Efficient Ghost Suppression by Analytical Gradient Energy Minimization
Sofia Chavez* and Qing-San Xiang
Depts of Radiology and Physics*, University of British Columbia, Vancouver, BC, Canada
INTRODUCTION
The feasibility of a two-point ghost suppression where Real [.] and Imag [.] are operators that take the real
technique has been previously demonstrated [1]. and imaginary parts of the complex numbers respectively.
Implementation of this technique was accomplished by an Summing the square of the partial derivatives (in x and y)
automated regional tuning procedure. However, this of Eq. (8) and Eq. (9) will give:
procedure is time consuming and inexact since the result
Eg(lw)=Eg(Ia)+Eg(Id)cot2(0/2)+2X(Ia,Id)cot(0/2)
(10)
depends on the step size of the tuning parameter. Also, the
results showed some discontinuities between regions with where Eg(Ia) and Eg(Id) are the gradient energies of Iaand
Id respectively and X(Ia,Id) is a cross-term given by:
signal intensity variations.
In this abstract, an improved, more efficient
X(Ia,Id) =
( Real[Id] alImag[Ia] )+(aReal[Id] Imag[Ia])'
implementation is presented. It is based on an analytical
ax ax ay ay
derivation of the dependence of the gradient energy on the
_(aReal[Ia]
mag[Id] ')_ (aReal[Ia] DImag[Idl) ()
tuning parameter. This improved implementation greatly
" Dx
ax a- Dy
ay
reduces the processing time and allows effective removal
of regional discontinuities making the two-point ghost where I indicates a summation over a given region.
Minimization of Eg(Iw) with respect to 0 yields:
suppression technique more practical.
- csc2(0/2) [2 cot(0/2) Eg(Id) +2X(Ia,Id)] = 0
(12)
METHOD
The two-point ghost suppression technique is which can be solved to give:
cot(0/2) = - X(Ia,Id) / Eg(Id)
(13)
based on combining two interleaved images, I and 12,
that have been acquired with slightly shifted phase This result, inserted in Eqs.(2,3) yields the desired
encoding steps. The combination is a summation weighted weighting factors that minimize the gradient energy:
by complex factors such that the ghosts of the two images
Wl=1/2 [1+ i X(Ia,Id)/Eg(Id)]
(14)
destructively interfere. The resulting weighted image, Iw,
W2=1/2 [1 - i X(Ia,Id)/Eg(Id)]
(15)
is given by:
Iw can thus be directly calculated in any given region.
Iw=WlI1 + W212
(1)
In order to remove the discontinuity between the
where:
regions, W1 and W2 were calculated in a small sliding
W 1=1/2 [1 - i cot(0/2)]
(2)
window and applied only to the central pixel.
(3)
W2=1/2 [1 + i cot(0/2)]
and hence the weighting factors are dependent on the
choice of 0. The two-point ghost suppression technique
relies on finding the value for 0 that will result in
destructive interference of the ghosts. In [1], such a 0,
was found by using a gradient energy, Eg, which, for any
complex field, C = R + i I, Eg(C) is given by.
Eg(C) =
DR 2 ' DI2
DR)2
) + (ax ) + (a ) ++ (a
a
aI2
)
RESULTS
Below are images of a stationary water-filled bottle
and moving hand. (a) and (b) show the magnitude of
ghosting patterns of I1 and 12. Regional discontinuities can
be seen in (c) which shows the deghostedimage resulting
from an 8X8 regional tuning procedure. (d) shows the
deghosted result obtained much faster via direct
(4) calculations of Eqs.(14,15) with a 7X7 sliding window.
where A indicates a summation over a given region.
Specifically, regional minimization of the gradient energy
of Iw, Eg(Iw), was shown to be appropriate for 0 selection
since it provides a balance between ghost suppression and
noise amplification [1,2]. The Eg(Iw) minimization was
achieved in [1] by regionally tuning 0 in steps where at
each step Eg(Iw) was calculated and compared. This
(a)
(b)
(c)
(d)
tedious, inefficient tuning procedure was recently found to
be unnecessary since an analytical calculation of 0 is DISCUSSION
possible.
A direct calculation of the weighting factors for
In order to calculate 0 which minimizes Eg(Iw),, two-point ghost suppression was demonstrated.
an expression for E (Iw), as a function of 0 must be Processing time and regional discontinuities were
derived. This can be done as follows. Define Ia and Id as
successfully reduced. It is believed that the Eg(Iw)vs
0
the average image and half the complex difference image curve will provide more information about the
of I1 and 12 respectively:
characteristics of ghosted images.
Ia = (I1 + 12)/2
Id =(I1 -12)/2
(6)
REFERENCES
[1] QS Xiang, MJ Bronskill and RM Henkelman.J.
Iw = Ia + i Id cot(0/2)
(7)
Magn.
Reson. Imag. 3, 900-906, 1993
Eq.(7) results in the following real and imaginary parts for
[2] B Madore and RM Henkelman Med. Phys. 23, 109Iw:
113, 1996
Real [Iw]=Real [Ia]- Imag[Id] cot(0/2)
(9)
Imag [Iw]=Imag[Ia]+ Real[Id] cot(0/2)
(9) Work supported in part by the Whitaker Foundation
Eqs.(1-3) will then give: