Technical noteCausal MRI reconstruction via Kalman prediction and compressed sensing correction
Introduction
This paper aims at causal reconstruction of dynamic Magnetic Resonance Images. There are broadly two classes of dynamic MRI reconstruction methods – offline and online. Offline methods reconstruct the images in retrospective, i.e. after all the data (pertaining to all time frames) has been acquired. Online methods reconstruct the images as and when each time frame is acquired.
Most previous studies in dynamic MRI reconstruction employed offline methods [1], [2], [3], [4], [5]; the reconstruction was retrospective. These studies exploited the spatio-temporal correlation of the MRI data in order to express it in a sparse fashion in some transform domain; Compressed Sensing (CS) based techniques were employed to recover the dynamic MRI sequence.
For most applications the offline reconstruction is satisfactory, but in cases where the frames need to be visualized in real-time, for example in any tracking application or image guided surgery, offline techniques cannot be employed. The focus of this work is on online/causal reconstruction. The images for each time-frame needs to be reconstructed individually, given the reconstructed frames till the previous instant. The number of studies on this topic is limited; they will be discussed in the next section.
In this work, we propose a new technique for causal dynamic MRI reconstruction. Since the frames till the previous instant are available, we use Kalman filter prediction for the current frame. If the difference between the predicted frame and the actual current frame is normally distributed, a Kalman filter based correction would be optimum. We will show that, such is not the case. The difference between the actual and the predicted frame is sparse. Thus we employ a CS based correction step.
The rest of the paper is organized into several sections. The following section discusses prior art on causal reconstruction. Section 3 details our proposed approach. The experimental results are given in Section 4. The conclusions of this work are discussed in Section 5.
Section snippets
Review of literature
We will not discuss offline dynamic MRI reconstruction techniques here; the interested reader can peruse chapter 4 of [6]. In the rest of this section we will discuss all the previous studies on causal dynamic MRI reconstruction.
The most straight-forward approach to causal dynamic MRI reconstruction is to apply a dynamic modelling technique on the sequence. This is exactly what has been done in [7], [8], [9]; basically all these studies employ the Kalman filtering technique. In [7], [8] the
Proposed approach
In this paper, we follow the basic approach outlined in [13], [14]. A reference frame is predicted for the current instant given the reconstructed frames till the previous instant. The difference between the predicted frame and the actual current frame is corrected using the k-space samples of the current instant.
Here xp denotes the predicted frame. In [13] this was simply kept as the previous frame. A simple linear predictor was used in [14] based on an auto-regressive model. In this
Experimental evaluation
The experiments were carried out on a laptop with an AMD 64 bit processor having 4GB of RAM. The simulations were done in Matlab 2009a environment running on Windows 7.
The experimental evaluation was performed on five sets of data. Two myocardial perfusion MRI datasets were obtained from [19]. We will call these two sequences as Cardiac Perfusion Sequence 1 and 2. The data was collected on a 3T Siemens scanner. Radial sampling trajectory was used; 24 radial sampling lines were acquired for each
Conclusion
In this work, we have proposed a causal real time dynamic MRI reconstruction method. In causal reconstruction, use can only be made of previous frames and their k-space data but not the k-space data for the future frames. Our method proceeds in two stages. In the prediction stage a Kalman filter based prediction is made of the current frame. In the correction stage the difference between the prediction and the correction is made; the difference image is sparse and hence a greedy sparse recovery
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