Zachary T. Harmany

This is SPIRAL-TAP: Sparse Poisson Intensity Reconstruction ALgorithms—Theory and Practice

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 effectively accomplished by minimizing a conventional penalized least-squares objective function. The problem addressed in this paper is the estimation of f* from y in an inverse problem setting, where the number of unknowns may potentially be larger than the number of observations and f* admits sparse approximation. The optimization formulation considered in this paper uses a penalized negative Poisson log-likelihood objective function with nonnegativity constraints (since Poisson intensities are naturally nonnegative). In particular, the proposed approach incorporates key ideas of using separable quadratic approximations to the objective function at each iteration and penalization terms related to l1 norms of coefficient vectors, total variation seminorms, and partition-based multiscale estimation methods.

Compressed Sensing Performance Bounds Under Poisson Noise

This paper describes performance bounds for compressed sensing (CS) where the underlying sparse or compressible (sparsely approximable) signal is a vector of nonnegative intensities whose measurements are corrupted by Poisson noise. In this setting, standard CS techniques cannot be applied directly for several reasons. First, the usual signal-independent and/or bounded noise models do not apply to Poisson noise, which is nonadditive and signal-dependent. Second, the CS matrices typically considered are not feasible in real optical systems because they do not adhere to important constraints, such as nonnegativity and photon flux preservation. Third, the typical l2 - l1 minimization leads to overfitting in the high-intensity regions and oversmoothing in the low-intensity areas. In this paper, we describe how a feasible positivity- and flux-preserving sensing matrix can be constructed, and then analyze the performance of a CS reconstruction approach for Poisson data that minimizes an objective function consisting of a negative Poisson log likelihood term and a penalty term which measures signal sparsity. We show that, as the overall intensity of the underlying signal increases, an upper bound on the reconstruction error decays at an appropriate rate (depending on the compressibility of the signal), but that for a fixed signal intensity, the error bound actually grows with the number of measurements or sensors. This surprising fact is both proved theoretically and justified based on physical intuition.

Compressive coded aperture imaging

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.

Performance Bounds for Expander-Based Compressed Sensing in Poisson Noise

This paper provides performance bounds for compressed sensing in the presence of Poisson noise using expander graphs. The Poisson noise model is appropriate for a variety of applications, including low-light imaging and digital streaming, where the signal-independent and/or bounded noise models used in the compressed sensing literature are no longer applicable. In this paper, we develop a novel sensing paradigm based on expander graphs and propose a maximum a posteriori (MAP) algorithm for recovering sparse or compressible signals from Poisson observations. The geometry of the expander graphs and the positivity of the corresponding sensing matrices play a crucial role in establishing the bounds on the signal reconstruction error of the proposed algorithm. We support our results with experimental demonstrations of reconstructing average packet arrival rates and instantaneous packet counts at a router in a communication network, where the arrivals of packets in each flow follow a Poisson process.

Sparse poisson intensity reconstruction algorithms

The observations in many applications consist of counts of discrete events, such as photons hitting a dector, 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 ℓ2 - ℓ1 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 approximation in some basis. 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.

Poisson image reconstruction with total variation regularization

This paper describes an optimization framework for reconstructing nonnegative image intensities from linear projections contaminated with Poisson noise. Such Poisson inverse problems arise in a variety of applications, ranging from medical imaging to astronomy. A total variation regularization term is used to counter the ill-posedness of the inverse problem and results in reconstructions that are piecewise smooth. The proposed algorithm sequentially approximates the objective function with a regularized quadratic surrogate which can easily be minimized. Unlike alternative methods, this approach ensures that the natural nonnegativity constraints are satisfied without placing prohibitive restrictions on the nature of the linear projections to ensure computational tractability. The resulting algorithm is computationally efficient and outperforms similar methods using wavelet-sparsity or partition-based regularization.

Poisson noise reduction with non-local PCA

Photon limitations arise in spectral imaging, nuclear medicine, astronomy and night vision. The Poisson distribution used to model this noise has variance equal to its mean so blind application of standard noise removals methods yields significant artifacts. Recently, overcomplete dictionaries combined with sparse learning techniques have become extremely popular in image reconstruction. The aim of the present work is to demonstrate that for the task of image denoising, nearly state-of-the-art results can be achieved using small dictionaries only, provided that they are learned directly from the noisy image. To this end, we introduce patch-based denoising algorithms which perform an adaptation of PCA (Principal Component Analysis) for Poisson noise. We carry out a comprehensive empirical evaluation of the performance of our algorithms in terms of accuracy when the photon count is really low. The results reveal that, despite its simplicity, PCA-flavored denoising appears to be competitive with other state-of-the-art denoising algorithms.

SPIRAL out of convexity: sparsity-regularized algorithms for photon-limited imaging

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 l2-l1 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.

Quantitative Segmentation of Fluorescence Microscopy Images of Heterogeneous Tissue: Application to the Detection of Residual Disease in Tumor Margins

Purpose To develop a robust tool for quantitative in situ pathology that allows visualization of heterogeneous tissue morphology and segmentation and quantification of image features. Materials and Methods Tissue excised from a genetically engineered mouse model of sarcoma was imaged using a subcellular resolution microendoscope after topical application of a fluorescent anatomical contrast agent: acriflavine. An algorithm based on sparse component analysis (SCA) and the circle transform (CT) was developed for image segmentation and quantification of distinct tissue types. The accuracy of our approach was quantified through simulations of tumor and muscle images. Specifically, tumor, muscle, and tumor+muscle tissue images were simulated because these tissue types were most commonly observed in sarcoma margins. Simulations were based on tissue characteristics observed in pathology slides. The potential clinical utility of our approach was evaluated by imaging excised margins and the tumor bed in a cohort of mice after surgical resection of sarcoma. Results Simulation experiments revealed that SCA+CT achieved the lowest errors for larger nuclear sizes and for higher contrast ratios (nuclei intensity/background intensity). For imaging of tumor margins, SCA+CT effectively isolated nuclei from tumor, muscle, adipose, and tumor+muscle tissue types. Differences in density were correctly identified with SCA+CT in a cohort of ex vivo and in vivo images, thus illustrating the diagnostic potential of our approach. Conclusion The combination of a subcellular-resolution microendoscope, acriflavine staining, and SCA+CT can be used to accurately isolate nuclei and quantify their density in anatomical images of heterogeneous tissue.

Gradient projection for linearly constrained convex optimization in sparse signal recovery

The ℓ2-ℓ1 compressed sensing minimization problem can be solved efficiently by gradient projection. In imaging applications, the signal of interest corresponds to nonnegative pixel intensities; thus, with additional nonnegativity constraints on the reconstruction, the resulting constrained minimization problem becomes more challenging to solve. In this paper, we propose a gradient projection approach for sparse signal recovery where the reconstruction is subject to nonnegativity constraints. Numerical results are presented to demonstrate the effectiveness of this approach.