The big data era poses great challenges as well as opportunities for researchers to develop efficient statistical approaches to analyze massive data. Sufficient dimension reduction is such an important tool in modern data analysis and has received extensive attention in both academia and industry. In this dissertation, we introduce inverse regression estimators using Fourier transforms, which is superior to the existing SDR methods in two folds, (1) it avoids the slicing of the response variable, (2) it can be readily extended to solve the high dimensional data problem. For the ultra-high dimensional problem, we investigate both eigenvalue decomposition and minimum discrepancy approaches to achieve optimal solutions and also develop a novel and efficient optimization algorithm to obtain the sparse estimates. We derive asymptotic properties of the proposed estimators and demonstrate its efficiency gains compared to the traditional estimators. The oracle properties of the sparse estimates are derived. Simulation studies and real data examples are used to illustrate the effectiveness of the proposed methods. Wavelet transform is another tool that effectively detects information from time-localization of high frequency. Parallel to our proposed Fourier transform methods, we also develop a wavelet transform version approach and derive the asymptotic properties of the resulting estimators.