Non-iterative reconstruction with a prior for undersampled radial MRI data

Abstract

This paper develops an FBP-MAP (filtered backprojection, maximum a posteriori) algorithm to reconstruct MRI images from undersampled data. An objective function is first set up for the MRI reconstruction problem with a data fidelity term and a Bayesian term. The Bayesian term is a constraint in the temporal dimension. This objective function is minimized using the calculus of variations. The proposed algorithm is non-iterative. Undersampled dynamic myocardial perfusion MRI data were used to test the feasibility of the proposed technique. It is shown that the non-iterative Fourier–Bayesian reconstruction method effectively incorporates the temporal constraint and significantly reduces the angular aliasing artifacts caused by undersampling. A significant advantage of the proposed non-iterative Fourier–Bayesian technique over the iterative techniques is its fast computation time and its ability to reach the optimal solution. © 2013 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 23, 53–58, 2013.

I. INTRODUCTION

The Fourier transform-based image reconstruction is standard practice in MRI. However, when constraints need to be incorporated in the image reconstruction, iterative algorithms are usually utilized. One obvious drawback of iterative reconstruction methods is the long computation time and the fact that the iteration number is always finite and the image is unable to reach the optimal solution. This situation is similar to that in x-ray CT, where the FBP (filtered backprojection) algorithm is the work horse and iterative algorithms are called for if constrained reconstruction is demanded.

We must point out that the main focus of this paper is not on compressed sensing MRI, which considers undersampled k-space data and uses iterative total variation (TV) constrained techniques for image reconstruction (Wajer2001; Chang et al.2006; Lustig et al.2006,2008; Block et al.2007; Ye et al.2007). This paper considers a less ill-conditioned problem, where the k-space is undersampled at each time frame, but the k-space data are available at other time frames. Data at other time frames can assist the image reconstruction at current time frame. These data assisting image reconstruction methods have been intensively studied in our group and other groups (Cao and Levin,1995; Fessler and Noll,2004; Chen et al.2008; Adluru et al.2009; Francois et al.2009; Adluru,2010; Ramani and Fessler,2011). In our group, iterative spatio-temporal constrained reconstruction (STCR) methods use the L₁ norm for the spatial constraint and the L₁ or L₂ norm for the temporal constraint (Adluru et al.2009).

Recently, we developed a non-iterative FBP-MAP (maximum a posteriori) algorithm for low-dose x-ray CT image reconstruction (Zeng,2012). The FBP-MAP algorithm in Zeng (2012) emulates an iterative algorithm and can provide almost the same result that the iterative algorithm achieves at a given iteration number. The current paper uses a different (the calculus of variations) method to derive an FBP-MAP algorithm and applies the algorithm to the field of undersampled MRI, in which the k-space is radially sampled.

Since we do not know how to deal with the L₁ norm in an analytical algorithm, the L₁ norm (or TV norm) spatial constraint is not considered; the temporal constraint is considered using the L₂ norm. The temporal constraint encourages the reconstruction to look like a reference image, which can be the image of the dynamic images using the averaged adjacent data. The derivation of the algorithm based on the calculus of variations will be presented in Sec. II. A dynamic cardiac MRI study is then used to illustrate the feasibility of the proposed method. Finally, some conclusions are drawn at the end.

II. METHODS

The radially sampled k-space MRI data are in the Fourier domain. The k-space data are first converted into the spatial domain by the one-dimensional (1D) Fourier transform. The spatial-domain data have a real part and an imaginary part. Our strategy is to reconstruct a real-part image, Img_R, using the real-part spatial-domain data and to reconstruct an imaginary-part image, Img_I, using the imaginary-part spatial-domain data, separately. The final image is the norm of the complex image Img_R + jImg_I, that is, . The method of reconstructing Img_R and Img_I are identical. In the following sections, we will assume that the projection data are given in the spatial domain and are real, without specifying whether they are the real- or the imaginary part. Therefore, the image reconstruction method is the same as that for the Radon transform, which is the line-integral of the two-dimensional (2D) object.

