Discrete Slices Research Articles

Recovering Missing Slices of the Discrete Fourier Transform Using Ghosts

The discrete Fourier transform (DFT) underpins the solution to many inverse problems commonly possessing missing or unmeasured frequency information. This incomplete coverage of the Fourier space always produces systematic artifacts called Ghosts. In this paper, a fast and exact method for deconvolving cyclic artifacts caused by missing slices of the DFT using redundant image regions is presented. The slices discussed here originate from the exact partitioning of the Discrete Fourier Transform (DFT) space, under the projective Discrete Radon Transform, called the discrete Fourier slice theorem. The method has a computational complexity of O(n log(2) n) (for an n=N×N image) and is constructed from a new cyclic theory of Ghosts. This theory is also shown to unify several aspects of work done on Ghosts over the past three decades. This paper concludes with an application to fast, exact, non-iterative image reconstruction from a highly asymmetric set of rational angle projections that give rise to sets of sparse slices within the DFT.

Slow subduction and buoyant exhumation of the Sanbagawa eclogite

The Western Iratsu body of the Sanbagawa belt (SW Japan) represents a <2km thick garnet–epidote amphibolite slice with minor intercalated occurrences of eclogite. Equilibrium phase assemblage diagrams (pseudosections) for eclogite and garnet–epidote amphibolite from this body can be successfully used to reconstruct the pressure (P)–temperature (T) path and the evolution of the associated mineral assemblages. Microstructural observations combined with calculated pseudosections reveal two distinct stages of subduction-related metamorphism referred to as M1 and M2. P–T conditions of M1 evolved from 0.9GPa, 570°C to 1.2GPa, 660°C, implying relatively high thermal gradients possibly developed shortly after the initiation of subduction. The available age data and the characteristics of the P–T path suggest that the Western Iratsu body was detached from the slab at 1.2GPa (~40km depth) and the subsequent subduction occurred at a very slow rate due to weak mechanical coupling to the downgoing slab. The prograde metamorphism associated with the second subduction phase (M2) reached the eclogite facies (~1.8GPa, 510–560°C), but mafic rock compositions recrystallized into either eclogite or garnet–epidote amphibolite assemblages, depending on the bulk rock (CaO+Na2O)/(FeO+MgO) ratio. The thermal peak of M2 (~620°C at 1.6GPa) was attained during the early phase of exhumation. Further decompression and re-equilibration (M3) took place in the epidote-amphibolite facies. The M2 to M3 P–T evolution is associated with a progressive increase in thermal gradient (T/P ratio), which can be explained by thermal modeling as the result of the inflow of a very young hot section of the slab. The bulk density of the whole Western Iratsu body at the point of onset of exhumation—the return point—is estimated to be ~3180kgm−3, which is less than the density of mantle, implying that buoyant rise is a viable mechanism for its initial exhumation. Buoyant rise is in accordance with the widespread early-exhumation related deformation that has a dip-parallel stretching lineation. Exhumation of the buoyant discrete mafic slice along the decoupled slab–mantle interface could have been triggered by the heating that lead to fluid liberation and a resultant reduction of the rock strength.

