R*2mapping is a useful tool in blood-oxygen-level dependent fMRI due to its quantitative-nature. However, like T*2-weighted imaging, standard R*2mapping based on multi-echo EPI suffers from geometric distortion, due to strong off-resonance near the air-tissue interface. Joint mapping of R*2and off-resonance can correct the geometric distortion and is less susceptible to motion artifacts. Single-shot joint mapping of R*2and off-resonance is possible with a rosette trajectory due to its frequent sampling of the k-space center. However, the corresponding reconstruction is nonlinear, ill-conditioned, large-scale, and computationally inefficient with current algorithms. In this paper, we propose a novel algorithm for joint mapping of R*2and off-resonance, using rosette k-space trajectories. The new algorithm, based on the alternating direction method of multipliers, improves the reconstruction efficiency by simplifying the original complicated cost function into a composition of simpler optimization steps. Compared with a recently developed trust region algorithm, the new algorithm achieves the same accuracy and an acceleration of threefold to sixfold in reconstruction time. Based on the new algorithm, we present simulation and in vivo data from single-shot, double-shot, and quadruple-shot rosettes and demonstrate the improved image quality and reduction of distortions in the reconstructed R*2map.

Joint estimation of spin density R2* decay and OFF-resonance frequency maps is very useful in many magnetic resonance imaging applications. The standard multi-echo approach can achieve high accuracy but requires a long acquisition time for sampling multiple k-space frames. There are many approaches to accelerate the acquisition. Among them, single-shot or multi-shot trajectory-based sampling has recently drawn attention due to its fast data acquisition. However, this sampling strategy destroys the Fourier relationship between k-space and images, leading to a great challenge for the reconstruction. In this paper, we present two trust region methods based on two different linearization strategies for the nonlinear signal model. A trust region is defined as a local area in the variable space where a local linear approximation is trustable. In each iteration, the method minimizes a local approximation within a trust region so that the step size can be kept in a suitable scale. A continuation scheme is applied to reduce the regularization gradually over the parameter maps and facilitates convergence from poor initializations. The two trust region methods are compared with the two other previously proposed methods--the nonlinear conjugate gradients and the gradual refinement algorithm. Experiments based on various synthetic data and real phantom data show that the two trust region methods have a clear advantage in both speed and stability.

Joint reconstruction of three parameter images — spin magnitude, T2∗decay and off-resonance field map — is an important technique in some applications of magnetic resonance imaging (MRI). Despite its importance, the reconstruction problem is very challenging due to its nonlinearity and ill-conditioning. The standard conjugate gradient (CG) method with regularization is usually very slow for this problem. In this work, we propose a novel regularized trust region (TR) method that reconstructs the three images by solving a constrained linear sub-problem in each iteration. In particular, this method employs a change-of-variable technique, making the sub-problem linear yet keeping the convergence fast. To solve the sub-problem, we use a penalty method with preconditioned conjugate gradient (PCG) algorithm. Based on a synthesized data set, we demonstrate that our algorithm stably converges even with a poor initial guess, and the convergence rate is much higher than that of the CG algorithm.

A color image meant for human consumption can be appropriately displayed only if at least three distinct color channels are present. Typical digital cameras acquire three-color images with only one sensor. A color filter array (CFA) is placed on the sensor such that only one color is sampled at a particular spatial location. This sparsely sampled signal is then reconstructed to form a color image with information about all three colors at each location. In this paper, we show that the wavelength sensitivity functions of the CFA color filters affect both the color reproduction ability and the spatial reconstruction quality of recovered images. We present a method to select perceptually optimal color filter sensitivity functions based upon a unified spatial-chromatic sampling framework. A cost function independent of particular scenes is defined that expresses the error between a scene viewed by the human visual system and the reconstructed image that represents the scene. A constrained minimization of the cost function is used to obtain optimal values of color-filter sensitivity functions for several periodic CFAs. The sensitivity functions are shown to perform better than typical RGB and CMY color filters in terms of both the s-CIELAB ∆E error metric and a qualitative assessment.

Single shot parameter assessment by retrieval from signal encoding (SS-PARSE) is a recently introduced method to obtain quantitative parameter maps from a single-shot (typically 65 ms) magnetic resonance imaging (MRI) signal. Because it explicitly models local magnetization decay and phase evolution occurring during the signal 1) it can provide quantitative estimates of local transverse magnetization magnitude and phase, frequency, and relaxation rate and 2) it is free of geometric distortion or blurring due to field nonuniformities within the tissues. These properties promise to be advantageous in functional brain MRI (fMRI) and other dynamic imaging applications. In this paper, the basic phenomena underlying the performance of SS-PARSE in practice are discussed. Basic sources of bias errors in the parameter estimates are discussed, and performance of the method is characterized in terms of parameter estimates from simulation, experimental phantoms, and in vivo studies. Characteristics of the sum-of-square-error cost function and the iterative search algorithm are discussed, and their relative roles in determining estimation accuracy are described. Practical guidelines for use of the method are presented and discussed. In vivo parameter maps are also presented.