A. Setup of the objective function

We consider the 2D Radon transform and its reconstruction. Let the image to be reconstructed be f(x, y) and its Radon transform be [Rf](s, θ), defined as

• (1)

where δ is the Dirac delta function, θ is the detector rotation angle, and s is the line-integral location on the detector. The Radon transform [Rf](s, θ) is the line-integral of the object f(x, y). The coordinate systems are shown in Figure 1. Image reconstruction is to solve for the object f(x, y) from its Radon transform [Rf](s, θ). A popular image reconstruction method is the FBP algorithm, which consists of two steps. In the first step of the FBP algorithm, [Rf](s, θ) is convolved with a kernel function h(s) with respect to variable s, obtaining q(s, θ). The Fourier transform of the kernel h(s) is the ramp-filter |ω|, where ω is the Fourier variable with respect to s. The second step performs backprojection, which maps the filtered projection q(s, θ) into the image domain as

• (2)

Figure 1. Coordinate systems for 2D Radon transform.¹ [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

A common concern of this FBP reconstruction is its poor noise performance and angular aliasing artifacts if insufficient view angles are available. To reduce the noise and artifacts, this paper enforces a prior term in the objective function, and this approach is commonly used in iterative algorithms.

Let the noisy line-integral measurements be p(s, θ). The objective function v depends on the function f(x, y) as follows:

• (3)

where the first term enforces data fidelity, the second term imposes prior information about the image f, the parameter β > 0 controls the level of influence of the prior information to the image f, and g is a reference image. If the k-space is sufficiently sampled, no extra information is needed to assist the reconstruction and parameter β can be chosen as 0. If the value of β is too large, the assisting secondary information may dominate the reconstruction and may washout the primary information carried by the data fidelity term. If the k-space is undersampled, a relatively larger value of β should be used to fill in some unmeasured information, and trial-and-error experiments may be needed in practice before a satisfactory value of β can be determined. Using integral expressions, v in Eq. (3) can be written as

• (4)

B. Optimization

This section uses the calculus of variations to find the optimal function f(x, y) that minimizes the objective function v(f) in Eq. (4). The initial step is to replace the function f(x, y) in Eq. (4) by the sum of two functions: f(x, y) + ϵ η(x, y) (van Brunt,2004). The next step is to evaluate and set it to zero (van Brunt,2004). That is,

• (5)

In practice, the function f(x, y) is compact, bounded, and continuous almost everywhere. After changing the order of integrals, we have

• (6)

Equation (6) is in the form of . Since η(x, y) can be any arbitrary function, according to the calculus of variations, one must have c(x, y) = 0, which is the Euler-Lagrange equation (van Brunt,2004). The Euler-Lagrange equation in our case is

• (7)

By moving some terms from the left-hand-side to the right-hand-side, (7) can be further rewritten as

• (8)

Notice that

• (9)

is the backprojection of the projection data p(s, θ), and the backprojection is denoted as b(x, y). It must be pointed out that b(x, y) is not the same as f(x, y) even when p(s, θ) is noiseless, because no ramp filter has been applied to p(s, θ).

Also notice that

• (10)

is the point spread function of the projection/backprojection operator at point (x, y) if the point source is at ( ) (Zeng,2009).

Using Eqs. (9) and (10), Eq. (8) becomes

• (11)

The left-hand-side of Eq. (11) is a 2D convolution. Taking the 2D Fourier transform of Eq. (11) yields

• (12)

or

• (13)

Here the capital letters are used to represent the Fourier transform of their spatial domain counterparts, which are represented in lowercase letters; ω_x and ω_y are the frequency variables for x and y, respectively.

Equation (13) is in the form of “backprojection first, then 2D filter” reconstruction approach and the Fourier domain 2D filter is (Zeng,2009). By using the central slice theorem (Zeng,2009), an FBP algorithm, which performs 1D filtering first, then backprojects, can be obtained, and the Fourier domain 1D filter is the central slice of the 2D filter , which gives

• (14)

When β = 0, Eq. (14) reduces to the ramp filter |ω| of the conventional FBP algorithm.

C. Implementation

In a typical FBP-type algorithm, the projection data are first filtered by the ramp filter |ω| and then are backprojected into the image domain. In our newly derived FBP-MAP algorithm, the conventional ramp filter |ω| is replaced by . Our backprojector is the same as that in the conventional FBP algorithm.

