Jeremias Sulam

Trainlets: Dictionary Learning in High Dimensions

Sparse representation has shown to be a very powerful model for real world signals, and has enabled the development of applications with notable performance. Combined with the ability to learn a dictionary from signal examples, sparsity-inspired algorithms are often achieving state-of-the-art results in a wide variety of tasks. These methods have traditionally been restricted to small dimensions mainly due to the computational constraints that the dictionary learning problem entails. In the context of image processing, this implies handling small image patches. In this work we show how to efficiently handle bigger dimensions and go beyond the small patches in sparsity-based signal and image processing methods. We build our approach based on a new cropped Wavelet decomposition, which enables a multi-scale analysis with virtually no border effects. We then employ this as the base dictionary within a double sparsity model to enable the training of adaptive dictionaries. To cope with the increase of training data, while at the same time improving the training performance, we present an Online Sparse Dictionary Learning (OSDL) algorithm to train this model effectively, enabling it to handle millions of examples. This work shows that dictionary learning can be up-scaled to tackle a new level of signal dimensions, obtaining large adaptable atoms that we call Trainlets.

Expected Patch Log Likelihood with a Sparse Prior

Image priors are of great importance in image restoration tasks. These problems can be addressed by decomposing the degraded image into overlapping patches, treating the patches individually and averaging them back together. Recently, the Expected Patch Log Likelihood (EPLL) method has been introduced, arguing that the chosen model should be enforced on the final reconstructed image patches. In the context of a Gaussian Mixture Model (GMM), this idea has been shown to lead to state-of-the-art results in image denoising and debluring. In this paper we combine the EPLL with a sparse-representation prior. Our derivation leads to a close yet extended variant of the popular K-SVD image denoising algorithm, where in order to effectively maximize the EPLL the denoising process should be iterated. This concept lies at the core of the K-SVD formulation, but has not been addressed before due the need to set different denoising thresholds in the successive sparse coding stages. We present a method that intrinsically determines these thresholds in order to improve the image estimate. Our results show a notable improvement over K-SVD in image denoising and inpainting, achieving comparable performance to that of EPLL with GMM in denoising.

Working Locally Thinking Globally: Theoretical Guarantees for Convolutional Sparse Coding

The celebrated sparse representation model has led to remarkable results in various signal processing tasks in the last decade. However, despite its initial purpose of serving as a global prior for entire signals, it has been commonly used for modeling low dimensional patches due to the computational constraints it entails when deployed with learned dictionaries. A way around this problem has been recently proposed, adopting a convolutional sparse representation model. This approach assumes that the global dictionary is a concatenation of banded circulant matrices. While several works have presented algorithmic solutions to the global pursuit problem under this new model, very few truly-effective guarantees are known for the success of such methods. In this paper, we address the theoretical aspects of the convolutional sparse model providing the first meaningful answers to questions of uniqueness of solutions and success of pursuit algorithms, both greedy and convex relaxations, in ideal and noisy regimes. To this end, we generalize mathematical quantities, such as the

Image denoising through multi-scale learnt dictionaries

\ell _0

Dynamical system classification with diffusion embedding for ECG-based person identification

norm, mutual coherence, Spark and restricted isometry property to their counterparts in the convolutional setting, intrinsically capturing local measures of the global model. On the algorithmic side, we demonstrate how to solve the global pursuit problem by using simple local processing, thus offering a first of its kind bridge between global modeling of signals and their patch-based local treatment.

Large Inpainting of Face Images With Trainlets

Over the last decade, a number of algorithms have shown promising results in removing additive white Gaussian noise from natural images, and though different, they all share in common a patch based strategy by locally denoising overlapping patches. While this lowers the complexity of the problem, it also causes noticeable artifacts when dealing with large smooth areas. In this paper we present a patch-based denoising algorithm relying on a sparsity-inspired model (K-SVD), which uses a multi-scale analysis framework. This allows us to overcome some of the disadvantages of the popular algorithms. We look for a sparse representation under an already sparsifying wavelet transform by adaptively training a dictionary on the different decomposition bands of the noisy image itself, leading to a multi-scale version of the K-SVD algorithm. We then combine the single scale and multi-scale approaches by merging both outputs by weighted joint sparse coding of the images. Our experiments on natural images indicate that our method is competitive with state of the art algorithms in terms of PSNR while giving superior results with respect to visual quality.

