A Fast Compressed Sensing Approach to 3D MR Image ... - IEEE Xplore

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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 5, MAY 2011

A Fast Compressed Sensing Approach to 3D MR Image Reconstruction Laura B. Montefusco*, Damiana Lazzaro, Serena Papi, and Carla Guerrini

Abstract—The problem of high-resolution image volume reconstruction from reduced frequency acquisition sequences has drawn significant attention from the scientific community because of its practical importance in medical diagnosis. To address this issue, several reconstruction strategies have been recently proposed, which aim to recover the missing information either by exploiting the spatio-temporal correlations of the image series, or by imposing suitable constraints on the reconstructed image volume. The main contribution of this paper is to combine both these strategies in a compressed sensing framework by exploiting the gradient sparsity of the image volume. The resulting constrained 3D minimization problem is then solved using a penalized forward–backward splitting approach that leads to a convergent iterative two-step procedure. In the first step, the updating rule accords with the sequential nature of the data acquisitions, in the second step a truly 3D filtering strategy exploits the spatio-temporal correlations of the image sequences. The resulting NFCS-3D algorithm is very general and suitable for several kinds of medical image reconstruction problems. Moreover, it is fast, stable and yields very good reconstructions, even in the case of highly undersampled image sequences. The results of several numerical experiments highlight the optimal performance of the proposed algorithm and confirm that it is competitive with state of the art algorithms. Index Terms—Compressed sensing (CS), medical image sequences, nonlinear filters, splitting methods, total variation minimization.

I. INTRODUCTION

T

HREE-DIMENSIONAL medical reconstruction is a powerful technique in medical diagnosis. It has been widely studied and developed, but there are some problems which still remain unresolved or can be improved. For example, the existing reconstruction algorithms for circular tomography require, in general, a great number of measurements to obtain high-resolution reconstructed images, with consequent high radiation dose given to the patients. Reducing the projection data leads to unpleasant artifacts in the reconstructions, when both filtered back-projection algorithms and algebraic reconstruction techniques are used. Magnetic resonance imaging (MRI) does not use dangerous radiations but needs a very long time to acquire the complete data set necessary for a detailed Manuscript received June 15, 2010; revised July 29, 2010; accepted August 09, 2010. Date of publication August 19, 2010; date of current version May 04, 2011. This work was supported in part by Miur and in part by R.F.O. projects. The work of S. Papi was supported by Spinner2013 funds . Asterisk indicates corresponding author. *L. B. Montefusco is with the Department of Mathematics, University of Bologna, 40125 Bologna, Italy (e-mail: [email protected]). D. Lazzaro, S. Papi, and C. Guerrini are with the Department of Mathematics, University of Bologna, 40125 Bologna, Italy. Digital Object Identifier 10.1109/TMI.2010.2068306

high-resolution reconstruction. Using accelerated 3D MRI scans implies the use of parallel imaging with multiple-channel receivers [24], [28] or data sharing methods [16], [29]. In the latter case, in order to increase temporal resolution, only a reduced set of k-space data is acquired and numerical strategies have been developed to obtain high-resolution image sequence reconstructions from the under-sampled data sets. In particular, high temporal resolution is critical to monitor dynamic processes such as cardiac motion and dynamic contrast-enhanced or functional MRI, where the dynamic events change on time scales in the order of a few milliseconds. Motivated by the need to further reduce the amount of acquired data without compromising image quality, both for tomographic and magnetic resonance imaging, new algorithms have been proposed in recent years that exploit the spatio-temporal correlations exhibited by the under-sampled image sequences. Examples of this approach are UNFOLD [20], k-t BLAST/SENSE [11], [26], [27], and SPEAR [30]. A new promising approach for under-sampled image sequence reconstruction is the introduction of compressed sensing (CS) theory, a new framework recently developed to reconstruct signals and images from significantly fewer Fourier measurements that were thought necessary by the Nyquist sampling criterion [2]–[4]. According to this theory, k-t SPARSE [18], [19] takes advantage of the fact that MRI images meet the two conditions required by CS to be successfully applied: to be sparse in a certain transform domain and to be acquired in a domain incoherent with the sparsifying basis. K-t SPARSE has been successfully used for cardiac imaging applications and for the MR angiography problem, but its drawback is the computational burden. Other applications of the CS theory to MRI are shown in [8], [13], [17], [21], [25]. In an attempt to unify the approaches of k-t BLAST and k-t SPARSE, while overcoming their drawbacks, a new algorithm has been proposed in [12] and [14], named k-t FOCUSS. The minimization problem of the CS framework was addressed using a reweighted quadratic optimization technique and an efficient prediction and residual encoding was inserted to better sparsify the residual and further improve the performance of the algorithm. In this optimized form k-t FOCUSS can achieve high spatio-temporal resolution even from severely under-sampled k-t measurements of cardiac Cine MRI. The limits of k-t FOCUSS as presented [14] is that it is specifically designed for dynamic processes with periodic motion and requires Cartesian trajectory acquisitions. Recently, a new efficient approach has been proposed in minimiza[22] and [23] to solve the compressed sensing tion problem for generally under-sampled signal and image

