Research on the equivalent model of l1-l2 minimization in signal reconstruction
Summary:Signal reconstruction is a data processing technology based on partial signal to recover the whole signal.It is widely used in optical communication, image processing, data compression and other fields.The main problem is to find the sparse solution of a group of underdetermined linear or nonlinear equations.When using the traditional signal reconstruction method to transform the original problem into an underdetermined system of equations,it needs to add some restrictive conditions such as sparsity,so as to achieve signal reconstruction.In order to solve the shortcomings of the traditional signal reconstruction model, such as low accuracy,slow efficiency, and excessive dependence on signal sparsity,it is of great significance to further study the new signal reconstruction model.Among them, the recently proposed nonconvex l1-l2 minimization norm method is one of the most effective and important models in signal reconstruction.In this paper,a class of important nonconvex l1-l2 minimazation norm models proposed in recent years are studied, and a l1-l2 minimazation norm signal reconstruction model with equivalent significance is proposed,and the signal reconstruction theory is analyzed theoretically.Based on the definition of l1-l2 minimazation norm model and the nature of zero space, the objective function in the original optimization model is relaxed,and the new objective function equivalent to the original optimization problem is derived by using the coherence function.Two equivalent mathematical models of l1-l2 minimazation norm are established, and the feasibility of signal reconstruction is studied in theory.Proposition 1:Suppose (A∈Rm×n(m<n)) is a measurement matrix, if the non-zero vector Ax=b satisfies(G(x,h)=||h||1-(||x+h||2-||x||2)﹥0)Then the signal x∈Rn can be reconstructed through l1-l2 minimization norm in Ax=b, where, h≠0 and (h∈ker(A) ={z∈Rn:Az=0}).Proposition 2:Suppose A∈Rm×n(m<n) is a measurement matrix,if it exists t∈(0,1] ,so that the non-zero vector x∈Rn satisfies(G(t;x,h)=t||h||1-(||x+h||2-||x||2)﹥0)Then the signal x∈Rn can be reconstructed through l1-l2 minimization normin Ax=b,where, h≠0 and (h∈ker(A)={z∈Rn:Az=0}).In order to explore the sufficient conditions for signal recovery using l1-l2 minimazation norm, the article based on proposition 1 and proposition 2 , researchs the relationship between signal reconstruction and n-dimensional inner product and coefficient in zero space.Combining with the necessary and sufficient conditions of signal reconstruction in the general form of the l1-l2 minimazation norm model, the sufficient conditions of signal complete reconstruction are established for the l1-l2 minimazation norm model in the equivalent case.The effects of parameters and zero space on signal reconstruction are analyzed, and two kinds of robust conditions that can guarantee signal reconstruction in the case of no noise are proved.Theorem 1:Assuming the measurement matrix A∈Rm×n(m<n) ,the signal x∈Rn in Ax=b can be reconstructed by l1-l2 minimization norm, a sufficient condition exists t∈(0,1] ,so that(||h||1-<x,h>/t||x||2﹥0)Where,(h∈ker(A)={z∈Rn:Az=0}),||h 2|| =1 .Theorem 2: Suppose A∈Rm×n(m <n) is a measurement matrix.If there is a real number λi, such that(||H||1-[x,H]/||x||2﹥0)Then the signal x∈Rn in Ax=bcan be reconstructed by l1-l2 minimization norm,where,(H=λ1(h)1+λ2(h)2+…+λj(h)j,(h)i∈ker(A)={z∈Rn:Az=0})The conclusion shows that when certain conditions are satisfied, the signal can be effectively reconstructed by the equivalent l1-l2 minimazation norm model.
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