rather fuzzy measures given above,
namely moderate size and enough gray levels, the simplified
model of images having quantized Gaussian IID (independent
identically distributed) pixels is considered. The probability of
equal pixels as a function of and is evaluated. Notice that
the variance of the Gaussian distribution is closely related to
the number of graylevels of the image: since the probability of
having values outside the range situated around the mean
of the Gaussian is almost zero.
APPLICATIONS
The immediate use of exact histogram equalization/specification
is to replace its classical counterpart in some applications
where improvements are expected as, for instance, exact image
normalization or image enhancement. Newspecific applications
are foreseen, for example, image watermarking or histogram
equalization inversion.
Other Specific Applications
1) Image Normalization: Exact histogram specification provides
a procedure for real image normalization. By specifying
a uniform histogram one obtains images normalized with respect
to i) histogram (uniform histogram), ii) graylevel average
(L/2), iii) energy, and iv) entropy (8 bits/pixel). Other distributions
could be of interest for image normalization such as, for
instance, Gaussian or mixture of Gaussians, Laplacian, etc.
2) Histogram Specification Inversion: In the framework of
classical histogram specification or equalization, the recovery of
the original image is an unsolved problem. With the proposed
approach, this problem turns out to be exact histogram specification
of the original histogram for the transformed image. The
solution is exactly the original image under the hypothesis that
ordering among pixels is preserved by exact histogram specification.
Since the hypothesis of order preservation does not
completely hold, we expect the reconstruction not to be identical
with the original. Obviously, the histogram of the recovered
image is exactly the original histogram
CONCLUSION
An ill-posed problem, exact histogram specification, is
solved. Our approach is based on the definition of an ordering
relation which induces almost strict ordering on image pixels.
Theoretical and experimental results on the existence of the
strict ordering are provided. Once ordering is achieved, pixels
are immediately separated into classes and assigned to the
desired graylevel. The proposed strict ordering is consistent
with the natural one and thus, the information content of images
is generally preserved.