Rotation Covariant Image Processing for Biomedical Applications
Figure 3
The theory in practice: Laguerre expansion of a volumetric image with and a Gaussian width of . For the experiments we use an image (size ) showing the -weighted MRT image of a human skull. In (a) we depict the center slice of the 3D volume showing the real-valued parts () of the expansion coefficients computed explicitly by convolution of the image with the kernel functions (). (b) Shows the same expansion coefficients obtained when using the proposed differential approach, with . (c) Shows that the choice of the discrete operator has a big influence of the result.
(a) Ground truth: an image is convolved with each basis function requiring convolutions!! The resulting symmetric (central ) spherical tensor components are shown
(b) Differential approach using the discrete operators shown in Figure 2. The image is initially convolved ones with the basis function . All further expansion coefficients are obtained by iteratively applying the spherical up-down derivatives using the Laplace-operator depicted in Figure 2
(c) Differential approach using the standard Laplace operator considering only six neighbors results in strong artifacts and leads to unusable results (lower images)