Gauss-Markov random field (GMRF) image models are commonly used in many Bayesian-based imaging techniques to define priors that lead to computationally tractable solutions for the image restoration problem. Recently, the color reconstruction literature has demonstrated, and effectively employed, the high correlation among the bands of a color image for color reconstruction. In this paper, we formulate a compound GMRF prior based on cross-channel spatial derivatives that reflects the smoothness in the color-difference space in addition to the often used intra-channel smoothness assumption. The proposed model is used to develop an effective method for restoring sparsely sampled color images in the presence of noise. The value of the proposed method is demonstrated on the problem of color reconstruction for single-sensor cameras.

Radial blurring (sometimes called zoom blurring) of an image is a challenging problem because it is a shift-variant blur. As one travels outward from the center of an image, the blur length increases linearly with distance from the center. This shift-variant characteristic precludes the use of other traditional FFT-based deblurring techniques. We propose a way around this problem by transforming the coordinate system of the blurred image into one in which the blur is LSI. Equivalently, the blurred image is sampled in a particular nonuniform fashion so that the blur becomes LSI in the new discrete-space coordinates. As a result, the blur can be modeled by convolution so that FFT-based deblurring can be used.

Typical consumer-level digital cameras use color filter arrays (CFA) overlaid on the optical sensor such that only one color is sampled at a particular spatial location. The full color image required for display is reconstructed from the sub-sampled data in an interpolation step commonly referred to as demosaicking. We present a method to select optimal color filter responses that minimizes both spatial reconstruction and color reproduction errors in a perceptually uniform color space. Optimal sensitivities for a three-color Bayer-type CFA and a four-color CFA are obtained and found to perform significantly better than RGB and CMY color filters in terms of the S-CIELab error metric.

The regularization of the least-squares criterion is an effective approach in image restoration to reduce noise amplification. To avoid the smoothing of edges, edge-preserving regularization using a Gaussian Markov random field (GMRF) model is often used to allow realistic edge modeling and provide stable maximum a posteriori (MAP) solutions. However, this approach is computationally demanding because the introduction of a non-Gaussian image prior makes the restoration problem shift-variant. In this case, a direct solution using fast Fourier transforms (FFTs) is not possible, even when the blurring is shift-invariant. We consider a class of edge-preserving GMRF functions that are convex and have nonquadratic regions that impose less smoothing on edges. We propose a decomposition-enabled edge-preserving image restoration algorithm for maximizing the likelihood function. By decomposing the problem into two subproblems, with one shift-invariant and the other shift-variant, our algorithm exploits the sparsity of edges to define an FFT-based iteration that requires few iterations and is guaranteed to converge to the MAP estimate.

Fast Fourier transform (FFT)-based restorations are fast, but at the expense of assuming that the blurring and deblurring are based on circular convolution. Unfortunately, when the opposite sides of the image do not match up well in intensity, this assumption can create significant artifacts across the image. If the pixels outside the measured image window are modeled as unknown values in the restored image, boundary artifacts are avoided. However, this approach destroys the structure that makes the use of the FFT directly applicable, since the unknown image is no longer the same size as the measured image. Thus, the restoration methods available for this problem no longer have the computational efficiency of the FFT. We propose a new restoration method for the unknown boundary approach that can be implemented in a fast and flexible manner. We decompose the restoration into a sum of two independent restorations. One restoration yields an image that comes directly from a modified FFT-based approach. The other restoration involves a set of unknowns whose number equals that of the unknown boundary values. By summing the two, the artifacts are canceled. Because the second restoration has a significantly reduced set of unknowns, it can be calculated very efficiently even though no circular convolution structure exists.

We consider the problem of selection of samples during image acquisition in a color digital camera. Due to the high cost of imaging sensors, digital cameras do not ordinarily employ separate sensors for each color channel. Instead, a single sensor overlaid with a mosaic of color filters is used such that a single color channel is sampled at each photosite. The full-color image must be reconstructed before the image may be displayed. In this paper, we propose a perceptually based sample selection scheme for color filter arrays. Regularization techniques are used to formulate an error criterion that reflects the error between the original image and the reconstructed image when viewed through the human visual system. A sequential selection algorithm is used to select sample locations that minimize the error criterion. The resulting array is found to compare favorably with existing array patterns in terms of the error criterion.

