Plane In 3D Space Research Articles

The Gamma-Ray Origin of RX J0852.0-4622 Quantifying the Hadronic and Leptonic Components: Further Evidence for the Cosmic-Ray Acceleration in Young Shell-type SNRs

Fukui et al. quantified the hadronic and leptonic gamma-rays in the young TeV gamma-ray shell-type supernova remnant (SNR) RX J1713.7-3946 (RX J1713), and demonstrated that gamma rays are a combination of hadronic and leptonic gamma-ray components with a ratio of ∼6: 4 in gamma-ray counts N g. This discovery, which adopted a new methodology of multi-linear gamma-ray decomposition, was the first quantification of the two gamma-ray components. In the present work, we applied the same methodology to another TeV gamma-ray shell-type SNR RX J0852.0-4622 (RXJ0852) in 3D space characterized by (the interstellar proton column density N p)-(the nonthermal X-ray count N x)-[N g], and quantified the hadronic and leptonic gamma-ray components as having a ratio of ∼5:5 in N g. The present work adopted the fitting of two/three flat planes in 3D space instead of a single flat plane, which allowed suppression of the fitting errors. This quantification indicates that hadronic and leptonic gamma-rays are of the same order of magnitude in these two core-collapse SNRs, verifying the significant hadronic gamma-ray components. We argue that the target interstellar protons, in particular their spatial distribution, are essential in any attempts to identify the type of particles responsible for gamma-ray emission. The present results confirm that cosmic-ray (CR) energy ≲100 TeV is compatible with a scheme in which SNRs are the dominant source of these Galactic CRs.

Target enclosing control for fixed-wing unmanned aerial vehicle in three dimensional space: An attractive vector approach

This paper addresses the target enclosing control problem for fixed-wing UAVs in three dimensional space with a novel attractive vector approach. Firstly, by analyzing the properties of the desired ring motion plane in 3D space with an arbitrary desired direction vector, an attractive vector approach that provides a control direction for solving the target enclosing problem is established. Then, based on the attractive vector, the desired course angle and flight path angle are constructed, respectively. Thereafter, the target enclosing control law that guarantees the desired ring motion plane is perpendicular to the arbitrary desired direction vector in 3D space is designed, and the corresponding stability is analyzed by the Lyapunov function approach. Finally, the effectiveness of the proposed approach is validated by both numerical simulations and experiments.

A novel interpolation-free sharp-interface immersed boundary method

This paper describes a novel 2nd order direct forcing immersed boundary method designed for simulation of 2D and 3D incompressible flow problems with complex immersed boundaries. In this formulation, each cell cut by the immersed boundary (IB) is reshaped to conform to the shape of the IB. IBs are modeled as a series of 2D planes in 3D space that connect seamlessly at the edges of the cut cells, in a way that mimics conformal grid. IBs are represented in a continuous and consistent fashion from one cell to another, thus eliminating spatial pressure oscillations originating from inconsistent description of the IB as well as the traditional stair-step problem, leading to a more accurate resolution of the boundary layer. Boundary conditions are enforced at the exact location of the IB devoid of interpolation, which guarantees sound simulations even on grids with high aspect ratio, and enables simulations of flow packed with multiple IBs in close proximity. Boundary conditions for each phase across the IB are enforced independently, yielding a unique capability to solve flows with zero-thickness IBs. Simulations of a large number of 2D and 3D test cases confirm the prowess of the devised immersed boundary method in solving flows over multiple loosely/closely-packed IBs; stationary, moving and highly morphing IBs; as well as IBs with zero-thickness. Results show that predictions from the proposed scheme agree remarkably well with theoretical models and experimental data for both integral variables and local flow fields and they are often with less than 1% deviation from solutions obtained by conformal grid of similar resolution.

Edge Detection in 3D Point Clouds Using Digital Images

This paper presents an effective and semi-automated method for detecting 3D edges in 3D point clouds with the help of high-resolution digital images. The effort aims to contribute towards addressing the unsolved problem of automated production of vector drawings from 3D point clouds of cultural heritage objects. Edges are the simplest primitives to detect in an unorganized point cloud and an algorithm was developed to perform this task. The provided edges are defined and measured on 2D digital images of known orientation, and the algorithm determines the plane defined by the edge on the image and its perspective center. This is accomplished by applying suitable transformations to the image coordinates of the edge points based on the Analytical Geometry relationships and properties of planes in 3D space. This plane inevitably contains the 3D points of the edge in the point cloud. The algorithm then detects and isolates those points which define the edge in the world system. Finally, the goal is to reliably locate the points that describe the desired edge in their true position in the geodetic space, using several constraints. The algorithm is firstly investigated theoretically for its efficiency using simulation data and then assessed under real conditions and under different image orientations and lengths of the edge on the image. The results are presented and evaluated.

