The nuclear norm minimization (NNM) is commonly used to approximate the matrix rank by shrinking all singular values equally. However, the singular values have clear physical meanings in many practical problems, and NNM may not be able to faithfully approximate the matrix rank. To alleviate the above-mentioned limitation of NNM, recent studies have suggested that the weighted nuclear norm minimization (WNNM) can achieve a better rank estimation than NNM, which heuristically set the weight being inverse to the singular values. However, it still lacks a rigorous explanation why WNNM is more effective than NMM in various applications. In this paper, we analyze NNM and WNNM from the perspective of group sparse representation (GSR). Concretely, an adaptive dictionary learning method is devised to connect the rank minimization and GSR models. Based on the proposed dictionary, we prove that NNM and WNNM are equivalent to L1-norm minimization and the weighted L1-norm minimization in GSR, respectively. Inspired by enhancing sparsity of the weighted L1-norm minimization in comparison with L1-norm minimization in sparse representation, we thus explain that WNNM is more effective than NMM. By integrating the image nonlocal self-similarity (NSS) prior with the WNNM model, we then apply it to solve the image denoising problem. Experimental results demonstrate that WNNM is more effective than NNM and outperforms several state-of-the-art methods in both objective and perceptual quality.
While the sparse coding principle can successfully model information processing in sensory neural systems, it remains unclear how learning can be accomplished under neural architectural constraints. Feasible learning rules must rely solely on synaptically local information in order to be implemented on spatially distributed neurons. We describe a neural network with spiking neurons that can address the aforementioned fundamental challenge and solve the L1-minimizing dictionary learning problem, representing the first model able to do so. Our major innovation is to introduce feedback synapses to create a pathway to turn the seemingly non-local information into local ones. The resulting network encodes the error signal needed for learning as the change of network steady states caused by feedback, and operates akin to the classical stochastic gradient descent method.
Given a dataset, the task of learning a transform that allows sparse representations of the data bears the name of dictionary learning. In many applications, these learned dictionaries represent the data much better than the static well-known transforms (Fourier, Hadamard etc.). The main downside of learned transforms is that they lack structure and therefore they are not computationally efficient, unlike their classical counterparts. These posse several difficulties especially when using power limited hardware such as mobile devices, therefore discouraging the application of sparsity techniques in such scenarios. In this paper we construct orthogonal and non-orthogonal dictionaries that are factorized as a product of a few basic transformations. In the orthogonal case, we solve exactly the dictionary update problem for one basic transformation, which can be viewed as a generalized Givens rotation, and then propose to construct orthogonal dictionaries that are a product of these transformations, guaranteeing their fast manipulation. We also propose a method to construct fast square but non-orthogonal dictionaries that are factorized as a product of few transforms that can be viewed as a further generalization of Givens rotations to the non-orthogonal setting. We show how the proposed transforms can balance very well data representation performance and computational complexity. We also compare with classical fast and learned general and orthogonal transforms.
We address the multi-focus image fusion problem, where multiple images captured with different focal settings are to be fused into an all-in-focus image of higher quality. Algorithms for this problem necessarily admit the source image characteristics along with focused and blurred features. However, most sparsity-based approaches use a single dictionary in focused feature space to describe multi-focus images, and ignore the representations in blurred feature space. We propose a multi-focus image fusion approach based on sparse representation using a coupled dictionary. It exploits the observations that the patches from a given training set can be sparsely represented by a couple of overcomplete dictionaries related to the focused and blurred categories of images and that a sparse approximation based on such coupled dictionary leads to a more flexible and therefore better fusion strategy than the one based on just selecting the sparsest representation in the original image estimate. In addition, to improve the fusion performance, we employ a coupled dictionary learning approach that enforces pairwise correlation between atoms of dictionaries learned to represent the focused and blurred feature spaces. We also discuss the advantages of the fusion approach based on coupled dictionary learning, and present efficient algorithms for fusion based on coupled dictionary learning. Extensive experimental comparisons with state-of-the-art multi-focus image fusion algorithms validate the effectiveness of the proposed approach.
Age progression is defined as aesthetically re-rendering the aging face at any future age for an individual face. In this work, we aim to automatically render aging faces in a personalized way. Basically, for each age group, we learn an aging dictionary to reveal its aging characteristics (e.g., wrinkles), where the dictionary bases corresponding to the same index yet from two neighboring aging dictionaries form a particular aging pattern cross these two age groups, and a linear combination of all these patterns expresses a particular personalized aging process. Moreover, two factors are taken into consideration in the dictionary learning process. First, beyond the aging dictionaries, each person may have extra personalized facial characteristics, e.g. mole, which are invariant in the aging process. Second, it is challenging or even impossible to collect faces of all age groups for a particular person, yet much easier and more practical to get face pairs from neighboring age groups. To this end, we propose a novel Bi-level Dictionary Learning based Personalized Age Progression (BDL-PAP) method. Here, bi-level dictionary learning is formulated to learn the aging dictionaries based on face pairs from neighboring age groups. Extensive experiments well demonstrate the advantages of the proposed BDL-PAP over other state-of-the-arts in term of personalized age progression, as well as the performance gain for cross-age face verification by synthesizing aging faces.
