# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

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## Inverse Problems & Imaging

May 2008 , Volume 2 , Issue 2

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2008, 2(2): 167-185 doi: 10.3934/ipi.2008.2.167 +[Abstract](1860) +[PDF](590.0KB)
Abstract:
Approximating non-Gaussian noise processes with Gaussian models is standard in data analysis. This is due in large part to the fact that Gaussian models yield parameter estimation problems of least squares form, which have been extensively studied both from the theoretical and computational points of view. In image processing applications, for example, data is often collected by a CCD camera, in which case the noise is a Guassian/Poisson mixture with the Poisson noise dominating for a sufficiently strong signal. Even so, the standard approach in such cases is to use a Gaussian approximation that leads to a negative-log likelihood function of weighted least squares type.
In the Bayesian point-of-view taken in this paper, a negative-log prior (or regularization) function is added to the negative-log likelihood function, and the resulting function is minimized. We focus on the case where the negative-log prior is the well-known total variation function and give a statistical interpretation. Regardless of whether the least squares or Poisson negative-log likelihood is used, the total variation term yields a minimization problem that is computationally challenging. The primary result of this work is the efficient computational method that is presented for the solution of such problems, together with its convergence analysis. With the computational method in hand, we then perform experiments that indicate that the Poisson negative-log likelihood yields a more computationally efficient method than does the use of the least squares function. We also present results that indicate that this may even be the case when the data noise is i.i.d. Gaussian, suggesting that regardless of noise statistics, using the Poisson negative-log likelihood can yield a more computationally tractable problem when total variation regularization is used.
2008, 2(2): 187-204 doi: 10.3934/ipi.2008.2.187 +[Abstract](2690) +[PDF](922.2KB)
Abstract:
The restoration of blurred images corrupted with impulse noise is a difficult problem which has been considered in a series of recent papers. These papers tackle the problem by using variational methods involving an L1-shaped data-fidelity term. Because of this term, the relevant methods exhibit systematic errors at the corrupted pixel locations and require a cumbersome optimization stage. In this work we propose and justify a much simpler alternative approach which overcomes the above-mentioned systematic errors and leads to much better results. Following a theoretical derivation based on a simple model, we decouple the problem into two phases. First, we identify the outlier candidates---the pixels that are likely to be corrupted by the impulse noise, and we remove them from our data set. In a second phase, the image is deblurred and denoised simultaneously using essentially the outlier-free data. The resultant optimization stage is much simpler in comparison with the current full variational methods and the outlier contamination is more accurately corrected. The experiments show that we obtain a 2 to 6 dB improvement in PSNR. We emphasize that our method can be adapted to deblur images corrupted with mixed impulse plus Gaussian noise, and hence it can address a much wider class of practical problems.
2008, 2(2): 205-224 doi: 10.3934/ipi.2008.2.205 +[Abstract](1536) +[PDF](549.4KB)
Abstract:
We present a new variational framework for simultaneous smoothing and estimation of apparent diffusion coefficient (ADC) profiles from High Angular Resolution Diffusion-weighted MRI. The model approximates the ADC profiles at each voxel by a 4th order spherical harmonic series (SHS). The coefficients in SHS are obtained by solving a constrained minimization problem. The smoothing with feature preserved is achieved by minimizing a variable exponent, linear growth functional, and the data constraint is determined by the original Stejskal-Tanner equation. The antipodal symmetry and positiveness of the ADC are accommodated in the model. We use these coefficients and variance of the ADC profiles from its mean to classify the diffusion in each voxel as isotropic, anisotropic with single fiber orientation, or two fiber orientations. The proposed model has been applied to both simulated data and HARD MRI human brain data . The experiments demonstrated the effectiveness of our method in estimation and smoothing of ADC profiles and in enhancement of diffusion anisotropy. Further characterization of non-Gaussian diffusion based on the proposed model showed a consistency between our results and known neuroanatomy.
2008, 2(2): 225-250 doi: 10.3934/ipi.2008.2.225 +[Abstract](1455) +[PDF](849.5KB)
Abstract:
Many imaging methods involve probing a material with a wave and observing the back-scattered wave. The back-scattered wave measurements are used to compute an image of the internal structure of the material. Many of the conventional methods make the assumption that the wave has scattered just once from the region to be imaged before returning to the sensor to be recorded. The purpose of this paper is to show how this restriction can be partially removed and also how its removal leads to an enhanced image, free of the artifacts often associated with the conventionally reconstructed image.
2008, 2(2): 251-269 doi: 10.3934/ipi.2008.2.251 +[Abstract](1710) +[PDF](427.5KB)
Abstract:
In this work we study localized electric potentials that have an arbitrarily high energy on some given subset of a domain and low energy on another. We show that such potentials exist for general $L^\infty_+$-conductivities in almost arbitrarily shaped subregions of a domain, as long as these regions are connected to the boundary and a unique continuation principle is satisfied. From this we deduce a simple, but new, theoretical identifiability result for the famous Calderón problem with partial data. We also show how to construct such potentials numerically and use a connection with the factorization method to derive a new non-iterative algorithm for the detection of inclusions in electrical impedance tomography.
2008, 2(2): 271-290 doi: 10.3934/ipi.2008.2.271 +[Abstract](1811) +[PDF](302.5KB)
Abstract:
We consider the inverse problem to identify coefficient functions in boundary value problems from noisy measurements of the solutions. Our estimators are defined as minimizers of a Tikhonov functional, which is the sum of a nonlinear data misfit term and a quadratic penalty term involving a Hilbert scale norm. In this abstract framework we derive estimates of the expected squared error under certain assumptions on the forward operator. These assumptions are shown to be satisfied for two classes of inverse elliptic boundary value problems. The theoretical results are confirmed by Monte Carlo simulations.
2008, 2(2): 291-299 doi: 10.3934/ipi.2008.2.291 +[Abstract](1976) +[PDF](145.3KB)
Abstract:
In this paper we derive convergence and convergence rates results of the quasioptimality criterion for (iterated) Tikhonov regularization. We prove convergence and suboptimal rates under a qualitative condition on the decay of the noise with respect to the spectral family of $T$$T$*. Moreover, optimal rates are obtained if the exact solution satisfies a decay condition with respect to the spectral family of $T$*$T$.
2008, 2(2): 301-315 doi: 10.3934/ipi.2008.2.301 +[Abstract](1512) +[PDF](235.5KB)
Abstract:
We consider an inverse scattering problem arising in target identification. We prove a local stability result of logarithmic type for the determination of a sound soft obstacle from the far field measurements associated to one single incident wave.

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