Visualisation of Parkinsonian, essential and physiological tremor planes in 3Dspace.

Based on the fact that tremors display some distinct 3D spatial characteristics, we decided to visualise tremor planes in 3D space. We obtained 3-axial linear accelerometer signals of hand tremors from 58 patients with Parkinson´s disease (PD), 37 with isolated resting tremor (iRT), 75 with essential tremor (ET), and 44 healthy volunteers with physiological tremor (Ph). For each group analysis was done with subsequent spatial 3D regression of the input data i.e. along the x, y and z axes; the projected vector lengths in the individual (vertical transversal XY, vertical longitudinal XZ and horizontal YZ) reference frame planes and their angles. Most meaningful and statistically significant differences were found in the analyses of the 3D vector lengths. The tremor of the PD and the iRT group was oriented mainly in the horizontal YZ plane. The tremors of the patients with ET and Ph were oriented approximately in the midway between the all three referential planes with less tilt toward the vertical longitudinal XZ plane.

Analysis of Three-Dimensional Circular Tracking Movements Based on Temporo-Spatial Parameters in Polar Coordinates

Motor control characteristics of the human visuomotor control system need to be analyzed in the three-dimensional (3D) space to study and imitate human movements. In this paper, we examined circular tracking movements on two planes in 3D space from a motor control perspective based on three temporospatial parameters in polar coordinates. Sixteen healthy human subjects participated in this study and performed circular target tracking movements rotating at 0.125, 0.25, 0.5, and 0.75 Hz in the frontal or sagittal planes in three-dimensional space. The results showed that two temporal parameter errors on each plane were proportional to the change in the target velocity. Furthermore, frontal plane circular tracking errors without depth for a spatial parameter were lower than those for sagittal plane circular tracking with depth. The experimental protocol and data analysis allowed us to analyze the motor control characteristics temporospatially for circular tracking movement with various depths and speeds in the 3D VR space.

AirContour

Recognizing in-air hand gestures will benefit a wide range of applications such as sign-language recognition, remote control with hand gestures, and “writing” in the air as a new way of text input. This article presents AirContour, which focuses on in-air writing gesture recognition with a wrist-worn device. We propose a novel contour-based gesture model that converts human gestures to contours in 3D space and then recognizes the contours as characters. Different from 2D contours, the 3D contours may have the problems such as contour distortion caused by different viewing angles, contour difference caused by different writing directions, and the contour distribution across different planes. To address the above problem, we introduce Principal Component Analysis (PCA) to detect the principal/writing plane in 3D space, and then tune the projected 2D contour in the principal plane through reversing, rotating, and normalizing operations, to make the 2D contour in right orientation and normalized size under a uniform view. After that, we propose both an online approach, AC-Vec, and an offline approach, AC-CNN, for character recognition. The experimental results show that AC-Vec achieves an accuracy of 91.6% and AC-CNN achieves an accuracy of 94.3% for gesture/character recognition, both outperforming the existing approaches.

Layer-based visualization and biomedical information exploration of multi-channel large histological data

Background and objectiveModern microscopes can acquire multi-channel large histological data from tissues of human beings or animals, which contain rich biomedical information for disease diagnosis and biological feature analysis. However, due to the large size, fuzzy tissue structure, and complicated multiple elements integrated in the image color space, it is still a challenge for current software systems to effectively calculate histological data, show the inner tissue structures and unveil hidden biomedical information. Therefore, we developed new algorithms and a software platform to address this issue. MethodsThis paper presents a multi-channel biomedical data computing and visualization system that can efficiently process large 3D histological images acquired from high-resolution microscopes. A novelty of our system is that it can dynamically display a volume of interest and extract tissue information using a layer-based data navigation scheme. During the data exploring process, the actual resolution of the loaded data can be dynamically determined and updated, and data rendering is synchronized in four display windows at each data layer, where 2D textures are extracted from the imaging volume and mapped onto the displayed clipping planes in 3D space. ResultsTo test the efficiency and scalability of this system, we performed extensive evaluations using several different hardware systems and large histological color datasets acquired from a CryoViz 3D digital system. The experimental results demonstrated that our system can deliver interactive data navigation speed and display detailed imaging information in real time, which is beyond the capability of commonly available biomedical data exploration software platforms. ConclusionTaking advantage of both CPU (central processing unit) main memory and GPU (graphics processing unit) graphics memory, the presented software platform can efficiently compute, process and visualize very large biomedical data and enhance data information. The performance of this system can satisfactorily address the challenges of navigating and interrogating volumetric multi-spectral large histological image at multiple resolution levels.

