THESIS
2022
1 online resource (xiv, 113 pages) : illustrations (some color)
Abstract
In this thesis, we study the sparse phase retrieval problem, which is to reconstruct
the sparse signal x
♮ with ∥x
♮∥
0 ≤ s from its intensity-only measurements
y
i = ∣a
iT
x
♮∣ for i = 1,2,…,m. The sparse phase retrieval problem arises in
many applications in scientific imaging, such as bio-imaging systems, astronomical
imaging, diffraction imaging and optics. The existing sparse phase retrieval
algorithms are mostly first-order methods and merit a linear convergence property.
However, there are only a few second-order algorithms for sparse phase
retrieval. To mitigate this gap, we propose two second-order algorithms in this
thesis.
The first approach is the Newton Hard Thresholding Pursuit (NHTP) algorithm,
which is a provable second-order algorithm. Theoretical analysis show
that NHTP conv...[
Read more ]
In this thesis, we study the sparse phase retrieval problem, which is to reconstruct
the sparse signal x
♮ with ∥x
♮∥
0 ≤ s from its intensity-only measurements
y
i = ∣a
iT
x
♮∣ for i = 1,2,…,m. The sparse phase retrieval problem arises in
many applications in scientific imaging, such as bio-imaging systems, astronomical
imaging, diffraction imaging and optics. The existing sparse phase retrieval
algorithms are mostly first-order methods and merit a linear convergence property.
However, there are only a few second-order algorithms for sparse phase
retrieval. To mitigate this gap, we propose two second-order algorithms in this
thesis.
The first approach is the Newton Hard Thresholding Pursuit (NHTP) algorithm,
which is a provable second-order algorithm. Theoretical analysis show
that NHTP converges at least linearly and merits a quadratic convergence after
O(log(∥x
♮∥/x
♮min))
iterations. The sparse spectral initialization method is employed
for a close initial guess. Numerical simulations confirm the efficiency of
the proposed NHTP algorithm in terms of sample complexity and running time.
The second approach is the Primal Dual Active Set algorithm with Continuation
(PDASC). Practically, the sparsity level of the underlying signal is unknown.
However, the development of most algorithms for sparse phase retrieval are based
on the knowledge of the sparsity. PDASC is developed to reconstruct the signal
without knowing sparsity level. We show that the update rule for PDASC
can be interpreted as a Newton's method. Also, a novel initialization method
which does not require the sparsity level as a prior knowledge is proposed. The
effectiveness of the new initialization method is guaranteed both empirically and
theoretically. Extensive experiments on both synthetic data and natural images confirm the efficiency of the proposed PDASC algorithm.
Post a Comment