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Mon, 22 May 2017 11:26:42 +0200Mon, 22 May 2017 11:26:42 +0200Convex Analysis for Processing Hyperspectral Images and Data from Hadamard Spaces
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4650
This thesis brings together convex analysis and hyperspectral image processing.
Convex analysis is the study of convex functions and their properties.
Convex functions are important because they admit minimization by efficient algorithms
and the solution of many optimization problems can be formulated as
minimization of a convex objective function, extending much beyond
the classical image restoration problems of denoising, deblurring and inpainting.
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At the heart of convex analysis is the duality mapping induced within the
class of convex functions by the Fenchel transform.
In the last decades efficient optimization algorithms have been developed based
on the Fenchel transform and the concept of infimal convolution.
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The infimal convolution is of similar importance in convex analysis as the
convolution in classical analysis. In particular, the infimal convolution with
scaled parabolas gives rise to the one parameter family of Moreau-Yosida envelopes,
which approximate a given function from below while preserving its minimum
value and minimizers.
The closely related proximal mapping replaces the gradient step
in a recently developed class of efficient first-order iterative minimization algorithms
for non-differentiable functions. For a finite convex function,
the proximal mapping coincides with a gradient step of its Moreau-Yosida envelope.
Efficient algorithms are needed in hyperspectral image processing,
where several hundred intensity values measured in each spatial point
give rise to large data volumes.
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In the \(\textbf{first part}\) of this thesis, we are concerned with
models and algorithms for hyperspectral unmixing.
As part of this thesis a hyperspectral imaging system was taken into operation
at the Fraunhofer ITWM Kaiserslautern to evaluate the developed algorithms on real data.
Motivated by missing-pixel defects common in current hyperspectral imaging systems,
we propose a
total variation regularized unmixing model for incomplete and noisy data
for the case when pure spectra are given.
We minimize the proposed model by a primal-dual algorithm based on the
proximum mapping and the Fenchel transform.
To solve the unmixing problem when only a library of pure spectra is provided,
we study a modification which includes a sparsity regularizer into model.
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We end the first part with the convergence analysis for a multiplicative
algorithm derived by optimization transfer.
The proposed algorithm extends well-known multiplicative update rules
for minimizing the Kullback-Leibler divergence,
to solve a hyperspectral unmixing model in the case
when no prior knowledge of pure spectra is given.
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In the \(\textbf{second part}\) of this thesis, we study the properties of Moreau-Yosida envelopes,
first for functions defined on Hadamard manifolds, which are (possibly) infinite-dimensional
Riemannian manifolds with negative curvature,
and then for functions defined on Hadamard spaces.
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In particular we extend to infinite-dimensional Riemannian manifolds an expression
for the gradient of the Moreau-Yosida envelope in terms of the proximal mapping.
With the help of this expression we show that a sequence of functions
converges to a given limit function in the sense of Mosco
if the corresponding Moreau-Yosida envelopes converge pointwise at all scales.
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Finally we extend this result to the more general setting of Hadamard spaces.
As the reverse implication is already known, this unites two definitions of Mosco convergence
on Hadamard spaces, which have both been used in the literature,
and whose equivalence has not yet been known.Martin J. Montagdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4650Mon, 22 May 2017 11:26:42 +0200