Machine Learning Overcomes Challenges of Selecting Locations for Infill Wells

This article, written by Special Publications Editor Adam Wilson, contains highlights of paper SPE 190101, “Infill-Well Location-Selection Procedures in Lost Hills Using Machine Learning,” by T.H. Kim, SPE, D.J. Crane, and E.F. Grijalva, SPE, Chevron, prepared for the 2018 SPE Western Regional Meeting, Garden Grove, California, USA, 22–27 April. The paper has not been peer reviewed. For most conventional reservoirs, numerical simulation is successful in forecasting and extracting valuable information regarding optimal locations for new wells. The results of numerical simulations for the Lost Hills field, however, were not successful because of the special characteristics of its diatomite reservoirs—low permeability but high porosity, weak rock strength, and strong imbibition. Machine learning (ML) has been considered because it does not require specific physical models but can provide good estimations with enough data. Introduction The Lost Hills field is approximately 45 miles northwest of Bakersfield, California, USA. It was discovered in 1910, and hydraulic fracturing was introduced in the late 1970s. Waterflooding was introduced in 1992 and has become the main production method. The main reservoir rock type is diatomite, which is formed by the accumulation of diatoms, single-cell organisms. The aver-age pay thickness is approximately 800 to 1,000 ft, and depth to the reservoir base is approximately 2,000 ft. The estimated original oil in place is more than 2 billion bbl. ML Algorithms and Voronoi Diagram K-Means Clustering. K-means clustering is one type of unsupervised learning algorithm. As opposed to supervised algorithms, unsupervised learning algorithms do not need answers supplied. K-means clustering algorithms identify the members of each cluster by iterating the following steps: Generate the centroid of each of the clusters randomly. The number of clusters, K, is a user input. Each data point is assigned to its nearest centroid so that K clusters are created. The new centroid of each cluster is updated so that the Euclidian distances to the centroid are minimized. Repeat Steps 2 and 3 until convergence has been reached. Support Vector Machine (SVM). SVM is a supervised ML algorithm. For supervised learning algorithms, the answers must be supplied to train and build a predictive model. SVM has two main applications, classification and regression, and classification is used in this study. SVM builds a hyperplane or set of hyperplanes that separate data into groups (classification) or form a regression line; a hyperplane is a subspace of one dimension less than its ambient space (e.g., a plane in 3D space). Fig. 1 shows an example of classification in 2D space. The red line is a hyperplane set to have the maximum gap between two classes. Data vectors in circles are borderline members and are called support vectors. When building a hyperplane is difficult, data are mapped into a higher-dimensional space to facilitate finding a hyperplane.

Research on medical applications of contrast sensitivity function to red–green gratings in 3D space

Contrast sensitivity function (CSF), which is a quick and valid index to measure human visual performance, can be applied into medical monitoring and various retinal disease diagnosis. However, there is a lack of researches on CSF in three-dimensional (3D) space. In this paper, CSF of human color vision to red–green chromatic gratings in 3D space is fully investigated. Based on the typical chromatic contrast sensitivity test system, in which the screen is parallel to human face, four inclined planes in 3D space are taken into consideration. Contrast thresholds for chromatic gratings of inclined planes are measured within 20 individual subjects. Experimental results show that the contrast sensitivity of each inclined plane is a low-pass function for red–green gratings. In order to fully utilize chromatic contrast sensitivity in 3D space, a visual model of red–green contrast sensitivity in 3D space is well fitted. Simulation results demonstrate that the proposed model achieves high consistency with human chromatic visual characteristics.

Tracking Individual Targets in High Density Crowd Scenes Analysis of a Video Recording in Hajj 2009

