A least-squares approach to fully constrained linear spectral mixture analysis using linear inequality constraints

Linear spectral mixture analysis (LSMA) is a widely used technique in remote sensing to estimate abundance fractions of materials present in an image pixel. In order for an LSMA-based estimator to produce accurate amounts of material abundance, it generally requires two constraints imposed on the linear mixture model used in LSMA, which are the abundance sum-to-one constraint and the abundance nonnegativity constraint. The first constraint requires the sum of the abundance fractions of materials present in an image pixel to be one and the second imposes a constraint that these abundance fractions be nonnegative. While the first constraint is easy to deal with, the second constraint is difficult to implement since it results in a set of inequalities and can only be solved by numerical methods. Consequently, most LSMA-based methods are unconstrained and produce solutions that do not necessarily reflect the true abundance fractions of materials. In this case, they can only be used for the purposes of material detection, discrimination, and classification, but not for material quantification. The authors present a fully constrained least squares (FCLS) linear spectral mixture analysis method for material quantification. Since no closed form can be derived for this method, an efficient algorithm is developed to yield optimal solutions. In order to further apply the designed algorithm to unknown image scenes, an unsupervised least squares error (LSE)-based method is also proposed to extend the FCLS method in an unsupervised manner.

A great challenge in hyperspectral image analysis is decomposing a mixed pixel into a collection of endmembers and their corresponding abundance fractions. This paper presents an improved implementation of Barycentric Coordinate approach to unmix hyperspectral images, integrating with the Most-Negative Remove Projection method to meet the abundance sum-to-one constraint (ASC) and abundance non-negativity constraint (ANC). The original barycentric coordinate approach interprets the endmember unmixing problem as a simplex volume ratio problem, which is solved by calculate the determinants of two augmented matrix. One consists of all the members and the other consist of the to-be-unmixed pixel and all the endmembers except for the one corresponding to the specific abundance that is to be estimated. In this paper, we first modified the algorithm of Barycentric Coordinate approach by bringing in the Matrix Determinant Lemma to simplify the unmixing process, which makes the calculation only contains linear matrix and vector operations. So, the matrix determinant calculation of every pixel, as the original algorithm did, is avoided. By the end of this step, the estimated abundance meet the ASC constraint. Then, the Most-Negative Remove Projection method is used to make the abundance fractions meet the full constraints. This algorithm is demonstrated both on synthetic and real images. The resulting algorithm yields the abundance maps that are similar to those obtained by FCLS, while the runtime is outperformed as its computational simplicity.

Currently used methods for the analysis of contingency tables with log-linear models and their extensions are essentially a way to subject the parameters of the saturated model to a set of linear constraints. However, in many contexts, expecially when a subset of the variables are ordinal, it would be very useful to be able to consider also linear inequality constraints. In this paper we show how to extend previous models for contingency tables by considering simultaneously both equality constraints. We also give a self contained discussion of the problem of testing equality and inequality constraints in the context of contingency tables analysis.

In this paper, we propose a proposed fully constrained least squares (p-FCLS) for spectral unmixing, which assumes that the endmember with the absolute maximum negative abundance by least squares is the disturbing endmember and that can be ignored subsequently, then the other endmembers are returned in the iteration until the abundance of all endmembers are non-negative. To evaluate the effectiveness of the proposed algorithm, hyperspectral data samples by ASD Field spectrometer and the EO-1 Hyperion image are employed. The results illustrate that p-FCLS could reduced time consumption and is more accurate in unmixing hyperspectral mixed spectral and pixel than fully constrained least squares (FCLS) and normalized non-negative matrix factorization (NNMF).

An optimal force distribution scheme of multiple cooperating robots is proposed. General formulations of the optimal force distribution problem are built into quadratic programming with linear equality and quadratic inequality constraints. The quadratic constraints are due to the norm constraints of grasping forces and the approximation of maximum joint torque constraints which is used for simplification of the problem. This paper presents a technique for solving these general optimal force distribution problems. The original problem is transformed into a compact problem by eliminating the linear equality constraints. Then, the compact problem is transformed into a dual problem to be solved by using Lagrange multipliers and nonlinear programming dual method. This technique reduces the problem size considerably and makes the problem almost unconstrained so that no initial feasible points are needed. These features exhibit the efficiency of this technique. Simulation results for two cooperating PUMA robots indicating the ability of real time applications of the proposed technique are presented.

We derive the asymptotic approximation for the posterior distribution when the data are multinomial and the prior is Dirichlet conditioned on satisfying a finite set of linear equality and inequality constraints so the posterior is also Dirichlet conditioned on satisfying these same constraints. When only equality constraints are imposed, the asymptotic approximation is normal. Otherwise it is normal conditioned on satisfying the inequality constraints. In both cases the posterior is a root-n-consistent estimator of the parameter vector of the multinomial distribution. As an application we consider the constrained Polya posterior which is a non-informative stepwise Bayes posterior for finite population sampling which incorporates prior information involving auxiliary variables. The constrained Polya posterior is a root-n-consistent estimator of the population distribution, hence yields a root-n-consistent estimator of the population mean or any other differentiable function of the vector of population probabilities.

