John K. Johnstone

The bisector of a point and a plane parametric curve

Abstract The bisector of a fixed point p and a smooth plane curve C—i.e., the locus traced by a point that remains equidistant with respect to p and C—is investigated in the case that C admits a regular polynomial or rational parameterization. It is shown that the bisector may be regarded as (a subset of) a “variable-distance” offset curve to C which has the attractive property, unlike fixed-distance offsets, of being generically a rational curve. This “untrimmed bisector” usually exhibits irregular points and self-intersections similar in nature to those seen on fixed-distance offsets. A trimming procedure, which identifies the parametric subsegments of this curve that constitute the true bisector, is described in detail. The bisector of the point p and any finite segment of the curve C is also discussed.

Variation of the Axial Location of Bruch's Membrane Opening With Age, Choroidal Thickness, and Race

PURPOSE This study explores variation in the axial location of Bruchs membrane opening (BMO) to determine if this reference plane varies with age and race. METHODS There were 168 spectral-domain optical coherence tomography (SDOCT) optic nerve head volumes that were obtained from healthy subjects and manually delineated within 24 axial slices to develop point clouds for Bruchs membrane and anterior scleral surfaces. A BMO-independent reference plane was generated based on the peripapillary sclera to measure BMO position. General estimating equations were used to determine the relationship of the axial position of BMO (BMO height) with choroidal thickness, age, and race (African Descent [AD] versus European Descent [ED]) controlling for variations in axial length. RESULTS The peripapillary choroid was thinner with increasing axial length (-14.9 μm/mm, P = 0.0096), advancing age (-1.1 μm/y, P = 0.00091), and in the ED group (20.2 μm, P = 0.019) in a multivariable model. Choroidal thickness was also strongly related to BMO height (P < 0.00001) independent of all covariates. Bruchs membrane opening position was more posterior relative to the sclera in older subjects (1.3 μm/y, P = 0.00017), independent of axial length and race. However, when choroidal thickness was included in the model, this association was lost (P = 0.225). There was no significant difference in BMO height between racial groups after adjustment for age and axial length. CONCLUSIONS Bruchs membrane opening is more posteriorly located in older individuals. These differences are largely due to differences in choroidal thickness and suggest that BMO migrates posteriorly with age due to age-related choroidal thinning. However, additional studies in longitudinal datasets are needed to validate these findings.

On the lower degree intersections of two natural quadrics

In general, two quadric surface intersect in a space quartic curve. However, the intersection frequently degenerates to a collection of plane curves. Degenerate cases are frequent in geometric/solid modeling because degeneracies are often required by design. Their detection is important because degenerate intersections can be computed more easily and allow simpler treatment of important problems. In this paper, we investigate this problem for natural quadrics. Algorithms are presented to detect and compute conic intersections and linear intersections. These methods reveal the relationship between the planes of the degenerate intersections and the quadrics. Using the theory developed in the paper, we present a new and simplified proof of a necessary and sufficient condition for conic intersection. Finally, we present a simple method for determining the types of conic in a degenerate intersection without actually computing the intersection, and an enumeration of all possible conic types. Since only elementary geometric routines such as line intersection are used, all of the above algorithms are intuitive and easily implementable.

A new intersection algorithm for cyclides and swept surfaces using circle decomposition

Abstract The present vocabulary of a solid modeler is canonically the plane, (some subset of) the quadrics, and the torus. The class of cyclides is also becoming important. Quadrics and cyclides lie in the more general class of ringed surfaces: surfaces that can be swept out by a circle. This class also contains the important class of revolute surfaces. We will present a method for the exact intersection of any ringed surface with any quadric or cyclide. This algorithm shows that it is feasible to expand the vocabulary of solid modeling primitives to include all ringed surfaces. In solid modeling, surface intersection is crucial to the design of solids and their subsequent analysis. Our intersection algorithm is exact: that is, the intersection is computed symbolically rather than numerically. For exact intersection, we must reduce to degree-4 computations. We do this by concentrating on the decomposition of a surface into simpler components. Previous algorithmic development has centered around the degree of an algebraic surface. Two keys to our algorithm are circle decomposition and inversion. Solutions are provided for the inversion of a cyclide to a torus, a torus, a torus to a cyclide, and the inversion of any circle.

Peripapillary Choroidal Thickness Variation With Age and Race in Normal Eyes

PURPOSE This study examined the association between peripapillary choroidal thickness (PCT) with age and race in a group of African descent (AD) and European descent (ED) subjects with normal eyes. METHODS Optic nerve head images from enhanced depth imaging spectral-domain optical coherence tomography of 166 normal eyes from 84 subjects of AD and ED were manually delineated to identify the principal surfaces of Bruchs membrane (BM), Bruchs membrane opening (BMO), and anterior sclera (AS). Peripapillary choroidal thickness was measured between BM and AS at increasing distance away from BMO. The mean PCT was compared between AD and ED subjects and generalized estimating equation (GEE) regression analysis was used to examine the association between race and PCT overall, in each quadrant, and by distance from BMO. Models were adjusted for age, BMO area, and axial length in the regression analysis. RESULTS Overall, the mean PCT increased from 63.9 μm ± 18.1 at 0 to 250 μm to 170.3 μm ± 56.7 at 1500 to 2000 μm from BMO. Individuals of AD had a greater mean PCT than those of ED at all distances from BMO (P < 0.05 at each distance) and in each quadrant (P < 0.05 in each quadrant). Results from multivariate regression indicate that ED subjects had significantly lower PCT compared to AD overall and in all quadrants and distances from BMO. Increasing age was also significantly associated with a lower PCT in both ED and AD participants. CONCLUSIONS Peripapillary choroidal thickness varies with race and age, as individuals of AD have a thicker peripapillary choroid than those of ED. (ClinicalTrials.gov number, NCT00221923.).

