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Methods
76
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Distance between the center of the fovea (cf in Figure 36) and the center of the wedge
(cw in Figure 36)
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Distance between the center of the fovea and the center of the ring (for the ring stimu-
lus atlas)
5.2.2 Representation of the Displacement Field
Once the initially aligned atlases have been created, they can be deformed to improve the
matching with the corresponding measured functional images. This is done with the help of
two displacement fields called u and v. They are 2D matrices and represent the full displace-
ment fields (or deformation fields). Each has the same size as the images that are to be de-
formed (atlas images). For each pixel, u holds a displacement value in x-direction and v the
corresponding one in y-direction. Together, u and v hold a 2D displacement vector for each
pixel. To decrease the computational effort, and for reasons that are explained later, the num-
ber of freedoms is reduced by assigning displacement values only to a subset of the pixels
(control points) in the atlas image. The displacement values for the control points lie on a
regular grid (constant distance between adjacent control points in x direction; the same for y
direction) and they are stored, again, in two 2D matrices, called cu (x-direction) and cv (y-
direction). Figure 37 shows control point locations (circles) and displacement values (dots). If
a dot is in the middle of its circle, the displacement values for this control point are both zero.
An arrow points from a dot to the corresponding circle. This means that the value at the posi-
tion of the dot in the undeformed image ends up at the position of the circle in the deformed
image. The left image shows an example where there has been only one non-zero displace-
ment value assigned to one control point in x direction. Matrix cu is therefore a matrix of ze-
ros except for one entry. Matrix cv has only zero entries. The right image depicts another ex-
ample. Matrix cu has only zeros; matrix cv stores one element that is not zero.
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