In this paper, we solve the ℓ<sub>2</sub>-ℓ<sub>1</sub> sparse recovery problem by transforming the objective function of this problem into an unconstrained differentiable function and applying a limited-memory trust-region method. Unlike gradient projection-type methods, which uses only the current gradient, our approach uses gradients from previous iterations to obtain a more accurate Hessian approximation. Numerical experiments show that our proposed approach eliminates spurious solutions more effectively while improving computational time.
Reconstructing high-dimensional sparse signals from low-dimensional low-count photon observations is a challenging nonlinear optimization problem. In this paper, we build upon previous work on minimizing the Poisson log-likelihood and incorporate recent work on the <i>generalized </i>nonconvex Shannon entropy function for promoting sparsity in solutions. We explore the effectiveness of the proposed approach using numerical experiments.
Proc. SPIE. 8165, Unconventional Imaging, Wavefront Sensing, and Adaptive Coded Aperture Imaging and Non-Imaging Sensor Systems
KEYWORDS: Imaging systems, Video, Fourier transforms, Video compression, Optical flow, Reconstruction algorithms, Coded apertures, Motion models, Motion measurement, Simulation of CCA and DLA aggregates
This paper describes an adaptive compressive coded aperture imaging system for video based on motion-compensated
video sparsity models. In particular, motion models based on optical flow and sparse deviations from optical flow (i.e.
salient motion) can be used to (a) predict future video frames from previous compressive measurements, (b) perform
reconstruction using efficient online convex programming techniques, and (c) adapt the coded aperture to yield higher
reconstruction fidelity in the vicinity of this salient motion.
The emerging field of compressed sensing has potentially powerful implications for the design of optical imaging devices. In particular, compressed sensing theory suggests that one can recover a scene at a higher resolution than is dictated by the pitch of the focal plane array. This rather remarkable result comes with some important caveats however, especially when practical issues associated with physical implementation are taken into account. This tutorial discusses compressed sensing in the context of optical imaging devices, emphasizing the practical hurdles related to building such devices, and offering suggestions for overcoming these hurdles. Examples and analysis specifically related to infrared imaging highlight the challenges associated with large format focal plane arrays and how these challenges can be mitigated using compressed sensing ideas.
Traditionally, optical sensors have been designed to collect the most directly interpretable and intuitive measurements possible.
However, recent advances in the fields of image reconstruction, inverse problems, and compressed sensing indicate
that substantial performance gains may be possible in many contexts via computational methods. In particular, by designing
optical sensors to deliberately collect "incoherent" measurements of a scene, we can use sophisticated computational
methods to infer more information about critical scene structure and content. In this paper, we explore the potential of
physically realizable systems for acquiring such measurements. Specifically, we describe how given a fixed size focal
plane array, compressive measurements using coded apertures combined with sophisticated optimization algorithms can
significantly increase image quality and resolution.
The observations in many applications consist of counts of discrete events, such as photons hitting a detector, which cannot
be effectively modeled using an additive bounded or Gaussian noise model, and instead require a Poisson noise model. As
a result, accurate reconstruction of a spatially or temporally distributed phenomenon (f*) from Poisson data (y) cannot be
accomplished by minimizing a conventional l<sub>2</sub>-l<sub>1</sub> objective function. The problem addressed in this paper is the estimation
of f* from y in an inverse problem setting, where (a) the number of unknowns may potentially be larger than the number
of observations and (b) f* admits a sparse representation. The optimization formulation considered in this paper uses a
negative Poisson log-likelihood objective function with nonnegativity constraints (since Poisson intensities are naturally
nonnegative). This paper describes computational methods for solving the constrained sparse Poisson inverse problem.
In particular, the proposed approach incorporates key ideas of using quadratic separable approximations to the objective
function at each iteration and computationally efficient partition-based multiscale estimation methods.
KEYWORDS: Staring arrays, Imaging systems, Cameras, Video, Fourier transforms, Video compression, Coded apertures, Simulation of CCA and DLA aggregates, Distributed interactive simulations, Compressed sensing
Nonlinear image reconstruction based upon sparse representations of images has recently received widespread attention
with the emerging framework of compressed sensing (CS). This theory indicates that, when feasible, judicious selection
of the type of distortion induced by measurement systems may dramatically improve our ability to perform image reconstruction.
However, applying compressed sensing theory to practical imaging systems poses a key challenge: physical
constraints typically make it infeasible to actually measure many of the random projections described in the literature, and
therefore, innovative and sophisticated imaging systems must be carefully designed to effectively exploit CS theory. In
video settings, the performance of an imaging system is characterized by both pixel resolution and field of view. In this
work, we propose compressive imaging techniques for improving the performance of video imaging systems in the presence
of constraints on the focal plane array size. In particular, we describe a novel yet practical approach that combines
coded aperture imaging to enhance pixel resolution with superimposing subframes of a scene onto a single focal plane
array to increase field of view. Specifically, the proposed method superimposes coded observations and uses wavelet-based
sparsity recovery algorithms to reconstruct the original subframes. We demonstrate the effectiveness of this approach by
reconstructing with high resolution the constituent images of a video sequence.