Variation of Laminar Depth in Normal Eyes With Age and Race

PURPOSE To determine if laminar depth (LD) and prelaminar tissue volume (PTV) are associated with age and race in healthy human eyes. METHODS Optic nerve head images from enhanced depth imaging spectral-domain optical coherence tomography of 166 normal eyes from 84 subjects of African descent (AD) and European descent (ED) were manually delineated to identify the principal surfaces: internal limiting membrane, Bruchs membrane (BM), anterior sclera (AS), and anterior surface of the lamina cribrosa. These four surfaces defined the LD and PTV using Bruchs membrane opening (BMO) and AS for reference structures. Generalized estimating equations were used to evaluate whether the effect of age on each outcome was differential by race. RESULTS When age was analyzed as a continuous variable, the interaction term between age and race was statistically significant for mean LDBMO (P = 0.015) and mean LDAS (P = 0.0062) after adjusting for axial length and BMO area. For every 1-year increase in age, the LDAS was greater on average by 1.78 μm in AD subjects and less by 1.71 μm in ED subjects. Mean PTV was lower in the older subjects (1248 × 10(6) μm(3) AD, 881 × 10(6) μm(3) ED) compared to the younger subjects (1316 × 10(6) μm(3) AD, 1102 × 10(6) μm(3) ED) in both groups. CONCLUSIONS With increasing age, the LD changes differently across racial groups in normal subjects. The LD in ED subjects showed a significantly decreasing slope suggesting that the lamina moves anteriorly with age in this group.

Computing the intersection of a plane and a natural quadric

Abstract A method of computing the intersection of a plane and a natural quadric surface is presented. This problem is basic in geometric areas such as solid modeling and descriptive geometry. Our method, arising out of recent work on lower degree intersections of quadrics, computes the directions of the axes of the intersection, and then computes their lengths using the Dandelin sphere. The method also gives all parallel plane sections of the natural quadric, with no added computation.

Displacement of the Lamina Cribrosa in Response to Acute Intraocular Pressure Elevation in Normal Individuals of African and European Descent

Purpose To assess if the in vivo mechanical displacement of the anterior laminar cribrosa surface (ALCS) as a response of an acute elevation in intraocular pressure (IOP) differs in individuals of European (ED) and African descent (AD). Methods Spectral-domain optical coherence tomography (SDOCT) scans were obtained from 24 eyes of 12 individuals of AD and 18 eyes of 9 individuals of ED at their normal baseline IOP and after 60 seconds IOP elevation using ophthalmodynamometry. Change in depth (displacement) of the LC and to the prelaminar tissue (PLT) were computed in association with the change (delta) in IOP (Δ IOP), race, age, corneal thickness, corneal rigidity (ocular response analyzer [ORA]), and axial. Results In the ED group for small IOP elevations (Δ IOP < 12 mm Hg), the ALCS initially displaced posteriorly but for larger increase of IOP an anterior displacement of the lamina followed. Inversely, in the AD group the ALCS did not show a significant posterior displacement for small Δ IOP, while for larger IOP increases the ALCS significantly displaced posteriorly. Posterior displacement of the lamina cribrosa (LC) was also significantly correlated with longer axial length, higher corneal thickness, and ORA parameters. Prelaminar tissue posteriorly displaced for any magnitude of Δ IOP, in both groups. Conclusions The African descent group demonstrated a greater acute posterior bowing of the LC after adjustment for age, axial length, Bruchs membrane opening (BMO) area, and ORA parameters. Greater PLT posterior displacement was also seen in the AD group with increasing IOP, which was tightly correlated with the displacement of the LC.

The convex hull of freeform surfaces

We present an algorithm for computing the convex hull of freeform rational surfaces. The convex hull problem is reformulated as one of finding the zero-sets of polynomial equations; using these zero-sets we characterize developable surface patches and planar patches that belong to the boundary of the convex hull.

Rational discrete generalized cylinders and their application to shape recovery in medical images

Generalized cylinders (GCs) are a popular representational tool in computer vision. In medical imaging, the curved axis GC is particularly applicable to a number of elongated physical structures such as vasculature, bone and bronchi. In many of these instances, it is necessary to recover curved-axis GCs with arbitrary cross-sections. It is also vital that these structures, once recovered, can be analyzed and visualized with off-the-shelf algorithms and software packages. Such tools are usually designed to operate on the domain of polynomial or rational surfaces. Unfortunately most extant, suitably versatile GC representations do not admit rational parameterizations. We develop an entirely rational B-spline representation for generalized cylinders with curved axes and arbitrary cross-section functions. We demonstrate how our representation can be used as a deformable model by extracting a rational GC from pre-segmented spinal data using a discrete dynamic surface fit.