There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

In the last few years, several new algorithms based on graph cuts have been developed to solve energy minimization problems in computer vision. Each of these techniques constructs a graph such that the minimum cut on the graph also minimizes the energy. Yet, because these graph constructions are complex and highly specific to a particular energy function, graph cuts have seen limited application to date. In this paper, we give a characterization of the energy functions that can be minimized by graph cuts. Our results are restricted to functions of binary variables. However, our work generalizes many previous constructions and is easily applicable to vision problems that involve large numbers of labels, such as stereo, motion, image restoration, and scene reconstruction. We give a precise characterization of what energy functions can be minimized using graph cuts, among the energy functions that can be written as a sum of terms containing three or fewer binary variables. We also provide a general-purpose construction to minimize such an energy function. Finally, we give a necessary condition for any energy function of binary variables to be minimized by graph cuts. Researchers who are considering the use of graph cuts to optimize a particular energy function can use our results to determine if this is possible and then follow our construction to create the appropriate graph. A software implementation is freely available.

/pdf/what-energy-functions-can-be-minimized-via-graph-cuts-inkqpz6gaf.pdf

What energy functions can be minimized via graph cuts

Given a set of mixed spectral (multispectral or hyperspectral) vectors, linear spectral mixture analysis, or linear unmixing, aims at estimating the number of reference substances, also called endmembers, their spectral signatures, and their abundance fractions. This paper presents a new method for unsupervised endmember extraction from hyperspectral data, termed vertex component analysis (VCA). The algorithm exploits two facts: (1) the endmembers are the vertices of a simplex and (2) the affine transformation of a simplex is also a simplex. In a series of experiments using simulated and real data, the VCA algorithm competes with state-of-the-art methods, with a computational complexity between one and two orders of magnitude lower than the best available method.

/pdf/vertex-component-analysis-a-fast-algorithm-to-unmix-1egulrpm76.pdf

Vertex component analysis: a fast algorithm to unmix hyperspectral data

Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher spectral resolution than multispectral cameras. Imaging spectrometers are therefore often referred to as hyperspectral cameras (HSCs). Higher spectral resolution enables material identification via spectroscopic analysis, which facilitates countless applications that require identifying materials in scenarios unsuitable for classical spectroscopic analysis. Due to low spatial resolution of HSCs, microscopic material mixing, and multiple scattering, spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus, accurate estimation requires unmixing. Pixels are assumed to be mixtures of a few materials, called endmembers. Unmixing involves estimating all or some of: the number of endmembers, their spectral signatures, and their abundances at each pixel. Unmixing is a challenging, ill-posed inverse problem because of model inaccuracies, observation noise, environmental conditions, endmember variability, and data set size. Researchers have devised and investigated many models searching for robust, stable, tractable, and accurate unmixing algorithms. This paper presents an overview of unmixing methods from the time of Keshava and Mustard's unmixing tutorial to the present. Mixing models are first discussed. Signal-subspace, geometrical, statistical, sparsity-based, and spatial-contextual unmixing algorithms are described. Mathematical problems and potential solutions are described. Algorithm characteristics are illustrated experimentally.

/pdf/hyperspectral-unmixing-overview-geometrical-statistical-and-2lb11fr9l4.pdf

Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches

Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher spectral resolution than multispectral cameras. Imaging spectrometers are therefore often referred to as hyperspectral cameras (HSCs). Higher spectral resolution enables material identification via spectroscopic analysis, which facilitates countless applications that require identifying materials in scenarios unsuitable for classical spectroscopic analysis. Due to low spatial resolution of HSCs, microscopic material mixing, and multiple scattering, spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus, accurate estimation requires unmixing. Pixels are assumed to be mixtures of a few materials, called endmembers. Unmixing involves estimating all or some of: the number of endmembers, their spectral signatures, and their abundances at each pixel. Unmixing is a challenging, ill-posed inverse problem because of model inaccuracies, observation noise, environmental conditions, endmember variability, and data set size. Researchers have devised and investigated many models searching for robust, stable, tractable, and accurate unmixing algorithms. This paper presents an overview of unmixing methods from the time of Keshava and Mustard's unmixing tutorial [1] to the present. Mixing models are first discussed. Signal-subspace, geometrical, statistical, sparsity-based, and spatial-contextual unmixing algorithms are described. Mathematical problems and potential solutions are described. Algorithm characteristics are illustrated experimentally.