Notice the right-hand-side of Eq. (11) or (12), there are two components contributing to the input of the FBP-MAP algorithm. In Eq. (11), b(x, y) is the backprojection of the primary measurements p(s, θ), and g(x, y) is the reference image. The reference image g(x, y) can be expressed as the conventional FBP reconstruction from a set of secondary measurements p_g(s, θ). After ramp-filtering, p_g(s, θ) becomes q_g(s, θ). The combined input data for the FBP-MAP algorithm is the sum of p(s, θ) + β q_g(s, θ).

The implementation procedure of the proposed FBP-MAP algorithm is as follows:

Step 1: Prepare two sets of projection data: the primary set p(s, θ) and the secondary set p_g(s, θ).

Step 2: Apply the conventional ramp filter |ω| to the secondary projection set p_g(s, θ), obtaining q_g(s, θ).

Step 3: Form the new projection data set: p(s, θ) + β q_g(s, θ).

Step 4: Apply the newly derived FBP-MAP filter to the combined data set.

Step 5: Perform conventional backprojection.

In our implementation, the primary data are the current time frame data, while the secondary data are the summation of the data from the current time frame and the data from four other time frames.

The above-mentioned procedure is used to obtain the reconstruction Img_R from the real-part of the spatial-domain data and to obtain the reconstruction Img_I from the imaginary-part of the spatial-domain data. The final image is the norm of these two reconstructions, that is, .

C. Perfusion MRI study

To illustrate the feasibility of the proposed FBP-MAP algorithm, a human cardiac perfusion study was used (Adluru et al.2009). The MRI data were acquired with a Siemens 3T Trio scanner, using a phased array of coils, one of which was chosen to demonstrate the proposed method. The scanner parameters for the radial acquisition were TR = 2.6 ms, TE = 1.1 ms, flip angle = 12°, Gd dose = 0.03 mmol/kg, and slice thickness = 6 mm. Reconstruction pixel size was 1.8 x 1.8 mm². Each image was acquired in a 62 ms readout. The acquisition matrix size for an image frame was 256 x 24, and 60 sequential images were obtained at 60 different times. At each time frame, the k-space is sampled with 24 uniformly spaced radial lines over an angular range of 180°; however, the 24-line sampling patterns of the adjacent time frames are offset by 180°/96. The k-space sampling pattern is illustrated in Figure 2, where there are 16 possible radial lines, and at each time frame four lines are measured. The time sequence follows the pattern of A-B-C-D-A-B-C-D-… .

Figure 2. Illustration of k-space sampling pattern using an example with 16 possible radial lines. At each time frame, four lines are sampled. The sampling pattern follows the sequence of A-B-C-D-A-B-C-D- ….

If one sums up the measurements from temporally adjacent four time frames, the summed k-space will have a 96-line sampling pattern, uniformly distributed over an angular range of 180°. In our image reconstruction method, each time frame requires current 24-line measurement P and associated time-averaged 96-line measurement , which uses the measurements from the current 24-line data, two “immediately after” 24-line measurements, and two “immediately before” 24-line measurements. A symbolic expression for is given as

• (15)

The fact that one image was acquired with 24 views (i.e., 24 radial lines in the k-space) makes the k-space undersampled. In cardiac imaging, the object is in constant motion. The number of time frames to be used in the secondary data set should be as small as possible while still cover as much the k-space as possible. In our example, there are 96 possible radial k-space lines and each time frame measures 24 of those lines. That is, four total time frames are required to cover the 96 lines. In addition to the current measurement, the secondary data set needs additional data from other three time frames. To balance the “before” and “after” time frames, we chose two “immediate before” time frames and two “immediate after” time frames.

Both the conventional FBP algorithm and the proposed non-iterative FBP-MAP algorithm were used to reconstruct the images. In our MRI data acquisition, each k-space radial readout had 256 samples. After zero-padding, the length N was chosen as 1024, and the frequency variable ω took discrete values of 2πn/N, for integers n. Parameter β was chosen as 0.07, which was selected by experience and the noise level of a data set. Parameter β controls the influence of the reference image, and β = 0 implies that the reference image is not used.