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MONTEFUSCO et al.: A FAST COMPRESSED SENSING APPROACH TO 3D MR IMAGE RECONSTRUCTION

reconstruction. In these papers an adaptive nonlinear filtering strategy is used to evaluate the optimal reconstruction which, in addition to satisfying the acquisition constraints, has a minimum property, namely has minimal norm or minimal total variation norm, according to the considered reconstruction problem. In particular, in [23] it has been shown that, since many real medical images can be approximated by piecewise smooth signals, they well meet the gradient sparsity requirement that guarantees the success of the previous approach, when applied for reconstructing such images from highly under-sampled frequency acquisitions. Motivated by the efficiency and speed of the nonlinear filtering approach, in this paper we have exploited its ability to take advantage of the correlations existing between neighboring frames to obtain high-resolution reconstructions from accelerated frequency acquisitions of medical image sequences. Since the measurements consist of a sequence of 2D data in the Fourier domain, the novelty here is to formulate a hybrid 2D-3D constrained minimization problem, whose approximate solution is obtained by solving a sequence of 3D total variation regularized unconstrained subproblems. Each of these subproblems is solved making use of a proximal forward–backward splitting approach, which leads to an iterative alternating minimization strategy, consisting in two main steps: a sequence of 2D updating steps and a 3D total variation filtering step. To carry out the latter step a novel 3D recursive median filter has been proposed that, with just one iteration, well approximates the true solution. Moreover the overall convergence of the algorithm has been stated and greatly improved by a suitable acceleration strategy. The resulting algorithm is completely independent from the data acquisition technique (e.g., cartesian, radial, spiral, etc.) since it exploits the intra and inter-frame correlations in the real domain, and is suitable for several kinds of medical image reconstruction problems. Moreover, it is fast, stable and yields very good reconstructions, even for severely under-sampled image sequences. The results of several numerical experiments highlight the optimal performance of the proposed algorithm and confirm that it is competitive with the state of the art algorithms, particularly in regarding the quality of the reconstructions. The paper is organized as follows: in the next section we present our CS 2D-3D formulation of the reconstruction problem and we describe the proposed reconstruction method. In Section III we deduce our 3D generalization of the recursive New Median Filter and in Section IV we present the proposed NFCS-3D algorithm, together with its accelerated and box constrained version. Finally, in Section V we present our experimental results and some comparisons with k-t FOCUSS, one of the best state of the art algorithms. II. COMPRESSED SENSING APPROACH Let be the image volume to be reconstructed, which can be expressed as an image sequence, namely

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The data of the reconstruction problem are represented in both cases by the image volume in the Fourier domain, formed by suitably selected frequency encodings of each frame. We denote this volume by (1) where , are acquisition dependent binary masks, denotes the 2D Fourier transform of and represents the image the under-sampling in the Fourier domain ( -space) obtained with . by means of the point product of the mask The CS reconstruction problem can be stated in this case as the following constrained minimization: given as in (1) and , find , the convex sparsifying function that solves (2) When the data are corrupted by random noise, namely (3) with

