Advanced Image Reconstruction Methods in MRI - Advance Image Processing in Magnetic Resonance Imaging - page 41

2.3 CONSTRAINED IMAGE RECONSTRUCTION

For years, the belief existed that information beyond the measurement cutoff frequency
was not recoverable, thus the Rayleigh resolution limit [1]. Although the information
is not apparent in the measured data, we now have learned how to look elsewhere for
the additional information required to restore those frequency contents [2,3]. Con-
strained methods are the mathematical tools developed to accomplish this objective
by using a priori information to compensate for the lack of high-frequency experi-
mental data in the reconstruction process. Although constrained data processing meth-
ods have been used extensively for decades in other fields, application of the con-
strained reconstruction concept to MRI is very recent. The first successful effort was
perhaps due to Smith [4] and, since then, research interest in this area has continued
to grow for at least two reasons: first, the rapid development of computing technology
has made it possible to use computation-intensive algorithms for practical applications
and, second, the advantages of modern constrained reconstruction methods have made
them worthwhile. In particular, the ability to reduce data truncation artifacts and
improve image resolution is very desirable and can produce effects unmatched by the
traditional unconstrained Fourier methods. Nonparametric constraints permit the use
of the conventional Fourier series model for image function, and reconstruction meth-
ods of this type usually involve explicit data extrapolation to recover some of the
unmeasured (presumably lost) high-spatial-frequency data so as to reduce truncation
artifact. Parametric modeling methods, on the other hand, represent the image function
in terms of a set of parameterized basis functions, rather than the nonparameterized
harmonic sinusoidal functions used in the Fourier series. These methods can, in
principle, generate images of infinite resolution from the model without explicitly
extrapolating the data to the infinite frequency range. In this sense, parametric model
constraints are often more powerful than nonparametric constraints, although some-
times they may not be as robust. Explicit data extrapolation is also possible and often
used with parametric methods by using the model to generate the unmeasured data.

2.3.1 N ONPARAMETRIC M ETHODS

A popular mathematical algorithm used in many nonparametric reconstruction
methods is alternate projection, or projection onto convex sets (POCS). The
principle of POCS has been discussed in great detail in the signal processing
literature. We review here only the central ideas and give a couple of examples
of its use in MRI.

Definition: A subset in the Hilbert space H is said to be convex if

together with any x 1 and x 2 , it also contains

Ω

µ x 1 + (1 − µ ) x 2 for all µ , 0 ≤

1.
Definition: For any x

µ ≤

∈ H , the projection P Ω x of x onto Ω is the element

in Ω closes to x. If Ω is closed and convex, P Ω exists and is uniquely

determined by x and Ω from the following minimality criterion

|| xPx − || = min || xy − || .

Ω

y ∈ Ω