Graphs are extensively used to represent networked data. In many applications, especially when considering large datasets, it is a desirable feature to focus the analysis onto specific subgraphs of interest. Slepian theory and its extension to graphs allows to do this and has been applied recently to analyze various types of networks. One limitation of this framework, however, is that the number of subgraphs of interest is typically limited to one. We introduce an extended Slepian design that allows to consider an arbitrary number of subgraphs of interest. This extension offers the possibility to encode prior information about multiple subgraphs in a two-dimensional plane. As a proof of concept and potential application, we demonstrate that this framework allows to perform time-resolved and spatio-temporal analyses of dynamic graphs.
KEYWORDS: Electronic filtering, Filtering (signal processing), Signal processing, Brain, Linear filtering, Digital filtering, Matrices, Functional magnetic resonance imaging, Signal detection
Joint localization of graph signals in vertex and spectral domain is achieved in Slepian vectors calculated by either maximizing energy concentration (μ) or minimizing modified embedded distance (ξ) in the subgraph of interest. On the other hand, graph Laplacian is extensively used in graph signal processing as it defines graph Fourier transform (GFT) and operators such as filtering, wavelets, etc. In the context of modeling human brain as a graph, low pass (smooth over neighboring nodes) filtered graph signals represent a valuable source of information known as aligned signals. Here, we propose to define GFT and graph filtering using Slepian orthogonal basis. We explored power spectrum density estimates of random signals on Erdős-Rényi graphs and determined local discrepancies in signal behavior which cannot be accessed by the graph Laplacian, but are detected by the Slepian basis. This motivated the application of Slepian guided graph signal filtering in neuroimaging. We built a graph from diffusion-weighed brain imaging data and used blood-oxygenation-level-dependent (BOLD) time series as graph signals residing on its nodes. The dataset included recordings of 21 subjects performing a working memory task. In certain brain regions known to exhibit activity negatively correlated to performing the task, the only method capable of identifying this type of behavior in the bandlimited framework was ξ-Slepian guided filtering. The localization property of the proposed approach provides significant contribution to the strength of the graph spectral analysis, as it allows inclusion of a priori knowledge of the explored graph's mesoscale structure.
Spectral approaches of network analysis heavily rely upon the eigendecomposition of the graph Laplacian. For instance, in graph signal processing, the Laplacian eigendecomposition is used to define the graph Fourier trans- form and then transpose signal processing operations to graphs by implementing them in the spectral domain. Here, we build on recent work that generalized Slepian functions to the graph setting. In particular, graph Slepi- ans are band-limited graph signals with maximal energy concentration in a given subgraph. We show how this approach can be used to guide network analysis; i.e., we propose a visualization that reveals network organization of a subgraph, but while striking a balance with global network structure. These developments are illustrated for the structural connectome of the C. Elegans.
Physiological Brain connectivity and spontaneous interaction between regions of interest of the brain can be represented by a matrix (full or sparse) or equivalently by a complex network called connectome. This representation of brain connectivity is adopted when comparing different patterns of structural and functional connectivity to null models or between groups of individuals. Two levels of comparison could be considered when analyzing brain connectivity: the global level and the local level. In the global level, the whole brain information is summarized by one summary statistic, whereas in the local analysis, each region of interest of the brain is summarized by a specific statistic. We show that these levels are mutually informatively integrative in some extent. We present different methods of analysis at both levels, the most relevant global and local network measures. We discuss as well the assumptions to be satisfied for each method; the error rates controlled by each method, and the challenges to overcome, especially, in the local case. We also highlight the possible factors that could influence the statistical results and the questions that have to be addressed in such analyses.
