Biomedical Engineering Reference
In-Depth Information
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1 prior to forward transform to ensure that the circular overlap
portions of the convolution result are null.
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8.3.3 Image Transformation
The collection of point correspondences generated by the precise matching pro-
cess provides the information needed to form a mapping that transforms the float
image into conformation with the base. A variety of nonrigid mappings are used
in practice, differing in computational burden, robustness to erroneous correspon-
dences, and existence of inverse form. High-performance implementation of the
transformation process is not a simple matter and is a separate issue not treated
in this work.
The Polynomial Transformation
In choosing a transformation type we seek something capable of correcting com-
plex distortions, that is robust to matching errors, that admits a closed inverse
form, and that is computationally reasonable to calculate and apply. Of the com-
monly used nonrigid mapping types such as thin-plate spline, local weighted mean,
affine, polynomial, and piece-wise variations, we choose polynomial mapping.
Thin-plate spline provides a minimum energy solution which is appealing for prob-
lems involving physical deformation; however, perfect conformity at correspon-
dence locations can potentially cause large distortion in other areas and excess
error if an erroneous correspondence exists. The lack of an explicit inverse form
means the transformed image is calculated in a forward direction, likely leaving
holes in the transformed result. Methods such as gradient search can be used to
overcome the inverse problem but at the cost of added computation which can be-
come astronomical when applied at each pixel in a gigapixel image. Kernel-based
methods such as local weighted mean require a uniform distribution of correspon-
dences. Given the heterogeneity of tissue features, this distribution cannot always
be guaranteed.
Polynomial warping admits an inverse form, is fast in application, and from
our experience is capable of satisfactorily correcting the distortion encountered
in sectioned images. Polynomial warping parameters can be calculated using least
squares or least squares variants which can mitigate the effect of matching errors.
Affine mapping offers similar benefits but is more limited in the complexity of the
warpings it can represent.
In our algorithm, we use second degree polynomials. Specifically, for a point
x
,
y
)
(
x
,
y
)
in the base image, the coordinate
(
of its correspondence in the float
image is
x
=
a
1
x
2
c
1
y
2
+
+
+
+
+
b
1
xy
d
1
x
e
1
y
f
1
(8.8)
y
=
a
2
x
2
c
2
y
2
+
b
2
xy
+
+
d
2
x
+
e
2
y
+
f
2
Since each pair of matched point correspondences provides two equations, we need
at least six pairs of point correspondences to solve for the coefficients in (8.8). In
practice, a much larger number of point correspondences is obtained.
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