A multiple instance dictionary learning approach, Dictionary Learning using Functions of Multiple Instances (DL-FUMI), is used to perform beat-to-beat heart rate estimation and to characterize heartbeat signatures from ballistocardiogram (BCG) signals collected with a hydraulic bed sensor. DL-FUMI estimates a "heartbeat concept" that represents an individual's personal ballistocardiogram heartbeat pattern. DL-FUMI formulates heartbeat detection and heartbeat characterization as a multiple instance learning problem to address the uncertainty inherent in aligning BCG signals with ground truth during training. Experimental results show that the estimated heartbeat concept found by DL-FUMI is an effective heartbeat prototype and achieves superior performance over comparison algorithms.
Computational imaging methods that can exploit multiple modalities have the potential to enhance the capabilities of traditional sensing systems. In this paper, we propose a new method that reconstructs multimodal images from their linear measurements by exploiting redundancies across different modalities. Our method combines a convolutional group-sparse representation of images with total variation (TV) regularization for high-quality multimodal imaging. We develop an online algorithm that enables the unsupervised learning of convolutional dictionaries on large-scale datasets that are typical in such applications. We illustrate the benefit of our approach in the context of joint intensity-depth imaging.
Recent research in image and video recognition indicates that many visual processes can be thought of as being generated by a time-varying generative model. A nearby descriptive model for visual processes is thus a statistical distribution that varies over time. Specifically, modeling visual processes as streams of histograms generated by a kernelized linear dynamic system turns out to be efficient. We refer to such a model as a System of Bags. In this work, we investigate Systems of Bags with special emphasis on dynamic scenes and dynamic textures. Parameters of linear dynamic systems suffer from ambiguities. In order to cope with these ambiguities in the kernelized setting, we develop a kernelized version of the alignment distance. For its computation, we use a Jacobi-type method and prove its convergence to a set of critical points. We employ it as a dissimilarity measure on Systems of Bags. As such, it outperforms other known dissimilarity measures for kernelized linear dynamic systems, in particular the Martin Distance and the Maximum Singular Value Distance, in every tested classification setting. A considerable margin can be observed in settings, where classification is performed with respect to an abstract mean of video sets. For this scenario, the presented approach can outperform state-of-the-art techniques, such as Dynamic Fractal Spectrum or Orthogonal Tensor Dictionary Learning.
We develop fast spectral algorithms for tensor decomposition that match the robustness guarantees of the best known polynomial-time algorithms for this problem based on the sum-of-squares (SOS) semidefinite programming hierarchy. Our algorithms can decompose a 4-tensor with $n$-dimensional orthonormal components in the presence of error with constant spectral norm (when viewed as an $n^2$-by-$n^2$ matrix). The running time is $n^5$ which is close to linear in the input size $n^4$. We also obtain algorithms with similar running time to learn sparsely-used orthogonal dictionaries even when feature representations have constant relative sparsity and non-independent coordinates. The only previous polynomial-time algorithms to solve these problem are based on solving large semidefinite programs. In contrast, our algorithms are easy to implement directly and are based on spectral projections and tensor-mode rearrangements. Or work is inspired by recent of Hopkins, Schramm, Shi, and Steurer (STOC'16) that shows how fast spectral algorithms can achieve the guarantees of SOS for average-case problems. In this work, we introduce general techniques to capture the guarantees of SOS for worst-case problems.
Sonar imaging has seen vast improvements over the last few decades due in part to advances in synthetic aperture Sonar (SAS). Sophisticated classification techniques can now be used in Sonar automatic target recognition (ATR) to locate mines and other threatening objects. Among the most promising of these methods is sparse reconstruction-based classification (SRC) which has shown an impressive resiliency to noise, blur, and occlusion. We present a coherent strategy for expanding upon SRC for Sonar ATR that retains SRC's robustness while also being able to handle targets with diverse geometric arrangements, bothersome Rayleigh noise, and unavoidable background clutter. Our method, pose corrected sparsity (PCS), incorporates a novel interpretation of a spike and slab probability distribution towards use as a Bayesian prior for class-specific discrimination in combination with a dictionary learning scheme for localized patch extractions. Additionally, PCS offers the potential for anomaly detection in order to avoid false identifications of tested objects from outside the training set with no additional training required. Compelling results are shown using a database provided by the United States Naval Surface Warfare Center.
