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148
Monotonic Algorithms for Transmission Tomography
- IEEE Tr. Med. Im
, 1999
"... Abstract — We present a framework for designing fast and monotonic algorithms for transmission tomography penalized-likelihood image reconstruction. The new algorithms are based on paraboloidal surrogate functions for the log-likelihood. Due to the form of the log-likelihood function, it is possible ..."
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Cited by 91 (40 self)
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Abstract — We present a framework for designing fast and monotonic algorithms for transmission tomography penalized-likelihood image reconstruction. The new algorithms are based on paraboloidal surrogate functions for the log-likelihood. Due to the form of the log-likelihood function, it is possible to find low curvature surrogate functions that guarantee monotonicity. Unlike previous methods, the proposed surrogate functions lead to monotonic algorithms even for the nonconvex log-likelihood that arises due to background events such as scatter and random coincidences. The gradient and the curvature of the likelihood terms are evaluated only once per iteration. Since the problem is simplified at each iteration, the CPU time is less than that of current algorithms which directly minimize the objective, yet the convergence rate is comparable. The simplicity, monotonicity and speed of the new algorithms are quite attractive. The convergence rates of the algorithms are demonstrated using real and simulated PET transmission scans.
Conjugate-Gradient Preconditioning Methods for Shift-Variant PET Image Reconstruction
- IEEE Tr. Im. Proc
, 2002
"... Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian mat ..."
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Cited by 79 (33 self)
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Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e. for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shiftvariant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shiftvariant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.
Grouped-coordinate ascent algorithms for penalized-likelihood transmission image reconstruction,”
- IEEE TT. Med. Im.,
, 1997
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Regularization for uniform spatial resolution properties in penalized-likelihood image reconstruction
- IEEE Tr. Med. Im
, 2000
"... Traditional space-invariant regularization methods in tomographic image reconstruction using penalized-likelihood estimators produce images with nonuniform spatial resolution properties. The local point spread functions that quantify the smoothing properties of such estimators are space-variant, as ..."
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Cited by 59 (26 self)
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Traditional space-invariant regularization methods in tomographic image reconstruction using penalized-likelihood estimators produce images with nonuniform spatial resolution properties. The local point spread functions that quantify the smoothing properties of such estimators are space-variant, asymmetric, and object-dependent even for space-invariant imaging systems. We propose a new quadratic regularization scheme for tomographic imaging systems that yields increased spatial uniformity and is motivated by the least-squares tting of a parameterized local impulse response to a desired global response. We have developed computationally e cient methods for PET systems with shift-invariant geometric responses. We demonstrate the increased spatial uniformity of this new method versus conventional quadratic regularization schemes in simulated PET thorax scans.
Statistical image reconstruction for polyenergetic X-ray computed tomography
- IEEE Transactions on Medical Imaging
, 2002
"... Abstract—This paper describes a statistical image reconstruction method for X-ray computed tomography (CT) that is based on a physical model that accounts for the polyenergetic X-ray source spectrum and the measurement nonlinearities caused by energydependent attenuation. We assume that the object c ..."
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Cited by 53 (11 self)
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Abstract—This paper describes a statistical image reconstruction method for X-ray computed tomography (CT) that is based on a physical model that accounts for the polyenergetic X-ray source spectrum and the measurement nonlinearities caused by energydependent attenuation. We assume that the object consists of a given number of nonoverlapping materials, such as soft tissue and bone. The attenuation coefficient of each voxel is the product of its unknown density and a known energy-dependent mass attenuation coefficient. We formulate a penalized-likelihood function for this polyenergetic model and develop an ordered-subsets iterative algorithm for estimating the unknown densities in each voxel. The algorithm monotonically decreases the cost function at each iteration when one subset is used. Applying this method to simulated X-ray CT measurements of objects containing both bone and soft tissue yields images with significantly reduced beam hardening artifacts. Index Terms—Beam hardening, penalized likelihood, statistical reconstruction, X-ray CT. I.
A Theoretical Study of the Contrast Recovery and Variance of MAP Reconstructions From PET Data
- IEEE Trans. Med. Imag
, 1999
"... We examine the spatial resolution and variance properties of PET images reconstructed using maximum a posteriori (MAP) or penalized-likelihood methods. Resolution is characterized by the contrast recovery coefficient (CRC) of the local impulse response. Simplified approximate expressions are derived ..."
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Cited by 46 (9 self)
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We examine the spatial resolution and variance properties of PET images reconstructed using maximum a posteriori (MAP) or penalized-likelihood methods. Resolution is characterized by the contrast recovery coefficient (CRC) of the local impulse response. Simplified approximate expressions are derived for the local impulse response CRCs and variances for each voxel. Using these results we propose a practical scheme for selecting spatially variant smoothing parameters to optimize lesion detectability through maximization of the local CRC-to-noise ratio in the reconstructed image. I. INTRODUCTION PET image reconstruction algorithms based on maximum likelihood (ML) or maximum a posteriori (MAP) principles can produce improved spatial resolution and noise properties in comparison to conventional filtered backprojection (FBP) methods. It is often important to be able to quantify this improvement in terms of the resolution (or bias) and variance of the resulting images. These measures can be...
