The acquisition of hundreds of images of a scene, each at a different wavelength, is known as hyperspectral imaging. This high amount of data allows the extraction of much more information from hyperspectral images compared with conventional color images. The forward-looking imaging approach emerged from remote sensing, but is still not very widespread in industrial and other practical applications. Spectral unmixing, in particular, aims at the determination of the components present in a scene as well as the abundance to which each component contributes. This information is valuable, for instance, when discrimination tasks are to be performed. Involving not only spectral, but also spatial information was found to have the potential to improve the unmixing results. Several publications use spatial first-order regularization (closely related to the total variation approach) to incorporate this spatial information. Like in classical image processing, this approach favors piecewise constant pixel transitions. This is why it was proposed in the literature to use second-order regularization instead of first order to approach piecewise-linear transitions. Therefore, we introduce Hessian-based regularization to hyperspectral unmixing and propose an algorithm to calculate the regularized result. We use simulated data and images measured in our laboratory to show that both the first- and second-order approaches share many properties and produce similar results. The second-order approach, however, is more robust and thus more accurate in finding the minimum. Both methods smoothen the images in the case of supervised unmixing (i.e., the component spectra are known beforehand) and enhance unsupervised unmixing (when the spectra are not known).

Hyperspectral images, in contrast to common RGB images, offer the possibility to not only determine the pure materials present in a scene, but also material abundances in mixtures. The calculation of the material fractions with the so-called linear mixing model is not unique, an infinite number of solutions exists. Therefore, additional constraints should be incorporated. Some algorithms involve spatial constraints explicitly, e. g., they assume that the abundances mostly do not change considerably from one pixel to another. Recently, we presented such algorithms. The calculation time with spatial constraints included, however, is rather long, so it was checked if there is a faster way to include the spatial information. In this paper, we extend the well-known alternating least-squares algorithm to implicitly include the previously used spatial information in a slightly different way, namely by adding an extra image denoising step to the calculation. The extended algorithm is called ALSmooth. We compare the computing time and the results of the ALSmooth and the previously presented algorithms. For this purpose, laboratory data of mixtures with known ground truth had been acquired. Both the previously investigated algorithms and the ALSmooth algorithm are quite sensitive towards parameter value changes; the ALSmooth algorithm is even more sensitive. For certain applications with defined environment and endmembers, however, it can be a faster alternative.Zusammenfassung: Im Gegensatz zu herkömmlichen Farbbildern bieten hyperspektrale Bilder die Möglichkeit, nicht nur die in einer Szene enthaltenen Reinstoffe, sondern auch deren Anteile in Materialmischungen festzustellen. Da die Berechnung der Materialanteile mittels des sogenannten Linearen Mischmodells nicht eindeutig ist, existieren für dieses Problem unendlich viele Lösungen. Aus diesem Grund sollten zusätzliche Nebenbedingungen mit eingebunden werden. Einige Algorithmen binden räumliche Nebenbedingungen auf explizite Weise ein, indem sie die Annahme ausnutzen, dass die Materialanteile sich von einem Pixel zu seinen Nachbarn nicht wesentlich ändern. Wir haben solche Algorithmen kürzlich untersucht. Da diese Algorithmen, die räumliche Nebenbedingungen einbeziehen, vergleichsweise viel Rechenzeit beanspruchen, wurde nach recheneffizienteren Wegen gesucht, um die räumliche Information einzubeziehen. In diesem Beitrag wird die Methode der alternierenden kleinsten Quadrate (Alternating Least-Squares, ALS) um einen zusätzlichen Bildglättungsschritt erweitert, um die räumliche Information auf diese Weise einzubeziehen. Wir nennen diesen erweiterten Algorithmus ALSmooth und vergleichen ihn mit den bereits vorgestellten Algorithmen hinsichtlich Rechenzeit und Güte der Ergebnisse. Zu diesem Zweck wurden hyperspektrale Bilder von Pulvermischungen mit bekanntem Mischungsverhältnis aufgenommen. Sowohl ALSmooth als auch die anderen Algorithmen reagieren sehr sensibel auf Änderungen von Parameterwerten, wobei ALSmooth noch sensibler ist. Für manche Anwendungen stellt er sich ...

No abstract

No abstract