, the reconstruction problem (2) can be cast as

(4) We remark that the above formalizations for the CS reconstruction problem are not truly 3D CS problems, since the measurements are not performed in the 3D frequency domain, but are only a sequence of 2D subsampled acquisitions. Nevertheless, in spite of this hybrid 2D-3D formulation, we can exploit the Compressed Sensing theoretical results to guarantee that, if all the images forming volume are sparse (or approximately sparse) in a certain domain and the given measurements are obtained in a domain incoherent with the sparsifying one, under suitable hypotheses on the measurement matrices, and with a , the solution of proper choice of the sparsifying function problem (2) or (4) gives an exact (or well approximated) reconstruction of the original volume . A. Reconstruction Method In the present context we have assumed that the measurement matrices are suitable subsamplings of the Fourier transform, and, since medical images can be well approximated by piece-wise smooth images, each frame can be considered to have sparse gradient. These two conditions meet the CS requirements and are the basis of our reconstruction method. Unlike the k-t Focuss approach, which finds the minimal norm reconstruction after applying the Fourier Transform along the temporal direction to sparsify the data, our reconstruction method exploits the sparsity of the gradient of the images and as chooses the convex, sparsity inducing function (5)

where the images , , represent both the unknown dynamic frames in the case of dynamic MRI and the adjacent slices of the static object in the case of 3D reconstruction.

namely, it minimizes the total variation of the whole volume. This choice is motivated by the observation that also the gradient components in the temporal direction can be assumed to be sparse for medical image volumes.

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The nondifferentiability of the choice (5) for makes problems (2) and (4) very difficult to solve directly, so we have preferred to convert them into unconstrained optimization problems. To this aim we have considered the sequential penalization approach (see [10] for details) that approximates the original constrained problems with a sequence of unconstrained subproblems of the form

possess the Restricted Isometry Property (see [2] and [4] for details). This assesses the convergence of the iterative scheme (9) . with the choice of Relation (9) can now be written as the following two step indicates the inverse Fourier iterative procedure, where , for any transform of the point product Algorithm FBS: Forward–Backward Splitting

(6) Given , , is a decreasing sequence of pewhere nalization parameter values, enforcing the constraints to be satisfied, that ensures the convergence of the penalization method to the solution of the original constrained problems. The penalized formulation (6) still presents intrinsic numerical difficulties, due to the huge dimensions of the minimizations involved and to the ill-conditioning of the unconstrained subproblems for small values of the penalization parameter . To overcome both difficulties we have approached the solution of the unconstrained minimization subproblems by making use of a fixed-point iterative scheme based on a special instance of the so-called forward–backward splitting method [7], and solving the resulting problems by taking advantage of their particular structure. Specifically, recalling that the proximal operator assois defined, for any scalar ciated with the convex function , by (7) , then, given the continuously differwhere with -Lipschitz continuous gradient, entiable function the splitting approach finds the minimum of (8) using the following iterative scheme (see [7])

The convergence of this scheme is proved in [7] for In our context, where

(9) .

.

Initialization for

do Updating step:

Minimization step:

The main advantage of this splitting approach is that the two steps of the iterative procedure reflect the two aspects of the minimization problem (6): the first sequence of 2D updating steps corresponds to the sequential 2D strategy of data acquisition, while the 3D minimization step inherits the truly 3D nature of the reconstruction problem. This 3D phase of the solution process is performed in the real domain and consists of a 3D total variation denoising step. Since the discrete gradient estimation involves many neighbors of each point value, this 3D minimization makes it possible to exploit the existing inter and intra-frame correlations necessary to obtain an optimal reconstruction, even from severly undersampled acquisitions. Nevertheless, the numerical solution of this minimization step may imply such a high computational burden that it makes the efficiency of the presented splitting approach still not sufficient for widespread usage in medical image problems. In the next section, we will present our approach to solve this basic step efficiently. III. 3D DISCRETE TOTAL VARIATION MINIMIZATION The minimization step (10)

we can easily see that

has -Lipschitz gradient with

where denotes the maximum modulus eigenand represents the th undervalue of the matrix sampled frequency acquisition matrix. Moreover, the Lipschitz since all acquisition matrices , constant satisfies satisfy the Compressed Sensing hypothesis to

is in fact the 3D version of the Rudin–Osher–Fatemi (ROF) model for total variation image denoising. This model, originally designed for continuous signals, when applied to a digital image must be carefully discretized, in order to overcome the problems due to the nonlinearity of its corresponding Euler–Lagrange equation. For the 2D case, in recent years some efficient iterative algorithms have been proposed [5], [6], [15] which, in order to obtain the optimal solution, apply a successive filtering process to the noisy image. In this paper we extended to


