Hyperspectral unmixing compressive sensing method based on three-dimensional total variation sparse prior

1. Based on three-dimensional full variation of a sparse a priori high spectrum xie Hun compression sensing method, which is characterized by comprising the following steps:

Step one, for high spectral Image Wherein each pixel spectrum x_i expressed into all endmember Linear combinations are as follows:

x_i =Wh_i   (1)

wherein n_p said space containing the number of picture on, n_b number of said band, The abundance value corresponding to the vector;

The whole data X represents the abundance value matrix The product of the and the endmember matrix W:

X=WH   (2)

In the H, row direction is spectrum Uygur , different pixels representative of each row of the spectrum at the same one endmember projection; is spatial Uygur the column direction, each row in a pixel representative of the spectrum of the projection of the different endmember on;

Step two, meet gaussianity random distribution of the normalized random observation matrix Random sampling the original data, compressed data Are as follows:

F=AX=AWH   (3)

The length is that wherein m n_b the length of the signal compression after, m<n_b;

Step three, for limited imaging scene, on the basis of the scene information extracted from the spectral library n_e spectral composition endmember matrix W;

Step four, (1) in the spectral dimension H of the one-dimensional application of sparse variation of a priori, the sparsity of the combined spatial Uygur H, H of the obtained three-dimensional full variation sparse a priori, are as follows:

wherein e_j and ε_j that and Section j in the space a unit vector; TV (x) described is Full variation, D_i (x) x gradient in said component i; formula (4) of the H part 1st of said two-dimensional full variation spatial Uygur sparse a priori, wherein the corresponding D_i (·) as a two-dimensional gradient; H 2nd part of said one-dimensional full variation spectrum Uygur sparse a priori, wherein the corresponding D_i (·) for the one-dimensional gradient;

(2) constructing a priori other work out abundance values; the linear xie Hun model commonly used in the introduction of the abundance of the a priori value, respectively in different end of the spectrum to be mixed on the abundance value and the projection nonnegative and 1 limitation, are as follows:

1neTH=1npT,H>0---(5)

Wherein and Is for all element 1, lengths are respectively n_e and n_p the vector;

(3) constructing the abundance value matrix of the reconstruction model H; combining formula (3), (4) and (5) get the following reconstruction model:

minHΣj=1npΣi=1ne|Di(Hej)|+Σj=1neΣi=1np|Di(εjTH)|s.t.AWH=F,1neTH=1npT,H>0---(6)

In order to facilitate the subsequent solution, to the formula (6) separating the variables introduced in v_ij = D_i (He_j), Get:

minH,υij,uijΣj=1npΣi=1ne|vij|+Σj=1neΣi=1np|uij|s.t.vij=Di(Hej),[!ForAll!]i,j;uij=Di(εjTH),[!ForAll!]i,j;AWH=F,1neTH=1npT,H>0---(7)

(4) solving the formula (7) is an estimate of the abundance of the H value matrix Specific solving process are as follows:

(1) to broaden their Lagrange method of use, according to the formula (7) construction of H to broaden their Lagrange equation

wherein α=2⁵, κ=2⁵, β=2¹³, γ=2⁵ to quadratic punishment coefficient, λ_ij, π_ij, Π, υ to the corresponding Lagrange multiplier, initialization of all elements of each multiplier for 0, || · ||_F expressed the Frobenius norm;

(2) fixing the Lagrange multiplier and H, update separating variable v_ij, u_ij; forms are as follows:

vij=max{|Di(Hej)-λijα|-1α,0}sgn(Di(Hej)-λijα)uij=max{|Di(εjTH)-πijκ|-1κ,0}sgn(Di(εjTH)-πijκ)---(9)

(3) fixing the Lagrange multiplier and separate variable v_ij, u_ij, update the H using gradient descent method; it is assumed that resolution k time update, by H^k obtain H^k+1, forms are as follows:

Wherein to On a first derivative that is H, the following forms:

In the formula, τ is the gradient down step size; the calculation is divided into initialization and refinement two-step; in the initialization process, when the 1st time update H⁰ time, the most τ of steepest descent for initialization, is upgraded after H^k, k=1, 2, ... At the time, the step of adopting two-point τ for initializing the gradient method of; two-point step specific forms the gradient method is as follows:

Said path of the matrix wherein tr (·); τ in the refining process are as follows:

(A) into the initialization of the τ, according to the formula (10) to obtain H^k+1, setting parameter δ =3.2 × 10^-4, η=0.6 and counter c=0;

(B) judging H^k+1 whether it satisfies the following conditions:

If not satisfied, updating counter c=c + 1;

If c <5, τ =τ·η narrowing step, judging whether to satisfy and continues circulating (13);

Otherwise, τ is determined by the the method of steepest descent, then the formula (13) to obtain the updated H^k+1;

Otherwise, to obtain the updated H^k+1;

(4) fixing the updated v_ij, u_ij and H, Lagrange multiplier is updated using the following formula:

λijk+1=λijk-α[Di(Hej)-vij],πijk+1=πijk-κ[Di(εjTH)-uij]Πk+1=Πk-β(AWH-F),υk+1=υk-γ(1neTH-1npT)T---(14)

(5) cycle step (2), (3) and (4) until the convergence, of the final estimate of the abundance of the obtained value matrix

Step five, combining selected endmember matrix W and linear mixed model formula (2) is the high spectral data reconstruction

X^=WH^---(15).