Nonnegativity Constraints in Numerical Analysis - CiteSeer

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$Math 733: Vector Fields, Differential Forms, and Cohomology$

Figure 3: Artist rendition of a JSAT satellite. Image obtained from the Boeing SatelliteDevelopment Center.0.09Aluminum0.086Mylar0.0850.0840.080.0750.070.0820.080.0780.0760.0650.0740.060 0.5 1 1.5 2Wavelength, microns0.0720 0.5 1 1.5 2Wavelength, microns0.35Solar Cell0.1White Paint0.30.250.080.20.060.150.040.10.050.0200 0.5 1 1.5 2Wavelength, microns00 0.5 1 1.5 2Wavelength, micronsFigure 4: Laboratory spectral signatures for aluminum, mylar, solar cell, and white paint.For details see [70].The objective is then, given a set of spectral measurements or traces of an object, todetermine i) the type of constituent materials and ii) the proportional amount in which thesematerials appear. The first problem involves the detection of material spectral signaturesor endmembers from the spectral data. The second problem involves the computation ofcorresponding proportional amounts or fractional abundances. This is known as the spectralunmixing problem in the hyperspectral imaging community.Recall that in In Nonnegative Matrix Factorization (NMF), an m × n (nonnegative)mixed data matrix X is approximately factored into a product of two nonnegative rankkmatrices, with k small compared to m and n, X ≈ WH. This factorization has theadvantage that W and H can provide a physically realizable representation of the mixeddata, see e.g. [68]. Two sets of factors, one as endmembers and the other as fractionalabundances, are optimally fitted simultaneously. And due to reduced sizes of factors, data24
Figure 5: A blurred and noisy simulated hyperspectral image above the original simulatedimage of the Hubble Space Telescope representative of the data collected by the Maui ASISsystem.compression, spectral signature identification of constituent materials, and determination oftheir corresponding fractional abundances, can be fulfilled at the same time.Spectral reflectance data of a space object can be gathered using ground-based spectrometers,such as the SPICA system located on the 1.6 meter Gemini telescope and the ASISsystem located on the 3.67 meter telescope at the Maui Space Surveillance Complex (MSSC),and contains essential information regarding the make up or types of materials comprisingthe object. Different materials, such as aluminum, mylar, paint, plastics and solar cell,possess characteristic wavelength-dependent absorption features, or spectral signatures, thatmix together in the spectral reflectance measurement of an object. A new spectral imagingsensor, capable of collecting hyperspectral images of space objects, has been installed on the3.67 meter Advanced Electrocal-optical System (AEOS) at the MSSC. The AEOS SpectralImaging Sensor (ASIS) is used to collect adaptive optics compensated spectral images ofastronomical objects and satellites. See Figure 4 for a simulated hyperspectral image of theHubble Space Telescope similar to that collected by ASIS.In [92] and [93] Zhang, et al. develop NTF methods for identifying space objects usinghyperspectral data. Illustrations of material identification, material abundance estimation,and data compression are demonstrated for data similar to that shown in Figure 5.6 SummaryWe have outlined some of what we consider the more important and interesting problemsfor enforcing nonnegativity constraints in numerical analysis. Special emphasis has beenplaced nonnegativity constraints in least squares computations in numerical linear algebraand in nonlinear optimization. Techniques involving nonnegative low-rank matrix and tensorfactorizations and their many applications were also given. This report also includes aneffort toward a literature survey of the various algorithms and applications of nonnegativityconstraints in numerical analysis. As always, such an overview is certainly incomplete, andwe apologize for omissions. Hopefully, this work will inform the reader about the importanceof nonnegativity constraints in many problems in numerical analysis, while pointing towardthe many advantages of enforcing nonnegativity in practical applications.25
Page 1: Nonnegativity Constraints in Numeri
Page 5: NNLS Problem:Given a matrix A ∈ R
Page 8 and 9: However, when applied in a straight
Page 10 and 11: the active set method fnnls. It als
Page 12 and 13: Block principal pivoting algorithm:
Page 14 and 15: 4.1 Nonnegative Matrix Factorizatio
Page 16 and 17: algorithms can be very fast. The im
Page 18 and 19: Furthermore, the product T (m×np)
Page 20 and 21: 5.2 Image Processing and Computer V
Page 22 and 23: 5.4 Environmetrics and Chemometrics
Page 26 and 27: References[1] S. Bellavia, M. Macco
Page 28 and 29: [29] V. Franc, V. Hlavč, and M. Na
Page 30 and 31: [60] J. G. Nagy, Z. Strakoš, Enfor
Page 32: [89] R. S. Varga, Matrix Iterative

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