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the 3D case the algorithm proposed in [15], since, as experimented in [23], it seems the fastest and therefore more suited to dealing with large amounts of data, as in the case of image volume processing. A. New Median Filters for 3D TV Denoising Median Filters are well known nonlinear filters widely used for signal processing. In the 3D case they act by shifting a mask voxels along the three directions of the window of image volume and replacing the central value with the median of the ordered set inside the window. These filters possess the following minimum property: the output of a weighted median can be defined filter acting on the set as the value minimizing

where are nonnegative weights. Let us now consider the more general minimization problem

p

Fig. 1. The 18-neighbors structure used for the anisotropic gradient evaluation: the dark-gray points have weight 1, the median-gray points have weight 1= 2.

Using the gradient discretization (14) we obtain problem (11), can be evaluated by means of the new median forso that mula as

(11) and use the main result of [15]. It states that, if the weights are non negative, the minimum of (11) is a median, namely (12) where the

values are given by (13)

We can use the previous result to generalize the approach in [15] and obtain an iterative algorithm for total variation image volume denoising. We consider the 3D ROF model (10) with the anisotropic gradient extimate , [9], and we try to find the solution of a discrete approximation on a 3D regular grid, point by point. We fix a set of 18 neighbors of the unknown using difcentral value , as shown in Fig. 1, and discretize ferent weights as follows:

where , are evaluated according to (13). By applying iteratively the median formula voxel-by-voxel over the whole image volume the iteration process converges to the global minimizer of the 3D ROF model. The convergence of the iterative algorithm in the 3D case can be proved as in [15] by observing that the global energy decreases (or at least does not increase) at every local minimization step and by using Lemma 1 of [15]. This iterative algorithm to find the minimum of the ROF model has two main drawbacks. The first is that it can stagnate at a nondifferentiable point, the second is the high computational burden of the median algorithm when applied iteratively to a 3D volume. In the present framework the first problem is not very important, since no great accuracy is required by the minimization step in the context of the penalized approach. To overcome the second problem and reduce the computational cost we have used a classical result on the median filters, namely, the fact that the action of an iterated median filter can be obtained using only one iteration of its recursive version. We have therefore used the following new recursive median filter defined as:

(16) (14) where and refer to the dark-gray and mediumgray circles in Fig. 1. Then, given the noisy central value we look for the optimal that minimizes the local ROF problem namely (15)

and we have applied this filter just once. To further reduce the computational complexity we have also used the cheaper sorting algorithm proposed in [15] that requires, in this particular case, only 111 comparisons. It is important to note that, in the practical application of the proposed algorithm, it is possible to use a hardware implementation of the median that would greatly reduce the computing time.

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IV. NFCS-3D ALGORITHM Following the theoretical approach presented in the previous sections, we have realized an algorithm characterized by two iteration levels. An outer iteration that updates the value of the penalization parameter and decides the stopping criterion, an inner iteration that, for each value of , iteratively solves the associated unconstrained minimization problem according to the proximal forward–backward splitting method. These two basic building blocks have then been enriched with suitable numerical tools in order to improve the reconstruction capabilities and the convergence speed of the whole algorithm.

The minimization step consists of a constrained total variation 3D filtering step. Thus, we add a projection phase in our recursive new median filter, substituting relation (16) with

where

and

A. Bound Constrained Reconstruction In image recovery problems, the original image intensity values are known to be nonnegative and less than or equal to a positive value . We can therefore insert additional information in the CS reconstruction problem by adding a bound constraint to problem (2), namely, by considering

(17) where

is the closed convex set

Since the penalized splitting approach used to solve problem (2) is quite general, it allows us to simply handle bound constraints. In fact, in the bound constrained case, the penalized formulation becomes: solve for each value of

