DescriptionIn this project, a non-trivial extension of the nonlinear framework for aligning planar shapes (Recovering Diffeomorfic Shape Deformations without Correspondences) is developed for 3D elastic objects. The basic idea is to set up a system of nonlinear equations whose solution directly provides the parameters of the aligning transformation. Each equation is generated by integrating a nonlinear function over the object’s volumes.
When registering a pair of objects, first
we have to characterize the possible deformations. From this
point of view, registration techniques can be classified into
two main categories: physical model-based and parametric or
functional representation. Herein, we deal with the latter
representation, which typically originate from interpolation
and approximation theory. Two broadly used classes of such
parametric models are polynomials and splines, in particular
thin plate splines (TPS)
Polynomial deformations are governed by fewer parameters
and are acting globally on the shapes, hence regularization is
not needed. Moreover, many non-polynomial transformation
can be approximated by a polynomial one e.g. via a Taylor
expansion.
Thin plate spline models are useful whenever a parametric
free-form registration is required but the underlying
physical model of the object deformation is unknown or too
complex. Furthermore, TPS models can be extended to include
various regularizations, such as the bending energy.
Syntethic tests
The algorithm had been quantitatively evaluated on large synthetic datasets generated by the corresponding transformation model. The database consisted 500 images for the polynomial test and 750 images for the thin plate spline tests. The sizes of the objects were between 0.1 and 2.5 megavoxels.
![](./img/elasticreg3d/results.jpg) |
Samples from the synthetic database. |
Robustness tests
The robustness against segmentation errors was also
evaluated on simulated data: we randomly added or removed
squares uniformly around the boundary of each slice of the
observations.
![](./img/elasticreg3d/segmentation.jpg) |
Sample surface errors on a slice. |
For the polynomial model, the surface error of 15%, 22% and 30% of
the original object volume were tested and we find that the proposed approach is quite robust up to as high as 22% surface error.
For the thin plate spline model, the surface error of 10%, 20%
and 30% of the original object volume were generated. Using this model the proposed approach is quite robust up to as high as 20% surface error.
Real images
Lung alignment is a crucial task in lung cancer diagnosis. During the treatment, changes in the tumor size
are determined by comparing follow-up PET/CT scans which
are taken at regular intervals depending on the treatment and
the size of the tumor. Due to respiratory motion, the lung
is subject to a nonlinear deformation between such followups,
hence it is hard to automatically find correspondences.
A common practice is to determine corresponding regions
by hand, but this makes the procedure time consuming and the obtained alignments may not be accurate enough for measuring changes.
Our algorithm has been successfully applied to align 3D
lung CT scans. Both of models proved to be a good approximation
of the underlying physical deformation. Promising
results were obtained on the available 8 image pairs. In order to align the inner parts also, proper regularization is very important for thin plate splines. Therefore we added the bending energy of the transformation to the algebric error of the equation system.
![](./img/elasticreg3d/lung_tmp.gif) |
![](./img/elasticreg3d/lung_obs.gif) |
![](./img/elasticreg3d/lung_reg.gif) |
![](./img/elasticreg3d/lung_slice_113.jpg) |
![](./img/elasticreg3d/lung_slice_140.jpg) |
![](./img/elasticreg3d/lung2_tmp.gif) |
![](./img/elasticreg3d/lung2_obs.gif) |
![](./img/elasticreg3d/lung2_reg.gif) |
![](./img/elasticreg3d/lung2_slice_100.jpg) |
![](./img/elasticreg3d/lung2_slice_143.jpg) |
Alignment of lung CT volumes and the combined slices of the original and the transformed images as an 8x8 checkerboard pattern. Segmented 3D lung images were generated by the InterView Fusion software of Mediso Ltd. |
|