A Generalization of Mordell to Ternary Quadratic Forms

Mordell in 1958 gave a new proof of the three squares theorem. We generalize those techniques to characterize the integers represented by the remaining six "Ramanujan-Dickson ternaries" as well as three other ternary forms.Comment: 17 page

How does the femoral cortex depend on bone shape? A methodology for the joint analysis of surface texture and shape.

In humans, there is clear evidence of an association between hip fracture risk and femoral neck bone mineral density, and some evidence of an association between fracture risk and the shape of the proximal femur. Here, we investigate whether the femoral cortex plays a role in these associations: do particular morphologies predispose to weaker cortices? To answer this question, we used cortical bone mapping to measure the distribution of cortical mass surface density (CMSD, mg/cm2) in a cohort of 125 females. Principal component analysis of the femoral surfaces identified three modes of shape variation accounting for 65% of the population variance. We then used statistical parametric mapping (SPM) to locate regions of the cortex where CMSD depends on shape, allowing for age. Our principal findings were increased CMSD with increased gracility over much of the proximal femur; and decreased CMSD at the superior femoral neck, coupled with increased CMSD at the calcar femorale, with increasing neck-shaft angle. In obtaining these results, we studied the role of spatial normalization in SPM, identifying systematic misregistration as a major impediment to the joint analysis of CMSD and shape. Through a series of experiments on synthetic data, we evaluated a number of registration methods for spatial normalization, concluding that only those predicated on an explicit set of homologous landmarks are suitable for this kind of analysis. The emergent methodology amounts to an extension of Geometric Morphometric Image Analysis to the domain of textured surfaces, alongside a protocol for labelling homologous landmarks in clinical CT scans of the human proximal femur

Independent measurement of femoral cortical thickness and cortical bone density using clinical CT.

The local structure of the proximal femoral cortex is of interest since both fracture risk, and the effects of various interventions aimed at reducing that risk, are associated with cortical properties focused in particular regions rather than dispersed over the whole bone. Much of the femoral cortex is less than 3mm thick, appearing so blurred in clinical CT that its actual density is not apparent in the data, and neither thresholding nor full-width half-maximum techniques are capable of determining its width. Our previous work on cortical bone mapping showed how to produce more accurate estimates of cortical thickness by assuming a fixed value of the cortical density for each hip. However, although cortical density varies much less over the proximal femur than thickness, what little variation there is leads to errors in thickness measurement. In this paper, we develop the cortical bone mapping technique by exploiting local estimates of imaging blur to correct the global density estimate, thus providing a local density estimate as well as more accurate estimates of thickness. We also consider measurement of cortical mass surface density and the density of trabecular bone immediately adjacent to the cortex. Performance is assessed with ex vivo clinical QCT scans of proximal femurs, with true values derived from high resolution HRpQCT scans of the same bones. We demonstrate superior estimation of thickness than is possible with alternative techniques (accuracy 0.12 ± 0.39 mm for cortices in the range 1-3mm), and that local cortical density estimation is feasible for densities >800 mg/cm(3).This is the accepted manuscript of an article published in Medical Image Analysis (GM Treece, AH Gee, Medical Image Analysis 2015, 20(1), 249–264

Changes in Body Measurements of Heifers at First Parturition

Author Institution: Department of Dairy Science, Ohio Agricultural Experiment Station, Wooste

Calibration of an orientation sensor for freehand 3D ultrasound and its use in a hybrid acquisition system

BACKGROUND: Freehand 3D ultrasound is a powerful imaging modality with many potential applications. However, its reliance on add-on position sensors, which can be expensive, obtrusive and difficult to calibrate, is a major drawback. Alternatively, freehand 3D ultrasound can be acquired without a position sensor using image-based techniques. Sensorless reconstructions exhibit good fine scale detail but are prone to tracking drift, resulting in large scale geometrical distortions. METHOD: We investigate an alternative position sensor, the Xsens MT9-B, which is relatively unobtrusive but measures orientation only. We describe a straightforward approach to calibrating the sensor, and we measure the calibration precision (by repeated calibrations) and the orientation accuracy (using independent orientation measurements). We introduce algorithms that allow the MT9-B potentially to correct both linear and angular drift in sensorless reconstructions. RESULTS: The MT9-B can be calibrated to a precision of around 1 degrees . Reconstruction accuracy is also around 1 degrees . The MT9-B was able to eliminate angular drift in sensorless reconstructions, though it had little impact on linear drift. In comparison, six degree-of-freedom drift correction was shown to produce excellent reconstructions. CONCLUSION: Gold standard freehand 3D ultrasound acquisition requires the synthesis of image-based techniques, for good fine scale detail, and position sensors, for good large scale geometrical accuracy. A hybrid system incorporating the MT9-B offers an attractive compromise between quality and ease of use. The position sensor is unobtrusive and the system is capable of faithful acquisition, with the one exception of linear drift in the elevational direction