Gaussian mixture diffusion

The problem of system classification consists of identifying the source system corresponding to a certain output signal. In the context of dynamical systems, the outputs are usually given in the form of time series, and this identification process includes determining the underlying states of the system or their intrinsic set of parameters. In this work we propose a general framework for classification and identification based on a manifold learning algorithm. This data-driven approach provides a low-dimensional representation of the systems intrinsic variables, which enables the natural organization of points in time as a function of their dynamics. By leveraging the diffusion maps algorithm, a particular manifold learning method, we are not only able to distinguish between different states of the same system but also to discriminate different systems altogether. We construct a classification scheme based on a notion of distance between the distributions of embedded samples for different classes, and propose three ways of measuring such separation. The proposed method is demonstrated on a synthetic example and later applied to the problem of person identification from ECG recordings. Our approach obtains a 97.25% recognition accuracy over a database of 90 subjects, the highest accuracy reported for this database. HighlightsA novel signal classification method for dynamical systems is proposed.Diffusion Maps is applied on stable representations of sensor observations.Three variations are proposed, with different motivations and implications.Person identification from ECG signals is studied on a free available database.Our method achieves the highest reported accuracy on this dataset.

Fusion of ultrasound harmonic imaging with clutter removal using sparse signal separation

Image inpainting is concerned with the completion of missing data in an image. When the area to inpaint is relatively large, this problem becomes challenging. In these cases, traditional methods based on patch models and image propagation are limited, since they fail to consider a global perspective of the problem. In this letter, we employ a recently proposed dictionary learning framework, coined Trainlets, to design large adaptable atoms from a corpus of various datasets of face images by leveraging the online sparse dictionary learning algorithm. We, therefore, formulate the inpainting task as an inverse problem with a sparse-promoting prior based on the learned global model. Our results show the effectiveness of our scheme, obtaining much more plausible results than competitive methods.

Maximizing AUC with Deep Learning for Classification of Imbalanced Mammogram Datasets

Most state-of-the-art denoising algorithms employ a patch-based approach by enforcing a local model or prior, such as self similarity, sparse representation, or Gaussian Mixture Model (GMM). While applying these models, these algorithms implicitly build a notion of similarity between the image pixels. This can be formulated as an image-adaptive linear-filter which is then used to denoise or restore the degraded image. In this work we focus on such a filter emerging from the GMM, study its properties and construct a graph Laplacian from it. Focusing on a variational denoising formulation, we incorporate a graph-based regularization term by leveraging the corresponding GMM Laplacian. The resulting algorithm extends and improves the non-local diffusion algorithm by replacing the Non-Local Means kernel with a GMM one. Our results indicate that this approach, termed Gaussian Mixtures Diffusion (GMD), consistently improves over both the original GMM scheme and the non-local diffusion algorithm. Furthermore, GMD is competitive or even better than the state-of-the-art method of EPLL.

Multilayer Convolutional Sparse Modeling: Pursuit and Dictionary Learning

In ultrasound, second harmonic imaging is usually preferred due to the higher clutter artifacts and speckle noise common in the first harmonic image. Typical ultrasound use either one or the other image, applying corresponding filters for each case. In this work we propose a method based on a joint sparsity model that fuses the first and second harmonic images while performing clutter mitigation and noise reduction. Our approach, Fused Morphological Component Analysis (FMCA), uses two adaptive dictionaries for characterizing the clutter components in each image, and a common dictionary for the tissue representation. Our results indicate that the obtained images contain less clutter artifacts, less speckle noise and as such enjoy of the benefits of both harmonic input images.