B. Acceleration Strategies The theoretical properties of the proximal forward–backward splitting method have been well studied in the literature (see e.g. [7] and the references therein). Its advantage consists in its simplicity, but it has been recognized that the corresponding iterative scheme (9) converges slowly. Recently, in [1] a new version of the FBS approach has been proposed, named FISTA, which preserves the computational simplicity of (9), but significantly improves its convergence speed. Specifically, the FISTA algorithm defines a sequence of positive numbers

and, at each iteration step, improves the estimate yielded by (9) performing a specific linear combination with the unmod. ified previous iterate The Fista-accelerated FBS algorithm is as follows: Algorithm AFBS: Accelerated Forward–Backward Splitting

(18) For the solution of this unconstrained minimization problem, the use of the proximal FBS method yields the following constrained FBS iterative procedure:

Given

and

and

Inizialization: for

. ,

do

Algorithm CFBS: Constrained Forward–Backward Splitting Given ,

,

and

.

INIZIALIZATION: for

do Updating step:

Minimization step:

We have inserted the Fista acceleration in the inner iteration loop of our reconstruction algorithm and, as was expected, the reduction of the iteration number to obtain a predefined reconstruction quality was noticeable, with an absolutely insignificant increase of computing time. We then found that the use of the Fista approach also in the outer iteration loop further improves the performance of NFCS-3D. Thus, even if not supported by theoretical arguments, we have maintained this double use of Fista acceleration in all our numerical experiments.


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Fig. 2. Test volumes: a) 3D Phantom; b) Cardiac MRI; c) MRA.

C. Parameter Settings The doubly iterative structure of the proposed NFCS-3D algorithm implies the need to set some free parameters. The outer iteration loop, implementing the penalization approach, requires and a reduction rule for the penalization paa starting value rameter, as well as a suitable stopping criterion. The inner loop needs to set the value of and, for each value of the penalization parameter, the precision requested for the solution of the corresponding minimization problem. Regarding the choice of , according to what has been done in [23], we have always used, for noise free data, , and for some noisy cases. The , rule for reducing the penalization parameter was and the choice for the reducing rate in the ma. The greater value jority of the cases has been was used for highly downsampled experiments, while was chosen to accelerate the convergence speed for the lowest acceleration factors (3 and 4). In our implementation, the stopping criterion of the outer loop is based on the relative precision with which the th frame meets the constraint, that is

In the numerical experiments the value of was chosen according to the absence or presence of noise in the data and to the characteristics of the original images. We remark that, since the stopping rule refers to each single frame, it may happen that some frames reach the requested precision before the others. In this case our algorithm no longer modifies those frame reconstructions, but works only to improve the reconstruction quality of the others. Regarding the parameters of the inner iteration, meets the convergence requirement of since the value [7] we have adopted this simple choice. For the precision of the unconstrained minimization problem, by considering the penalized framework in which we are working, we have used an exit criterion that depends on the value of the penalization parameter, namely (19) where . The parameter in (19) is crucial for the speed of the algorithm and is used to reduce the accuracy of the inner iteration

when the overall precision is not very high. This happens, for example, when we deal with approximately sparse gradient image volumes or with noisy data. We remark that, since the inner loop is the truly 3D phase of the algorithm, a reduction of the inner iteration number can significantly reduce the computing time. V. NUMERICAL RESULTS In order to demonstrate the effectiveness of the proposed algorithm we tested it on the three different data sets shown in Fig. 2: a synthetic volume, named 3D Phantom, and two real MRI image sequences, named Cardiac MRI and MRA. The 3D Phantom is a 3D analogue of the popular Shepp–Logan phantom. It has been generated by a Matlab function, written by Schabel of Utah University, which is available online. The size of the volume generated for our experimentation is . The Cardiac MRI test volume is a part of a cardiac cine MRI acquired by the Bio Imaging Signal Processing Lab (Republic of Korea) and available online. This volume has a and we have selected the first dimension of sixteen frames. The magnetic resonance angiography (MRA) consists of gadolinium-contrast-enhanced coronal slices used to visualize the cardiopulmonary vasculature. It is available onframes. We have selected line and contains sixteen frames (from 51 to 66) to form our test volume of size . In all our experiments the gray level range of the images has been transformed to be [0,1] and we have considered subsampled frequency acquisitions. We have simulated two kinds of accelerated acquisitions: Cartesian acquisitions and Radial acquisitions. In the first case, for all considered acceleration factors (4, 8, 16, 32), a Gaussian random sampling pattern has been generated for each frame, obtaining a random set of fully sampled lines, parallel to the read out direction, that are dense more in the low frequency region. Fig. 3(a) shows a cartesian Gaussian sampling pattern used for the eight-fold acceleration experiment, while Fig. 3(b) illustrates the whole volume acquisition pattern corresponding to the same acceleration factor. When using radial trajectories we have considered as fully sampled acquisition that obtained with 180 views. The -fold views for each frame, was acquisitions, consisting of interleaved under-sampling then obtained by considering 1http://www.mathworks.com/matlabcentral/fileexchange/9416 2http://wwwbisp.kaist.ac.kr/ktFOCUSS.html 3http:///physionet.fri.uni-lj.si/physiobank/database/images

