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, ImgR, using the real-part spatial-domain data and to reconstruct an imaginary-part image, ImgI, using the imaginary-part spatial-domain data, separately. The final image is the norm of the complex image ImgR + jImgI, that is, . The method of reconstructing ImgR and ImgI 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
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
Figure 1. Coordinate systems for 2D Radon transform.1 [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
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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:
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
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,
In practice, the function f(x, y) is compact, bounded, and continuous almost everywhere. After changing the order of integrals, we have
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
By moving some terms from the left-hand-side to the right-hand-side, (7) can be further rewritten as
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
is the point spread function of the projection/backprojection operator at point (x, y) if the point source is at ( ) (Zeng,2009).
The left-hand-side of Eq. (11) is a 2D convolution. Taking the 2D Fourier transform of Eq. (11) yields
Here the capital letters are used to represent the Fourier transform of their spatial domain counterparts, which are represented in lowercase letters; ωx and ωy are the frequency variables for x and y, respectively.
When β = 0, Eq. (14) reduces to the ramp filter |ω| of the conventional FBP algorithm.
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 pg(s, θ). After ramp-filtering, pg(s, θ) becomes qg(s, θ). The combined input data for the FBP-MAP algorithm is the sum of p(s, θ) + β qg(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 pg(s, θ).
Step 2: Apply the conventional ramp filter |ω| to the secondary projection set pg(s, θ), obtaining qg(s, θ).
Step 3: Form the new projection data set: p(s, θ) + β qg(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 ImgR from the real-part of the spatial-domain data and to obtain the reconstruction ImgI 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- ….
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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.