Special relativity derived from spacetime magma.

We present a derivation of relativistic spacetime largely untethered from specific physical considerations, in constrast to the many physically-based derivations that have appeared in the last few decades. The argument proceeds from the inherent magma (groupoid) existing on the union of spacetime frame components [Formula: see text] and Euclidean [Formula: see text] which is consistent with an "inversion symmetry" constraint from which the Minkowski norm results. In this context, the latter is also characterized as one member of a class of "inverse norms" which play major roles with respect to various unital [Formula: see text]-algebras more generally.

Susceptibility weighted imaging and cerebrovascular disorders.

To investigate the correlation between the SWI findings and prognosis of the cerebrovascular disorders. From July 2008 to July 2010, 299 ischemic stroke patients were found in our hospital. The gender ratio is as male and female being 157 to 142. The mean age of all patients is 65.4, mean female age is 69.1, and mean male age is 62.6. There were 86 patients who had satisfactory pre-and post-treatment of CT, MRI with SWI. 23 of these 86 patients had catheter cerebral angiography. 50 of these 86 patients had MR angiogram or CT angiogram. 13 of these 86 patients did not have angiogram. We have also collected 7 severe cardiac arrested and cessation of cerebral circulation and 2 patients with chronic venous hypertension. Among the 86 patients, 23 patients who had negative with deoxygenated vessel on SWI were with small infarction on DWI. Thirty-one patients had negative on initial CT head scan. CT finding did not accord with presence of hypointense vessel on SWI. Sixty-three patients had varied degree of abnormal hypointense vessels on SWI as deoxygenated vessels. The initial small foci on DWI may result with a larger infarction if there were with prominent hypointense vessels.

Susceptibility-weighted imaging for differential diagnosis of cerebral vascular pathology: a pictorial review.

Susceptibility-weighted imaging (SWI) is a high-spatial resolution, three-dimensional, gradient-echo (GRE) magnetic resonance (MR) technique. This fully velocity-compensated pulse sequence utilizes the magnetic susceptibility differences of various tissues or substances, such as blood products, iron, and calcification. By postprocessing the magnitude images using a phase mask, it emphasizes the magnetic properties of different susceptibility effects. Generated minimal intensity projection (minIP) images can further demonstrate tortuous vasculature and the continuity of vessels or abnormalities across slices. SWI has been used to improve the diagnosis of neurological trauma, brain neoplasm, neurodegenerative disorders, and cerebrovascular disease because of its ability to demonstrate microbleeds and conspicuity of the veins and other sources with susceptibility effects. We have used SWI to identify cerebrovascular lesions which may be obscured on other MR sequences to aid in the differential diagnosis. We present a review with selected cases to demonstrate the usefulness of this new neuroimaging technique in improving the diagnosis of cerebral vascular pathology.

Use of group theory in the selection and description of regularization methods for functional source imaging.

Commonly, functional source imaging problems are "partial" rather than "ordinary" inverse problems--wherein the defining operator consists of component operators that individually do not address all variables of the unknown. When this ordinary-to-partial transition is minimally constrained, algebraic principles can be used to derive a favored methodology--which we do here. The resulting Isotropy method is compared to two other regularization methods proposed for functional source imaging (Kalman and Joint Regularization). This theoretical support for the favored status of the Isotropy method is consistent with its favorable computational performance in low prior information settings, as indicated in recent publications.

Things arising from electrocardiographic imaging: toward a theory of partial inverse problems.

The task of Electrocardiographic Imaging is an ill-posed inverse problem, requiring regularization. However, it has special features, firstly because it is a "non-stationary" inverse problem, and secondly because the inherent dynamical variety (e.g., epicardial breakthroughs, arrhythmias, ischemic changes) may preclude a fruitful nontrivial process model. Importantly, its structure places it in the category of "partial inverse problems" - a theory that arises from this setting. Surprising features of the resulting regularization methodology include the ability to fashion nontrivial regularization matrices in part (and sometimes entirely) from the data. There is evidence that these theoretical results can have significant practical benefits.

Dynamic medical imaging as a partial inverse problem.

Dynamic medical imaging problems are typically structured as "partial" inverse problems: the desired time series of images (a spatiotemporal matrix) is subject to a purely spatial transformation (its product with a transfer matrix) - so that the forward operator does not address all of the variables of the function to be transformed (analogous to partial differentiation, which is also a partial inverse problem). Given that the transfer matrix is ill-conditioned, the problem of providing the image sequence requires regularization. Under rather general conditions applicable to the setting of partial inverse problems, the regularization parameter of Tikhonov regularization generalizes to a regularization parameter operator, which in proper combination with a standard (spatial) Tikhonov regularizing operator describes the minimum-mean-square-error estimate analogously to that of the Bayesian interpretation of Tikhonov regularization as applied to a "complete" inverse problem. This is in distinction to usual methodology employed for dynamic imaging problems, including usual applications of Tikhonov regularization in this realm, which cannot similarly supply the minimum-mean-square-estimate (under the stated general conditions). The potential power of the implied methodology is illustrated with a numerical example - indicating that substantial improvements in solution estimation are possible.

The temporal prior in bioelectromagnetic source imaging problems.

The multiplicity of temporal priors proposed for regularization of the bioelectromagnetic source imaging problems [e.g., the inverse electrocardiogram (ECG) and inverse electroencephalogram (EEG) problems], is discordant with the fact that fundamental statistical principles sharply limit the choice. Thus, our objective is to derive the form of the prior consistent with the general unavailability of temporal constraints. Writing linear formulations of the inverse ECG and inverse EEG problems as H = FG + N (where the ith columns of matrices H, G, and N, are data, signal, and noise vectors at time step i, and F is the transfer matrix), and using the noninformative principle that features of the spatiotemporal prior not supplied a posteriori should be invariant under temporal transformations, we show that the implied spatiotemporal signal autocovariance matrix (of the vector formed by the entries of G) is given in block matrix form [equation in text] where Cg is a matrix of unit trace proportional to the autocovariance matrix of any column of G (representing supplied information regarding the spatial prior), epsilon[.] denotes expectation, superscript ' indicates transpose, [symbol in text] is the Kronecker product, [symbol in text] is Frobenius norm, and the "matrix scalar product" [symbol in text] indicates the inner product of the two vectors formed by the entries of the two adjacent matrices (i.e., A [symbol in text] B [triple bond] trace[A'B]). This result eliminates some uncertainties and ambiguities that have characterized spatiotemporal regularization methods--including eight methods previously introduced in this transactions. Ultimately, the result derives from an implied symmetry principle under which the form of a nontrivial noninformative temporal component of the prior can be identified. Among other things, separability of the spatiotemporal prior in terms of the above Kronecker product can be thought of as the expression of the lack of "entanglement" of the spatial and temporal contributions (a consequence of noninformativity). The approach is generalized to the important cases of non-Gaussian spatial priors, and signal and noise that are not independent (transfer matrix noise). We also demonstrate a means for computational complexity reduction, related to the application of a particular orthogonal transformation, having features dependent on whether or not the transfer matrix represents a surjective mapping.