12,660 research outputs found
Learning Sparse Wavelet Representations
In this work we propose a method for learning wavelet filters directly from
data. We accomplish this by framing the discrete wavelet transform as a
modified convolutional neural network. We introduce an autoencoder wavelet
transform network that is trained using gradient descent. We show that the
model is capable of learning structured wavelet filters from synthetic and real
data. The learned wavelets are shown to be similar to traditional wavelets that
are derived using Fourier methods. Our method is simple to implement and easily
incorporated into neural network architectures. A major advantage to our model
is that we can learn from raw audio data.Comment: 7 pages, 5 figure
On learning with shift-invariant structures
We describe new results and algorithms for two different, but related,
problems which deal with circulant matrices: learning shift-invariant
components from training data and calculating the shift (or alignment) between
two given signals. In the first instance, we deal with the shift-invariant
dictionary learning problem while the latter bears the name of (compressive)
shift retrieval. We formulate these problems using circulant and convolutional
matrices (including unions of such matrices), define optimization problems that
describe our goals and propose efficient ways to solve them. Based on these
findings, we also show how to learn a wavelet-like dictionary from training
data. We connect our work with various previous results from the literature and
we show the effectiveness of our proposed algorithms using synthetic, ECG
signals and images
Cross-scale predictive dictionaries
Sparse representations using data dictionaries provide an efficient model
particularly for signals that do not enjoy alternate analytic sparsifying
transformations. However, solving inverse problems with sparsifying
dictionaries can be computationally expensive, especially when the dictionary
under consideration has a large number of atoms. In this paper, we incorporate
additional structure on to dictionary-based sparse representations for visual
signals to enable speedups when solving sparse approximation problems. The
specific structure that we endow onto sparse models is that of a multi-scale
modeling where the sparse representation at each scale is constrained by the
sparse representation at coarser scales. We show that this cross-scale
predictive model delivers significant speedups, often in the range of
10-60, with little loss in accuracy for linear inverse problems
associated with images, videos, and light fields.Comment: 12 page
An Adaptive Markov Random Field for Structured Compressive Sensing
Exploiting intrinsic structures in sparse signals underpins the recent
progress in compressive sensing (CS). The key for exploiting such structures is
to achieve two desirable properties: generality (\ie, the ability to fit a wide
range of signals with diverse structures) and adaptability (\ie, being adaptive
to a specific signal). Most existing approaches, however, often only achieve
one of these two properties. In this study, we propose a novel adaptive Markov
random field sparsity prior for CS, which not only is able to capture a broad
range of sparsity structures, but also can adapt to each sparse signal through
refining the parameters of the sparsity prior with respect to the compressed
measurements. To maximize the adaptability, we also propose a new sparse signal
estimation where the sparse signals, support, noise and signal parameter
estimation are unified into a variational optimization problem, which can be
effectively solved with an alternative minimization scheme. Extensive
experiments on three real-world datasets demonstrate the effectiveness of the
proposed method in recovery accuracy, noise tolerance, and runtime.Comment: 13 pages, submitted to IEEE Transactions on Image Processin
Signal Representations on Graphs: Tools and Applications
We present a framework for representing and modeling data on graphs. Based on
this framework, we study three typical classes of graph signals: smooth graph
signals, piecewise-constant graph signals, and piecewise-smooth graph signals.
For each class, we provide an explicit definition of the graph signals and
construct a corresponding graph dictionary with desirable properties. We then
study how such graph dictionary works in two standard tasks: approximation and
sampling followed with recovery, both from theoretical as well as algorithmic
perspectives. Finally, for each class, we present a case study of a real-world
problem by using the proposed methodology
Graph Wavelets via Sparse Cuts: Extended Version
Modeling information that resides on vertices of large graphs is a key
problem in several real-life applications, ranging from social networks to the
Internet-of-things. Signal Processing on Graphs and, in particular, graph
wavelets can exploit the intrinsic smoothness of these datasets in order to
represent them in a both compact and accurate manner. However, how to discover
wavelet bases that capture the geometry of the data with respect to the signal
as well as the graph structure remains an open question. In this paper, we
study the problem of computing graph wavelet bases via sparse cuts in order to
produce low-dimensional encodings of data-driven bases. This problem is
connected to known hard problems in graph theory (e.g. multiway cuts) and thus
requires an efficient heuristic. We formulate the basis discovery task as a
relaxation of a vector optimization problem, which leads to an elegant solution
as a regularized eigenvalue computation. Moreover, we propose several
strategies in order to scale our algorithm to large graphs. Experimental
results show that the proposed algorithm can effectively encode both the graph
structure and signal, producing compressed and accurate representations for
vertex values in a wide range of datasets (e.g. sensor and gene networks) and
significantly outperforming the best baseline
Understanding Deep Convolutional Networks
Deep convolutional networks provide state of the art classifications and
regressions results over many high-dimensional problems. We review their
architecture, which scatters data with a cascade of linear filter weights and
non-linearities. A mathematical framework is introduced to analyze their
properties. Computations of invariants involve multiscale contractions, the
linearization of hierarchical symmetries, and sparse separations. Applications
are discussed.Comment: 17 pages, 4 Figure
Multi-Focus Image Fusion Using Sparse Representation and Coupled Dictionary Learning
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.Comment: 25 pages, 15 figures, 2 tabl
Superresolution of Noisy Remotely Sensed Images Through Directional Representations
We develop an algorithm for single-image superresolution of remotely sensed
data, based on the discrete shearlet transform. The shearlet transform extracts
directional features of signals, and is known to provide near-optimally sparse
representations for a broad class of images. This often leads to superior
performance in edge detection and image representation when compared to
isotropic frames. We justify the use of shearlets mathematically, before
presenting a denoising single-image superresolution algorithm that combines the
shearlet transform with sparse mixing estimators (SME). Our algorithm is
compared with a variety of single-image superresolution methods, including
wavelet SME superresolution. Our numerical results demonstrate competitive
performance in terms of PSNR and SSIM.Comment: 5 pages (double column). IEEE copyright adde
Overcomplete Frame Thresholding for Acoustic Scene Analysis
In this work, we derive a generic overcomplete frame thresholding scheme
based on risk minimization. Overcomplete frames being favored for analysis
tasks such as classification, regression or anomaly detection, we provide a way
to leverage those optimal representations in real-world applications through
the use of thresholding. We validate the method on a large scale bird activity
detection task via the scattering network architecture performed by means of
continuous wavelets, known for being an adequate dictionary in audio
environments
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