### Abstract

- Top of page
- Abstract
- INTRODUCTION
- THEORY
- METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- REFERENCES
- APPENDIX A: NONLINEAR CONJUGATE-GRADIENT SOLUTION OF THE CS OPTIMIZATION PROCEDURE
- APPENDIX B: DERIVATION OF THE INTERFERENCE STANDARD DEVIATION FORMULA

The sparsity which is implicit in MR images is exploited to significantly undersample *k*-space. Some MR images such as angiograms are already sparse in the pixel representation; other, more complicated images have a sparse representation in some transform domain–for example, in terms of spatial finite-differences or their wavelet coefficients. According to the recently developed mathematical theory of compressed-sensing, images with a sparse representation can be recovered from randomly undersampled *k*-space data, provided an appropriate nonlinear recovery scheme is used. Intuitively, artifacts due to random undersampling add as noise-like interference. In the sparse transform domain the significant coefficients stand out above the interference. A nonlinear thresholding scheme can recover the sparse coefficients, effectively recovering the image itself. In this article, practical incoherent undersampling schemes are developed and analyzed by means of their aliasing interference. Incoherence is introduced by pseudo-random variable-density undersampling of phase-encodes. The reconstruction is performed by minimizing the ℓ^{1} norm of a transformed image, subject to data fidelity constraints. Examples demonstrate improved spatial resolution and accelerated acquisition for multislice fast spin-echo brain imaging and 3D contrast enhanced angiography. Magn Reson Med, 2007. © 2007 Wiley-Liss, Inc.

### INTRODUCTION

- Top of page
- Abstract
- INTRODUCTION
- THEORY
- METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- REFERENCES
- APPENDIX A: NONLINEAR CONJUGATE-GRADIENT SOLUTION OF THE CS OPTIMIZATION PROCEDURE
- APPENDIX B: DERIVATION OF THE INTERFERENCE STANDARD DEVIATION FORMULA

Imaging speed is important in many MRI applications. However, the speed at which data can be collected in MRI is fundamentally limited by physical (gradient amplitude and slew-rate) and physiological (nerve stimulation) constraints. Therefore, many researches are seeking for methods to reduce the amount of acquired data without degrading the image quality.

When *k*-space is undersampled, the Nyquist criterion is violated, and Fourier reconstructions exhibit aliasing artifacts. Many previous proposals for reduced data imaging try to mitigate undersampling artifacts. They fall in three groups: (a) Methods generating artifacts that are incoherent or less visually apparent, at the expense of reduced apparent SNR (1–5); (b) Methods exploiting redundancy in *k*-space, such as partial-Fourier, parallel imaging, etc. (6–8); (c) Methods exploiting either spatial or temporal redundancy or both (9–13).

In this article we aim to exploit the sparsity which is implicit in MR images, and develop an approach combining elements of approaches *a* and *c*. By implicit sparsity we mean transform sparsity, i.e., the underlying object we aim to recover happens to have a sparse representation in a known and fixed mathematical transform domain. To begin with, consider the identity transform, so that the transform domain is simply the image domain itself. Here sparsity means that there are relatively few significant pixels with nonzero values. For example, angiograms are extremely sparse in the pixel representation. More complex medical images may not be sparse in the pixel representation, but they do exhibit transform sparsity, since they have a sparse representation in terms of spatial finite differences, in terms of their wavelet coefficients, or in terms of other transforms.

Sparsity is a powerful constraint, generalizing the notion of finite object support. It is well understood why support constraints in image space (i.e., small FOV or band-pass sampling) enable sparser sampling of *k*-space. Sparsity constraints are more general because nonzero coefficients do not have to be bunched together in a specified region. Transform sparsity is even more general because the sparsity needs only to be evident in some transform domain, rather than in the original image (pixel) domain. Sparsity constraints, under the right circumstances, can enable sparser sampling of *k*-space as well (14, 15).

The possibility of exploiting transform sparsity is motivated by the widespread success of data compression in imaging. Natural images have a well-documented susceptibility to compression with little or no visual loss of information. Medical images are also compressible, though this topic has been less thoroughly studied. Underlying the most well-known image compression tools such as JPEG, and JPEG-2000 (16) are the discrete cosine transform (DCT) and wavelet transform. These transforms are useful for image compression because they transform image content into a vector of sparse coefficients; a standard compression strategy is to encode the few significant coefficients and store them, for later decoding and reconstruction of the image.

The widespread success of compression algorithms with real images raises the following questions: Since the images we intend to acquire will be compressible, with most transform coefficients negligible or unimportant, is it really necessary to acquire all that data in the first place? Can we not simply measure the compressed information directly from a small number of measurements, and still reconstruct the same image which would arise from the fully sampled set? Furthermore, since MRI measures Fourier coefficients, and not pixels, wavelet, or DCT coefficients, the question is whether it is possible to do the above by measuring only a subset of *k*-space.

A substantial body of mathematical theory has recently been published establishing the possibility to do exactly this. The formal results can be found by searching for the phrases *compressed sensing* (CS) or *compressive sampling* (14, 15). According to these mathematical results, if the underlying image exhibits transform sparsity, and if *k*-space undersampling results in incoherent artifacts in that transform domain, then the image can be recovered from randomly undersampled frequency domain data, provided an appropriate nonlinear recovery scheme is used.

In this article we aim to develop a framework for using CS in MRI. To keep the discussion as short and simple as possible, we focus this work only on Cartesian sampling. Since most product pulse sequences in the clinic today are Cartesian, the impact of Cartesian CS can be substantial. We keep in mind though, that non-Cartesian CS has great potential and may be even more advantageous than Cartesian for some applications. Some very promising results for radial and spiral imaging have been presented by (17–21).