10 research outputs found
Hierarchical Bayesian sparse image reconstruction with application to MRFM
This paper presents a hierarchical Bayesian model to reconstruct sparse
images when the observations are obtained from linear transformations and
corrupted by an additive white Gaussian noise. Our hierarchical Bayes model is
well suited to such naturally sparse image applications as it seamlessly
accounts for properties such as sparsity and positivity of the image via
appropriate Bayes priors. We propose a prior that is based on a weighted
mixture of a positive exponential distribution and a mass at zero. The prior
has hyperparameters that are tuned automatically by marginalization over the
hierarchical Bayesian model. To overcome the complexity of the posterior
distribution, a Gibbs sampling strategy is proposed. The Gibbs samples can be
used to estimate the image to be recovered, e.g. by maximizing the estimated
posterior distribution. In our fully Bayesian approach the posteriors of all
the parameters are available. Thus our algorithm provides more information than
other previously proposed sparse reconstruction methods that only give a point
estimate. The performance of our hierarchical Bayesian sparse reconstruction
method is illustrated on synthetic and real data collected from a tobacco virus
sample using a prototype MRFM instrument.Comment: v2: final version; IEEE Trans. Image Processing, 200
Semi-blind Sparse Image Reconstruction with Application to MRFM
We propose a solution to the image deconvolution problem where the
convolution kernel or point spread function (PSF) is assumed to be only
partially known. Small perturbations generated from the model are exploited to
produce a few principal components explaining the PSF uncertainty in a high
dimensional space. Unlike recent developments on blind deconvolution of natural
images, we assume the image is sparse in the pixel basis, a natural sparsity
arising in magnetic resonance force microscopy (MRFM). Our approach adopts a
Bayesian Metropolis-within-Gibbs sampling framework. The performance of our
Bayesian semi-blind algorithm for sparse images is superior to previously
proposed semi-blind algorithms such as the alternating minimization (AM)
algorithm and blind algorithms developed for natural images. We illustrate our
myopic algorithm on real MRFM tobacco virus data.Comment: This work has been submitted to the IEEE Trans. Image Processing for
possible publicatio
Sampling and Recovery of Pulse Streams
Compressive Sensing (CS) is a new technique for the efficient acquisition of
signals, images, and other data that have a sparse representation in some
basis, frame, or dictionary. By sparse we mean that the N-dimensional basis
representation has just K<<N significant coefficients; in this case, the CS
theory maintains that just M = K log N random linear signal measurements will
both preserve all of the signal information and enable robust signal
reconstruction in polynomial time. In this paper, we extend the CS theory to
pulse stream data, which correspond to S-sparse signals/images that are
convolved with an unknown F-sparse pulse shape. Ignoring their convolutional
structure, a pulse stream signal is K=SF sparse. Such signals figure
prominently in a number of applications, from neuroscience to astronomy. Our
specific contributions are threefold. First, we propose a pulse stream signal
model and show that it is equivalent to an infinite union of subspaces. Second,
we derive a lower bound on the number of measurements M required to preserve
the essential information present in pulse streams. The bound is linear in the
total number of degrees of freedom S + F, which is significantly smaller than
the naive bound based on the total signal sparsity K=SF. Third, we develop an
efficient signal recovery algorithm that infers both the shape of the impulse
response as well as the locations and amplitudes of the pulses. The algorithm
alternatively estimates the pulse locations and the pulse shape in a manner
reminiscent of classical deconvolution algorithms. Numerical experiments on
synthetic and real data demonstrate the advantages of our approach over
standard CS
Sampling and recovery of pulse streams
Compressive Sensing (CS) is a new technique for the efficient acquisition of signals, images, and other data that have a sparse representation in some basis, frame, or dictionary. By sparse we mean that the N -dimensional basis representation has just K << N significant coefficients; in this case, the CS theory maintains that just M = O (K log N) random linear signal measurements will both preserve all of the signal information and enable robust signal reconstruction in polynomial time. In this paper, we extend the CS theory to pulse stream data, which correspond to S-sparse signals/images that are convolved with an unknown F-sparse pulse shape. Ignoring their convolutional structure, a pulse stream signal is K = SF sparse. Such signals figure prominently in a number of applications, from neuroscience to astronomy. Our specific contributions are threefold. First, we propose a pulse stream signal model and show that it is equivalent to an infinite union of subspaces. Second, we derive a lower bound on the number of measurements M required to preserve the essential information present in pulse streams. The bound is linear in the total number of degrees of freedom S + F, which is significantly smaller than the naive bound based on the total signal sparsity K = SF. Third, we develop an efficient signal recovery algorithm that infers both the shape of the impulse response as well as the locations and amplitudes of the pulses. The algorithm alternatively estimates the pulse locations and the pulse shape in a manner reminiscent of classical deconvolution algorithms. Numerical experiments on synthetic and real data demonstrate the advantages of our approach over standard CS
Reconstruction, Classification, and Segmentation for Computational Microscopy
This thesis treats two fundamental problems in computational microscopy: image reconstruction for magnetic resonance force microscopy (MRFM) and image classification for electron backscatter diffraction (EBSD). In MRFM, as in many inverse problems, the true point spread function (PSF) that blurs the image may be only partially known. The image quality may suffer from this possible mismatch when standard image reconstruction techniques are applied. To deal with the mismatch, we develop novel Bayesian sparse reconstruction methods that account for possible errors in the PSF of the microscope and for the inherent sparsity of MRFM images. Two methods are proposed: a stochastic method and a variational method. They both jointly estimate the unknown PSF and unknown image. Our proposed framework for reconstruction has the flexibility to incorporate sparsity inducing priors, thus addressing ill-posedness of this non-convex problem, Markov-Random field priors, and can be extended to other image models. To obtain scalable and tractable solutions, a dimensionality reduction technique is applied to the highly nonlinear PSF space. The experiments clearly demonstrate that the proposed methods have superior performance compared to previous methods.
In EBSD we develop novel and robust dictionary-based methods for segmentation and classification of grain and sub-grain structures in polycrystalline materials. Our work is the first in EBSD analysis to use a physics-based forward model, called the dictionary, to use full diffraction patterns, and that efficiently classifies patterns into grains, boundaries, and anomalies. In particular, unlike previous methods, our method incorporates anomaly detection directly into the segmentation process. The proposed approach also permits super-resolution of grain mantle and grain boundary locations. Finally, the proposed dictionary-based segmentation method performs uncertainty quantification, i.e. p-values, for the classified grain interiors and grain boundaries. We demonstrate that the dictionary-based approach is robust to instrument drift and material differences that produce small amounts of dictionary mismatch.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/102296/1/seunpark_1.pd