Notes on reconstructing the data cube in hyperspectral image processing
Abstract
The hyperspectral imaging technique described in (12) leads to the inter- esting problem of reconstructing a three-dimensional data cube from measured data. This problem involves three separate steps in which we must estimate values of a function from values of its Fourier transform. Depending on which of the two functions involved at each step has bounded support, that is, is zero off of a bounded set, the estimation process can take one of two forms, which we call Type One and Type Two problems. We discuss these two types and suggest techniques for solving both of them. For Type Two problems there is a good opportunity to incorporate whatever prior information we may have about the shape of the function being reconstructed. Since the data sets are large we recommend the use of iterative methods, such as the algebraic reconstruction technique (ART).
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From finite complex spectral data one can construct a continuous object with a given support that is consistent with the data. Given Fourier magnitude data only, one can choose the phases arbitrarily in the above construction. The energy in the extrapolated spectrum is phase dependent and provides a cost function to be used in phase retrieval. The minimization process is performed iteratively, using an algorithm that can be viewed as a combination of Gerchberg-Papoulis and Fienup error reduction.
We present a new algorithm for image restoration in limited-angle chromotomography. The algorithm is a generalization of the technique considered previously by the authors, based on a hybrid of a direct method of inversion and the iterative method of projections onto convex sets. The generalization is achieved by introducing a new object domain constraint. This constraint takes advantage of hyperspectral data redundancy and is realized by truncating the singular-value decomposition of the spatial–chromatic image matrix. As previously, the transform domain constraint is defined in terms of nonzero singular values of the system transfer function matrix. The new algorithm delivers high image fidelity, converges rapidly, and is easy to implement. Results of experiments on real data are included.
The ill-posed problem of restoring object information from finitely many measurements of its spectrum can be solved by using the best approximation in Hilbert spaces appropriately designed to include a priori information about object extent and shape and noise statistics. The procedures that are derived are noniterative, the linear ones extending the minimum-energy band-limited extrapolation methods (and thus related to Gerchberg–Papoulis iteration) and the nonlinear ones generalizing Burg’s maximum-entropy reconstruction of nonnegative objects.
A spectral imager constructs a three-dimensional (two spatial and one spectral)image from a series of two-dimensional images. We discuss a technique for spectral imaging that multiplexes the spatial and spectral information on a staring focal plane and then demultiplexes the resulting imagery to obtain the spectral image. The spectral image consists of 100×100 spatial pixels and 25 spectral bands. The current implementation operates over the 3–5-μm band, but can easily be applied to other spectral regions. This approach to spectral imaging has high optical throughput and is robust to focal plane array nonuniformities. A hardware description, the mathematical development, and experimental results are presented.
The problem of reconstructing a non-negative function from finitely many values of its Fourier transform is a problem of approximating one function by another and, as such, is analogous to the design of finite-impulse-response approximations to the Wiener filter. Using this analogy the authors obtain reconstruction methods that are computationally simpler approximations of entropy-based procedures. Their linear estimators allow for the inclusion of prior information about the oversampling rate, i.e. support information, as well as other prior knowledge of the general shape of the object. Their nonlinear methods, designed to recover spiky objects, make use of prior information about non-uniformity in the background to avoid bias in the estimation of peak locations.
The simultaneous MART algorithm (SMART) and the expectation maximization method for likelihood maximization (EMML) are extended to block-iterative versions, BI-SMART and BI-EMML, that converge to a solution in the feasible case, for any choice of subsets. The BI-EMML reduces to the "ordered subset" EMML of Hudson et al. (1992, 1994) when their "subset balanced" property holds.
Analysis of convergence of the algebraic reconstruction technique
(ART) shows it to be predisposed to converge to a solution faster than
simultaneous methods, such as those of the Cimmino-Landweber (1992)
type, the expectation maximization maximum likelihood method for the
Poisson model (EMML), and the simultaneous multiplicative ART (SMART),
which use all the data at each step. Although the choice of ordering of
the data and of relaxation parameters are important, as Herman and Meyer
(1993) have shown, they are not the full story. The analogous
multiplicative ART (MART), which applies only to systems y=Px in which
y>0, P⩾0 and a nonnegative solution is sought, is also sequential
(or “row-action”), rather than simultaneous, but does not
generally exhibit the same accelerated convergence relative to its
simultaneous version, SMART. By dividing each equation by the maximum of
the corresponding row of P, we find that this rescaled MART (RMART) does
converge faster, when solutions exist, significantly so in cases in
which the row maxima are substantially less than one. Such cases arise
frequently in tomography and when the columns of P have been normalized
to have sum one. Between simultaneous methods, which use all the data at
each step, and sequential (or row-action) methods, which use only a
single data value at each step, there are the block-iterative (or
ordered subset) methods, in which a single block or subset of the data
is processed at each step. The ordered subset EM (OSEM) of Hudson et al.
(see IEEE Trans. Med. Imag., vol.13, p.601-9, 1994) is significantly
faster than the EMML, but often fails to converge. The “rescaled
block-iterative” EMML (RBI-EMML) is an accelerated block-iterative
version of EMML that converges, in the consistent case, to a solution,
for any choice of subsets; it reduces to OSEM when the restrictive
“subset balanced” condition holds. Rescaled block-iterative
versions of SMART and MART also exhibit accelerated convergence
It has been shown that convergence to a solution can be
significantly accelerated for a number of iterative image reconstruction
algorithms, including simultaneous Cimmino-type algorithms, the
“expectation maximization” method for maximizing likelihood
(EMML) and the simultaneous multiplicative algebraic reconstruction
technique (SMART), through the use of rescaled block-iterative (BI)
methods. These BI methods involve partitioning the data into disjoint
subsets and using only one subset at each step of the iteration. One
drawback of these methods is their failure to converge to an approximate
solution in the inconsistent case, in which no image consistent with the
data exists; they are always observed to produce limit cycles (LCs) of
distinct images, through which the algorithm cycles. No one of these
images provides a suitable solution, in general. The question that
arises then is whether or not these LC vectors retain sufficient
information to construct from them a suitable approximate solution; we
show that they do. To demonstrate that, we employ a
“feedback” technique in which the LC vectors are used to
produce a new “data” vector, and the algorithm restarted.
Convergence of this nested iterative scheme to an approximate solution
is then proven. Preliminary work also suggests that this feedback method
may be incorporated in a practical reconstruction method
