Subhadip Mukherjee

An iterative algorithm for phase retrieval with sparsity constraints: application to frequency domain optical coherence tomography

We address the problem of phase retrieval, which is frequently encountered in optical imaging. The measured quantity is the magnitude of the Fourier spectrum of a function (in optics, the function is also referred to as an object). The goal is to recover the object based on the magnitude measurements. In doing so, the standard assumptions are that the object is compactly supported and positive. In this paper, we consider objects that admit a sparse representation in some orthonormal basis. We develop a variant of the Fienup algorithm to incorporate the condition of sparsity and to successively estimate and refine the phase starting from the magnitude measurements. We show that the proposed iterative algorithm possesses Cauchy convergence properties. As far as the modality is concerned, we work with measurements obtained using a frequency-domain optical-coherence tomography experimental setup. The experimental results on real measured data show that the proposed technique exhibits good reconstruction performance even with fewer coefficients taken into account for reconstruction. It also suppresses the autocorrelation artifacts to a significant extent since it estimates the phase accurately.

Fienup Algorithm With Sparsity Constraints: Application to Frequency-Domain Optical-Coherence Tomography

We address the problem of reconstructing a sparse signal from its DFT magnitude. We refer to this problem as the sparse phase retrieval (SPR) problem, which finds applications in tomography, digital holography, electron microscopy, etc. We develop a Fienup-type iterative algorithm, referred to as the Max- K algorithm, to enforce sparsity and successively refine the estimate of phase. We show that the Max- K algorithm possesses Cauchy convergence properties under certain conditions, that is, the MSE of reconstruction does not increase with iterations. We also formulate the problem of SPR as a feasibility problem, where the goal is to find a signal that is sparse in a known basis and whose Fourier transform magnitude is consistent with the measurement. Subsequently, we interpret the Max- K algorithm as alternating projections onto the object-domain and measurement-domain constraint sets and generalize it to a parameterized relaxation, known as the relaxed averaged alternating reflections (RAAR) algorithm. On the application front, we work with measurements acquired using a frequency-domain optical-coherence tomography (FDOCT) experimental setup. Experimental results on measured data show that the proposed algorithms exhibit good reconstruction performance compared with the direct inversion technique, homomorphic technique, and the classical Fienup algorithm without sparsity constraint; specifically, the autocorrelation artifacts and background noise are suppressed to a significant extent. We also demonstrate that the RAAR algorithm offers a broader framework for FDOCT reconstruction, of which the direct inversion technique and the proposed Max- K algorithm become special instances corresponding to specific values of the relaxation parameter.

ℓ 1 -K-SVD

We develop a new dictionary learning algorithm called the ?1-K-SVD, by minimizing the ?1 distortion on the data term. The proposed formulation corresponds to maximum a posteriori estimation assuming a Laplacian prior on the coefficient matrix and additive noise, and is, in general, robust to non-Gaussian noise. The ?1 distortion is minimized by employing the iteratively reweighted least-squares algorithm. The dictionary atoms and the corresponding sparse coefficients are simultaneously estimated in the dictionary update step. Experimental results show that ?1-K-SVD results in noise-robustness, faster convergence, and higher atom recovery rate than the method of optimal directions, K-SVD, and the robust dictionary learning algorithm (RDL), in Gaussian as well as non-Gaussian noise. For a fixed value of sparsity, number of dictionary atoms, and data dimension, ?1-K-SVD outperforms K-SVD and RDL on small training sets. We also consider the generalized ? p , 0 < p < 1 , data metric to tackle heavy-tailed/impulsive noise. In an image denoising application, ?1-K-SVD was found to result in higher peak signal-to-noise ratio (PSNR) over K-SVD for Laplacian noise. The structural similarity index increases by 0.1 for low input PSNR, which is significant and demonstrates the efficacy of the proposed method. HighlightsWe propose an algorithm, which we refer to as ?1-K-SVD, to learn data-adaptive dictionaries in the presence of non-Gaussian noise. The fundamental idea behind the algorithm is to replace the usual ?2-norm-based data-fidelity metric with ?2-norm, and minimize it using iteratively reweighted least-squares (IRLS).In the dictionary update stage of ?1-K-SVD, we adopt a simultaneous updating strategy similar to K-SVD, that is found to result in faster convergence.We elucidate how the proposed idea can be extended to minimize the ?p data error, where 0 < p < 1 , in scenarios where one has to deal with sparse/impulsive noise contamination.We demonstrate experimentally that the ?1-K-SVD algorithm results in faster convergence and more accurate atom detection performance compared with the state-of-the-art algorithms. It is also shown that ?1-K-SVD is more suitable than the competing algorithms, when the training dataset contains fewer examples.As an application, we deploy the algorithm for image denoising. It is found that ?1-K-SVD results in peak signal-to-noise ratio (PSNR) values that are on par with the K-SVD algorithm, but the improvement in structural similarity index (SSIM) over K-SVD is approximately 0:08-0:10, indicating its efficacy in preserving the structural content of images.