Tracking of cardiac motion using magnetic resonance tagging has attracted increasing attention in recent years. Several methods for tagging the cardiac tissue and tracking the motion of the tags have been developed. However, the choice of tag pattern that minimizes tracking error has received less attention. In this paper, we are concerned with the optimal tagging and acquisition of MR tagged images for cardiac motion analysis. We formulate the measurement of tissue deformation as a multidimensional parametric estimation problem which can be solved using the nonlinear least squares estimator. Along with this, we derive the Cramer-Rao lower bound (CRLB) on the average estimation error variance. We then show that under certain conditions a complex sinusoidal tag shape minimizes the CRLB. We validate our results with computer simulations. Finally, based on the previous findings, we make recommendations concerning the most desirable imaging strategy for images tagged with a complex sinusoidal tag pattern.

Many problems in signal and image processing require the solution of systems with a Toeplitz-block-Toeplitz structure in which both the Toeplitz blocks and the block structure are banded. We present a fast algorithm that exploits both the bandedness of the blocks and the block-bandedness of the block Toeplitz structure to solve the system of equations.

In some types of imaging, the signal is strictly limited in one domain while sampling takes places in another. If sampling is done in a rectangular array pattern at sub-Nyquist density, the array must be dithered to sample the image at the Nyquist density in each dimension. However, the Nyquist density oversamples the image due to the nonrectangular support in the transform domain. We present an efficient forward selection algorithm for optimizing the dithering pattern so that the image can be reconstructed as reliably as possible from a periodic nonuniform set of samples, which can be obtained from a dithered rectangular-grid array. Our examples show that this new algorithm makes selective sampling possible in a real-time image acquisition setting for MR spectroscopic imaging.

One of the primary drawbacks in the application of magnetic resonance spectroscopic imaging is the long acquisition times required to obtained the desired resolution. When region of support information is available, the number of phase-encoding steps and thus time can be reduced without loss of information if the kappa-space locations are chosen well. We propose to select locations using a rectangular sampling array that is shifted to various positions in kappa-space to obtain the necessary sampling density. This method allows multiple samples to be selected simultaneously and reduces the computation required to evaluate the selection criterion. We present an efficient forward selection algorithm for optimizing the shift pattern so that the image can be reconstructed as reliably as possible from a periodic nonuniform set of samples. The proposed algorithm has important practical potential in that it can finish the selection in less than half a minute for typical image sizes and can reconstruct the image with fewer samples than regular sampling. With appropriate imaging hardware, this new algorithm makes selective sampling possible in a real-time image acquisition setting.

In some types of imaging, the signal is strictly limited in one domain while sampling takes places in another. If sampling is done in a rectangular array pattern at sub-Nyquist density, the array must be dithered to sample the image at the Nyquist density in each dimension. However, the Nyquist density oversamples the image due to the nonrectangular support in the transform domain. We present an efficient algorithm for optimizing the dithering pattern so that the image can be reconstructed as reliably as possible from a periodic nonuniform set of samples, which can be obtained from a dithered rectangular-grid array. Taking into account the transform support of the image, we sequentially eliminate the least informative array recursively until the minimal number of arrays remain.

Magnetic resonance spectroscopic imaging (MRSI) requires a relatively long time to sample k-space (the spatial frequency domain), effectively lowpass filtering the resulting reconstructed image. Ringing is especially problematic when a region of interest (ROI) is close to a bright region outside the ROI, since the bright region tends to create a ringing artifact into the ROI due to the lowpass nature of the data. In this paper, we propose a method that reduces the effect of a stronger signal region on a weaker signal in a nearby ROI by designing a postprocessing filter that steers the strong interference away from the ROI. The proposed method is computationally simple both in the design stage and in applying it to images. We present experiments that illustrate the value of the technique.

Magnetic resonance spectroscopic imaging requires a great deal of time to gather the data necessary to achieve satisfactory resolution. When the image has a limited region of support (ROS), it is possible to reconstruct the image from a subset of k-space samples. Therefore, we desire to choose the best possible combination of a small number of k-space samples to guarantee the quality of the reconstructed image. Sequential forward selection (SFS) is appealing as an optimization method because the previously selected sample can be observed while the next sample is selected. However, when the number of selected k-space samples is less than the number of unknowns at the beginning of the selection process, the optimality criterion is undefined and the resulting SFS algorithm cannot be used. In this paper, we present a modified form of the criterion that overcomes this problem and develop an SFS algorithm for the new criterion. Then we develop an efficient computational strategy for this algorithm as well as for the standard SFS algorithm. The combined algorithm efficiently selects a reduced set of k-space samples from which the ROS can be reconstructed with minimal noise amplification.