In this paper we present a number of methods (manual, semi-automatic and automatic) for tracking individual targets in high density crowd scenes where thousands of people are gathered. The necessary data about the motion of individuals and a lot of other physical information can be extracted from consecutive image sequences in different ways, including optical flow and block motion estimation. One of the famous methods for tracking moving objects is the block matching method. This way to estimate subject motion requires the specification of a comparison window which determines the scale of the estimate. In this work we present a real-time method for pedestrian recognition and tracking in sequences of high resolution images obtained by a stationary (high definition) camera located in different places on the Haram mosque in Mecca. The objective is to estimate pedestrian velocities as a function of the local density. The resulting data of tracking moving pedestrians based on video sequences are presented in the following section. Through the evaluated system the spatiotemporal coordinates of each pedestrian during the Tawaf ritual are established. The pilgrim velocities as function of the local densities in the Mataf area (Haram Mosque Mecca) are illustrated and very precisely documented. Tracking in such places where pedestrian density reaches 7 to 8 persons/m2 is extremely challenging due to the small number of pixels on the target, appearance ambiguity resulting from the dense packing, and severe inter-object occlusions. The tracking method which is outlined in this paper overcomes these challenges by using a virtual camera which is matched in position, rotation and focal length to the original camera in such a way that the features of the 3D-model match the feature position of the filmed mosque. In this model an individual feature has to be identified by eye, where contrast is a criterion. We do know that the pilgrims walk on a plane, and after matching the camera we also have the height of the plane in 3D-space from our 3D-model. A point object is placed at the position of a selected pedestrian. During the animation we set multiple animation-keys (approximately every 25 to 50 frames which equals 1 to 2 seconds) for the position, such that the position of the point and the pedestrian overlay nearly at every time. By combining all these variables with the available appearance information, we are able to track individual targets in high density crowds.

Increasing Reality and Educational Merits of a Virtual Game

Linear algebra, especially vector equations in 3-dimensional (3D) space, was the least performed topic of fundamental mathematics for Japanese engineering students. For most of them, symbolic representations in textbooks were too abstract, 3D graphic objects took too much time to draw on notebooks, and too few verbal explanations were given in the lectures. Because they didn’t feel the reality in vectors and didn’t connect symbolic vector equations with graphic representations in their mind, the learnt knowledge evaporated easily just after the exams. A virtual game with interactively changeable 3D graphics was developed to improve this situation. The game was conducted in a classroom with a projector displaying the on-going results in front of the class. Students’ avatars had graphic icons and characteristic vectors, and team of avatars fought each other in battles. The winners of the battles were calculated using a scalar product of 3D vectors. The game was combined with a web-based learning system, where students could learn the connection between symbolic and graphic representations of lines or planes in 3D space interactively. Students earn learning points by using the system, and the points could be used advantageously in the battles. Young students felt some reality in the game, and increased experiences of handling 3D graphic objects using vectors. They learnt how an inner product of two vectors decided the winners of battles graphically through the game. And interested students increased the amount of out-of-class activities on the learning system. However, they testified that the game was too simple and had little space for them to devise their winning tactics. That is why we added several new features to the game for increasing reality of the battles and also helping the students to deepen their understanding on vector operations. Students’ reflections to the game and the learning system were collected through interviews to some students or measured through their learning histories on the system and the out-of-class exercises. According to the interviews, the game and the learning system helped them to feel more reality in linear algebra and to start learning it meaningfully.

Human Tracking based on Multiple View Homography

We propose a method for detection and tracking for objects under multiple cameras system. To track objects, one need to establish correspondence objects among multiple views. We apply the principal axis of objects and the homography constraint to match objects across multiple cameras. The principal axis belongs to the silhouette of objects that is extracted by the background subtraction. We use the multiple back- ground model to the background subtraction. In an image sequence, many changes happen with respect to pixel intensity. This cannot be characterized by the single background model so that is necessary to use the multiple background model. Also, we use the median background model reducing some noises. The silhouette is detected by difference with background models and current image which includes moving objects. For calculating homography, we use landmarks on the ground plane in 3D space. The homography means the relation between two correspondence between two coinciding points from different views. The intersection of principal axes and ground plane in 3D space are the same point shown in each view. The intersection occurs when a principal axis in an image crosses to the transformed ground plane from another image. We con- struct the correspondence which means the relationship between intersection in current image and transformed intersection from the other image by homography constraint. Those correspondences confirm within a short distance measuring in the top viewed plane. Thus, we track a person by these corresponding points on the ground plane.

사람의 움직임 추적을 위한 다중 카메라 기반의 지면 위 발의 대응

In this paper, we describe correspondence among multiple images taken by multiple cameras. The correspondence among multiple views is an interesting problem which often appears in the application like visual surveillance or gesture recognition system. We use the principal axis and the ground plane homography to estimate foot of human. The principal axis belongs to the subtracted silhouette-based region of human using subtraction of the predetermined multiple background models with current image which includes moving person. For the calculation of the ground plane homography, we use landmarks on the ground plane in 3D space. Thus the ground plane homography means the relation of two common points in different views. In the normal human being, the foot of human has an exactly same position in the 3D space and we represent it to the intersection in this paper. The intersection occurs when the principal axis in an image crosses to the transformed ground plane from other image. However the positions of the intersection are different depend on camera views. Therefore we construct the correspondence that means the relationship between the intersection in current image and the transformed intersection from other image by homography. Those correspondences should confirm within a short distance measuring in the top viewed plane. Thus, we track a person by these corresponding points on the ground plane. Experimental result shows the accuracy of the proposed algorithm has almost 90% of detecting person for tracking based on correspondence of intersections.

