We call a restriction semigroup *almost perfect* if it is proper and the least congruence that identifies all its projections is perfect. We show that any such semigroup is isomorphic to a ‘*W *-product’ W(T,Y)W(T,Y), where *T* is a monoid, *Y* is a semilattice and there is a homomorphism from *T * into the inverse semigroup *TI*_{Y}TIY of isomorphisms between ideals of *Y*. Conversely, all such *W*-products are almost perfect. Since we also show that every restriction semigroup has an easily computed cover of this type, the combination yields a ‘McAlister-type’ theorem for all restriction semigroups. It is one of the theses of this work that almost perfection and perfection, the analogue of this definition for restriction monoids, are the appropriate settings for such a theorem. That these theorems do *not* reduce to a general theorem for inverse semigroups illustrates a second thesis of this work: that restriction (and, by extension, Ehresmann) semigroups have a rich theory that does not consist merely of generalizations of inverse semigroup theory. It is then with some ambivalence that we show that all the main results of this work easily generalize to encompass *all* proper restriction semigroups.

The notation W(T,Y)W(T,Y) recognizes that it is a far-reaching generalization of a long-known similarly titled construction. As a result, our work generalizes Szendrei's description of almost factorizable semigroups while at the same time including certain classes of free restriction semigroups in its realm.

]]>**Purpose**

Achieving a reduction in scan time with minimal inter-slice signal leakage is one of the significant obstacles in parallel MR imaging. In fMRI, multiband-imaging techniques accelerate data acquisition by simultaneously magnetizing the spatial frequency spectrum of multiple slices. The SPECS model eliminates the consequential inter-slice signal leakage from the slice unaliasing, while maintaining an optimal reduction in scan time and activation statistics in fMRI studies.

**Materials and Methods**

When the combined *k*-space array is inverse Fourier reconstructed, the resulting aliased image is separated into the un-aliased slices through a least squares estimator. Without the additional spatial information from a phased array of receiver coils, slice separation in SPECS is accomplished with acquired aliased images in shifted FOV aliasing pattern, and a bootstrapping approach of incorporating reference calibration images in an orthogonal Hadamard pattern.

**Result**

The aliased slices are effectively separated with minimal expense to the spatial and temporal resolution. Functional activation is observed in the motor cortex, as the number of aliased slices is increased, in a bilateral finger tapping fMRI experiment.

**Conclusion**

The SPECS model incorporates calibration reference images together with coefficients of orthogonal polynomials into an un-aliasing estimator to achieve separated images, with virtually no residual artifacts and functional activation detection in separated images.

]]>**Purpose**

To develop a linear matrix representation of correlation between complex-valued (CV) time-series in the temporal Fourier frequency domain, and demonstrate its increased sensitivity over correlation between magnitude-only (MO) time-series in functional MRI (fMRI) analysis.

**Materials and Methods**

The standard in fMRI is to discard the phase before the statistical analysis of the data, despite evidence of task related change in the phase time-series. With a real-valued isomorphism representation of Fourier reconstruction, correlation is computed in the temporal frequency domain with CV time-series data, rather than with the standard of MO data. A MATLAB simulation compares the Fisher-*z* transform of MO and CV correlations for varying degrees of task related magnitude and phase amplitude change in the time-series. The increased sensitivity of the complex-valued Fourier representation of correlation is also demonstrated with experimental human data. Since the correlation description in the temporal frequency domain is represented as a summation of second order temporal frequencies, the correlation is easily divided into experimentally relevant frequency bands for each voxel's temporal frequency spectrum. The MO and CV correlations for the experimental human data are analyzed for four voxels of interest (VOIs) to show the framework with high and low contrast-to-noise ratios in the motor cortex and the supplementary motor cortex.

**Results**

The simulation demonstrates the increased strength of CV correlations over MO correlations for low magnitude contrast-to-noise time-series. In the experimental human data, the MO correlation maps are noisier than the CV maps, and it is more difficult to distinguish the motor cortex in the MO correlation maps after spatial processing.

**Conclusions**

Including both magnitude and phase in the spatial correlation computations more accurately defines the correlated left and right motor cortices. Sensitivity in correlation analysis is important to preserve the signal of interest in fMRI data sets with high noise variance, and avoid excessive processing induced correlation.

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