Saliency detection has drawn a lot of attention of researchers in various fields over the past several years. Saliency is the perceptual quality that makes an object, person to draw the attention of humans at the very sight. Salient object detection in an image has been used centrally in many computational photography and computer vision applications like video compression, object recognition and classification, object segmentation, adaptive content delivery, motion detection, content aware resizing, camouflage images and change blindness images to name a few. We propose a method to detect saliency in the objects using multimodal dictionary learning which has been recently used in classification and image fusion. The multimodal dictionary that we are learning is task driven which gives improved performance over its counterpart (the one which is not task specific).
Recognition of low resolution face images is a challenging problem in many practical face recognition systems. Methods have been proposed in the face recognition literature for the problem which assume that the probe is low resolution, but a high resolution gallery is available for recognition. These attempts have been aimed at modifying the probe image such that the resultant image provides better discrimination. We formulate the problem differently by leveraging the information available in the high resolution gallery image and propose a dictionary learning approach for classifying the low-resolution probe image. An important feature of our algorithm is that it can handle resolution change along with illumination variations. Furthermore, we also kernelize the algorithm to handle non-linearity in data and present a joint dictionary learning technique for robust recognition at low resolutions. The effectiveness of the proposed method is demonstrated using standard datasets and a challenging outdoor face dataset. It is shown that our method is efficient and can perform significantly better than many competitive low resolution face recognition algorithms.
Despite the fact that different objects possess distinct class-specific features, they also usually share common patterns. This observation has been exploited partially in a recently proposed dictionary learning framework by separating the particularity and the commonality (COPAR). Inspired by this, we propose a novel method to explicitly and simultaneously learn a set of common patterns as well as class-specific features for classification with more intuitive constraints. Our dictionary learning framework is hence characterized by both a shared dictionary and particular (class-specific) dictionaries. For the shared dictionary, we enforce a low-rank constraint, i.e. claim that its spanning subspace should have low dimension and the coefficients corresponding to this dictionary should be similar. For the particular dictionaries, we impose on them the well-known constraints stated in the Fisher discrimination dictionary learning (FDDL). Further, we develop new fast and accurate algorithms to solve the subproblems in the learning step, accelerating its convergence. The said algorithms could also be applied to FDDL and its extensions. The efficiencies of these algorithms are theoretically and experimentally verified by comparing their complexities and running time with those of other well-known dictionary learning methods. Experimental results on widely used image datasets establish the advantages of our method over state-of-the-art dictionary learning methods.
We consider the problem of efficiently approximating and encoding high-dimensional data sampled from a probability distribution $\rho$ in $\mathbb{R}^D$, that is nearly supported on a $d$-dimensional set $\mathcal{M}$ - for example supported on a $d$-dimensional Riemannian manifold. Geometric Multi-Resolution Analysis (GMRA) provides a robust and computationally efficient procedure to construct low-dimensional geometric approximations of $\mathcal{M}$ at varying resolutions. We introduce a thresholding algorithm on the geometric wavelet coefficients, leading to what we call adaptive GMRA approximations. We show that these data-driven, empirical approximations perform well, when the threshold is chosen as a suitable universal function of the number of samples $n$, on a wide variety of measures $\rho$, that are allowed to exhibit different regularity at different scales and locations, thereby efficiently encoding data from more complex measures than those supported on manifolds. These approximations yield a data-driven dictionary, together with a fast transform mapping data to coefficients, and an inverse of such a map. The algorithms for both the dictionary construction and the transforms have complexity $C n \log n$ with the constant linear in $D$ and exponential in $d$. Our work therefore establishes adaptive GMRA as a fast dictionary learning algorithm with approximation guarantees. We include several numerical experiments on both synthetic and real data, confirming our theoretical results and demonstrating the effectiveness of adaptive GMRA.
We propose a novel denoising framework for task functional Magnetic Resonance Imaging (tfMRI) data to delineate the high-resolution spatial pattern of the brain functional connectivity via dictionary learning and sparse coding (DLSC). In order to address the limitations of the unsupervised DLSC-based fMRI studies, we utilize the prior knowledge of task paradigm in the learning step to train a data-driven dictionary and to model the sparse representation. We apply the proposed DLSC-based method to Human Connectome Project (HCP) motor tfMRI dataset. Studies on the functional connectivity of cerebrocerebellar circuits in somatomotor networks show that the DLSC-based denoising framework can significantly improve the prominent connectivity patterns, in comparison to the temporal non-local means (tNLM)-based denoising method as well as the case without denoising, which is consistent and neuroscientifically meaningful within motor area. The promising results show that the proposed method can provide an important foundation for the high-resolution functional connectivity analysis, and provide a better approach for fMRI preprocessing.