Comparison of China/USA science and technology performance

The relative performance of science and technology (S&T) in the USA and PRC was compared in terms of quantity and quality, as reflected in their technical literatures. Three databases (Science Citation Index (SCI), INSPEC, Ei Compendex) were selected for the quantity comparison, and citation analysis in the SCI was used for the quality comparison. Thirty technology and research areas were compared for quantity production, and are presented in this paper. These 30 areas were selected based on our previous assessment of PRC S&T output, and represented areas of emphasis by the PRC in physical, environmental, engineering, and life sciences. In almost all technical areas, the USA had the quantity (number of papers) lead (for the period 2002–2007) based on the SCI results, although the PRC has made dramatic strides to overtake the USA. In most of the technical areas, by 2007 PRC had attained parity with, or exceeded, the S&T literature production of the USA in the INSPEC database. The major exceptions were the biomedical field and some aspects of environmental science, where the USA still had a large lead. For most technical areas, by 2007 the PRC had even higher relative S&T literature production, based on the Ei Compendex, compared to the INSPEC results. Moreover, the USA production appears to have peaked (in the Ei Compendex) in the 2005 time frame, despite increasing amounts of funding for S&T research. The PRC challenge in non-biomedical research and technology sectors becomes apparent in those databases that do not contain substantial biomedical research papers, and therefore remove a substantial intrinsic USA advantage. For quality computations, the publication and citation results were normalized to discrete slices of time, and are presented for nanotechnology only (for the period 1998–2003). While the USA held a commanding lead in quality over the PRC (and the other major nanotechnology producer nations as well) during the past decade, the PRC has increased the quality of its publications monotonically, and now appears to be competitive with France, Italy, Japan, and Australia, using the quality metric in this paper.

Poster 93: Standardized Magnetic Resonance Image Signal Intensity in Tibialis Posterior Tendinopathy

Objectives: To test the hypothesis that signal intensity from tibialis posterior tendon is significantly different in patients with tendinopathy and normals and to test whether discrete anatomical locations along the posterior tibialis tendon are sufficiently divergent in signal intensity to determine areas of greatest pathology. Design: Cross-sectional masked analytical study. Setting: Biomedical research facility and image-processing laboratory. Participants: 12 subjects with tibialis posterior tendinopathy include 6 with inhomogeneous signal (partial tear), 4 with tendonitis, 2 with peritendonitis, and 5 control subjects confirmed by magnetic resonance imaging. Interventions: Not applicable. Main Outcome Measures: Postgadolinium-spoiled gradient-echo axial magnetic resonance images with fat suppression of the tibialis posterior tendon were obtained for each subject. Image processing was undertaken to normalize each image on a standard scale (0–4095) to give the same tissue-specific intensity, independent of the scanning protocol and patient. The mean signal intensity was derived for each slice and the peak value chosen as the primary outcome measure and between-group comparisons made at discrete anatomical locations. Results: Throughout the length of the tendon, and at every discrete tendon slice the mean signal intensity was higher for abnormal tendons for 14 slices (mean ± SD abnormal tendon group vs control group, 999±355 and 745±189 at insertion; mean, 833±480 and 569±137 at midlength; mean, 724±294 and 534±175 at ankle wrap-around; mean, 496212 and 379±77 at the proximal ankle region). Mean between-group differences (95% CI) were 254 (−31 to 540) at insertion, 263 (−61 to 588) at tendon midlength, 190 (−60 to 440) at the ankle wrap-around region, and 117 (−33 to 266) at the proximal ankle region. Conclusions: Tendinopathy is quantitatively distinguished from normal tendon using this technique. At discrete anatomic locations including insertion, midlength, ankle wrap-around, and proximal ankle regions, signal intensities are higher than normal values.

Intensity-modulated Radiotherapy Planning from Limited Anatomical Information: Is Sim-CT Sufficient for Planning Women with Breast Cancer Receiving Intensity-modulated Radiotherapy?