An optimum shrinkage estimator based on minimum-probability-of-error criterion and application to signal denoising

We address the problem of designing an optimal pointwise shrinkage estimator in the transform domain, based on the minimum probability of error (MPE) criterion. We assume an additive model for the noise corrupting the clean signal. The proposed formulation is general in the sense that it can handle various noise distributions. We consider various noise distributions (Gaussian, Students-t, and Laplacian) and compare the denoising performance of the estimator obtained with the mean-squared error (MSE)-based estimators. The MSE optimization is carried out using an unbiased estimator of the MSE, namely Steins Unbiased Risk Estimate (SURE). Experimental results show that the MPE estimator outperforms the SURE estimator in terms of SNR of the denoised output, for low (0-10 dB) and medium values (10-20 dB) of the input SNR.

Joint dictionary training for bandwidth extension of speech signals

We address the problem of extending the bandwidth of speech signals, which is of importance to enhance the quality and intelligibility of the telephone speech. The low-pass filtering effect of the telephone communication channels eliminate the high-frequency components of the speech signal, and it is necessary to retrieve those to maintain the speech quality. We adopt a joint-dictionary training approach to recover the missing spectral information. By exploiting the sparsity of the spectrogram frames, the dictionaries for the wide-band (WB) and the corresponding narrow-band (NB) spectrogram frames are trained in a coupled manner in order to learn the mapping from NB to WB frames. We refer to this approach as the joint dictionary training for bandwidth extension (JDTBE). To ensure that the reconstructed bandwidth-extended speech is consistent with the measurement, we propose to apply a suitable affine transformation that depends on the properties of the telephone channel. We study the effect of the choice of sparsity on the quality of the reconstructed speech, for both male and female speakers. A comparison of the proposed JDTBE algorithm with a bandwidth extension technique based on stochastic modeling reveals the superiority of the JDTBE approach in terms of subjective listening test scores.

A divide-and-conquer dictionary learning algorithm and its performance analysis

We address the problem of learning a sparsifying synthesis dictionary over large datasets that occur in numerous signal and image processing applications, such as inpainting, super-resolution, etc. We develop a dictionary learning algorithm that exploits the similarity of the training examples to reduce the training time. Training datasets containing correlated examples typically occur in image processing applications, as the datasets contain the patches extracted from natural images as training vectors. Our algorithm employs a divide- and-conquer approach, where one leverages the correlation within the training examples to segment the dataset into clusters containing similar examples, and learn local dictionaries for each of them. This constitutes the divide step of the algorithm. In the conquer step, a global dictionary is trained using the atoms of the local dictionaries as the training examples. We analyze the run-time complexity and the representation error of the proposed divide-and-conquer dictionary learning algorithm, and compare the performance with the batch and online dictionary learning algorithms, both on synthesized dataset and natural images. The analysis reveals that the proposed algorithm has an asymptotic complexity that is linear and logarithmic in the number of training examples, corresponding to sequential and parallel implementations, respectively.

CONVERGENCE RATE ANALYSIS OF SMOOTHED LASSO

The LASSO regression has been studied extensively in the statistics and signal processing community, especially in the realm of sparse parameter estimation from linear measurements. We analyze the convergence rate of a first-order method applied on a smooth, strictly convex, and parametric upper bound on the LASSO objective function. The upper bound approaches the true non-smooth objective as the parameter tends to infinity. We show that a gradient-based algorithm, applied to minimize the smooth upper bound, yields a convergence rate of O (1/K), where K denotes the number of iterations performed. The analysis also reveals the optimum value of the parameter that achieves a desired prediction accuracy, provided that the total number of iterations is decided a priori. The convergence rate of the proposed algorithm and the amount of computation required in each iteration are same as that of the iterative soft thresholding technique. However, the proposed algorithm does not involve any thresholding operation. The performance of the proposed technique, referred to as smoothed LASSO, is validated on synthesized signals. We also deploy smoothed LASSO for estimating an image from its blurred and noisy measurement, and compare the performance with the fast iterative shrinkage thresholding algorithm for a fixed run-time budget, in terms of the reconstruction peak signal-to-noise ratio and structural similarity index.

A Split-and-Merge Dictionary Learning Algorithm for Sparse Representation: Application to Image Denoising

In big data image/video analytics, we encounter the problem of learning an over-complete dictionary for sparse representation from a large training dataset, which cannot be processed at once because of storage and computational constraints. To tackle the problem of dictionary learning in such scenarios, we propose an algorithm that exploits the inherent clustered structure of the training data and make use of a divide-and-conquer approach. The fundamental idea behind the algorithm is to partition the training dataset into smaller clusters, and learn local dictionaries for each cluster. Subsequently, the local dictionaries are merged to form a global dictionary. Merging is done by solving another dictionary learning problem on the atoms of the locally trained dictionaries. This algorithm is referred to as the split-and-merge algorithm. We show that the proposed algorithm is efficient in its usage of memory and computational complexity, and performs on par with the standard learning strategy, which operates on the entire data at a time. As an application, we consider the problem of image denoising. We present a comparative analysis of our algorithm with the standard learning techniques that use the entire database at a time, in terms of training and denoising performance. We observe that the split-and-merge algorithm results in a remarkable reduction of training time, without significantly affecting the denoising performance.

Phase retrieval for a class of 2-D signals characterized by first-order difference equations

We address the problem of signal reconstruction from the Fourier transform magnitude of a certain class of two-dimensional (2-D) signals that are characterized by first-order difference equations. We show that when such a signal has a Z-transform that includes the unit sphere in the region of convergence, it can be reconstructed uniquely from the Fourier magnitude. We employ the annihilating filter approach to find the parameters of the rational transfer function.