Applied Physics Laboratory

Delft University of Technology

P.O. Box 5046

2600 GA Delft

The Netherlands

The acquired informative samples are *non*uniformly distributed
over a uniform grid [1] We seek a solution in the Fourier domain that
optimizes some quality criterion while the inverse FFT of the solution
is kept compatible with the acquired samples. This approach is similar
to that of MEM [2], but our quality criterion is not derived from the
entropy of an image. In fact, ours is based on the finding that the
histogram of {\em differences} of grey values of adjacent pixels
(*edges*) has approximately Lorentzian shape [3]. This was
successfully applied to a real-world human brain scan [1].

Using Bayesian probability theory, we have enhanced the method of Ref. [1]. This allows concurrent handling of noise and ringing. The procedure is applied to each row in the incompletely sampled phase encoding space, after FFT in the acquisition space (i.e., columnwise).

Equation 1

in which **I** is a row of the image, **S**
the attendant row of samples. **p(I|S)** is the
probability of an image given the samples. **p(S|I)**,
the probability of the samples given an image, relates to the noise
distribution function. **p(I)** comprises prior knowledge
about the image, *e.g.*, the Lorentzian shape of the histogram
mentioned above, and low intensity outside the boundaries of the
object. **p(S)** is usually just a scaling constant,
once the samples have been acquired [4].

The task is to find the image that maximises
**p(I|S)**. Assuming Gaussian measurement noise with
standard deviation **\sigma**, and a unitary FFT matrix,
one finds for each row

Equation 2

the index **l** pertaining to the informative samples
alluded to above.
The prior knowledge term **p(I)** can be split into two
parts, one for the actual object **O**, and one for the
background **B** beyond the perimeter of
**O**. After phasing and retaining only the real part of
**I**, we write for the object image
**I_{O}**

Equation 3

with

,

which has Lorentzian shape, **2a** being the width at
half height. For the background image we write

Equation 4

Furthermore, **p(I) = p(I_{O})p(I_B)**. One may choose
to ignore more knowledge of object boundaries. This extends
Eq.3 to the entire image and obviates Eq.4.

In actual practice, we optimise the natural logarithm of
**p(I_S)**, choosing relative weights for the
contributing terms **\ln p(S|I)**,
**\ln p(I_O)**, and **\ln p(I_B)**.
This is an iterative process, based on steepest descent. Various
combinations of weights can be used, depending on the application.

Ringing as a result of zero-filling of skipped less informative samples can be very substantially reduced. Furthermore, combination of the method with the partial (one-sided) scan technique [5] appeared feasible.

[2] Constable, R.T., Henkleman, R.M., Magn. Reson. Med., 14, 12 (1990)

[3] Fuderer, M., Proc. SPIE, 1137, 84 (1989)

[4] Norton, J.P., {\em An Introduction to Identification}, Academic Press, London (1986)

[5] Cuppen J., Van Est, A., Book of Abstracts, Topical Conf. on Fast Imag., Cleveland, May 15--17, 1987.

MaxEnt 94 Abstracts / mas@mrao.cam.ac.uk