Image analysis – Image transformation or preprocessing – Changing the image coordinates
Reexamination Certificate
1998-11-05
2001-05-01
Tran, Phuoc (Department: 2721)
Image analysis
Image transformation or preprocessing
Changing the image coordinates
C382S131000, C382S151000, C382S291000
Reexamination Certificate
active
06226418
ABSTRACT:
TECHNICAL FIELD
The present invention relates to image processing systems and methods, and more particularly to image registration systems that combine two or more images into a composite image in particular the fusion of anatomical manifold based knowledge with volume imagery via large deformation mapping which supports both kinds of information simultaneously, as well as individually, and which can be implemented on a rapid convolution FFT based computer system.
BACKGROUND ART
Image registration involves combining two or more images, or selected points from the images, to produce a composite image containing data from each of the registered images. During registration, a transformation is computed that maps related points among the combined images so that points defining the same structure in each of the combined images are correlated in the composite image.
Currently, practitioners follow two different registration techniques. The first requires that an individual with expertise in the structure of the object represented in the images label a set of landmarks in each of the images that are to be registered. For example, when registering two MRI images of different axial slices of a human head, a physician may label points, or a contour surrounding these points, corresponding to the cerebellum in two images. The two images are then registered by relying on a known relationship among the landmarks in the two brain images. The mathematics underlying this registration process is known as small deformation multi-target registration.
In the previous example of two brain images being registered, using a purely operator-driven approach, a set of N landmarks identified by the physician, represented by x
i
, where i=1 . . . N, are defined within the two brain coordinate systems. A mapping relationship, mapping the N points selected in one image to the corresponding N points in the other image, is defined by the equation u(x
i
)=k
i
, where i=1 . . . N. Each of the coefficients, k
i
, is assumed known.
The mapping relationship u(x) is extended from the set of N landmark points to the continuum using a linear quadratic form regularization optimization of the equation:
u
=
arg
min
u
∫
&LeftDoubleBracketingBar;
Lu
&RightDoubleBracketingBar;
2
(
1
)
subject to the boundary constraints u(x
i
)=k
i
,. The operator L is a linear differential operator. This linear optimization problem has a closed form solution. Selecting L=&agr;∇
2
+&bgr;∇(∇.) gives rise to small deformation elasticity.
For a description of small deformation elasticity see S. Timoshenko,
Theory of Elasticity
, McGraw-Hill, 1934 (hereinafter referred to as Timoshenko) and R. L. Bisplinghoff, J. W. Marr, and T. H. H. Pian,
Statistics of Deformable Solids
, Dover Publications, Inc., 1965 (hereinafter referred to as Bisplinghoff). Selecting L=∇
2
gives rise to a membrane or Laplacian model. Others have used this operator in their work, see e.g., Amit, U. Grenander, and M. Piccioni, “Structural image restoration through deformable templates,”
J American Statistical Association.
86(414):376-387, June 1991, (hereinafter referred to as Amit) and R. Szeliski,
Bayesian Modeling of Uncertainty in Low-Level Vision
, Kluwer Academic Publisher, Boston, 1989 (hereinafter referred to as Szeliski) (also describing a bi-harmonic approach). Selecting L=∇
4
gives a spline or biharmonic registration method. For examples of applications using this operator see Grace Wahba, “
Spline Models for Observational Data
,” Regional Conference Series in Applied Mathematics. SIAM, 1990, (hereinafter referred to as Whaba) and F. L. Bookstein,
The Measurement of Biological Shape and Shape Change
, volume 24, Springer-Verlag: Lecture Notes in Biomathematics, New York, 1978 (hereinafter referred to as Bookstein).
The second currently-practiced technique for image registration uses the mathematics of small deformation multi-target registration and is purely image data driven. Here, volume based imagery is generated of the two targets from which a coordinate system transformation i:; constructed. Using this approach, a distance measure, represented by the expression D(u), represents the distance between a template T(x) and a target image S(x). The optimization equation guiding the registration of the two images using a distance measure is:
u
=
arg
min
u
∫
&LeftDoubleBracketingBar;
Lu
&RightDoubleBracketingBar;
2
+
D
(
u
)
(
2
)
The distance measure D(u) measuring the disparity between imagery has various forms, e.g., the Gaussian squared error distance ∫|T(h(x))−S(x)|
2
dx, a correlation distance, or a Kullback Liebler distance. Registration of the two images requires finding a mapping that minimizes this distance.
Other fusion approaches involve small deformation mapping coordinates x &egr; &OHgr; of one set of imagery to a second set of imagery. Other techniques include the mapping of predefined landmarks and imagery, both taken separately such as in the work of Bookstein, or fused as covered via the approach developed by Miller-Joshi-Christensen-Grenander the '212 patent described in U.S. Pat. No. 6,009,212 (hereinafter “the '212 patent”) herein incorporated by reference mapping coordinates x &egr; &OHgr; of one target to a second target: The existing state of the art for small deformation matching can be stated as follows:
Small Deformation Matching: Construct h(x)=x−u(x) according to the minimization of the distance D(u) between the template and target imagery subject to the smoothness penalty defined by the linear differential operator L:
h
(
·
)
=
·
-
u
(
·
)
where
u
^
=
arg
min
u
∫
&LeftDoubleBracketingBar;
Lu
&RightDoubleBracketingBar;
2
+
∑
i
=
1
N
D
(
u
)
.
(
3
)
The distance measure changes depending upon whether landmarks or imagery are being matched.
1. Landmarks alone. The first approach is purely operator driven in which a set of point landmarks x
i
, i=1, . . . N are defined within the two brain coordinate systems by, for example, an anatomical expert, or automated system from which the mapping is assumed known: u(x
i
)=k
i
, i=1, . . . , N. The field u(x) specifying the mapping h is extended from the set of points {x
i
} identified in the target to the points {y
i
} measured with Gaussian error co-variances &Sgr;
i
:
u
^
=
arg
min
u
∫
&LeftDoubleBracketingBar;
Lu
&RightDoubleBracketingBar;
2
+
∑
i
=
1
N
(
y
i
-
x
i
-
u
(
x
i
)
)
t
Σ
-
1
(
y
i
-
x
i
-
u
(
x
i
)
)
T
(
4
)
Here (.)
&tgr;
denotes transpose of the 3×1 real vectors, and L is a linear differential operator giving rise to small deformation elasticity (see Timoshenko and Bisplinghoff), the membrane of Laplacian model (see Amit and Szeliski), bi-harmonic (see Szeliski), and many of the spline methods (see Wahba and Bookstein). This is a linear optimization problem with closed form solution.
2. The second approach is purely volume image data driven, in which the volume based imagery is generated of the two targets from which the coordinate system transformation is constructed. For this a distance measure between the two images being registered I
0
, I
1
is defined, D(u)=∫|I
0
(x−u(x))−I
1
(x)|
2
dx, giving the optimization by
h
(
·
)
=
·
-
u
(
·
)
where
u
=
arg
min
u
∫
&LeftDoubleBracketingBar;
Lu
&RightDoubleBracketingBar;
2
+
∫
&LeftBracketingBar;
I
0
(
x
-
u
(
x
)
)
-
I
1
(
x
)
&RightBracketingBar;
2
ⅆ
x
.
(
5
)
The data function D(u) measures the disparity between imagery and has been taken to be of various forms. Other distances are used besides just th
Christensen Gary E.
Joshi Sarang C.
Miller Michael I.
Finnegan Henderson Farabow Garrett & Dunner L.L.P.
Mariam Daniel G.
Tran Phuoc
Washington University
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