Image analysis – Applications – Motion or velocity measuring
Reexamination Certificate
1999-12-22
2003-04-22
Ahmed, Samir (Department: 2623)
Image analysis
Applications
Motion or velocity measuring
C382S131000, C382S277000, C382S297000, C348S699000, C378S069000
Reexamination Certificate
active
06553132
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention is directed to a method for motion correction given series of two-dimensional images of a rigid body, wherein the rigid body does not lie entirely in the coverage area of the images and is cut off at an image edge, or at two image edges that do not lie opposite one another.
2. Description of the Prior Art
A number of methods of motion correction for series of time-successive images (scenes) are known from the literature. Such scenes can, for example, be acquired by imaging systems of medical diagnostics such as computed tomography or magnetic resonance tomography, and include movements of an examination subject, for example a patient, during the registration of the scene, which can cause disturbances in the image that are intended to be eliminated by these motion correction methods.
Iterative methods are frequently employed for this purpose, in which all parameters of the motion are simultaneously involved, but these require a high computational outlay.
A Fourier transformation of the image data with a following phase correlation is often applied as an auxiliary technique for the motion determination and correction. For example, European Application 0 368 746 and British Specification 2 187 059 disclose such techniques. German PS 195 00 338 discloses a method for determining the transformation parameters for automatic image registration, wherein the translation vector and rotational angle are determined by one-dimensional correlation of radial projections of selected search image areas from a current image and from a reference image.
The publication of Ohm, J.-R., Digita'e Bildcodierung, Springer-Verlag, 1995, pages 32-34, 42-50, is concerned with the problem of “edge continuation of image signals”, which arises when a filter mask is used that does not coincide with the image coverage area, and thus values exist that lie outside the image coverage region. Various methods are proposed for definition of the signal values at these positions, namely periodic continuation, symmetrical continuation and value-constant continuation.
A further approach is the k-space method for motion correction. This shall be explained in greater detail for the two-dimensional case of a rigid body that lies in the xy-plane. Each movement is then unambiguously identified by the parameters representing translation in the x-direction, translation in the y-direction and rotation around the z-axis. By contrast to the iterative methods, the k-space method decouples the individual parameters from one another. This has the advantage that the sought parameters can be identified more simply by splitting the identification into discrete problems.
Since the Fourier transformation with its properties is the basis of this motion correction method, the relevant definitions and properties of Fourier transformation are first discussed herein, as follows.
Fourier transformation is a linear, mapping rom one mathematical domain to another that is defined as follows in the continuous, two-dimensional case:
f
∼
(
k
→
)
=
∫
X
→
ϵ
R
2
f
(
x
→
)
·
ⅇ
2
m
k
→
x
→
ⅆ
x
→
(
1
)
f
~
(
X
→
)
=
∫
X
→
ϵ
R
2
f
(
x
→
)
·
ⅇ
-
2
m
k
→
x
→
ⅆ
x
→
(
2
)
The target domain of the Fourier transformation is called k-space and the source domain is called the spatial domain. When f(x) is real, then the k-space becomes point-symmetrical relative to the zero point upon application of complex conjugation:
∫
x
→
ϵ
R
2
f
(
x
→
)
·
(
ⅇ
2
m
(
-
k
→
x
→
)
*
ⅆ
x
→
=
∫
x
_
ϵ
R
2
f
(
x
→
)
·
ⅇ
2
m
k
→
x
→
ⅆ
x
→
⇔
f
~
(
k
~
)
=
f
~
(
-
k
→
)
*
(
3
)
When x is shifted by an arbitrary vector, then the function f(x+a) can be calculated using the Fourier transform of f(x):
f
(
X
→
+
a
→
)
=
∫
k
→
∈
R
2
f
→
(
k
→
)
·
ⅇ
-
2
m
k
→
(
x
→
=
a
→
)
ⅆ
k
→
=
∫
k
→
∈
R
2
f
→
(
k
→
)
·
ⅇ
-
2
m
k
→
a
→
⏟
ϕ
(
k
→
)
·
ⅇ
-
2
m
k
→
x
→
ⅆ
k
→
(
4
)
A shift of the image in the spatial domain is thus expressed in an additional phase &phgr;({right arrow over (k)}) of the Fourier transform. A rotation in the location space by an angle &thgr; expressed by multiplication with a rotation matrix (R&thgr;), yields:
f
(
R
(
θ
)
·
X
→
)
=
∫
k
→
∈
R
2
f
→
(
k
→
)
·
ⅇ
-
2
m
k
→
R
(
θ
)
x
→
ⅆ
k
→
=
∫
k
→
∈
R
2
f
→
(
k
→
)
·
ⅇ
-
2
m
R
-
1
(
θ
)
k
→
x
→
·
ⅇ
-
2
m
k
→
x
→
ⅆ
k
→
with
R
(
θ
)
=
(
cos
(
θ
)
-
sin
(
θ
)
sin
(
θ
)
cos
(
θ
)
)
(
5
)
and with the substitution {right arrow over (k)}′=R
−1
(&thgr;){right arrow over (k)}=R(−&thgr;){right arrow over (k)}:
ⅆ
k
z
′
ⅆ
k
→
=
(
cos
(
-
θ
)
sin
(
-
θ
)
-
sin
(
-
θ
)
cos
(
-
θ
)
)
ⅆ
k
y
′
ⅆ
k
→
=
(
cos
(
-
θ
)
-
sin
(
-
θ
)
sin
(
-
θ
)
cos
(
-
θ
)
)
⇒
ⅆ
k
→
′
:=
ⅆ
k
x
′
·
ⅆ
k
y
′
=
(
cos
(
-
θ
)
sin
(
-
θ
)
-
sin
(
-
θ
)
cos
(
-
θ
)
)
·
(
cos
(
-
θ
)
-
sin
(
-
ϑ
)
sin
(
-
θ
)
cos
(
-
θ
)
)
·
ⅆ
k
→
⇒
f
(
R
(
θ
)
·
X
→
=
∫
k
→
∈
R
2
f
∼
(
R
(
θ
)
k
→
′
)
·
ⅇ
-
2
m
k
→
x
→
ⅆ
k
→
′
(
6
)
A rotation of the location space by the angle &thgr; effects the same rotation of the Fourier transform.
Overall, thus, for an arbitrary motion:
Location Space
k-space
Translation
additional phase
rotation
rotation
Given the shift of two functions f and g, R→R with f(x)−g(x+a),
the following is valid with respect to its Fourier transform:
{tilde over (ƒ)}(
k
)=
{tilde over (g)}
(
k
)·
e
−2mka
(7)
{tilde over (ƒ)}(
k
)·
{tilde over (g)}
(
k
)*=|
{tilde over (g)}
(
k
)|
2
·e
−2mka
(8)
f
∼
(
k
)
·
g
∼
(
k
)
*
&LeftBracketingBar;
g
∼
(
k
)
&RightBracketingBar;
2
=
ⅇ
-
2
mka
(
9
)
If the motion parameters of a rigid body in a plane that is arbitrarily shifted and rotated in two successive images is to be found, then th
Ahmed Samir
Bhatnagar Anand
Schiff & Hardin & Waite
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