  In this paper, we propose two novel algorithms for histogram specification
and quantile transformation of data without local information. These are core
techniques that can serve as building blocks for applications that require
specifying the sample distribution of a given set of data. Histogram
specification is best known for its image enhancement applications, whereas
quantile transformation is typically employed in data preprocessing for data
normalization. In signal processing, methods often require temporal or spatial
information; in data preprocessing, methods work by interpolation or by
approximation, drawing from results in computational statistics, and have a
trade-off between speed and quality. It is nontrivial to accommodate for cases
that do not have local information (e.g., tabular data) while also providing a
fast, exact solution. For that, we take up a concept in image processing called
group mapping law and propose an extension. The proposed extension allows us to
formulate a convex functional where we look for the best approximation between
the output unique values and the reference histogram. Then, we apply the
ordered assignment solution, a result in optimal transport, to reconstruct the
output from the optimal unique values. Two sets of results show the
effectiveness of the proposed algorithms when compared to traditional and
state-of-the-art methods. The proposed algorithms are fast, exact, and least
$p$-norm optimal. Further, we define the algorithms as generic data processing
methods. Thus, contributions from this paper can be easily incorporated in
applications spanning many disciplines, especially in applied data science.