To investigate intensity-modulated radiotherapy (IMRT) plans for women with carcinoma of the breast, using a small number of Sim-CT slices, thus avoiding changing the patient's position and potential problems with CT capacity. Ten CT scans of women with breast cancer were obtained for use in the study. IMRT plans based on an open tangent pair and additional top-up segment fields were created using the full CT scan, and represented the gold standard treatment plan for comparison purposes. Five-slice CT simulator scans were artificially created by omitting intermediate slices from the full CT scans. Additionally, the intermediate CT slices were recreated via interpolation of the five slices using a standard interpolation algorithm. IMRT plans were created in the same way as for the full CT scans. To allow a suitable plan comparison to be made, the beam segments and monitor units were transferred to the full CT scans, and the dose distribution calculated. The interpolated five-slice plans showed no significant difference in the volume of tissue receiving dose outside the range 95-105%, compared with the IMRT plans created using the full CT data set (1.3 +/- 2.2%, P = 0.092). In contrast, the discrete slice CT simulator plans increased by 6.3 +/- 5.4%, P = 0.0054, showing a statistically significant difference in the dose distribution produced and a clinically inferior plan. Plans created using five discrete slice CT scans were inferior to full CT-derived IMRT treatment plans, and are therefore not acceptable for IMRT. However, interpolating five CT simulator slices provides adequate anatomical information to produce comparable IMRT plans to those created by full CT scans of the patient. This allows the introduction of IMRT for this patient group without the need to change treatment position to accommodate CT scanning.

3D discrete X-ray transform

The analysis of 3D discrete volumetric data becomes increasingly important as computation power increases. 3D analysis and visualization applications are expected to be especially relevant in areas like medical imaging and nondestructive testing, where elaborated continuous theory exists. However, this theory is not directly applicable to discrete datasets. Therefore, we have to establish theoretical foundations that will replace the existing inexact discretizations, which have been based on the continuous regime. We want to preserve the concepts, properties, and main results of the continuous theory in the discrete case. In this paper, we present a discretization of the continuous X-ray transform for discrete 3D images. Our definition of the discrete X-ray transform is shown to be exact and geometrically faithful as it uses summation along straight geometric lines without arbitrary interpolation schemes. We derive a discrete Fourier slice theorem, which relates our discrete X-ray transform with the Fourier transform of the underlying image, and then use this Fourier slice theorem to derive an algorithm that computes the discrete X-ray transform in O ( n 4 log n ) operations. Finally, we show that our discrete X-ray transform is invertible.

3D Fourier based discrete Radon transform

The Radon transform is a fundamental tool in many areas. For example, in reconstruction of an image from its projections (CT scanning). Recently A. Averbuch et al. [SIAM J. Sci. Comput., submitted for publication] developed a coherent discrete definition of the 2D discrete Radon transform for 2D discrete images. The definition in [SIAM J. Sci. Comput., submitted for publication] is shown to be algebraically exact, invertible, and rapidly computable. We define a notion of 3D Radon transform for discrete 3D images (volumes) which is based on summation over planes with absolute slopes less than 1 in each direction. Values at nongrid locations are defined using trigonometric interpolation on a zero-padded grid. The 3D discrete definition of the Radon transform is shown to be geometrically faithful as the planes used for summation exhibit no wraparound effects. There exists a special set of planes in the 3D case for which the transform is rapidly computable and invertible. We describe an algorithm that computes the 3D discrete Radon transform which uses O( Nlog N) operations, where N= n 3 is the number of pixels in the image. The algorithm relies on the 3D discrete slice theorem that associates the Radon transform with the pseudo-polar Fourier transform. The pseudo-polar Fourier transform evaluates the Fourier transform on a non-Cartesian pointset, which we call the pseudo-polar grid. The rapid exact evaluation of the Fourier transform at these non-Cartesian grid points is possible using the fractional Fourier transform.

Orthogonal discrete periodic Radon transform. Part II: applications

In this paper, we study the properties and possible applications of the newly proposed orthogonal discrete periodic Radon transform (ODPRT). Similar to its previous version, the new ODPRT also possesses the useful properties such as the discrete Fourier slice theorem and the circular convolution property. They enable us to convert a 2-D application into some 1-D ones such that the computational complexity is greatly reduced. Two examples of using ODPRT in the realization of 2-D circular convolution and blind image resolution are illustrated. With the fast ODPRT algorithm, efficient realization of 2-D circular convolution is achieved. For the realization of blind image restoration, we convert the 2-D problem into some 1-D ones that reduces the computation time and memory requirement. Besides, ODPRT adds more constraints to the restoration problem in the transform domain that makes the restoration solution better. Significant improvement is obtained in each case when comparing with the traditional approaches in terms of quality and computation complexity. They illustrate the potentially widespread applications of the proposed technique.