Distinct texture classes are often sharing several visual concepts. Texture instances from different classes are sharing regions in the feature hyperspace, which results in ill-defined classification configurations. In this work, we detect rotation-covariant visual concepts using steerable Riesz wavelets and bags of visual words. In a first step, K-means clustering is used to detect visual concepts in the hyperspace of the energies of steerable Riesz wavelets. The coordinates of the clusters are used to construct templates from linear combinations of the Riesz components that are corresponding to visual concepts. The visualization of these templates allows verifying the relevance of the concepts modeled. Then, the local orientations of each template are optimized to maximize their response, which is carried out analytically and can still be expressed as a linear combination of the initial steerable Riesz templates. The texture classes are learned in the feature space composed of the concatenation of the maximum responses of each visual concept using support vector machines. An experimental evaluation using the Outex TC 00010 test suite allowed a classification accuracy of 97.5%, which demonstrates the feasibility of the proposed approach. An optimal number K = 20 of clusters is required to model the visual concepts, which was found to be fewer than the number of classes. This shows that higher-level classes are sharing low-level visual concepts. The importance of rotation-covariant visual concept modeling is highlighted by allowing an absolute gain of more than 30% in accuracy. The visual concepts are modeling the local organization of directions at various scales, which is in accordance with the bottom{up visual information processing sequence of the primal sketch in Marr's theory on vision.
Photoacoustic tomography (PAT) is a hybrid imaging method, which combines ultrasonic and optical imaging modalities, in order to overcome their respective weaknesses and to combine their strengths. It is based on the reconstruction of optical absorption properties of the tissue from the measurements of a photoacoustically generated pressure field. Current methods consider laser excitation, under thermal and stress confinement assumptions, which leads to the generation of a propagating pressure field. Conventional reconstruction tech niques then recover the initial pressure field based on the boundary measurements by iterative reconstruction algorithms in time- or Fourier-domain. Here, we propose an application of a new sensing principle that allows for efficient and non-iterative reconstruction algorithm for imaging point absorbers in PAT. We consider a closed volume surrounded by a measurement surface in an acoustically homogeneous medium and we aim at recovering the positions and the amount of heat absorbed by these absorbers. We propose a two-step algorithm based on proper choice of so-called sensing functions. Specifically, in the first step, we extract the projected positions on the complex plane and the weights by a sensing function that is well-localized on the same plane. In the second step, we recover the remaining z-location by choosing a proper set of plane waves. We show that the proposed families of sensing functions are sufficient to recover the parameters of the unknown sources without any discretization of the domain. We extend the method for sources that have joint-sparsity; i.e., the absorbers have the same positions for different frequencies. We evaluate the performance of the proposed algorithm using simulated and noisy sensor data and we demonstrate the improvement obtained by exploiting joint sparsity.
We discuss a novel sampling theorem on the sphere developed by McEwen & Wiaux recently through an association
between the sphere and the torus. To represent a band-limited signal exactly, this new sampling theorem
requires less than half the number of samples of other equiangular sampling theorems on the sphere, such as
the canonical Driscoll & Healy sampling theorem. A reduction in the number of samples required to represent
a band-limited signal on the sphere has important implications for compressive sensing, both in terms of the
dimensionality and sparsity of signals. We illustrate the impact of this property with an inpainting problem on
the sphere, where we show superior reconstruction performance when adopting the new sampling theorem.
KEYWORDS: Functional magnetic resonance imaging, Hemodynamics, Associative arrays, Data modeling, Brain mapping, Motion models, Scanners, Signal detection, Visualization, Data acquisition
Functional magnetic resonance imaging (fMRI) is a non-invasive imaging technique that maps the brain's response
to neuronal activity based on the blood oxygenation level dependent (BOLD) effect. This work proposes
a novel method for fMRI data analysis that enables the decomposition of the fMRI signal in its sources based
on morphological descriptors. Beyond traditional fMRI hypothesis-based or blind data-driven exploratory approaches,
this method allows the detection of BOLD responses without prior timing information. It is based
on the deconvolution of the neuronal-related haemodynamic component of the fMRI signal with paradigm free
mapping and also furnishes estimates of the movement-related effects, instrumental drifts and physiological fluctuations.
Our algorithm is based on an overcomplete representation of the fMRI voxel time series with an additive
linear model that is recovered by means of a L1-norm regularized least-squares estimators and an adapted block
coordinate relaxation procedure. The performance of the technique is evaluated with simulated data and real
experimental data acquired at 3T.
