Chapter 14

Sparse Sampling in MRI

Philip J. Bones and Bing Wu

14.1 Introduction

The significant time necessary to record each resonance echo from the volume being imaged in magnetic resonance imaging (MRI) has led to much effort to develop methods which take fewer measurements. Faster methods mean less time for the patient in the scanner, increased efficiency in the use of expensive scanning facilities, improved temporal resolution in studies involving moving organs or flows, and they lessen the probability that patient motion adversely affects the quality of the images. Images like those of the human body possess the property of sparsity, that is the property that in some transform space they can be represented much more compactly than in image space. The technique of compressed sensing, which aims to exploit sparsity, has therefore been adapted for use in MRI. This, coupled with the use of multiple receiving coils (parallel MRI) and the use of various forms of prior knowledge (e.g., support constraints in space and time), has resulted in significantly faster image acquisitions with only a modest penalty in the computational effort required for reconstruction. We describe the background motivation for adopting sparse sampling and show evidence of the sparse nature of biological image data sets. We briefly present the theory behind parallel MRI reconstruction, compressed sensing and the application of various forms of prior knowledge to image reconstruction. We summarize the work of other groups in applying these concepts to MRI and our own contributions. We finish with a brief conjecture on the possibilities for future development in the area.

P.J. Bones ( )

University of Canterbury, Christchurch, New Zealand e-mail: [e-mail]

G. Dougherty (ed.), Medical Image Processing: Techniques and Applications, Biological

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and Medical Physics, Biomedical Engineering, DOI 10.1007/978-1-4419-9779-1 14, © Springer Science+Business Media, LLC 2011

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14.1.1 Magnetic Resonance Imaging

MRI is the term used to represent the entire set of methods which apply the principles first developed for chemistry as nuclear magnetic resonance in such a way that the spatial variation of a property within an object is observed. A strong and extremely uniform magnetic field is applied to the object. Under the influence of the field those atoms in the object which have a magnetic spin property align in the direction of the magnetic field in one of two orientations such that a small net magnetization occurs. The atoms exhibit a resonance at a frequency, which is linearly dependent on the magnetic field strength. The resonance can be excited by means of a radiofrequency (RF) pulse at the appropriate frequency and the atoms which have been excited precess about an axis aligned with the direction of the magnetic field. After excitation, the precessing atoms relax back to equilibrium and in the process generate a small, but measurable, RF field – an “echo.”

By imposing a gradient in magnetic field strength as a linear function of one of the Cartesian space coordinates, z say, it is possible to encode that spatial coordinate in the resonance frequencies of the spins. By considerable extension of this basic idea, signal processing of the signals recovered from echoes after a specific sequence of gradient impositions with respect to the x, y, and z directions, coupled with RF excitations, and echo signal acquisitions, allows the formation of an image of the interior of the object. Many excitation and acquisition sequences have been devised. Because of the relationship between resonance frequency and magnetic field strength, virtually all of them make measurements in spatial frequency space, or “k-space” as the MRI community generally refers to it. Moreover, tissues in the body can be characterized in terms of the time constants associated with the atomic resonances, known as “T1” and “T2”. The differences between tissue responses help to make MRI effective in distinguishing between them. For a good overview of the basis of MRI, see [1] and for a comprehensive review of MRI sequences and algorithms, see [2].

While MRI also has applications in biological science and in the study of materials, it is its role in medicine that has led to a whole industry. The size of the annual meeting of the International Society for Magnetic Resonance in Medicine (ISMRM, http://ismrm.org) is testament to the extraordinary interest in the imaging modality. The reason that MRI has had such a profound effect on the practice of modern medicine is because of the exquisite detail that it has achieved in images of soft tissues within the human body. In this, it is quite complementary to X-ray computed tomography, which is particularly good at imaging harder tissues, notably bone. The two imaging modalities thus coexist and many patients are imaged using both for some diagnoses. Note that there are no known detrimental effects on the human body caused by MRI scanners of the sort in regular use, while the use of X-ray computed tomography is strictly limited by the dose of ionizing radiation the patient receives in such a scanner.

The use of MRI is restricted in one specific way: the physical processes involved in the excitation and reception of MR signals are inherently quite slow. Thus, the time taken to invoke a specific sequence and to measure the signals that are

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generated is measured in a matter of milliseconds per pulse. Note that this has nothing to do with the electronics associated with the scanner – faster hardware does not solve the problem. To take a complete set of measurements for, say, a 3D imaging study of the brain may take many minutes; some acquisition sequences require periods approaching 1 h [2]. As well as reducing the throughput of an expensive facility, the slowness of acquisition limits how useful MRI is in imaging organs where motion is intrinsic, most notably the heart and circulatory system; even when these organs are not specifically the target for a particular imaging study, their activity may affect the success of imaging nearby or associated organs. The efforts of the authors and of many others involved in MRI research are directed toward developing smarter algorithms to attempt to reconstruct useful images from fewer measurements and therefore in less time.

14.1.2 Compressed Sensing

The conventional wisdom in signal processing is that the sampling rate for any signal must be at twice the maximum frequency present in the signal. The Sampling Theorem is variously attributed to Whittaker, Nyquist, Kotelnikov, and Shannon and its conception represents a very significant landmark in the history of signal processing. However in the work performed in recent years related to signal compression, it has become obvious that the total amount of information which is needed to represent a signal or image to high accuracy is in many cases much less than that implied by the “Nyquist limit.” This is nowhere more apparent than in the modern digital camera where quite acceptable images can be stored and recreated from a small fraction of the data volume that was associated with the original image sampling. The total amount of information acquired, 4 megapixels at 24-bit per pixel for example, may often be compressed to several hundred thousand bytes by the JPEG compression method without appreciable loss of image quality. The image property that lies behind this compressibility is “sparsity”: the fact that under some transformation many of the data values in the space associated with the transform can be set to zero and the image reconstructed from the rest of the data values without appreciable effect. An image which is very sparse (under that transformation) is one for which the number of nonzero values is relatively low.

The technique of compressed sensing (also known as “compressive sensing”) was introduced to exploit image sparsity [3, 4]. Consider a 2D image with N pixels represented by the vector x and suppose that it can be accurately represented by K N data values under the linear transformation y = Φ x. Rather than measuring the N pixel values and then performing the transformation, we seek to make just M measurements m, where K ≤ M N. Thus, m = Ψy, where Ψ is a measurement matrix of dimension M × K. While this might be of little direct benefit in the case cited above of a modern digital camera, for which the design of the sensor is most straightforwardly implemented as a regular 2D array of individual pixel detectors, there are many other applications, notably including MRI, for which making fewer

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Fig. 14.1 Comparing (a) conventional image sensing and compression to (b) compressed sensing. In (a), the image is sampled at the Nyquist sampling rate and stored. Then all of the smallest coefficients in the image’s wavelet transform are discarded to reduce the storage volume. In (b), the significant coefficients of the wavelet transform are directly estimated from a lesser number of samples of the image

measurements does offer an advantage. In the remainder of this chapter, we show how the exploitation of sparsity by means of compressed sensing methods has considerable potential in speeding up MRI image acquisition.

An illustration of how compressed sensing might work for the general optical imaging case is shown in Fig. 14.1. Suppose that the well-known cameraman image is being acquired with a conventional digital camera in Fig. 14.1a and a full set