Registration is the attempt to compute the difference (rotation and translation) between two images. One of these images is the so called fixed image. It is the original one. The other image is called moving image. It is a somehow rotated and translated image.
Feature-based Registration
Planar Registration
Spatial Registration
Unfortunately the complex number trick does not work in 3-dimensional space. But we can also start with the least squares approach. Each zero-mean set is converted to a matrix. The rotation is then the matrix multiplication Q \approx RP with R is an orthogonal matrix (RR^T=I_{3\times3} and det(R) = 1). To solve this the singular value decomposition (SVD) is used which makes use of the square root of matices. In 1978 Sibson rewrote the problem as QQ^T \approx R\left[ PQ^T \right]. He defined the data matrix H=PQ^T and inverse rotation matrix W = R^T to calculate argmax(W^T H). With this maximized value he could find an optimal R^\diamond=\left[ H \left[ H^TH \right]^{-1/2} \right]^T by computing the eigenvalue decomposition. Arun who may be unaware of Sibson, computed the SVD with the idea that H=U\Sigma V^T for orthogonal U and V. The optimal rotation is thenR^\diamond=UV^T. With a known R^\diamond the optimal \vec{d}^\diamond can be found. The spatial registration can also be used for the registration of a part to the whole. Therefore, the ICP algorithm is used. It uses the spatial registration to register the data as well as a nearest neighbor approach to solve the subset problem [1].
2D/3D-Registration
This registration method is especially used to register pre-interventional 3D images to n 2D intra-interventional (live) images. For registration there are different approaches. First, you can project the 3D data to a 2D plane and register the projected data to the 2D data. Second, you can back-project the n 2D images to a 2.5D image which can be registered with the original 3D data. The third and last option is to reconstruct a 3D image from the n 2D images and also register the 3D images (see table). In table 2, the feasibility of the methods is shown [1].
3D | n x 2D | |||
---|---|---|---|---|
projection |
|
2D/2D | ||
3D/3D |
back-projection | |||
3D/3D |
reconstruction |
Dimensional correspondence | ||||
---|---|---|---|---|
Projection | Back-projection | Reconstruction | ||
| Feature | 2D segmentation is problematic | only one 2D image is available | |
Intensity | ||||
Gradient | feasible |
Bibliography
Lecture Slides from CAMP I, Lecture 8, 11 and 12, 2016
Lecture Slides from CAMP II, Lecture 3, 2017
Kommentar
Unbekannter Benutzer (ga38gis) sagt:
08.Juni 2017Hey
I like that everyone can set his own focus and so I like that you set your focus on the mathematical part and describe the registration very detailed. There's only one thing to criticize: In my opinion lectures are not good as content sources, for images I think it is okay (sometimes it's hard to find an equivalent one), but for content it should be possible to find a primary source.
Your articles are nice to read and clearly structured and I like that you always visualize the content with images.