Recent work in sparse signal reconstruction has shown that the backward greedy algorithm can select the optimal subset of unknowns if the perturbation of the data is sufficiently small. We propose an efficient implementation of the backward greedy algorithm that yields a significant improvement in computational efficiency over the standard implementation. Furthermore, we propose an efficient algorithm for the case in which the transform matrix is too large to be stored. We analyze the computational complexity and compare the algorithms, and we illustrate the improved efficiency with examples.

we determine the optimal pulse shape for estimating positions of superimposed pulses by deriving the Cramer-Rao lower bound (CRLB) on the average estimation error variance and optimizing it with respect to the pulse shape. Our results show that a significant improvement in estimation error variance can be achieved relative to Gaussian and rectangular pulse shapes.

Some signal reconstruction problems allow for flexibility in the selection of observations and hence the signal formation equation. In such cases, we have the opportunity to determine the best combination of observations before acquiring the data. We analyze the computational complexity of various forms of sequential backward selection (SBS) to select observations. In light of this analysis, we present a computationally improved algorithm for large-scale observation selection problems.

The authors introduce a new phase-based magnetic resonance (MR) tagging method for a direct and noninvasive measurement of the 2-D myocardial strain tensor. Their high-frequency complex sinusoidal tags can be conveniently generated with the SPAMM tagging pulse sequence, followed by one-sided band-pass imaging. In the authors' method, the instantaneous spatial phase data of the deformed tags carry the essential information with regard to local myocardial activity from which the strain tensor can be computed directly. Segmentation of tags and/or epicardial and endocardial contours is not required. As a result, the authors' proposed method is relatively simple to implement on current clinical MRI scanners, cheap in terms of computational cost, and can be generalized, at least conceptually, for 3-D imaging. The authors present computer simulation results for an idealized 2-D myocardial contraction to demonstrate the key idea as well as the performance of their method.

Some signal reconstruction problems allow for flexibility in the selection of observations and hence the signal formation equation. In such cases, we have the opportunity to determine the best combination of observations before acquiring the data. In previous work, we defined a mean-squared error (MSE) selection criterion based on the known signal formation equation. Furthermore, we proposed the use of sequential backward selection (SBS) to optimize this criterion. Although SBS is suboptimal, it performs consistently well. In this paper, we analyze the performance of SBS and develop an upper bound on the MSE of the SBS-optimized solution. This upper bound provides analytical justification for the claim that SBS performs well in spite of being suboptimal. Furthermore, we derive a lower bound on the MSE of preliminary combinations of rows. This lower bound can be used as a point of reference for evaluating the upper performance bound of the SBS algorithm.

This paper addresses the use of H/sub /spl infin// filtering to deconvolution, in particular, to the problem of image restoration. The proposed H/sub /spl infin// deconvolution filter has some advantages in the image restoration such as it can deal with unknown boundary problem and spatially varying blurs. The H/sub /spl infin// filter is compared with the inverse Wiener filter and a regularized restoration. The experimental results show that the H/sub /spl infin// filter deals with the unknown boundary problem better than the Wiener filter. Compared with the regularization method, it gives a sharper restored image, especially, when the original image contains many details.

Magnetic resonance spectroscopic imaging (MRSI) is a type of MRI in which both spatial and spectral information are gathered. Unfortunately, the time required to acquire a high-resolution image is prohibitive. Thus, we desire to gather the most informative data possible in the limited time available. The particular choice of a limited set of k-space samples has a tremendous impact on the quality of the reconstructed image. In previous work, we demonstrated a technique for choosing k-space samples under the assumption of a least-squares reconstruction given knowledge of the region of support of the image. In this work, we extend our criterion to a Wiener filter reconstruction. Furthermore, we exploit the properties of the Wiener filter criterion to derive a new optimization strategy. This optimization strategy allows us to begin the k-space acquisition process before the choice of k-space samples is complete. Results are demonstrated on an acquired MRI phantom.

A formal perceptual study shows that the perceptually optimal choice of regularization parameter varies from the minimum mean-square error parameter as a function of the original image, the point-spread function, and the signal-to noise ratio (SNR). However, further studies show that this variation has only a slight effect on the perceived quality of the restoration.

