4.3 Homology Error

For a constant m and r, the variability of the error in the registrations only depends on the homology error between point pairs. The homology error in each modality may be thought of as an effective Gaussian error envelope. The use of is preferred over because it allows a more direct comparison to the description of quantity which gives rise to homology error the most, the finite spatial resolution, , and, perhaps less intuitively, the contribution of human interaction, [Hil88]. An operator unfamiliar with the functional nature of the information provided by Tc-HMPAO SPECT images will undoubtedly introduce greater homology errors and hence larger registrations errors. From linear systems theory, and characterise the components of a cascaded linear system (see figure 2.6) which combine in quadrature (Appendix G in [SP87] assuming Gaussian distributions of errors) to convolute the exact location of an anatomical point within some Gaussian distribution of uncertainty as described by

The homology error in a SPECT/MR registration similarly combines the homology error contributions of each modality in quadrature such that

Due to SPECT's anisotropic spatial resolution, possible distortions in MR and the interpretive faculties of the operator involved, is in general a likewise anisotropic Gaussian distribution and non-homogeneous in space because of its dependence on and the general intra-observer anatomical variability of [Sor]. Additionally, the approximation may be made wherein

so that

may be measured for a specific anatomical location in practice by picking the same point N times and calculating the standard deviations (equal to for normal distributions) in the three ordinates. With interactive display software which includes on the fly magnification control and inter-plane display updates toggled to the cursor position, it is found that is very small so that may further be approximated by (henceforth simply referred to as from the approximation) when a knowledgeable operator is involved. Point simulations may produce various and for different combinations of m, r and , where N un-registered point sets are created by randomly perturbing the same target point set within the homogeneous 3-D Gaussian error envelope of . The data from the point simulations study may serve as a calibration file representing ideal spherically symmetric configurations for different r, m, and . Similarly, real scan studies with the 3-D ``Hoffman'' brain phantom (Data Spectrum Corporation, Chapel Hill, NC) with external fiducials as a basis for exact registrations may be registered N times for specific m, the modality's , and configuration (unequal r in each ordinate because of the brain's roughly oblong shape). and for each ordinate and about each plane from the real scan studies are interpreted in the context of the results from the point simulation studies.