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TABLE I 3D PHANTOM DATA SET

Fig. 3. a) Cartesian Gaussian sampling pattern corresponding to an acceleration factor of 8. b) Scheme of the Gaussian sampling pattern for the whole volume.

The results shown in the next subsections are obtained by running a Matlab implementation of the bound constrained, doubly Fista accelerated version of NFCS-3D on a PC with an Intel Pentium4 HT 3.4-GHz processor and 3 GB of RAM under Windows XP. For comparison purposes we have also run on the same PC the free Matlab code implementing the cartesian k-t FOCUSS [14], available online. A. Exactly Sparse Gradient Image Volume

Fig. 4. Two consecutive radial sampling patterns used for the acceleration factor of 16.

patterns applied to adjacent frames. For example, when using only 30 views per frame, corresponding to an acceleration factor of 6, the view ordering was as follows: the first frame views, the second frame had had the views, and so on. Fig. 4 shows two the consecutive reduced radial sampling patterns at the acceleration factor of 16. The quality of the reconstruction has been objectively evaluvalue, namely the mean of the ated using the values obtained for each frame. Specifically, given the original and the reconstruction , , we deframe fine

and

where range, then

is the maximum value of the image gray level

The value that we give in the different experiments refers to the first minimum energy reconstruction. It allows us to better appreciate the improvement obtained with our reconstruction algorithm.

In this section we present the experiments we have performed using the 3D-Phantom test volume. This data set represents a synthetic exactly sparse gradient volume that exactly meets the theoretical requirements of the CS theory, allowing us to properly evaluate the capabilities of the proposed reconstruction method. In the first set of experiments we have considered unperturbed acquisitions in the frequency domain obtained using several random Cartesian subsampling patterns as well as interleaved radial patterns (see, as examples, Figs. 2 and 3). This exact theoretical context has allowed us to highlight both the reconstruction capability of the presented algorithm and the improved efficiency obtained using Fista. We have considered down-sampling factors ranging from 3 to 32 and in all cases NFCS-3D has reached numerically perfect reconstruction (with numerically perfect reconstruction we which corresponds to a relative intend ). In order to obtain this high reconstruction error approximation level we have set the parameter and to reach a sufficient precision in the inner iteration loop. Table I displays the iteration numbers necessary to obtain perfect reconstruction for the different acceleration factors. It is clear that the double use of Fista greatly improves the convergence speed of the algorithm making viable for practical applications also very high down-sampling factors such as 16 of the and 32. In Fig. 5 we show the behavior of the reconstructed image volume when we use the proposed algorithm with or without FISTA. The Fista accelerated algorithm with less than 30 iterations reach a very good reconstruction , while the original algorithm needs quality about 150 iterations to obtain a similar result. Since the Fista accelerated version of NFCS-3D is clearly better, from now on we will consider only this version. The second set of experiments with the 3D-Phantom data set is devoted to show the stability properties of the proposed algorithm. To this aim we have perturbed the data by adding white 4http://bisp.kaist.ac.kr/ktFOCUSS.html


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Fig. 6. Original images: a) MRA, frame 13; b) Cardiac MRI, frame 5.

