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Methods
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In mrFindBorders, the positions in (the assumed) X and Y correspond to the positions of the
pixels. The pixels do not move, i.e., the grid on which the pixels lie does not change when
warping. Only the pixel values change.
The function interp2 gets the new values at (x, y) from the locations (x+u, y+v). The left half
of Figure 40 shows a measured phase-wedge image and the corresponding initially aligned
atlas image. On the right side, the 2D displacement field for the control points (from which
the full displacement fields are calculated as explained above) and the deformed atlas can be
seen. The atlas was deformed by using the full displacement fields u and v. Please note that
the arrows point from certain locations, which can be between control points, to the control
points. This shows that the control points lie on a fixed grid, like the pixels. They do not
move. Only the values, e.g., phase values, change.
5.2.4 Estimating Displacement Fields
The aim is to deform the atlas images so that they match their corresponding measured im-
ages. In other words, appropriate displacement fields have to be estimated. The general idea is
to create test displacement fields for the control points (cu and cv). The full displacement
fields u and v can then be derived and the atlas images can be deformed with the full dis-
placement fields.
Then, the deformed atlases have to be evaluated. The evaluation is done by an energy
function, which decreases if the deformation becomes better. The calculation of the energy is
described in more detail in section 5.2.5. To improve the warping the energy is minimized
while different deformations are tested. Therefore, a minimization (optimization) algorithm
had to be found that minimizes the energy in an efficient way and therefore estimates the dis-
placement fields. This is explained in section 5.2.6.
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