Modeling of cutting geometry and forces for 5-axis sculptured surface machining

5-Axis sculptured surface machining is simulated using discrete geometric models of the tool and workpiece to determine the tool contact area, and a discrete mechanistic model to estimate the cutting forces. An extended Z-buffer model represents the workpiece, while a discrete axial slice model represents the cutting tool. Determination of the contact area for a given tool move requires a swept envelope (SWE) of the tool path. The SWE is used to find the intersections of the tool envelope with Z-buffer elements (ZDVs) representing the workpiece. A 3-axis approximation of the 5-axis tool movement is used to simplify the calculations while maintaining a desired level of accuracy. The intersection of the SWE with each ZDV yields segments which are used to find the contact area between the cutter and the workpiece for a given tool path. The contact area is subsequently used with the discrete force model to calculate the vector cutting force acting on the tool.

Single kick approximations for beam-beam deflections

A six-dimensional symplectic beam-beam interaction map using finite discrete slices of a strong beam is extended to infinitesimal slices. The new map is calculated under the assumption of a longitudinal Gaussian distribution with approximations. A round Gaussian beam is simulated to demonstrate accuracies of the approximations.

DISCRETE TIME PROCESS ALGEBRA: ABSOLUTE TIME, RELATIVE TIME AND PARAMETRIC TIME

We discuss the key notions of discrete time process algebra in the setting of ACP. Time is measured in discrete slices. The emphasis is on absolute, relative and parametric time notation. Note: Partial support received from ESPRIT Basic Research Action 7166, C0NCUR2.

The discrete periodic Radon transform

In this correspondence, a discrete periodic Radon transform and its inversion are developed. The new discrete periodic Radon transform possesses many properties similar to the continuous Radon transform such as the Fourier slice theorem and the convolution property, etc. With the convolution property, a 2-D circular convolution can be decomposed into 1-D circular convolutions, hence improving the computational efficiency. Based on the proposed discrete periodic Radon transform, we further develop the inversion formula using the discrete Fourier slice theorem. It is interesting to note that the inverse transform is multiplication free. This important characteristic not only enables fast inversion but also eliminates the finite-word-length error that may be generated in performing the multiplications.

Petrology of the crossite schist from Fruška Gora Mts (Yugoslavia), relic of a subducted slab of the Tethyan oceanic crust

The mineral assemblage formed during the main stage of metamorphic recrystallization in the crossite schist from the Fruška Gora Mts corresponds to the epidote-blueschist subfacies or high- T epidote-bearing segment of blueschist facies ( P = 7–9 kbar and T = ± 400 °C). Geochemical investigations indicate a primary alkaline character of the crossite schist, which suggests a volcanic arc or mid ocean ridge basalt tectonic setting. These rocks most probably represent a small relic of the former Mezozoic Tethyan oceanic crust, a discrete slice of mafic rocks which was metamorphosed under subducting conditions in the Early Cretaceous.

The multi‐load method of determining discrete system parameters in a loaded three dimensional photoelastic model

AbstractThe determination of the secondary principal stress difference and their orientations for a discrete slice of thickness dz along a particular light path in a loaded three dimensional photoelastic model is discussed in this paper. It is well known that once the geometry of the loaded member and the geometry of loading on the member are fixed, the secondary principal stress directions at any particular point along any line of consideration remain constant. The secondary principal stress values will get multiplied by the factor through which the loading on the component is increased. This has been made use of in the present method and from the characteristic parameters obtained at the end of any light path for different loadings on the three dimensional model, the secondary principal stress difference and their orientations are obtained. The method proposed is a fully non‐destructive method of testing and involves measurements only on a transmission polariscope.

