Statistical methods


Sparse Representation of Signals:

A signal is said to be sparse if it can be represented in a basis or frame (Fourier, Wavelets, Curvelets, etc.) in which the curve obtained by plotting the obtained coefficients, sorted by their decreasing absolute values, exhibits a polynomial decay. The basis or frame is called the dictionary. Note that most natural signals and images are compressible in an appropriate dictionary. Faster is the decay, better it is, since a very good approximation of the signal can be obtained from a few coefficients. For instance, for a signal composed of a sine, the Fourier dictionary is optimal from a sparse point of view since all information is contained in a single coefficient. Wavelets have been extremely successful to represent images, most natural images present a sparse behavior in the wavelet domain, and this explains why wavelets have been chosen in the JPG2000 image compression norm. Other representations, such as curvelets, are more adequate when the data contains filaments. We have been working on several ill posed inverse problems where we have shown that sparsity is a very efficient way to regularize the problem in order to get a unique and stable solution (Starck et al, Cambridge University Press, book, 2010):

  • Blind Source Separation (BSS): Exceptional results were obtained (Bobin, Starck et al, IEEE Trans. on Image Processing, 2007), (Bobin, Starck, et al, Journal of Mathematical Imaging and Vision, 2009) when sparsity is used to recover sources from a set of multichannel observations, each channel containing a mixture of the different sources (classic BSS problem). This framework was also extended so as to deal with non-negative mixtures of non-negative sources (non-negative matrix factorization, NMF, Rapin, Bobin, Larue, Starck, IEEE Transactions on Signal Processing, 2013 and Rapin, Bobin, Larue, Starck, SIAM Journal on Imaging Sciences, 2014). An overview of our activities in BSS can be found at this location, which includes code samples.
  • Inpainting: we have shown that missing data could be interpolated in very efficient way using sparsity (Fadili, Starck, Murtagh, Computer Journal, 2009).
  • Deconvolution: We have studied the recent proximal theory in optimization theory, and shown that it provides very elegant solutions for image restoration (DupĂ©, Starck, et al, IEEE Trans. on Image Processing, 2009).
  • Structured Sparsity: Using a sparse representation, such as wavelet or curvelet decomposition, there are some correlations between neighboring pixels that can be captured and used to improve denoising results. (Chesneau, Fadili, and Starck, Applied and Computational Harmonic Analysis, 2010).
  • 3D Sparse Representations: We have extended to the third dimension recent sparse 2D decompositions, such as ridgelet or curvelet (Woiselle, Starck and Fadili, Applied and Computational Harmonic Analysis, 2010), (Woiselle, Starck, Fadili, J. of Mathematical Imaging and Vision, 2011).
  • Compressed Sensing (CS): CS is a theory which links the data acquisition principle to the sparsity concept. We have investigated how this kind of new idea could be useful for the transfer of astronomical data from satellite, such as Herschel (Bobin, Starck, and R. Ottensamer, IEEE Journal of Selected Topics in Signal Processing, 2008), and we have developed algorithms to recover the solution from compressed sensing data (Donoho, Tsaig, Drori, Starck, IEEE Transactions on Information Theory, 2012). We have shown using a Herschel data set, especially acquired to test the CS approach, that CS could indeed be a very practical solution for astronomical data transfer from a satellite to earth (Barbey, Sauvage, Starck, Ottensamer, A&A, 2011).
  • Compressive Video Sensing (CVS): We tackled the problem of designing efficient codecs for lightweight remote imaging systems by embedding compressed sensing in already existing video compression standards. First, we modified an MPEGx-based imaging system and we showed that our proposed CVS method achieves a comparable, or an even superior, performance when compared with MPEGx, but at significantly reduced bit rates, especially for noisy videos. Then, in order to satisfy the limited power, memory, and bandwidth resources of a lightweight remote imaging system, we combined the simplicity of an MJPEG-based encoder, with the efficiency of an MPEGx-based decoder, in the framework of CS. We showed that the proposed CVS system is able to achieve a high-quality reconstruction at even lower bit rates, with this reduction in the necessary bit rate to increase by introducing an efficient compressed measurements allocation scheme (Tzagkarakis, Woiselle, Tsakalides, Starck, VISAPP, 2012). Moreover, we developed algorithms for the classification of video sequences by exploiting directly the highly reduced set of compressed measurements. The proposed techniques improved the classification accuracy of previous commonly used classifiers, without requiring access to the original full-resolution data (Tzagkarakis et al, PCS, 2012), (Tzagkarakis et al, EUSIPCO, 2012).
  • Range Imaging: We have introduced a novel approach for Time-of-Flight (ToF)-based range imaging, which utilizes the recently introduced theory of compressed sensing to dramatically reduce the number of necessary frames required for the reconstruction of a depth map. Our technique employs a random gating function along with state-of-the-art minimization techniques in order to estimate the location of a returning laser pulse and subsequently to infer the distance. Our experimental results have shown that sampling rates at the order of 20% of the frames that traditional ToF cameras require, can achieve almost perfect reconstruction in low-resolution regimes, while the proposed method is also robust to realistic noise models (Tsagkatakis, Woiselle, Tzagkarakis, Bousquet, Starck, Tsakalides, SPIE Security+Defence, 2012).
  • Wireless Sensor Networks (WSN): We have shown that accurate joint reconstruction of a sparse signal ensemble can be achieved in a decentralized fashion, by exchanging a minimum amount of information among the sensors of a WSN. The reconstruction is performed by developing a novel distributed Bayesian Matching Pursuit algorithm, which was shown to be superior in terms of reconstruction accuracy, when compared with previous centralized approaches, while employing a small number of random incoherent projections, thus satisfying potential resource constraints (Tzagkarakis, Starck, Tsakalides, EUSIPCO, 2011). In addition, motivated by the recent wide use of wireless networks in the application of estimating the position of a mobile user, we developed efficient localization algorithms by working directly with a compressed set of signal-strength values received from a set of access points. Then, these compressed measurements are employed in the framework of compressed sensing to recover a sparse position-indicator vector. Our proposed approach, which is also shown to be very robust in noisy environments, results in a significant improvement of the localization accuracy when compared with previous state-of-the-art methods, while using a highly reduced set of data (compressed signal-strength values), thus increasing the system’s lifetime (Milioris, Tzagkarakis, et al, J. of Ad Hoc Networks, 2012), (Milioris, Tzagkarakis, et al, EUSIPCO, 2011).