## Superresolution of magnetic resonance images

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##### Author

##### Date

2000##### Permanent Link

http://hdl.handle.net/10092/6882##### Thesis Discipline

Electrical Engineering##### Degree Grantor

University of Canterbury##### Degree Level

Doctoral##### Degree Name

Doctor of PhilosophyMagnetic resonance imaging (MRI) is used to image parts of the body using only electromagnetic interaction with the body's own atomic nuclei (hydrogen protons, in particular). The complex amplitude is directly measured at arbitrary locations in 2-D spatial frequency space; usually the locations are arranged in a square grid to allow reconstruction using the fast Fourier transform. The resolution of the obtained image is proportional to the number of locations and hence to the time allowed for data acquisition; and the bandlimit imposed by restricting the acquisition time is more severe in one of the coordinate directions than the other. To increase the efficiency of using expensive MR scanners and to reduce the time during which a patient must remain still, computational methods are sought to superresolve MR images. Superresolution is commonly defined as the recovery of spatial frequency information of the object beyond the bandlimit imposed by the transfer function of the imaging system [Hunt 1995]. The well-known and straightforward Gerchberg-Papoulis (G-P) algorithm [Gerchberg 1974] was used to superresolve two sets of data, a simulated MR brain image and a phantom head image. These images were blurred by bandlimiting the spectrum (i.e. removing some of the high frequencies). Images resulting from applying the G-P algorithm were then compared to the corresponding original images. The algorithm was modified by making use of available information regarding the image, e.g. positivity, upper and lower bounds, and energy constraints, etc. The G-P algorithm and its modifications are special cases of the projection onto convex sets algorithm (POCS). The basic idea of POCS is that any prior information is used as a constraint on the image to lie in a closed convex set [Sezan and Stark 1982]. A more general form of the restoration algorithm, known as the method of generalized projections, extends the POCS method to utilizing nonconvex constraints, such as single level and neighbourhood based quantization. The different constraints were studied, and in some cases considerable improvement in the performance (as indicated by the improvement in the signal-to-noise ratio (ISNR)) and visually sharper images were achieved. Since the superresolution problem is ill-posed [Hunt 1995] (i.e. trivial perturbation in the recorded data may lead to nontrivial perturbations in the solution), regularization methods are required to transform an ill-posed problem to a well-posed one, whose solution is an approximation to that of the ill-posed problem. In the case where no information regarding the nature of the image was assumed, the well-known Tikhonov-Miller regularization method [Tikhonov and Arsenin 1977] was used to stabilize the problem by using a penalty function that represents a bound on the energy of the image. Other regularization methods such as constraining smoothness or limiting total variation (which either smooth out the solution or preserve edges) were studied. The choice of applying these methods depends on prior knowledge of the nature of the object. The second part of the research concentrated on recovery of undersampled images, where the noisy spectra were undersampled by multiplying them with different undersampling patterns. The algorithm, when applied on the undersampled images, in some cases resulted in useful recovery of even quite severely aliased regions. Both the determinacy of the system and the shape of the undersampling pattern used affected the amount of recovery achievable. A study was also made on the recovery of a region of interest, which may be applied in situations where a follow-up image or dynamic imaging is required. The iterative region of interest (iROI) algorithm was presented in two versions. In the first version, information from a high resolution reference image is used to superresolve a low resolution dynamic image. This required some prior approximate knowledge of the location and extent of the region of interest. The total variation method was also used to improve the edges of the region of interest whenever accurate information regarding the region of interest was unavailable. The second version of the iROI method superresolves a low resolution dynamic image using a low resolution reference image; the resultant image is not as good as for the other version but it can be achieved with less prior knowledge. Two other superresolving methods that utilize prior knowledge were investigated. The boxcar estimation method models the image by a series of boxcar functions with varying widths, locations and amplitudes. It results in sharp images with reduced Gibb's oscillation when applied to piecewise images. The nonconvex level penalty function utilizes prior knowledge of the levels present in the solution to superresolve an image. Both methods are shown to be useful when the image genuinely contains piecewise homogeneous regions.