Cenozoic thrust emplacement of a Devonian batholith, northeastern Brooks Range: Involvement of crystalline rocks in a foreland fold-and-thrust belt

Involvement of crystalline rocks in thrusting near the foreland basin of a fold-and-thrust belt is relatively uncommon. In the northeastern Brooks Range, the Devonian Okpilak batholith was thrust northward and structurally elevated above adjacent foreland basin deposits during Cenozoic fold-and-thrust deformation. The batholith may have acted initially as a regional structural buttress, but a drop in the basal detachment surface to greater depth south of the batholith resulted in northward transport of the batholith. Shortening within the batholith was accommodated by (1) the development of discrete thrust slices bounded by ductile shear zones, (2) simple shear and development of penetrative mesoscopic and microscopic fabrics throughout the batholith, or both. The Mississippian Kayak Shale, a regional detachment horizon at the base of the overlying cover sequence, is depositionally thin or absent adjacent to the batholith. Thus, most of the cover sequence remained structurally coupled to the batholith during thrusting and was shortened by the development of penetrative structures.

Metamorphic evolution of the Central Metamorphic Belt, Klamath Province, California: an inverted metamorphic gradient beneath the Trinity peridotite

ABSTRACT Metabasalts and metasedimentary rocks of the Devonian Central Metamorphic Belt comprise the lower plate of the east‐dipping Trinity thrust system in the Klamath province. An inverted metamorphic gradient is preserved in the Central Metamorphic Belt; metamorphic conditions decrease from amphibolite facies adjacent to the Trinity thrust, through albite‐epidote amphibolite facies, to upper greenschist facies at the base of the Central Metamorphic Belt. Mineral chemistry, mineral assemblages and limited geothermometry suggest that peak metamorphic conditions decrease structurally downward from 650 ± 50° C at the Trinity thrust to 500 ± 50° C at the base of the Central Metamorphic Belt, under pressures of 5 ± 3 kbar. Synmetamorphic Ab + Qtz veins, up to 1 m thick, increase in abundance towards the Trinity thrust. Infiltration of H2O‐CO2 fluids derived from prograde devolatilization reactions in the Central Metamorphic Belt caused extensive hydration and metasomatism of the Trinity peridotite; the hanging wall block of the Trinity thrust zone.Geological relationships and the preserved inverted metamorphic gradient suggest that the Central Metamorphic Belt formed in an east‐dipping Devonian subduction zone in an oceanic environment. The Central Metamorphic Belt appears to represent a discrete slice of accreted oceanic crust several km thick, rather than progressively accreted material. Metamorphic pressures recorded by the Central Metamorphic Belt are intermediate between the ∼2 kbar pressures recorded in dynamothermal aureoles beneath obducted ophiolites and the 7–10 kbar preserved in subduction‐related inverted metamorphic gradients. The lack of blueschist facies mineral assemblages in the Central Metamorphic Belt may possibly be explained by an anomalously warm geotherm prior to subduction or early shear heating prior to the arrival of wet rocks at depth.

Stop identification using hidden Markov models

A series of experiments has been undertaken to assess the power of discrete spectral slices for automatically discriminating between the voiceless plosives /p,t,k/ in CV syllables. These experiments involve a 936‐token data base consisting of 52 instances of each of the 18 syllables /p,t,k/ × /i,e,ae,a,ow,u/ spoken by 13 male and 13 female talkers. In one experiment, identification was attempted using a single 12‐pole LPC onset spectrum. The onset spectra from each talker were compared, using a log‐likelihood distance measure, with the 900 onset spectra of the remaining 25 talkers. An overall classification accuracy of 92% was achieved using a k‐nearest‐neighbor decision strategy. A second experiment involves classifiers which use a series of LPC spectra computed during the first 50 ms of the stop release. Each CV syllable is modeled as a hidden Markov process which generates a spectrum every 5 ms. Classification is performed using either a Viterbi or forward‐backward decoding strategy.