Edge-Preserving Tomographic Reconstruction with Nonlocal Regularization
- IEEE Trans. on Medical Imaging
"... We propose a new objective function for the image reconstruction problem, where the image is comprised of piecewise smooth regions separated by sharp boundaries. We use alternating minimization to minimize our objective function. We use the level set technique to minimize with regard to the boudary. ..."
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Cited by 42 (6 self)
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We propose a new objective function for the image reconstruction problem, where the image is comprised of piecewise smooth regions separated by sharp boundaries. We use alternating minimization to minimize our objective function. We use the level set technique to minimize with regard to the boudary. The advantage of this new approach is shown through the biashariance analysis of a hot spot. 1
Statistical approaches in quantitative positron emission tomography
- Statistics and Computing
"... Positron emission tomography is a medical imaging modality for producing 3D images of the spatial distribution of biochemical tracers within the human body. The images are reconstructed from data formed through detection of radiation resulting from the emission of positrons from radioisotopes tagged ..."
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Cited by 40 (5 self)
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Positron emission tomography is a medical imaging modality for producing 3D images of the spatial distribution of biochemical tracers within the human body. The images are reconstructed from data formed through detection of radiation resulting from the emission of positrons from radioisotopes tagged onto the tracer of interest. These measurements are approximate line integrals from which the image can be reconstructed using analytical inversion formulae. However these direct methods do not allow accurate modeling either of the detector system or of the inherent statistical fluctuations in the data. Here we review recent progress in developing statistical approaches to image estimation that can overcome these limitations. We describe the various components of the physical model and review different formulations of the inverse problem. The wide range of numerical procedures for solving these problems are then reviewed. Finally, we describe recent work aimed at quantifying the quality of the resulting images, both in terms of classical measures of estimator bias and variance, and also using measures that are of more direct clinical relevance.
A penalized-likelihood image reconstruction method for emission tomography, compared to postsmoothed maximum-likelihood with matched spatial resolution
- IEEE Trans Med Imaging
"... Abstract—Regularization is desirable for image reconstruction in emission tomography. A powerful regularization method is the penalized-likelihood (PL) reconstruction algorithm (or equiva-lently, maximum a posteriori reconstruction), where the sum of the likelihood and a noise suppressing penalty te ..."
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Cited by 39 (14 self)
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Abstract—Regularization is desirable for image reconstruction in emission tomography. A powerful regularization method is the penalized-likelihood (PL) reconstruction algorithm (or equiva-lently, maximum a posteriori reconstruction), where the sum of the likelihood and a noise suppressing penalty term (or Bayesian prior) is optimized. Usually, this approach yields position-de-pendent resolution and bias. However, for some applications in emission tomography, a shift-invariant point spread function would be advantageous. Recently, a new method has been pro-posed, in which the penalty term is tuned in every pixel to impose a uniform local impulse response. In this paper, an alternative way to tune the penalty term is presented. We performed positron emission tomography and single photon emission computed tomography simulations to compare the performance of the new method to that of the postsmoothed maximum-likelihood (ML) approach, using the impulse response of the former method as the postsmoothing filter for the latter. For this experiment, the noise properties of the PL algorithm were not superior to those of postsmoothed ML reconstruction. Index Terms—Bayesian reconstruction, PET, regularization, SPECT, tomography.
A paraboloidal surrogates algorithm for convergent penalized-likelihood emission image reconstruction
- In Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf
, 1998
"... We present a new algorithm for penalized-likelihood emission image reconstruction. The algorithm monotonically increases the objective function, converges globally to the unique maximizer, and easily accommodates the nonnegativity constraint and nonquadratic but convex penalty functions. The algorit ..."
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Cited by 37 (22 self)
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We present a new algorithm for penalized-likelihood emission image reconstruction. The algorithm monotonically increases the objective function, converges globally to the unique maximizer, and easily accommodates the nonnegativity constraint and nonquadratic but convex penalty functions. The algorithm is based on finding paraboloidal surrogate functions for the log-likelihood at each iteration: quadratic functions that are tangent to the log-likelihood at the current image estimate, and lie below the log-likelihood over the entire nonnegative orthant. These conditions ensure monotonicity. The paraboloidal surrogates are maximized easily using existing algorithms such as coordinate ascent. Simulation results show that the proposed algorithm converges faster than the SAGE algorithm, yet the new algorithm is somewhat easier to implement. I.
