Methods 85 much. In simple words, this energy component is a cost that increases with non-uniform de- formation. It is determined by calculating derivatives of the full displacement fields in x and y directions. Then, the single matrix elements are squared and summed up.   { } 2 2 2 2 (  ,    )                     ij ij ij ij x y x y D x y x y all    i   j u u u u x y v v v v x y E u u v v = = = = ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ = + + + Equation 9: Deformation Energy In some cases where there is a lot of noise or where the initially aligned atlas image does not match the measured image well, the user can specify boundary points (landmarks) within the measured phase image. Since the boundaries in the phase-wedge atlas are known, a minimal squared distance between a specified boundary point and the corresponding boundary in the atlas image can be calculated. The distance energy is based on this measure. When the user specifies the boundary points, it is possible to assign a confidence value to each point. In addi- tion to the total weight for the distance energy, each minimal squared distance that is calcu- lated for each point is weighted with this confidence value. It is therefore possible to almost force the algorithm to deform the atlas in a way so that specified boundary points in the meas- ured image lie on corresponding boundaries in the atlas image. 5.2.6 Optimization Techniques After the basic principle of the energy minimization technique has already been described in section 5.2.4, more details are given next. Different algorithms have been developed and can be used to match functional images.