Analogical Basis Deconstruction Theory for Signal Restoration and MRI Reconstruction

A new so-called Analogical Basis Deconstruction (ABD) theory for noisy signal restoration has been developed in this paper. The noise we consider here contaminates partial frequency bandwidth of the signal strongly. Intrinsically, the ABD theory treats any linear transform (like Fourier Transform and Wavelet Transform) as the transform of basis, not the weights of basis, so the signal can be reconstructed only based on the partial pure frequency data of noisy signal. The key part of the ABD theory is the process of deconstructing the partial frequency signal to the sum of weighted analogical basis which depend on the form of pure or nearly pure frequency bandwidth. We propose the Asymptotic Iterative Estimate (AIE) algorithm to deconstruct the signal. We also apply the ABD theory to accomplish the image reconstruction from partial k-space data in MRI where only partial frequency data are valid.