The three functions implemented in this library are:

• Vertex Component Analysis (VCA). Related article is J. Nascimento and J. Dias, "Vertex Component Analysis: A fast algorithm to unmix hyperspectral data", IEEE Transactions on Geoscience and Remote Sensing, vol. 43, no. 4, pp. 898-910, 2005.
• Simplex Identification via Split Augmented Lagrangian (SISAL). Related article is J. Bioucas-Dias, "A variable splitting augmented Lagrangian approach to linear spectral unmixing", in First IEEE GRSS Workshop on Hyperspectral Image and Signal Processing-WHISPERS'2009, Grenoble, France, 2009.
• Sparse Unmixing via variable Splitting and Augmented Lagrangian methods(SUNSAL). Related article is Bioucas-Dias, J. M., & Figueiredo, M. A., "Alternating direction algorithms for constrained sparse regression: Application to hyperspectral unmixing." in Workshop in Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), 2010 (pp. 1-4).

Matlab versions of these codes are available in the Jose Bioucas Dias website.

This function were translated by Adrien Lagrange (view his github page).

### Usage

Ae, indice, Yp = vca(Y,R,verbose = True,snr_input = 0)

Y - matrix with dimensions L(channels) x N(pixels) each pixel is a linear mixture of R endmembers signatures Y = M x s, where

• s = gamma x alpha
• gamma is a illumination perturbation factor and
• alpha are the abundance fractions of each endmember.

R - positive integer number of endmembers in the scene

Ae - estimated mixing matrix (endmembers signatures)

indice - pixels that were chosen to be the most pure

Yp - Data matrix Y projected.

snr_input - (float) signal to noise ratio (dB)

v - [True | False]

Author: Adrien Lagrange ([email protected])

This code is a translation of a matlab code provided by Jose Nascimento ([email protected]) and Jose Bioucas Dias ([email protected]) available at https://www.lx.it.pt/~bioucas/code.htm under the GNU General Public License 2.0.

Translation of last version at 22-February-2018 (Matlab version 2.1 (7-May-2004)).

M,Up,my,sing_values = sisal(Y,p,**kwargs)

### Description

Simplex identification via split augmented Lagrangian (SISAL) estimates the vertices M={m_1,...m_p} of the (p-1)-dimensional simplex of minimum volume containing the vectors [y_1,...y_N], under the assumption that y_i belongs to a (p-1) dimensional affine set.

For details see José M. Bioucas-Dias, "A variable splitting augmented lagrangian approach to linear spectral unmixing", First IEEE GRSS Workshop on Hyperspectral Image and Signal Processing - WHISPERS, 2009. (https://arxiv.org/abs/0904.4635v1)

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

p - number of independent columns of M. Therefore, M spans a (p-1)-dimensional affine set. p is the number of endmembers.

mm_iters - Maximum number of constrained quadratic programs. Default: 80

tau - Regularization parameter in the problem

             Q^* = arg min_Q  -\log abs(det(Q)) + tau*|| Q*yp ||_h
                   subject to np.ones((1,p))*Q=mq
             where mq = ones(1,N)*yp'inv(yp*yp) and ||x||_h is the "hinge" induced norm.

   Default: 1

mu - Augmented Lagrange regularization parameter. Default: 1

spherize - {True, False} Applies a spherization step to data such that the spherized data spans over the same range along any axis. Default: True

tolf - Tolerance for the termination test (relative variation of f(Q)). Default: 1e-2

M0 - Initial M, dimension L x p. Defaults is given by the VCA algorithm.

verbose - {0,1,2,3}

• 0 - work silently
• 1 - display simplex volume
• 2 - display figures
• 3 - display SISAL information
• 4 - display SISAL information and figures Default: 1

M - estimated endmember signature matrix L x p

Up - isometric matrix spanning the same subspace as M, imension is L x p

my - mean value of Y

sing_values - (p-1) eigenvalues of Cy = (y-my)*(y-my)/N. The dynamic range of these eigenvalues gives an idea of the difficulty of the underlying problem

The identified affine set is given by

{z\in R^p : z=Up(:,1:p-1)*a+my, a\in R^(p-1)}

Author: Etienne Monier ([email protected])

This code is a translation of a matlab code provided by Jose Nascimento ([email protected]) and Jose Bioucas Dias ([email protected]) available at https://www.lx.it.pt/~bioucas/code.htm under the GNU General Public License 2.0.

Translation of last version at 20-April-2018 (Matlab version 2.1 (7-May-2004))

x = sunsal_v2(M,Y,**kwargs)

SUNSAL (sparse unmixing via variable splitting and augmented Lagrangian methods) algorithm implementation. Accepted constraints are:

• 1. Positivity: X >= 0
• 2. Addone: np.sum(X,axis=0) = np.ones(N)

For details see 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 Signal Processing-WHISPERS'2010, Raykjavik, Iceland, 2010.

M - endmember signature matrix with dimensions L(channels) x p(endmembers)

Y - matrix with dimensions L(channels) x N(pixels). Each pixel is a linear mixture of p endmembers signatures

al_iters - Minimum number of augmented Lagrangian iterations. Default: 100

lambda_p - 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. Default: False

addone - {True, False} Enforces the addone constraint. Default: False

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

verbose = {True, False}

• False - work silently
• True - display iteration info Default: True

X - estimated abundance matrix of size p x N

Author: Etienne Monier ([email protected])

This code is a translation of a matlab code provided by Jose Nascimento ([email protected]) and Jose Bioucas Dias ([email protected]) available at https://www.lx.it.pt/~bioucas/code.htm under the GNU General Public License 2.0.

Translation of last version at 20-April-2018 (Matlab version 2.1 (7-May-2004))

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

Initial matlab author: Jose Bioucas-Dias, 2009

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