Sparse unmixing via variable splitting and augmented Lagrangian methods (SUNSAL)

Description

SUNSAL solves the following l2-l1 optimization problem [size(M) = (L,p); size(X) = (p,N)]; size(Y) = (L,N)]

min_X  (1/2) ||M X-y||^2_F + lambda ||X||_1

where ||X||_1 = sum_i sum_j |X_{i,j}|.

CONSTRAINTS ACCEPTED:

1. POSITIVITY: X >= 0;
2. ADDONE: sum_j X_{i,j} = 1 for all i;

NOTES:

1. The optimization w.r.t each column of X is decoupled. Thus, SUNSAL solves N simultaneous problems.

2. SUNSAL solves the following problems:

   a) BPDN - Basis pursuit denoising l2-l1 (lambda > 0, POSITIVITY = False, ADDONE, False)

   b) CBPDN - Constrained basis pursuit denoising l2-l1 (lambda > 0, POSITIVITY = True, ADDONE, False)

   c) CLS - Constrained least squares (lambda = 0, POSITIVITY = True, ADDONE, False)

   c) FCLS - Fully constrained least squares (lambda >=0 , POSITIVITY = True, ADDONE, True) In this case, the regularizer ||X||_1 plays no role, as it is constant.

Line of Attack

SUNSAL solves the above optimization problem by introducing a variable splitting and then solving the resulting constrained optimization with the augmented Lagrangian method of multipliers (ADMM).

min_{X,Z}  (1/2) ||M X-y||^2_F + lambda ||Z||_1

subject to: sum_j X_{i,j} for all i; Z >= 0; X = Z

Augmented Lagrangian (scaled version):

L(X,Z,D) = (1/2) ||M X-y||^2_F + lambda ||Z||_1 + mu/2||X-Z-D||^2_F

where D are the scale Lagrange multipliers

ADMM:

do
    X  <-- arg min L(X,Z,D)
               X, s.t: sum(X) = ones(1,N));
    Z  <-- arg min L(X,Z,D)
               Z, s.t: Z >= 0;
    D  <-- D - (X-Z);
while ~stop_rule

More details on the method:

J. Bioucas-Dias and M. Figueiredo, "Alternating direction algorithms for constrained sparse regression: Application to hyperspectral unmixing", in 2nd IEEE GRSS Workshop on Hyperspectral Image and SignalProcessing-WHISPERS'2010, Raykjavik, Iceland, 2010.

Usage

x,res_p,res_d,i = sunsal(M,y,AL_iters=1000,lambda_0=0.,positivity=False,addone=False,tol=1e-4,x0 = None,verbose=False)

Required inputs

M - [L(channels) x p(endmembers)] endmembers matrix

y - pixels matrix with L(channels) x N(pixels). each pixel is a linear mixture of p endmembers signatures y = M*x + noise,

Optional inputs

AL_ITERS - Minimum number of augmented Lagrangian iterations - Default: 1000

lambda_0 - regularization parameter. lambda is either a scalar or a vector with N components (one per column of x) - Default: 0.

positivity = {True, False}; Enforces the positivity constraint: X >= 0 - Default: False

addone = {True, False}; Enforces the positivity constraint: X >= 0 - Default: False

tol - tolerance for the primal and dual residuals - Default: 1e-4;

verbose = {True, False}; Default: False

Output variables

x - estimated mixing matrix [pxN] res_p - primal residual res_d - dual residual i - number of iteration until convergence

Requirements

Scipy needs to be installed.

Authors

Software translated from matlab to python by Adrien Lagrange ([email protected]), 2018.

Initial matlab author: Jose Bioucas-Dias, 2009

License

SUNSAL is distributed under the terms of the GNU General Public License 2.0.