This letter proposes a multiple parametric dictionary learning algorithm for direction of arrival (DOA) estimation in presence of array gain-phase error and mutual coupling. It jointly solves both the DOA estimation and array imperfection problems to yield a robust DOA estimation in presence of array imperfection errors and off-grid. In the proposed method, a multiple parametric dictionary learning-based algorithm with an steepest-descent iteration is used for learning the parametric perturbation matrices and the steering matrix simultaneously. It also exploits the multiple snapshots information to enhance the performance of DOA estimation. Simulation results show the efficiency of the proposed algorithm when both off-grid problem and array imperfection exist.
Dictionary learning and component analysis are part of one of the most well-studied and active research fields, at the intersection of signal and image processing, computer vision, and statistical machine learning. In dictionary learning, the current methods of choice are arguably K-SVD and its variants, which learn a dictionary (i.e., a decomposition) for sparse coding via Singular Value Decomposition. In robust component analysis, leading methods derive from Principal Component Pursuit (PCP), which recovers a low-rank matrix from sparse corruptions of unknown magnitude and support. However, K-SVD is sensitive to the presence of noise and outliers in the training set. Additionally, PCP does not provide a dictionary that respects the structure of the data (e.g., images), and requires expensive SVD computations when solved by convex relaxation. In this paper, we introduce a new robust decomposition of images by combining ideas from sparse dictionary learning and PCP. We propose a novel Kronecker-decomposable component analysis which is robust to gross corruption, can be used for low-rank modeling, and leverages separability to solve significantly smaller problems. We design an efficient learning algorithm by drawing links with a restricted form of tensor factorization. The effectiveness of the proposed approach is demonstrated on real-world applications, namely background subtraction and image denoising, by performing a thorough comparison with the current state of the art.
Person re-identification is best known as the problem of associating a single person that is observed from one or more disjoint cameras. The existing literature has mainly addressed such an issue, neglecting the fact that people usually move in groups, like in crowded scenarios. We believe that the additional information carried by neighboring individuals provides a relevant visual context that can be exploited to obtain a more robust match of single persons within the group. Despite this, re-identifying groups of people compound the common single person re-identification problems by introducing changes in the relative position of persons within the group and severe self-occlusions. In this paper, we propose a solution for group re-identification that grounds on transferring knowledge from single person re-identification to group re-identification by exploiting sparse dictionary learning. First, a dictionary of sparse atoms is learned using patches extracted from single person images. Then, the learned dictionary is exploited to obtain a sparsity-driven residual group representation, which is finally matched to perform the re-identification. Extensive experiments on the i-LIDS groups and two newly collected datasets show that the proposed solution outperforms state-of-the-art approaches.
Symmetric positive definite (SPD) matrices are useful for capturing second-order statistics of visual data. To compare two SPD matrices, several measures are available, such as the affine-invariant Riemannian metric, Jeffreys divergence, Jensen-Bregman logdet divergence, etc.; however, their behaviors may be application dependent, raising the need of manual selection to achieve the best possible performance. Further and as a result of their overwhelming complexity for large-scale problems, computing pairwise similarities by clever embedding of SPD matrices is often preferred to direct use of the aforementioned measures. In this paper, we propose a discriminative metric learning framework, Information Divergence and Dictionary Learning (IDDL), that not only learns application specific measures on SPD matrices automatically, but also embeds them as vectors using a learned dictionary. To learn the similarity measures (which could potentially be distinct for every dictionary atom), we use the recently introduced alpha-beta-logdet divergence, which is known to unify the measures listed above. We propose a novel IDDL objective, that learns the parameters of the divergence and the dictionary atoms jointly in a discriminative setup and is solved efficiently using Riemannian optimization. We showcase extensive experiments on eight computer vision datasets, demonstrating state-of-the-art performances.
In recent years, the creation of block-structured dictionary has attracted a lot of interest. Learning such dictionaries involve two step process: block formation and dictionary update. Both these steps are important in producing an effective dictionary. The existing works mostly assume that the block structure is known a priori while learning the dictionary. For finding the unknown block structure given a dictionary commonly sparse agglomerative clustering (SAC) is used. It groups atoms based on their consistency in sparse coding with respect to the unstructured dictionary. This paper explores two innovations towards improving the reconstruction as well as the classification ability achieved with the block-structured dictionary. First, we propose a novel block structuring approach that makes use of the correlation among dictionary atoms. Unlike the SAC approach, which groups diverse atoms, in the proposed approach the blocks are formed by grouping the top most correlated atoms in the dictionary. The proposed block clustering approach is noted to yield significant reductions in redundancy as well as provides a direct control on the block size when compared with the existing SAC-based block structuring. Later, motivated by works using supervised \emph{a priori} known block structure, we also explore the incorporation of class information in the proposed block formation approach to further enhance the classification ability of the block dictionary. For assessment of the reconstruction ability with proposed innovations is done on synthetic data while the classification ability has been evaluated in large variability speaker verification task.