With the increasing amount of raw data generation produced every day, it

has become pertinent to develop new techniques for data representation, analyses,

and interpretation. Motivated by real-world applications, there is a trending interest

in techniques such as dimensionality reduction, wavelet decomposition, and

classication methods that allow for better understanding of data. This thesis details

the development of a new non-linear dimension reduction technique based on

transport model by advection. We provide a series of computational experiments,

and practical applications in hyperspectral images to illustrate the strength of our

algorithm. In wavelet decomposition, we construct a novel Haar approximation

technique for functions f in the Lp-space, 0 < p < 1, such that the approximants

have support contained in the support of f. Furthermore, a classification algorithm

to study tissue-specific deoxyribonucleic acids (DNA) is constructed using the support

vector machine. In magnetic resonance imaging, we provide an extension of

the T2-store-T2 magnetic resonance relaxometry experiment used in the analysis

of magnetization signal from 2 to N exchanging sites, where N >= 2.