Analytic sensing has recently been proposed for source localization from boundary measurements using a generalization
of the finite-rate-of-innovation framework. The method is tailored to the quasi-static electromagnetic
approximation, which is commonly used in electroencephalography. In this work, we extend analytic sensing
for physical systems that are governed by the wave equation; i.e., the sources emit signals that travel as waves
through the volume and that are measured at the boundary over time. This source localization problem is highly
ill-posed (i.e., the unicity of the source distribution is not guaranteed) and additional assumptions about the
sources are needed. We assume that the sources can be described with finite number of parameters, particularly,
we consider point sources that are characterized by their position and strength. This assumption makes
the solution unique and turns the problem into parametric estimation. Following the framework of analytic
sensing, we propose a two-step method. In the first step, we extend the reciprocity gap functional concept to
wave-equation based test functions; i.e., well-chosen test functions can relate the boundary measurements to
generalized measure that contain volumetric information about the sources within the domain. In the second
step-again due to the choice of the test functions - we can apply the finite-rate-of-innovation principle; i.e., the
generalized samples can be annihilated by a known filter, thus turning the non-linear source localization problem
into an equivalent root-finding one. We demonstrate the feasibility of our technique for a 3-D spherical geometry.
The performance of the reconstruction algorithm is evaluated in the presence of noise and compared with the
theoretical limit given by Cramer-Rao lower bounds.
One very influential method for texture synthesis is based on the steerable pyramid by alternately imposing
marginal statistics on the image and the pyramid's subbands. In this work, we investigate two extensions to this
framework. First, we exploit the steerability of the transform to obtain histograms of the subbands independent
of the local orientation; i.e., we select the direction of maximal response as the reference orientation. Second,
we explore the option of multidimensional histogram matching. The distribution of the responses to various
orientations is expected to capture better the local geometric structure. Experimental results show how the
proposed approach improves the performance of the original pyramid-based synthesis method.
Based on the class of complex gradient-Laplace operators, we show the design of a non-separable two-dimensional
wavelet basis from a single and analytically defined generator wavelet function. The wavelet decomposition is
implemented by an efficient FFT-based filterbank. By allowing for slight redundancy, we obtain the Marr
wavelet pyramid decomposition that features improved translation-invariance and steerability. The link with
Marr's theory of early vision is due to the replication of the essential processing steps (Gaussian smoothing,
Laplacian, orientation detection). Finally, we show how to find a compact multiscale primal sketch of the image,
and how to reconstruct an image from it.
During open brain surgery we acquire perfusion images non-invasively using laser Doppler imaging. The regions of
brain activity show a distinct signal in response to stimulation providing intraoperative functional brain maps of
remarkably strong contrast.
We present multiresolution spaces of complex rotation-covariant functions, deployed on the 2-D hexagonal lattice.
The designed wavelets, which are complex-valued, provide an important phase information for image analysis,
which is missing in the discrete wavelet transform with real wavelets. Moreover, the hexagonal lattice allows to
build wavelets having a more isotropic magnitude than on the Cartesian lattice. The associated filters, defined
in the Fourier domain, yield an efficient FFT-based implementation.
FMRI time course processing is traditionally performed using linear regression followed by statistical hypothesis
testing. While this analysis method is robust against noise, it relies strongly on the signal model. In this paper, we
propose a non-parametric framework that is based on two main ideas. First, we introduce a problem-specific type
of wavelet basis, for which we coin the term "activelets". The design of these wavelets is inspired by the form of
the canonical hemodynamic response function. Second, we take advantage of sparsity-pursuing search techniques
to find the most compact representation for the BOLD signal under investigation. The non-linear optimization
allows to overcome the sensitivity-specificity trade-off that limits most standard techniques. Remarkably, the
activelet framework does not require the knowledge of stimulus onset times; this property can be exploited to
answer to new questions in neuroscience.
