Synopsis

Incomplete data sampling is an attractive approach to accelerate MRI but it requires prior knowledge-driven image reconstruction. Sparsity is a powerful concept that allows linking many different types of prior knowledge to the mathematical apparatus adopted in MR image reconstruction. Compressed sensing theory establishes conditions for optimal use of sparse representations for high quality MR image reconstruction from undersampled data. In this talk, we will cover the aforementioned concepts of advanced image reconstruction and demonstrate real examples of accelerated structural and dynamic MRI. We will also discuss both theoretical requirements of compressed sensing and essential aspects of its practical implementation.

Highlights

• Incomplete data sampling is an attractive approach to accelerate MRI but requires prior knowledge-driven MR image reconstruction.
• Sparsity is a powerful concept that allows linking many different types of prior knowledge to the mathematical apparatus adopted in MR image reconstruction.
• Compressed sensing theory establishes criteria relating image sparsity and admissible undersampling strategies, and allows establishing feasible image reconstruction algorithms from incomplete data.

Target Audience

Outcome/Objectives

• Develop a basic understanding of how image acquisition can be accelerated by undersampling strategies and how employing prior knowledge of object sparsity can be utilized to restore missing data and obtain MR images of diagnostic quality
• Develop a basic understanding of essential results of the compressed sensing theory

Introduction

In MRI, a digital image needs to be estimated from a discrete set of measurements, each sample representing an integral of the image modulated by encoding function composed of magnetic field gradient and coil sensitivity contributions. In the matrix form, the set of measurements (i.e., signal vector) is modeled by

\begin{equation}s=Ef+\epsilon,\qquad\qquad\qquad[1]\end{equation}

where $$$f$$$ is the column vector of length $$$M$$$ corresponding to the image to be estimated, $$$E$$$ is the encoding matrix, and $$$\epsilon$$$ is measurement noise. Assuming that the problem is overdetermined (i.e., the number of linearly independent rows, or row rank of $$$E$$$, is higher than the number of unknowns $$$M$$$) and assuming independent noise, an SNR-optimized image reconstruction can be accomplished in the least squares sense:

$$f=\arg\min_f\|Ef-s\|_2^2,\qquad\qquad[2]$$

where $$$\|\cdot\|_2$$$ is $$$l_2$$$ norm defined as

$$\|x\|_p=\left(\sum_k|x|^p\right)^{1/p}\qquad\qquad[3]$$

with $$$p=2$$$.

Achieving higher spatial and/or temporal resolutions in MRI often necessitates incomplete data sampling, which, in turn, leads to poorly conditioned and, in the limiting case, underdetermined system of equations in Eq. [1] (row rank ($$$E$$$) < $$$M$$$), which generally have an infinite number of possible solutions. A common approach to isolate a feasible solution is to incorporate additional prior information about the problem at hand to restore images of diagnostic quality from the undersampled data (1-3). This, in turn, requires modifications to the least square estimation problem in Eq. [2] and leads to specialized reconstruction algorithms. In this presentation, we will describe a subset of such algorithms based on principles of sparse optimization.

Sparsity in MRI

Mathematical Measures of Sparsity and Their Use in Image Reconstruction

Ideally, the sparsity of an image is measured by its $$$l_0$$$ norm that counts the number of non-zero pixels. (Note that despite the name $$$l_0$$$ norm is technically not a norm as it does not satisfy all the necessary axioms). If we know in advance that the underlying image or its transformation is expected to be sparse, then the problem of MR image estimation from undersampled data may be modified by combining data consistency (Eq. [2]) and sparsity information in the form of additional regularization term as follows:

$$f=\arg\min_f\left(\|Ef-s\|_2^2+\lambda\|\Phi f\|_0\right).\qquad\qquad[4]$$

The ability of $$$l_0$$$ norm regularization to isolate a sparse solution is illustrated in Fig. 2a. However, the practical implementation of $$$l_0$$$-based problems is complicated due to combinatorial complexity of suitable optimization algorithms. The use of alternative norms may not always lead to sparse solutions. For example, Figure 2b demonstrates that using a more traditional $$$l_2$$$ norm results in a non-sparse solution. However, as can be seen from Figure 2c, optimization based on $$$l_1$$$ norm also produces a sparse solution and, importantly, is computationally tractable. Establishing a set of conditions warranting equivalency of $$$l_0$$$ and $$$l_1$$$ norm optimization was one of the milestones of the compressed sensing theory described below.

Compressed Sensing Theory

Recently, a novel mathematical theory named compressed sensing (CS) has been developed (4) that states that sparse images can be accurately reconstructed from undersampled datasets, provided the encoding matrix $$$E$$$ satisfies certain conditions. The first of these conditions, called Restricted Isometry Principle (RIP), requires that most of the energy of a sparse signal should be preserved with measurements performed via the encoding matrix $$$E$$$. As such, RIP guarantees that different sparse images produce different sets of measurements and thus can be differentiated in the reconstruction process. In practice, RIP can be achieved by incoherence of the sampling pattern and, in MRI applications, can be well approximated by randomized or non-Cartesian acquisitions. The second theoretical condition of CS relates the sparsity level of the signal with admissible acceleration factors. Under these conditions, CS theory proves that the $$$l_0$$$ and $$$l_1$$$ minimization problems result in equivalent solutions, i.e. we can solve the following problem instead of the one in Eq. [4]:

$$f=\arg\min_f\left(\|Ef-s\|_2^2+\lambda\|\Phi f\|_1\right)\qquad\qquad[5]$$

While historically a number of algorithms in the form of Eq. [5] were proposed earlier, including MRI algorithms (5-7), the compressed sensing theory was the first to formulate sufficient conditions on the number of samples and the encoding matrix to guarantee accurate sparse signal recovery, and later was applied in the context of MRI (8).

Conclusion

Acknowledgements

References

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