PixP2 y

where ^1(x) and p2(y) are probability distributions in respective images, ^1>2(x, y) is the joint probability distribution, and x and y are intensity values. Mutual information has been found very effective in multimodal image registration between modalities such as MR, PET, and CT.

Intensity-based registration has the advantage that the entire image volume is used in establishing the correspondence. The methods can be automatic but are usually very slow because of large amount of information involved in the computation.

Nonrigid Registration

Nonlinear spatial transformations are often needed for interpatient registration and intrapatient registration over time. The registration needs to reflect the variability in the anatomic features between different individuals and changes with time, including differences in shape and size as well as differences in internal density distribution.

One class of nonrigid registration is based on the deformation field computed from landmarks identified within the images. The process is accomplished in two steps—global matching achieved by alignment of landmarks, followed by local deformation based on viscous fluid, piecewise affine transformation, or optical flow (43-45).

Another class of nonrigid registration is based on statistical models derived from a population of patients. In this method, a statistical model consisting of an average instance and its statistical variability is first computed from training datasets (7,46). The registration is accomplished by a statistical optimization of the state of the statistical model. The average instance is warped within its statistical range to match the structure of interest. The statistical model serves as a common reference for the registration. Once both images are registered to the statistical model, the transformation between them can be derived.

2D-3D Registration

2D-3D registration is a special category of image registration where 3D images such as CT, MR are brought into alignment with 2D images such as X-ray fluoroscopy, ultrasound images, and video images. CT and MR images are frequently used in clinical diagnosis and surgical planning, but their use as interventional imaging modalities has been limited because of the expense and space within the operating room. Common modalities for guiding surgical interventions are X-ray fluoroscopes or ultrasound devices. These images are acquired in real time, but only present 2D information. A number of important anatomical features cannot be visualized well in 2D images, such as relative 3D location and orientation of anatomic structures. One method to provide 3D information during the intervention is to register and fuse preoperative 3D images with 2D intraoperative images (47).

In typical 2D-3D registration, the 3D images are projected to the 2D planes where the 2D images are taken. The projection images are then compared with 2D images to evaluate the alignment. The position and orientation of the 3D image volume is then updated until its projections reach optimal match with the 2D images.

Similar to 3D-3D registrations, landmark points, surfaces, and pixel intensities can be used for the 2D-3D registration (48).

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