In this thesis, we study the numerical computation of Adaptive Fourier decompo-sition. Furthermore, its application to image representation and FFT formulation are proposed, Adaptive Fourier decomposition (AFD) has been found to be among the most effective greedy algorithms. That is in particular in signal analysis and system identification. As the compensation of effectiveness, the computation complexity is great, that is especially due to the maximal selections of the parameters. It is necessary to study the numerical computations of AFD and corresponding fast algorithms with lower complexity. As the foundation theory of adaptive Fourier decomposition and a new type of series expansion, one-dimensional adaptive Fourier decomposition and its variations, abbreviated as 1D-AFDs, have been proven to be effective in applications. 1D-AFDsalso have considerable impact to the rational approximation of one complex variable and phase retrieving problems, etc, The numerical computation of 1D-AFD is first proposed at the first time of 1D-AFD's appearance with the computation complexity O(M N²). In the first part of the thesis, we first proposed the numerical computation models of 2D-AFDs, which contains 2D-AFD of product TM system type (Product AFD) and Pre-OGA of product Szegö dictionary type (Pre-OGA). Comparative exper-iments are conducted with Fourier decomposition, greedy algorithm and orthogonal greedy algorithm. In the second part, we proposed the numerical algorithm of Product AFD for real-valued signals, Additionally, we define the AFD spectrum of a 2D signal as an image by defining the amplitude and phase of the AFD spectrum. Through experiments swapping the AFD phase and the AFD amplitude of two images, it isundoubted the AFD phase makes sense on image content rather than the AFD ampli-tude. Comparatively speaking, AFD phase can express image phase better. In the third part, we proposed new and sharper estimations for the convergence rates of orthogonal greedy algorithm and pre-orthogonal greedy algorithm. In the last part, we study fast numerical computation of AFD starting from the numerical computation of 1D-AFD. It is shown that we have three ways to compute the discretization of the 1D-AFD inte-gration via with discrete Fourier transform (DFT), incorporating fast Fourier transform (FFT). The first two methods construct butterfly data flows for 1D-AFD respectively by conforming common factors and by expanding to Taylor series. For reusing FFT codes in various applications, the numerical algorithm of AFD implemented directly by FFT is also proposed as the third one. We show that the new algorithms, called FFT-AFD, reduces the computational complexity from O(M N²) to O(M N log N), the latter being the same as FFT, The idea of FFT-AFD is naturally applied to the nu-merical aspect of multi-dimensional AFDs, and in particular 2D-AFDs. Upon these numerical models, FFT formulations of 2D-AFDs are established. The proposed FFT-based algorithms for AFD lays a foundation for its practical applications. For verifying the effectiveness, accuracy, and robustness of the proposed numerical algorithms, we illustrate toy and real image experiments. From the numerical experiments and the estimation of error bounds, we prove that2D-AFDs is much more efficient than Fourier decomposition and any of the existing greedy algorithms in the general reproducing kernel Hilbert space context. Moreover, the proposed FFT formulations for AFDs lays a foundation for its practical applica-tions.