The Concept of Spatial Correspondence

Image registration establishes spatial correspondence. Almost all of the images that are used in image registration are digital and three dimensional (3D), consisting of an array of volume elements or voxels. To each voxel is assigned a number, the image intensity at that position. The registration transformation is a mapping, T, that transforms a position xA in image A to a point xB in image B,or

where the two images A and B are mappings of points in the patient, within their field of view or domain Q, to intensity values:

preted with care. Spatial correspondence may not exist when:

(i) Tissue is gained or lost between source and target image acquisition, for example, due to tumor growth or surgical removal.

(ii) Organ contents change (for example, bladder or bowel filling and emptying).

(iii) One (or both) of the images are corrupted, for example, by motion artifacts.

(iv) Structure present in one individual is absent in another (for example, detailed sulcal and gyral patterns in the cerebral cortex).

In inter-subject registration, correspondence may be defined structurally (i.e., geometrically), functionally, or histologically. Current non-rigid registration algorithms implicitly establish correspondence that does not necessarily conform to any of these definitions.

Images A and B represent one object X - the patient. Image A maps position x <£ X to xA and image B maps x to xB. The registration process derives T so that both A(xA) and BT(xA) represent the same location in the object (within some error depending on the accuracy of T). Strictly, T defines a spatial mapping while we need a mapping that maps an accurate estimate of intensity, not just position, by including interpolation [2]. The registration process recovers T over the domain of overlap between the two images, QTA,B , which depends on the domains of the original images A and B as well as the spatial transformation T.

By spatial correspondence between two images, we mean that a voxel in one image corresponds to the same physical location in the patient as the corresponding voxel in an image that has been registered to it. Although this may seem an obvious definition, problems can easily arise which may lead to errors in interpretation. For example, a voxel in a PET image will usually be much larger than one in MRI and the effective spatial resolution may be up to an order of magnitude coarser in PET than in MRI. One PET voxel may therefore contain information that is spread over many hundreds of MRI voxels. This is often called the partial volume effect and registered images must be inter

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