Sparse Reconstruction Techniques in Magnetic Resonance Imaging: Methods, Applications, and Challenges to Clinical Adoption

Abstract

The family of sparse reconstruction techniques, including the recently introduced compressed sensing framework, has been extensively explored to reduce scan times in magnetic resonance imaging (MRI). While there are many different methods that fall under the general umbrella of sparse reconstructions, they all rely on the idea that a priori information about the sparsity of MR images can be used to reconstruct full images from undersampled data. This review describes the basic ideas behind sparse reconstruction techniques, how they could be applied to improve MRI, and the open challenges to their general adoption in a clinical setting. The fundamental principles underlying different classes of sparse reconstructions techniques are examined, and the requirements that each make on the undersampled data outlined. Applications that could potentially benefit from the accelerations that sparse reconstructions could provide are described, and clinical studies using sparse reconstructions reviewed. Lastly, technical and clinical challenges to widespread implementation of sparse reconstruction techniques, including optimization, reconstruction times, artifact appearance, and comparison with current gold standards, are discussed.

Figure 1

(a) Fully sampled k-space is converted to an image via the Fourier Transform. (b) Cartesian undersampling by factor of 2, where every other line of k-space is missing. The resulting image is corrupted by aliasing or fold-over artifacts. (c) Random Cartesian undersampling. If k-space points are skipped in such random fashion, the resulting aliasing artifacts no longer appear as distinct replicas of the image (as in (b)), but instead as blurring or noise-like artifacts. This type of sampling is typically not realizable in 2D. (d) Radial undersampling by factor of 9. The resulting image contains streak artifacts due to the undersampling, but the bulk of the image is recognizable since the center of k-space is well sampled.

Figure 2

(a) An example of a sparse MR image, specifically a single partition subtraction image from a 3D abdominal and pelvic contrast enhanced MR angiogram, where few of the image pixels contain signal (and are bright), but most of the image pixels are near zero. (b) When the k-space data are undersampled, the resulting image is no longer sparse due to the presence of aliasing artifacts. If it is known a priori that the image should be sparse, a sparse reconstruction can be used to recover the original image.

Figure 3

(a) An axial brain image. (b) Horizontal finite differences transform of the image shown in (a). This transform shows the differences between neighboring pixels, and thus highlights edges in the image. Many images are sparse, i.e. contain more pixels with near-zero values, after such a spatial finite differences transform. (c) Wavelet transform of the image shown in (a), where the transformed images are again sparse compared to the untransformed image (a).

Figure 4

(a) A time series of dynamic cardiac images is shown, where the x-y plane shows the images, and the t axis depicts the time dimension (left). Three different images of the heart show the motion that occurs through time (right). (b) When one frame from the dynamic dataset is subtracted from another, the result is a sparser image because many tissues are stationary from frame to frame. Only pixels near the heart change, and these are reflected in the temporal difference image. (c) The 3D space-time data can be examined along one slice of the x-t plane (left), designated by the white dotted line. The areas with significant motion can be clearly seen in the “x-t space” image (middle). If a Fourier transform is applied to the x-t space in the time direction, the result is the image representation in the spatiotemporal domain (x-f space, right). Static areas of the image will only have a single bright pixel in the x-f space (orange arrow), and only pixels with significant motion will contribute many non-zero pixels (purple arrow), making x-f space sparse in many dynamic applications.

Figure 5

(top) By sampling data using an undersampled radial scheme, dynamic images of low spatial resolution but high temporal resolution can be collected. These images show the enhancement in different regions over time, but are not usable because of their low resolution. (middle) By gathering all acquired radial projections from each timeframe, a high spatial resolution composite image can be created. This single static image displays all enhanced vessels at a high resolution, but does not show the order in which they enhance. HYPR works by combining spatial information from the high-resolution composite image with temporal information from the low-resolution dynamic images. This reconstruction yields a time series of high spatial and high temporal resolution images (bottom row) that shows dynamic enhancement of the vessels.

Figure 6

A schematic of the k-t BLAST reconstruction technique. Both undersampled data with high spatial and temporal resolution (a, top) and fully-sampled low resolution data (a, bottom) are collected. The undersampled accelerated data are used to generate the image, and the low resolution data serve as training data for the k-t BLAST reconstruction. After using Fourier transform to convert the datasets into the image domain (b), the aliasing in the undersampled data can be seen (b, top). By applying a Fourier transform in time, the data are converted to the x-f domain (c). If the undersampled data are collected using the k-t interleaved sampling pattern, the duplicates seen in the x-f domain will be offset in the undersampled data (c, top). However, the low resolution training data do not show aliasing in the x-f domain because they are fully-sampled (c, bottom). A knowledge of the structure of the x-f training data (indicated by the dashed shape in x-f space) can be used to remove aliasing in the high resolution aliased x-f space (d). The final images are generated by applying an inverse Fourier transform to the reconstructed high resolution x-f data (e).

Figure 7

A schematic of a simplified compressed sensing reconstruction. Randomly undersampled k-space data are collected (bottom left), leading to an image which exhibits noise-like aliasing artifacts (top left). A sparse representation of this undersampled image can be obtained by applying a sparsifying transform such as finite differences, which highlights edges. Because the artifacts look like noise, it is possible to retain most of the significant pixels in the sparse image while removing some noise-like artifacts by thresholding this image. After thresholding, only true edges remain, although some may have been lost in the thresholding process. An updated image is generated by “undoing” the sparsifying transform, and this updated image with reduced aliasing artifacts is converted back to k-space. This k-space may contain data that is different from the collected k-space data after the thresholding step. Therefore, to ensure data consistency, the original k-space data are reinserted into the k-space of the updated image. This updated k-space is then converted into an image, which is now both consistent with the original data and contains reduced aliasing artifacts, and the loop begins again. When the error between the previous iteration of the updated image and the current iteration reaches the stopping criteria set by the user, the iteration loop is broken and the final image is output.

Figure 8

(a) From a set of rapidly collected undersampled T₁-weighted images, one pixel is selected and examined through time (b). With a knowledge of the underlying physics, a model can be used to calculate all possible signal timecourses for different values of T₁. In this case, the dictionary consists of signals which could arise from T₁ values of 300ms, 500ms, or 1000ms. The actually measured signal is then compared to the signals in the dictionary to find the closest match (d), in this case, the curve for T₁=500ms. (e) The value of 500ms is then assigned as the T₁ value for this pixel. The process is repeated for every pixel of interest to produce a T₁ map.

Figure 9

(left) An image generated using a sparse reconstruction with properly tuned parameters. The image appears clear and with no obvious aliasing artifacts remaining. (right) An image generated with the same data and sparse reconstruction but with a larger regularization term. The overemphasis on the sparsifying transform, in this case total variation, leads to a blurry reconstruction.