Probably the most important property of wavelets for signal processing is their multiscale derivative-like behavior
when applied to functions. In order to extend the class of problems that can profit of wavelet-based techniques, we
propose to build new families of wavelets that behave like an arbitrary scale-covariant operator. Our extension is
general and includes many known wavelet bases. At the same time, the method takes advantage a fast filterbank
decomposition-reconstruction algorithm. We give necessary conditions for the scale-covariant operator to admit
our wavelet construction, and we provide examples of new wavelets that can be obtained with our method.
KEYWORDS: Sensors, Data modeling, Inverse problems, Electromagnetism, Electroencephalography, Biomedical optics, 3D modeling, Mathematical modeling, Statistical analysis, Signal to noise ratio
Inverse problems play an important role in engineering. A problem that often occurs in electromagnetics (e.g.
EEG) is the estimation of the locations and strengths of point sources from boundary data.
We propose a new technique, for which we coin the term "analytic sensing". First, generalized measures are
obtained by applying Green's theorem to selected functions that are analytic in a given domain and at the same
time localized to "sense" the sources. Second, we use the finite-rate-of-innovation framework to determine the
locations of the sources. Hence, we construct a polynomial whose roots are the sources' locations. Finally, the
strengths of the sources are found by solving a linear system of equations. Preliminary results, using synthetic
data, demonstrate the feasibility of the proposed method.
Magnetic resonance spectroscopy imaging (MRSI) is a promising and developing tool in medical imaging. Because of various difficulties imposed by the imperfections of the scanner and the reconstruction algorithms, its applicability in clinical practice is rather limited. In this paper, we suggest an extension of the constrained
reconstruction technique (SLIM). Our algorithm, named B-SLIM, takes into account the the measured field inhomogeneity map, which contains both the scanner's main field inhomogeneity and the object-dependent magnetic susceptibility effects. The method is implemented and tested both with synthetic and physical two-compartment phantom data. The results demonstrate significant performance improvement over the SLIM technique. At the same time, the algorithm has the same computational complexity as SLIM.
We propose a method for sub-resolution axial localization of particles in fluorescence microscopy, based on
maximum-likelihood estimation. Given acquisitions of a defocused fluorescent particle, we can estimate its axial position with nanometer range precision.
The approximate behavior of wavelets as differential operators is often considered as one of their most fundamental properties. In this paper, we investigate how we can further improve on the wavelet's behavior as differentiator. In particular, we propose semi-orthogonal differential wavelets. The semi-orthogonality condition ensures that wavelet spaces are mutually orthogonal. The operator, hidden within the wavelet, can be chosen as a generalized differential operator ∂γτ, for a γ-th order derivative with shift τ. Both order of derivation and shift can be chosen fractional. Our design leads us naturally to select the fractional B-splines as scaling functions. By putting the differential wavelet in the perspective of a derivative of a smoothing function, we find that signal singularities are compactly characterized by at most two local extrema of the wavelet coefficients in each subband. This property could be beneficial for signal analysis using wavelet bases. We show that this wavelet transform can be efficiently implemented using FFTs.
Statistical Parametric Mapping (SPM) is a widely deployed tool for detecting and analyzing brain activity from fMRI data. One of SPM's main features is smoothing the data by a Gaussian filter to increase the SNR. The subsequent statistical inference is based on the continuous Gaussian random field theory. Since the remaining spatial resolution has deteriorated due to smoothing, SPM introduces the concept of "resels" (resolution elements) or spatial information-containing cells. The number of resels turns out to be inversely proportional to the size of the Gaussian smoother. Detection the activation signal in fMRI data can also be done by a wavelet approach: after computing the spatial wavelet transform, a straightforward coefficient-wise statistical test is applied to detect activated wavelet coefficients. In this paper, we establish the link between SPM and the wavelet approach based on two observations. First, the (iterated) lowpass analysis filter of the discrete wavelet transform can be chosen to closely resemble SPM's Gaussian filter. Second, the subsampling scheme provides us with a natural way to define the number of resels; i.e., the number of coefficients in the lowpass subband of the wavelet decomposition. Using this connection, we can obtain the degree of the splines of the wavelet transform that makes it equivalent to SPM's method. We show results for two particularly attractive biorthogonal wavelet transforms for this task; i.e., 3D fractional-spline wavelets and 2D+Z fractional quincunx wavelets. The activation patterns are comparable to SPM's.