Fig. 5. Behavior of the PSNR mean value as a function of the total iteration number: a comparison between NFCS-3D original (dashed line) and its FISTA accelerated version (solid line). TABLE II PERTURBED 3D PHANTOM DATA SET,

= 0:05

At first glance we note that only very few iterations are needed to obtain very good reconstruction levels (PSNR values between 36 and 40) and therefore correspondingly low computing times. If we then compare the relative perturbation and reconstruction errors, we can appreciate the great stability of the proposed algorithm, which succeeds in solving an inverse ill posed problem not only without a larger increase in the perturbation error, but in most cases with an error reduction. This very stable behaviour is partially due to the filtering approach that further reduces the reconstruction error. B. Real Medical Image Volumes

Gaussian noise of standard deviation to each subsamas in (3). The corresponding pled frequency acquisition , have been estimated, according noise levels , to [2], as (20) being the total number of frequency encodings of the th frame. Since (20) usually overstimates the true error norm, we have considered the following stopping criterion in the outer iteration that accords with the problem formulation (4), that is

where the role of the parameter is to reduce the estimate in order to approach the true error norm. Regarding in the exit criterion of the inner iterations, we have used (19) since no high precision is necessary in this noisy setting. The results obtained by NFCS-3D in this set of experiments are presented in Table II, where, to better appreciate the stability properties of the algorithm, we give also the relative perturbation error, (R.P.E.), and the relative reconstruction error, (R.R.E.), defined as

Medical images are known to be well approximated by piecewise smooth images. They can therefore be considered approximately sparse gradient images. Since they only approximately meet the CS requirements, the solution of the reconstruction problem (2) in this case cannot reach numerically perfect reconstruction. It yields only an approximation that is increasingly accurate according to how well approximated the original image is (by a sparse gradient image). For volumes of medical images this still holds. We have experimented the reconstruction capability of NFCS-3D on the two real medical image volumes of Fig. 2. Due to the approximate nature of the reconstruction problem we have used in the stopping criterion of the outer iteration and in the inner iteration exit test (19). For both data sets we have considered undersampled frequency acquisitions obtained using cartesian subsampling patterns, as well as interleaved radial patterns. The results of the experiments, presented in Tables III and IV, and in Figs. 7and 8, highlight the reconstruction capabilities of NFCS-3D and its speed. In particular, Figs. 7 and 8 show the reconstructions of the original frames displayed in Fig. 6 at the 4, 8, and 16 fold acceleration factors. In Fig. 7 we have also shown the zero-padded reconstructions in order to highlight their poor quality and the aliasing patterns presented along the downsampling direction, particularly for the 8 and 16 fold acceleration factors. These aliasing artifacts are completely suppressed in our reconstructions, which show well defined contours and small details even for the highest acceleration factors. C. Comparisons With the k-t FOCUSS Algorithm In order to evaluate the competitive performance of the proposed NFCS-3D algorithm, we have run the k-t FOCUSS software on the same set of experiments as in Table IV, for the

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Fig. 7. Reconstruction results of NFCS-3D for the MRA data set, cartesian acquisition, frame 13. The top row shows the inverse Fourier transforms of the zeropadded data at the 4, 8, 16 fold acceleration factors. The bottom row shows the corresponding reconstructions.

Fig. 8. Reconstruction results of NFCS-3D for the Cardiac MRI data set, cartesian acquisition, frame 5. The top row shows the reconstructions corresponding at the 4, 8, 16 fold acceleration factors. The bottom row shows the corresponding difference images (multiplied by an amplification factor of 5) with respect to the original frame 5. TABLE III MRA DATA SET

TABLE IV CARDIAC MRI VOLUME DATA SET


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Fig. 9. Reconstruction results of k-t FOCUSS for the Cardiac MRI data set, cartesian acquisition, frame 5. The top row shows the reconstructions corresponding at the 4, 8, 16 fold acceleration factors. The bottom row shows the corresponding difference images (multiplied by an amplification factor of 5) with respect to the original frame 5.

