4,371 research outputs found
Recovering edges in ill-posed inverse problems: optimality of curvelet frames
We consider a model problem of recovering a function from noisy Radon data. The function to be recovered is assumed smooth apart from a discontinuity along a curve, that is, an edge. We use the continuum white-noise model, with noise level .
Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model mean squared errors (MSEs) that tend to zero with noise level only as as . A recent innovation--nonlinear shrinkage in the wavelet domain--visually improves edge sharpness and improves MSE convergence to . However, as we show here, this rate is not optimal.
In fact, essentially optimal performance is obtained by deploying the recently-introduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build "curvelet shrinkage" estimators based on thresholding of the noisy curvelet coefficients. In effect, the estimator detects edges at certain locations and orientations in the Radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain.
We prove that the curvelet shrinkage can be tuned so that the estimator will attain, within logarithmic factors, the MSE as noise level . This rate of convergence holds uniformly over a class of functions which are except for discontinuities along curves, and (except for log terms) is the minimax rate for that class. Our approach is an instance of a general strategy which should apply in other inverse problems; we sketch a deconvolution example
Spatial Smoothing for Diffusion Tensor Imaging with low Signal to Noise Ratios
Though low signal to noise ratio (SNR) experiments in DTI give key information about tracking and anisotropy, e.g. by measurements with very small voxel sizes, due to the complicated impact of thermal noise such experiments are up to now seldom analysed. In this paper Monte Carlo simulations are presented which investigate the random fields of noise for different DTI variables in low SNR situations. Based on this study a strategy for spatial smoothing, which demands essentially uniform noise, is derived. To construct a convenient filter the weights of the nonlinear Aurich chain are adapted to DTI. This edge preserving three dimensional filter is then validated in different variants via a quasi realistic model and is applied to very new data with isotropic voxels of the size 1x1x1 mm3 which correspond to a spatial mean SNR of approximately 3
Confidence driven TGV fusion
We introduce a novel model for spatially varying variational data fusion,
driven by point-wise confidence values. The proposed model allows for the joint
estimation of the data and the confidence values based on the spatial coherence
of the data. We discuss the main properties of the introduced model as well as
suitable algorithms for estimating the solution of the corresponding biconvex
minimization problem and their convergence. The performance of the proposed
model is evaluated considering the problem of depth image fusion by using both
synthetic and real data from publicly available datasets
DTI denoising for data with low signal to noise ratios
Low signal to noise ratio (SNR) experiments in diffusion tensor imaging (DTI) give key information about tracking and anisotropy, e. g., by measurements with small voxel sizes or with high b values. However, due to the complicated and dominating impact of thermal noise such data are still seldom analysed. In this paper Monte Carlo simulations are presented which investigate the distributions of noise for different DTI variables in low SNR situations. Based on this study a strategy for the application of spatial smoothing is derived. Optimal prerequisites for spatial filters are unbiased, bell shaped distributions with uniform variance, but, only few variables have a statistics close to that. To construct a convenient filter a chain of nonlinear Gaussian filters is adapted to peculiarities of DTI and a bias correction is introduced. This edge preserving three dimensional filter is then validated via a quasi realistic model. Further, it is shown that for small sample sizes the filter is as effective as a maximum likelihood estimator and produces reliable results down to a local SNR of approximately 1. The filter is finally applied to very recent data with isotropic voxels of the size 1Ć1Ć1mm^3 which corresponds to a spatially mean SNR of 2.5. This application demonstrates the statistical robustness of the filter method. Though the Rician noise model is only approximately realized in the data, the gain of information by spatial smoothing is considerable
The generalized cusp in ABJ(M) N = 6 Super Chern-Simons theories
We construct a generalized cusped Wilson loop operator in N = 6 super
Chern-Simons-matter theories which is locally invariant under half of the
supercharges. It depends on two parameters and interpolates smoothly between
the 1/2 BPS line or circle and a pair of antiparallel lines, representing a
natural generalization of the quark-antiquark potential in ABJ(M) theories. For
particular choices of the parameters we obtain 1/6 BPS configurations that,
mapped on S^2 by a conformal transformation, realize a three-dimensional
analogue of the wedge DGRT Wilson loop of N = 4. The cusp couples, in addition
to the gauge and scalar fields of the theory, also to the fermions in the
bifundamental representation of the U(N)xU(M) gauge group and its expectation
value is expressed as the holonomy of a suitable superconnection. We discuss
the definition of these observables in terms of traces and the role of the
boundary conditions of fermions along the loop. We perform a complete two-loop
analysis, obtaining an explicit result for the generalized cusp at the second
non-trivial order, from which we read off the interaction potential between
heavy 1/2 BPS particles in the ABJ(M) model. Our results open the possibility
to explore in the three-dimensional case the connection between localization
properties and integrability, recently advocated in D = 4.Comment: 53 pages, 10 figures, added references, this is the version appeared
on JHE
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