The number of terminals that have access to multimedia content by means of a network is rapidly increasing. More and more, the characteristics of different terminals are increasing in variety. In addition, their users can have different preferences. Therefore, the adaptation of multimedia content to a specific terminal and/or its user has become an important research issue. Such an adaptation is mainly based on two aspects: the description of the multimedia content and the description of the user environment. Both can be considered as metadata, and can be formatted in an XML language (e.g., MPEG-7 and CC/PP). However, it is not yet clear how we can realize a generic mapping mechanism between two such vocabularies. We feel that such a mechanism is necessary to accomplish a mature content adaptation framework. This paper describes how such a mechanism can be achieved. We attach requirements and preferences of the user environment to specific aspects of the description of multimedia content. Based on this information, we try to maximize the value of the adapted content, while making it appropriate for the terminal. We also take into account the extensibility of the existing vocabularies we focus on, because this means our mechanism will also be extensible.
A comprehensive approach to the access of archival collections necessitates the interplay of various types of metadata standards. Each of these standards fulfills its own part within the context of a 'metadata infrastructure'. Besides this, it should be noted that present-day digital libraries are often limited to the management of mainly textual and image-based material. Archival Information Systems dealing with various media types are still very rare. There is a need for a methodology to deal with time-dependant media within an archival context. The aim of our research is to investigate and implement a number of tools supporting the content management multimedia data within digital collections. A flexible and extendible framework is proposed, based on the emerging Metadata Encoding and Transmission Standard (METS). Firstly, we will focus on the description of archival collections according to the archival mandates of provenance for the benefit of an art-historical research in an archive-theoretically correct manner. Secondly, we will examine the description tools that represent the semantics and structure of multimedia data. In this respect, an extension of the present archival metadata framework has been proposed to time-based media content delivered via standards such as the MPEG-7 multimedia content description standard.
The current explosive expansion of mobile communication systems will lead to an increased demand for multimedia applications. However, due to the large variety of mobile terminals (such as mobile phones, laptops .) and, because of this, a wide collection of different terminal possibilities and terminal characteristics, it is difficult to create a mobile multimedia application which can be used on mobile devices of different types. In this paper, we propose a mobile multimedia application that adapts its content to the possibilities of the mobile terminal and to the end-user preferences. Also the application takes changing device characteristics into account. To make this possible, a software framework is set up to enable negotiation between the mobile terminal and the content server. During the initial negotiation, the concept of the Universal Multimedia Access framework is used. Subsequent negotiations take place after changing terminal characteristics or end-user preferences, and this by means of time-dependent metadata. This newly created flexible and extendable framework makes it possible that multimedia applications interact with the content provider in order to deliver an optimal multimedia presentation for any arbitrary mobile terminal at any given time.
A new fuzzy filter is presented for the noise reduction of images corrupted with additive Gaussian noise. The filter consists of two stages. The first stage computes a fuzzy gradient for eight different directions around the currently processed pixel. The second stage uses the fuzzy gradient to perform fuzzy smoothing by taking different contributions of neighboring pixel values. The two stages are both based on fuzzy rules which make use of membership functions. The filter can be applied iteratively to effectively reduce heavy noise. The shape of the membership functions is adapted according to the remaining noise level after each iteration, making use of the distribution of the homogeneity in the image. The fuzzy operators are implemented by the classical min/max. Experimental results are obtained to show the feasibility of the proposed approach. These results are also compared to other filters by numerical measures and visual inspection.
Moire formation is often a major problem in the printing applications. These artifacts introduce new low frequency components which are very disturbing. Some printing techniques, e.g. gravure printing, are very sensitive to moire. The halftoning scheme used for gravure printing can basically be seen as a 2D non-isotropic subsampling process. The more problem is much more important in gravure printing than in conventional digital halftoning since the degree of freedom in constructing halftone dots is much more limited due to the physical constraints of the engraving mechanism.
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