Compressive Acquisition - Sparse Representations and Compressive Sensing for Imaging and Vision - page 18

Fig. 3.3 512 × 512 Shepp-Logan Phantom image and its edges

One of the major limitations of the MRI is the linear relation between the
number of measured data and scan time. As a result MRI machines tend to be slow,
claustrophobic, and generally uncomfortable for patients. It would be beneficial for
patients if one could significantly reduce the number of measurements that these
devices take in order to generate a high quality image. Hence, methods capable
of reconstructing from such partial sample sets would greatly reduce a patient's
exposure time.
The theory of CS can be used to reduce the scan time in MRI acquisition by
exploiting the transform domain sparsity of the MR images [79], [80], [108]. The
standard techniques result in aliasing artifacts when the partial Fourier measure-
ments are acquired. However, using sparsity as a prior in the MR images, one
can reconstruct the image using the sparse recovery methods without the aliasing
artifacts. While most MR images are not inherently sparse, they are sparse with
respect to the total variation (TV). Most CS approaches to recovering such images
utilize convex programs similar to that of Basis Pursuit. Instead of minimizing the

1 norm of the image subject to Fourier constraints, such programs minimize the
TV semi-norm which enforces the necessary total variational sparsity of the solution
(see [23,35,151], and others). While this methodology yields a significant reduction
in the number of Fourier samples required to recover a sparse-gradient image, it
does not take advantage of additional sparsity that can be exploited by utilizing the
two horizontal and vertical directional derivatives of the image. An example of a
sparse-gradient image along with an image of its edges is shown in Fig. 3.3 .
An interesting approach to the problem of recovering a sparse gradient image
from a small set of Fourier measurements was proposed in [108]. By using the
fact the Fourier transform of the gradients of an image are precisely equal to a
diagonal transformation of the Fourier transform of the original image, they utilize
CS methods to directly recover the horizontal and vertical differences of the desired
image. Then, integration techniques are performed to recover the original image
from the edge estimates.