Diffeomorphic Registration of Images with Variable Contrast Enhancement
Figure 1
Comparison between additive and compositive field accumulations. Warping is implemented using linear interpolation. In (a) and (b), two different displacement fields are defined on the plane (for visual clarity). In (c), the field warped by , that is, . In (d), the field is applied on the pixel grid. In (e), the grid is warped by the field resulting from an addition-based accumulation of and . In (f), the grid is warped by the displacement field arising by the composition of and , which is the sum of the dark blue and gray arrows (given by ). This composition is really the accumulation that matters since it corresponds to the way displacement fields are iteratively applied to an image (see Section 3). Since , the summed vectors in (f) correspond to the vectors in (a) and (c). In (g), a moving object , divided in 4 colors (regions between pixel centers). In (h), the result of the warping of by the sum of the fields. Clearly, the surfaces are inverted (mirror effect, visible because of the inversion of colors), leading to nonphysical deformations (negative Jacobians). In (i), the result of the warping of by the composition of the fields. One can notice that in spite of the deformation of the shape of the object, the location of the colors is conserved.