TABLE V CARDIAC MRI DATA SET; CARTESIAN ACQUISITIONS

cartesian acquisitions. We have experimented various combinations for the numbers, requested by the algorithm, of inner and outer iterations and, finally, we have chosen the best compromise between greatest accuracy and lowest computing time. The results of this experimentation are displayed in Table V, where we denote with (in,out) the inner and outer iteration numbers used in k-t FOCUSS. For comparison, in addition to our best results, we have shown the iteration number and computing time used by NFCS-3D to obtain a reconstruction quality similar to the best quality reached by k-t FOCUSS (indicated with NFCS-3D stopped). The reconstruction results of k-t FOCUSS for frame 5 of the Cardiac MRI test volume, at the 4, 8, and 16 fold acceleration factors are presented in Fig. 9. In order to compare these reconstructions with those given by NFCS-3D, shown in Fig. 8, we have also displayed the difference images (multiplied by an amplification factor

of 5) between the original image and the best reconstruction results obtained by both algorithms. A direct comparison of the reconstructed images shows that NFCS-3D succeeds in clearly reconstructing the image edges, even for severely downsampled data (16 fold acceleration). This is not the case for the k-t FOCUSS reconstructions, which present smoothed edges and unclear small details. The superiority of the NFCS-3D reconstruction quality is also confirmed by the analysis of the difference images. In fact, they show that, in the NFCS-3D case, some aliasing artifacts are still visible along the undersampling direction at the 16 fold acceleration factor, but the heart boundaries are always perfectly reconstructed, while in the k-t FOCUSS reconstructions most of the difference energy is concentrated along the heart boundaries. To better evidentiate the difference between the two methods, we have considered a ROI in frame 5 of the cardiac MRI test volume, indicated by the white box in Fig. 10, and, in the same Figure, we have displayed the better reconstructions of this ROI obtained with NFCS-3D at the 8 and 16 fold acceleration factors, enlarged with Matlab by a 2.5 factor as the original ROI. In Fig. 11 we have then displayed the reconstructions, at the same acceleration factors, obtained at comparable PSNR values by NFCS-3D stopped and k-t FOCUSS, togheter with the amplified difference images. In spite of the similar PSNR values, the reconstructed images are quite different: the results of NFCS-3D stopped algorithm present well defined contours even at the 16 fold acceleration factor, while the k-t FOCUSS reconstruction severely smooths the hearth boundaries, as it is clearly shown in the difference images. We are aware that we have experimented an early version of k-t FOCUSS and that improvements to this algorithm have been presented in [14], but the prediction methods proposed in that

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Fig. 10. The considered ROI indicated by a white box (a). The 2.5 zoom of the ROI (b). The NFCS-3D best reconstructions at the 8 and 16 fold acceleration factors (c, d, respectively).

Fig. 11. Zoomed reconstruction results at the 8 and 16 fold acceleration factors and corresponding difference images (multiplied by an amplification factor of 5). The top row shows the best k-t FOCUSS results. The bottom row shows the results of NFCS-3D stopped.

paper require either the knowledge of two fully sampled reference frames or the realization of a time-consuming motion estimation/compensation scheme. Moreover, the efficiency of k-t FOCUSS is strictly related to the use of a fast transform along the temporal direction. For static image volume, or for image sequences with non periodic motion, the use of other temporal transforms (like KLT) will probably increase the computing time. On the contrary, the NFCS-3D algorithm is very general and simply exploits a widely recognized property of medical images, namely, to be well approximated by sparse gradient signals. The proposed filtering strategy produces minimal total variation reconstructions, namely, well approximate reconstructions with well defined boundaries. Moreover, the computing time of NFCS-3D can be drastically reduced by a hardware implementation of the 3D recursive new median filter.

VI. CONCLUSION We have proposed an efficient algorithm for medical image volume reconstruction from sequentially acquired, undersampled frequency samples. It finds the solution for a hybrid 2D-3D constrained minimization problem, derived by applying the Compressed Sensing theory to the reconstruction problem, using a penalized splitting approach and an adaptive nonlinear filtering strategy. Its convergence property has been established in the general context of the forward–backward splitting methods, and its performance, in terms of accuracy, stability and speed, is illustrated by the results of several numerical experiments. Comparisons with the results obtained with k-t FOCUSS algorithm confirm that NFCS-3D is competitive with the state of the art algorithms.


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