Independent component analysis (ICA) has shown much success in blind source separation and channel equalization. Its applications to remotely sensed images are investigated in recent years. Linear spectral mixture analysis (LSMA) has been widely used for subpixel detection and mixed pixel classification in remote sensing image processing and already studied in Chapters 3, 8–10, . It models the spectral signature of an image pixel vector as a linear mixture of spectral signatures of targets present in the image data and the target abundance fractions are assumed to be unknown, but nonrandom constants. This chapter combines these two approaches to one, called ICA-based linear spectral random mixture analysis (LSRMA), which describes an image pixel vector as a random process resulting from a random composition of multiple spectral signatures of distinct targets in the image data. It differs from LSMA in that the abundance fractions of the target signatures in LSRMA are considered to be unknown random independent signal sources. Two major advantages can be benefited from LSRMA. First, LSRMA does not require any prior knowledge of the targets used in the linear mixture model. Second and most importantly, LSRMA models each of target signatures as an independent random signal source so that the spectral variability of target signatures can be captured more effectively in a stochastic manner. The only required knowledge for LSRMA is the number of signal sources, p assumed to be present in the image data. This number is estimated by Neyman-Pearson detection theory-based eigen-thresholding methods to be investigated in Chapter 17.

We consider the problem of minimizing a convex quadratic function over a region defined by a linear equality or inequality constraint and two-sided bounds on the variables (box constraints) in R n . Such problems are interesting from both theoretical and practical points of view because they arise as subproblems of some mathematical programming problems as well as in various practical problems. Polynomial algorithms are proposed for solving problems of this form and their convergence is proved. Some examples and results of numerical experiments are also presented.

This paper presents a new method for generating motion profiles of rotating bodies, for example, the joint angle motion of mechanical systems such as space manipulators and antennas. In the planning of a motion from the given initial conditions to the final conditions, there are many constraints that should be considered, for example, actuator torque limits, rate limits, jerk limits, etc. Furthermore, flexibility of a system cannot be ignored in general. Vibration suppression is of great importance in some applications. For stationary boundary conditions (rest-to-rest cases), if the only constraint is the torque limit, a simple bang-bang type profile is the optimal one. However, if we consider more general boundary conditions, complex constraints, and vibration suppression conditions, the problem becomes considerably complicated. In this study, motion planning for a fixed motion time is considered. The control input or the acceleration of the motion is expressed as a linear superposition of triangle waves whose weight coefficients are determined in order to optimize the motion profile. In this formulation, all the boundary conditions and several constraints, including vibration suppression conditions, are expressed as linear equality and inequality constraints. Therefore, by appropriately setting the performance index, the problem becomes a linear programming problem and can be solved efficiently.

We consider the problem of optimal allocation of measurement resources, when: (1) the total measurement budget and time duration of measurements are fixed, and (2) the cost of an individual measurement varies inversely with the (controllable) measurement accuracy. The objective is to determine the time-distribution of measurement variances that minimizes a measure of error in estimating a discrete-time, vector stochastic process with known auto-correlation matrix using a linear estimator. The metric of estimation error is the trace of weighted sum of estimation error covariance matrices at various time indices. We show that this problem reduces to a nonlinear optimization problem with linear equality and inequality constraints. The solution to this problem is obtained via a variation of the projected Newton method. For the special case when the vector stochastic process is the state of a linear, finite-dimensional stochastic system, the problem reduces to the solution of a nonlinear optimal control problem. In this case, the gradient and Hessian with respect to the measurement costs are obtained via the solution of a two-point boundary value problem and the resulting optimization problem is solved via a variation of the projected Newton method. The proposed method is illustrated using four examples. >

We consider least squares semidefinite programming (LSSDP) where the primal matrix variable must satisfy given linear equality and inequality constraints, and must also lie in the intersection of the cone of symmetric positive semidefinite matrices and a simple polyhedral set. We propose an inexact accelerated block coordinate descent (ABCD) method for solving LSSDP via its dual, which can be reformulated as a convex composite minimization problem whose objective is the sum of a coupled quadratic function involving four blocks of variables and two separable nonsmooth functions involving only the first and second block, respectively. Our inexact ABCD method has the attractive $O(1/k^2)$ iteration complexity if the subproblems are solved progressively more accurately. The design of our ABCD method relies on recent advances in the symmetric Gauss--Seidel technique for solving a convex minimization problem whose objective is the sum of a multiblock quadratic function and a nonsmooth function involving only th...

Theorems of the alternative for linear algebraic equations and inequalities are considered in this paper. Classical theorems of the alternative, such as Gordan’s theorem, Farkas’ theorem, Stiemke’s theorem, Motzkin’s theorem, Slater’s theorem, Tucker’s theorem, Duffin’s theorem, Gale’s theorems, etc. are recalled. Theorems of this form are important for both linear algebra and mathematical programming, especially for mathematical programming problems with linear equality and/or inequality constraints. Some new theorems of the alternative are formulated and proved.

1-bit compressing sensing (CS) is an important class of sparse optimization problems. This paper focuses on the stability theory for 1-bit CS with quadratic constraint. The model is rebuilt by reformulating sign measurements by linear equality and inequality constraints, and the quadratic constraint with noise is approximated by polytopes to any level of accuracy. A new concept called restricted weak RSP of a transposed sensing matrix with respect to the measurement vector is introduced. Our results show that this concept is a sufficient and necessary condition for the stability of 1-bit CS without noise and is a sufficient condition if the noise is available.

A VARIABLE METRIC METHOD FOR LINEARLY CONSTRAINED MINIMIZATION PROBLEMS

This paper presents an analog neural network model to recover sparse signals. In the original constrained optimization task for recovering sparse signals, the objective function is not differentiable. Hence, we recast the original nonlinear programming problem as a linear programming problem with linear inequality constraints and equality constraints. However, the second order gradient of the objective function is not convex at an equilibrium point. To solve this problem, we further modify the objective function such that the second order gradient is convex at the equilibrium point. This paper presents two sets of network dynamics. One is for the standard recovery of sparse signals. Another one is for the noisy situation.KeywordsOptimizationSparse signal