Passage of light through a general three dimensional model—equations linking characteristic parameters with system parameters

The determination of the secondary principal stress difference and their orientations for a discrete slice of thickness dz along a particular light path in a three dimensional photoelastic model is discussed in this paper. These quantities are determined in terms of the characteristic parameters1 and their variations at discrete points distance z and z + dz along the light path. The variations of the characteristic parameters are obtained by deriving the equations linking the characteristic parameters for the light path and the system parameters of the discrete element. The validity of the equations so derived has also been checked by applying these equations to the solution of the problem of a uniformly twisted prismatic bar.

Imbricate low-angle faulting in uppermost Franciscan rocks, South Yolla Bolly area, northern California

The South Yolla Bolly area of northern California contains the intersection of the Franciscan complex, the Klamath Mountains, and the Great Valley sequence. Four distinct lithic units, among them the South Fork Mountain Schist, as well as several subunits, have been discerned in the Franciscan rocks of this area. These units are composed mostly of coherent or semi-coherent layered sedimentary and mafic igneous rocks and their metamorphic equivalents rather than melange, and are arranged in largely upright, parallel imbricate fault slices. Four major low-angle faults have been clearly recognized. Paleontologic data indicate that at least one fault places older material over younger. Lithic units exhibit various degrees of differences in metamorphic mineral assemblages, textural grade, and amount of internal deformation across the faults, with more intensely deformed and metamorphosed material occupying successively higher structural positions. These contrasts indicate that some of these faults served as surfaces by which deformed rocks were emplaced differentially upward during and/or after metamorphism. Original bedding and compositional layering in all of the coherent units generally parallel the thrust faults. Two fault-bounded units are each composed of a coherent sequence of basalt at the base overlain by bedded chert, and further overlain by mudstone and graywacke or their metamorphic equivalents. The faults which underlie these two sequences seem to have originated as decollements localized at or near a sedimentary rock – mafic-crust interface, perhaps as discrete underthrust slices. The distinct contrasts in original rock composition among these units plus the aforementioned characteristics seem to indicate that these faults had an early history in the accretionary prism apart from their later use as convenient surfaces for differential upward emplacement of metamorphosed material.

Caledonian deformation in Norway

Geological research and mapping over the last decade has led to a reassessment of the Caledonian orogen in Norway. Two principal stages of orogenesis are now established: (1) a late Cambrian to early Ordovician Finnmarkian or Grampian event recognized mainly in Finnmark and probably also developed in western Norway; and (2) the widespread Middle-Upper Silurian Scandinavian stage of central and southern areas. Another significant finding, aided by radiometric studies, is that of incorporation of Precambrian crystallines as discrete tectonic slices within the Caledonian nappe complexes, reminiscent of the Lewisian-within-Moine relationship of Scotland. In the extensive gneiss terrain of western districts and of Lofoten the effects of Caledonian tectonometamorphic reconstitution have been minimal. Biostratigraphic and geochronological indicators provide evidence that diachroneity is a major feature common to both the above evorogenic stages, with deformation younging progressively towards the southeastern quadrant. In S Norway, folding and local thrusting continued into the Devonian. In terms of geometry and strain, individual nappes are generally extensive, thin, sheet or westward-thinning elements in which protracted high strains in basal zones have produced complex, banded mylonitic and pseudo-psammitic lithologies. Indications of an overall, primary, progressive simple shear deformation are ubiquitous, commonly with evidence of fold rotation into the stretching lineation trend in areas of high ductile strain; this is masked by important episodic increments of flattening and extensional strain producing lensoid, mega-boudin structures and leading to excision of some nappe units. This vertical shortening and gravitational spreading is recognized in both the Cambrian and the Silurian stages of the orogenesis and affects calculations of nappe translation, estimated in hundreds of km for central and southern areas. In conclusion, some comparisons are outlined between the Scandinavian and East Greenland Caledonides; in these two segments of the orogen, nappe displacement is in opposite directions.