III. RESULTS

Figure 3 shows the reconstruction results from the undersampled MRI data using both conventional FBP algorithm and proposed FBP-MAP algorithm. The images obtained by the convention method (see the 1st row of Fig. 3) suffer from severe angular aliasing artifacts and are noisy. The angular aliasing artifacts are significantly reduced and the noise is suppressed in the images obtained by the proposed FBP-MAP method (see the 2nd row of Fig. 3).

Figure 3. Comparison of conventional FBP reconstruction (1st row) and proposed FBP-MAP reconstruction (2nd row), using undersampled dynamic MRI data. The 3rd row shows the difference images between the 1st and 2nd rows. Six time frames are shown from left to right.

There are 60 time frames in the study, and the results from a subset of six time frames are shown in Figure 3, where the time spacing is 10 time frames. Because of the nature of dynamic imaging, the image intensity of a series of image changes over time. To view each individual reconstruction properly, each image in the 1st and 2nd rows is displayed from zero to its maximum value.

The 3rd row of Figure 3 shows the difference between the 1st row and the 2nd row. The purpose of the difference image is to check whether the constrained reconstruction is able to track the dynamic changes and shapes overtime. The images in the 3rd row are obtained as follows. Both of the FBP images and FBP-MAP images are first normalized. If one treats the time as the third dimension, the normalization procedure is three-dimensional, with the first two dimensions as spatial and the third dimension as temporal. The normalized “3D” FBP image has a maximum value of unity and so does the normalized “3D” FBP-MAP image. The “3D” difference image is calculated as the normalized FBP image minus the normalized FBP-MAP image. The difference image for all time frames is displayed using a common gray-scale interval of [−0.2, 0.2] in the 3rd row of Figure 3. In the cardiac region, the difference image is almost zero, implying that the constrained reconstruction faithfully follows the intensity change and the cardiac motion as represented by the FBP reconstruction.

As a comparison study, the secondary data were used to directly reconstruct the image. This approach is the simple interpolation method, which interpolates the undersampled k-space using neighbor time frames. The results from this comparison study are shown in Figure 4. The format of Figure 4 is the same as that in Figure 3. The image intensity of the reconstructions using this simple interpolation method does not follow the true intensity as indicated by the direct reconstruction method from one time frame to another. This can be seen from the difference images in Figure 4, where cardiac region has an obvious darker region.

Figure 4. Comparison of conventional FBP reconstruction (1st row) and simple interpolation direct reconstruction (2nd row). The 3rd row shows the difference images between the 1st and 2nd rows, and the cardiac region appears darker (i.e., not accurate). Six time frames are shown from left to right.

IV CONCLUSIONS

This paper introduces an FBP-MAP algorithm, based on minimizing an objective function that contains a data fidelity term and a constraint (Bayesian) term. The constraint encourages the image solution to look like a short-term time-average image in a dynamic data acquisition. Since no spatial constraint is used in the proposed FBP-MAP algorithm, the FBP-MAP images do not look as sharp as those obtained by the iterative algorithm that also contains the TV constraint. It is our future research goal to include the spatial constraint into the FBP-MAP algorithm. This new algorithm is Fourier transform-based and is non-iterative. The most distinguished advantage of this new algorithm over the iterative algorithms is its fast computation time and its capability to reach the optimal solution.

This paper also makes an important observation when the primary and secondary projection data sets are to be combined. These two data sets cannot be combined by a simple weighted summation. The secondary (that is, less important) data set needs to be filtered by a ramp filter before the weighted summation can be performed, in which the DC component is removed and the low frequency components are significantly suppressed.

Because the object is in constant motion, we propose to use the minimum number of time frames in the secondary data set that can still cover the possible k-space lines. The parameter β should be small enough not to let the secondary data dominate the reconstruction, and large enough to fill in some missing information that the primary data lack. Unfortunately, there is no explicit formula to determine β. Trial-and-error experiments may be needed to select a satisfactory value of β.

Finally, we point out that the proposed approach is different from the HYPR-type methods (Mistretta et al.2006), which were derived in an ad hoc manner and do not work well when the object is in constant motion. However, our method is analytically derived to be the optimal solution of a minimization problem and is able to track the object motion.