Computational and Mathematical Methods in Medicine / 2013 / Article / Fig 3

Research Article

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)

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