It has been shown that space-variant regularization in image restoration provides better results than space-invariant regularization. However, the optimal choice of the regularization parameter is usually unknown a priori. In previous work, the generalized cross-validation (GCV) criterion was shown to provide accurate estimates of the optimal regularization parameter. The author introduces a modified form of the GCV criterion that incorporates space-variant regularization and data error terms. Furthermore, he presents an efficient method for estimating the GCV criterion for the space-variant case using iterative image restoration techniques. This method performs nearly as well as the exact criterion for the image restoration problem. In addition, he proposes a Wiener filter interpretation for choosing the local weighting of the regularization. This interpretation suggests the use of a multistage estimation procedure to estimate the optimal choice of the local regularization weights. Experiments confirm the value of the modified GCV estimation criterion as well as the multistage procedure for estimating the local regularization weights.

In the above-titled paper (see ibid., vol. 2, p. 91-5, Mar. 1992), R. Rosenholtz and A. Zakhor proposed an effective method for reducing blocking in transform coded images. Their method uses two projection operators and the theory of projection onto convex sets (POCS) to guarantee convergence of the iteration. One projection operator is defined from the known quantization levels used to code the transform coefficients. The other projection is based on the set of bandlimited images with a given cutoff frequency. The algorithm they actually implemented is only tenuously related to the theory of POCS. A different basis for justifying their algorithm is offered, one that provides an exact formal basis for establishing convergence and a more flexible theory for elucidating the possibilities of the algorithm.>

A method for estimating the optimal degree of regularization in the presence of constraints on the restored image is presented. While regularization is a powerful method for obtaining satisfactory solutions to the image restoration problem, the optimal degree of regularization is seldom known ahead of time. Furthermore, the optimal degree of regularization changes when constraints are enforced in the restoration process. The generalized cross-validation criterion is modified to estimate the optimal degree of regularization in the presence of constraints on the solution. In addition, an efficient estimation procedure is used that makes it possible to evaluate the criterion with a computational complexity on the same order as the constrained restoration. The benefit of this procedure is demonstrated with examples.>

Constraints provide an important means of incorporating a priori information into the image restoration process. However, much of the information available for constructing constraints is of a tentative nature. If the validity of this tentative information can be assessed before it is incorporated into the solution, helpful constraints can be retained while harmful ones can be discarded. Cross validation is introduced as a technique for assessing the validity of such constraint sets. Because the full cross validation procedure is computationally burdensome, a modification is suggested that allows a more feasible implementation without substantially sacrificing the performance of the full procedure. Experimental results demonstrate the excellent performance of both the full and modified procedures.

The point spread function (PSF) of a blurred image is often unknown a priori; the blur must first be identified from the degraded image data before restoring the image. Generalized cross-validation (GCV) is introduced to address the blur identification problem. The GCV criterion identifies model parameters for the blur, the image, and the regularization parameter, providing all the information necessary to restore the image. Experiments are presented which show that GVC is capable of yielding good identification results. A comparison of the GCV criterion with maximum-likelihood (ML) estimation shows the GCV often outperforms ML in identifying the blur and image model parameters.

Cross-validation is introduced as a technique for assessing the validity of constraints imposed on the image restoration process before the process has begun. Because the full cross-validation procedure is computationally burdensome, a modification is suggested that allows a more feasible implementation without substantially sacrificing the performance of the full procedure. Experimental results demonstrate the excellent performance of both the full and modified procedures.>

A method for obtaining optimal estimates of the regularization parameter for iterative image restoration in a weighted Hilbert space is presented. The method is a modified form of the cross-validation technique. It requires no a priori knowledge of either the amount of noise or its distribution. Results are presented which demonstrate the effectiveness of the technique for different weighting matrices. Cross-validation is based on the concept of prediction error. For a fixed value of the regularization parameter, a restored image is determined using all the values from the observed image but one, and the restored image is reblurred in order to predict the blurred and noisy observation that was left out of the restoration. A different restored image is formed for each observation. The regularization parameter which minimizes the mean-square prediction error over all the observations is chosen as the estimated optimal parameter. A simplified version of cross-validation that is computationally manageable is presented.>

Because of the presence of noise in blurred images, an image restoration algorithm must constrain the solution to achieve stable restoration results. Such constraints are often introduced by biasing the restoration toward the minimizer of a given functional. However, a proper choice of the degree of bias is critical to the success of this approach. Generally, the appropriate bias cannot be chosen a priori and must be estimated from the blurred and noisy image. Cross-validation is introduced as a method for estimating the optimal degree of bias for a general form of the constraint functional. Results show that this constraint is capable of improving restoration results beyond the capabilities of the traditional Tikhonov constraint.