2,289 research outputs found
A new steplength selection for scaled gradient methods with application to image deblurring
Gradient methods are frequently used in large scale image deblurring problems
since they avoid the onerous computation of the Hessian matrix of the objective
function. Second order information is typically sought by a clever choice of
the steplength parameter defining the descent direction, as in the case of the
well-known Barzilai and Borwein rules. In a recent paper, a strategy for the
steplength selection approximating the inverse of some eigenvalues of the
Hessian matrix has been proposed for gradient methods applied to unconstrained
minimization problems. In the quadratic case, this approach is based on a
Lanczos process applied every m iterations to the matrix of the most recent m
back gradients but the idea can be extended to a general objective function. In
this paper we extend this rule to the case of scaled gradient projection
methods applied to non-negatively constrained minimization problems, and we
test the effectiveness of the proposed strategy in image deblurring problems in
both the presence and the absence of an explicit edge-preserving regularization
term
A discrepancy principle for Poisson data: uniqueness of the solution for 2D and 3D data
This paper is concerned with the uniqueness of the solution of a nonlinear
equation, named discrepancy equation. For the restoration problem of data corrupted
by Poisson noise, we have to minimize an objective function that combines a
data-fidelity function, given by the generalized Kullback–Leibler divergence, and a
regularization penalty function. Bertero et al. recently proposed to use the solution
of the discrepancy equation as a convenient value for the regularization parameter.
Furthermore they devised suitable conditions to assure the uniqueness of this solution
for several regularization functions in 1D denoising and deblurring problems.
The aim of this paper is to generalize this uniqueness result to 2D and 3D problems
for several penalty functions, such as an edge preserving functional, a simple case of
the class of Markov Random Field (MRF) regularization functionals and the classical
Tikhonov regularization
Iterative image restoration with non negativity constraints
In many image restoration applications the nonnegativity of the computed solution is required. General regularizationmethods, such as iterative semiconvergent methods, seldom compute nonnegative solutions even when the data are nonnegative. Some methods can be modified in order to enforce the nonnegativity constraint. Other methods, which can be derived from the Kuhn-Tucker conditions of a constrained maximization problem, naturally embed nonnegativity. In this note we aim to compare the performances of different iterative regularization methods which produce nonnegative images from various point of view, i.e. the computational cost, the efficiency and consistency with the discrepancy principle as standard technique for choosing the best regularization parameter and the sensitivity to this choice. An extensive experimentation on both 1D and 2D images has shown that the most noteworthy methods are truncated CGLS from the point of view of the computational cost and EM for the reconstruction efficiency. Both methods appear to be consistent with the discrepancy principle and not too sensitive to the choice of the number of iterations suggested by this principle
Solving ill-posed inverse problems using iterative deep neural networks
We propose a partially learned approach for the solution of ill posed inverse
problems with not necessarily linear forward operators. The method builds on
ideas from classical regularization theory and recent advances in deep learning
to perform learning while making use of prior information about the inverse
problem encoded in the forward operator, noise model and a regularizing
functional. The method results in a gradient-like iterative scheme, where the
"gradient" component is learned using a convolutional network that includes the
gradients of the data discrepancy and regularizer as input in each iteration.
We present results of such a partially learned gradient scheme on a non-linear
tomographic inversion problem with simulated data from both the Sheep-Logan
phantom as well as a head CT. The outcome is compared against FBP and TV
reconstruction and the proposed method provides a 5.4 dB PSNR improvement over
the TV reconstruction while being significantly faster, giving reconstructions
of 512 x 512 volumes in about 0.4 seconds using a single GPU
Refraction-corrected ray-based inversion for three-dimensional ultrasound tomography of the breast
Ultrasound Tomography has seen a revival of interest in the past decade,
especially for breast imaging, due to improvements in both ultrasound and
computing hardware. In particular, three-dimensional ultrasound tomography, a
fully tomographic method in which the medium to be imaged is surrounded by
ultrasound transducers, has become feasible. In this paper, a comprehensive
derivation and study of a robust framework for large-scale bent-ray ultrasound
tomography in 3D for a hemispherical detector array is presented. Two
ray-tracing approaches are derived and compared. More significantly, the
problem of linking the rays between emitters and receivers, which is
challenging in 3D due to the high number of degrees of freedom for the
trajectory of rays, is analysed both as a minimisation and as a root-finding
problem. The ray-linking problem is parameterised for a convex detection
surface and three robust, accurate, and efficient ray-linking algorithms are
formulated and demonstrated. To stabilise these methods, novel
adaptive-smoothing approaches are proposed that control the conditioning of the
update matrices to ensure accurate linking. The nonlinear UST problem of
estimating the sound speed was recast as a series of linearised subproblems,
each solved using the above algorithms and within a steepest descent scheme.
The whole imaging algorithm was demonstrated to be robust and accurate on
realistic data simulated using a full-wave acoustic model and an anatomical
breast phantom, and incorporating the errors due to time-of-flight picking that
would be present with measured data. This method can used to provide a
low-artefact, quantitatively accurate, 3D sound speed maps. In addition to
being useful in their own right, such 3D sound speed maps can be used to
initialise full-wave inversion methods, or as an input to photoacoustic
tomography reconstructions
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