Independent component analysis (ICA) has recently been proposed as a tool to unmix hyperspectral data. ICA is founded on two assumptions: 1) the observed spectrum vector is a linear mixture of the constituent spectra (endmember spectra) weighted by the correspondent abundance fractions (sources); 2)sources are statistically independent. Independent factor analysis (IFA) extends ICA to linear mixtures of independent sources immersed in noise. Concerning hyperspectral data, the first assumption is valid whenever the multiple scattering among the distinct constituent substances (endmembers) is negligible, and the surface is partitioned according to the fractional abundances. The second assumption, however, is violated, since the sum of abundance fractions associated to each pixel is constant due to physical constraints in the data acquisition process. Thus, sources cannot be statistically independent, this compromising the performance of ICA/IFA algorithms in hyperspectral unmixing. This paper studies the impact of hyperspectral source statistical dependence on ICA and IFA performances. We conclude that the accuracy of these methods tends to improve with the increase of the signature variability, of the number of endmembers, and of the signal-to-noise ratio. In any case, there are always endmembers incorrectly unmixed. We arrive to this conclusion by minimizing the mutual information of simulated and real hyperspectral mixtures. The computation of mutual information is based on fitting mixtures of Gaussians to the observed data. A method to sort ICA and IFA estimates in terms of the likelihood of being correctly unmixed is proposed.

/pdf/does-independent-component-analysis-play-a-role-in-unmixing-4iby1xzh1b.pdf

Does independent component analysis play a role in unmixing hyperspectral data

Presents a new method for endocardial (inner) and epicardial (outer) contour estimation from sequences of echocardiographic images. The framework herein introduced is fine-tuned for parasternal short axis views at the papillary muscle level. The underlying model is probabilistic; it captures the relevant features of the image generation physical mechanisms and of the heart morphology. Contour sequences are assumed to be two-dimensional noncausal first-order Markov random processes; each variable has a spatial index and a temporal index. The image pixels are modeled as Rayleigh distributed random variables with means depending on their positions (inside endocardium, between endocardium and pericardium, or outside pericardium). The complete probabilistic model is built under the Bayesian framework. As estimation criterion the maximum a posteriori (MAP) is adopted. To solve the optimization problem, one is led to (joint estimation of contours and distributions' parameters), the authors introduce an algorithm herein named iterative multigrid dynamic programming (IMDP). It is a fully data-driven scheme with no ad-hoc parameters. The method is implemented on an ordinary workstation, leading to computation times compatible with operational use. Experiments with simulated and real images are presented.

/pdf/wall-position-and-thickness-estimation-from-sequences-of-4kfglvtd40.pdf

Wall position and thickness estimation from sequences of echocardiographic images

Independent Component Analysis (ICA) have recently been proposed as a tool to unmix hyperspectral data. ICA is founded on two assumptions: i) The observed data vector is a linear mixture of the sources (abundance fractions); ii) sources are independent. Concerning hyperspectral data, the first assumption is valid whenever the constituent substances are surface distributed. The second assumption, however, is violated, since the sum of abundance fractions associated to each pixel is constant due to physical constraints in the data acquisition process. Thus, sources cannot be independent. This paper gives evidence that ICA, at least in its canonical form, is not suited to unmix hyperspectral data. We arrive to this conclusion by minimizing the mutual information of simulated hyperspectral mixtures. The hyperspectral data model includes signature variability, abundance perturbation, sensor Point Spread Function (PSF), abundance constraint and electronic noise. Mutual information computation is based on fitting mixtures of Gaussians to the observed data.

/pdf/does-independent-component-analysis-play-a-role-in-unmixing-1cywly7k1w.pdf

Does Independent Component Analysis Play a Role in Unmixing Hyperspectral Data

This paper presents an effective algorithm for absolute phase (not simply modulo-2-/spl pi/) estimation from incomplete, noisy and modulo-2/spl pi/ observations in interferometric aperture radar and sonar (InSAR/InSAS). The adopted framework is also representative of other applications such as optical interferometry, magnetic resonance imaging and diffraction tomography. The Bayesian viewpoint is adopted; the observation density is 2-/spl pi/-periodic and accounts for the interferometric pair decorrelation and system noise; the a priori probability of the absolute phase is modeled by a compound Gauss-Markov random field (CGMRF) tailored to piecewise smooth absolute phase images. We propose an iterative scheme for the computation of the maximum a posteriori probability (MAP) absolute phase estimate. Each iteration embodies a discrete optimization step (Z-step), implemented by network programming techniques and an iterative conditional modes (ICM) step (/spl pi/-step). Accordingly, the algorithm is termed Z/spl pi/M, where the letter M stands for maximization. An important contribution of the paper is the simultaneous implementation of phase unwrapping (inference of the 2/spl pi/-multiples) and smoothing (denoising of the observations). This improves considerably the accuracy of the absolute phase estimates compared to methods in which the data is low-pass filtered prior to unwrapping. A set of experimental results, comparing the proposed algorithm with alternative methods, illustrates the effectiveness of our approach.

/pdf/the-z-spl-pi-m-algorithm-a-method-for-interferometric-image-4sph96w4jd.pdf

J.M.B. Dias

Papers

Vertex component analysis: a fast algorithm to unmix hyperspectral data

Does independent component analysis play a role in unmixing hyperspectral data

Wall position and thickness estimation from sequences of echocardiographic images

Does Independent Component Analysis Play a Role in Unmixing Hyperspectral Data

The Z/spl pi/M algorithm: a method for interferometric image reconstruction in SAR/SAS