Diagnostic Value of Left Ventricular Outflow Area in Patients with Hypertrophic Cardiomyopathy: A Real-Time Three-Dimensional Echocardiographic Study

Earlier studies demonstrated the ability of real-time 3-dimensional (3D) echocardiography (3DE) to measure left ventricular outflow tract (LVOT) area (A(LVOT)) in patients with hypertrophic cardiomyopathy (HCM). However, its clinical value is unknown. We sought to investigate the feasibility and accuracy of real-time 3DE-derived A(LVOT) to diagnose significant LVOT obstruction in a large number of patients with HCM. A total of 162 patients with HCM had 3DE by using a volumetric system. The smallest A(LVOT) during systole was determined by moving a 2-dimensional plane in 3D space. The pressure gradient across LVOT was assessed by continuous wave Doppler method. Provocation was performed in patients without significant LVOT obstruction (pressure gradient across LVOT < 50 mm Hg) at rest. Twenty (12%) patients with poor image quality of 3DE were excluded; 16 (28%) patients with a volumetric system, but only 4 (4%) patients with commercial equipment (P < .001). In the remaining 142 patients, A(LVOT) inversely correlated with pressure gradient across LVOT both at rest (r = 0.82, P < .001) and after provocation (r = 0.60, P < .001). The value of A(LVOT) less than 0.85 cm(2) and less than 2.0 cm(2) predicted resting and provokable LVOT obstruction with sensitivity of 87% and 81%, and specificity of 77% and 90%, respectively. Real-time 3DE measurement of A(LVOT) was successful in diagnosing and quantifying LVOT obstruction at rest and after provocation in a large number of patients with HCM.

Turtle's Navigation and Manipulation of Geometrical Figures Constructed by Variable Processes in a 3d Simulated Space

Issues related to 3d turtle's navigation and geometrical figures' manipulation in the simulated 3d space of a newly developed computational environment, MaLT, are reported and discussed here. The joint use of meaningful formalism and the dynamic manipulation of graphically represented 3d figures seem to offer new resources and to pose new challenges as far as geometrical activities and construction of meanings are concerned, which are strongly related to the representational infrastructure of MaLT. Abilities such as spatial orientation and spatial visualisation come into play and are interwoven with the software's functionalities and semantics. Although the body-syntonic metaphor remains critical while navigating the turtle in the 3d simulated space, it seems that it has to be co-ordinated with other - often conflicting one another - frames of reference. The strong link between spatial graphical and geometrical aspects, that was accentuated by the dragging functionalities of the software, helped students go beyond an immediate perceptual approach, relating geometrical figures with real 3d objects and the change of planes in 3d space with physical angle situations. In this framework the concept of angle as turn and measure with emphasis on directionality but also as a relationship between the planes defined by 2d figures has arisen as central.

Spatial term apprehension with a reference object’s rotation in three-dimensional space

It has been reported that the overall shapes of spatial categorical patterns of projective spatial terms such as above and below are not influenced by the rotation of a reference object on a two-dimensional (2D) upright plane. However, is this also true in three-dimensional (3D) space? This study shows the dynamic aspects of the apprehension of projective spatial terms in 3D space by detailing how the rotation of a reference object with an inherent front influences the apprehension of projective spatial terms on a level plane by mapping their spatial categorical patterns. The experiment was designed to examine how spatial categorical patterns on a level plane changed with the rotation of a reference object with an inherent front in 3D computer graphics space. We manipulated the rotation of a reference object with an inherent front at three levels (0 degrees , 90 degrees , and 180 degrees rotations) and examined how such manipulation changed the overall spatial categorical patterns of four basic Japanese projective spatial terms: mae, ushiro, hidari, and migi (similar to in front of, behind, to the left of, and to the right of in English, respectively). The results show that spatial term apprehension was affected by the rotation of the reference object in 3D space. In particular, rotation influenced the mae-ushiro and hidari-migi systems differently. The results also imply that our understanding of projective spatial terms on a level plane in 3D space is affected dynamically by visual information from 3D cues.

Streamline-Based Production-Data Integration in Naturally Fractured Reservoirs

This article, written by Technology Editor Dennis Denney, contains highlights of paper SPE 89914, "Streamline-Based Production-Data Integration in Naturally Fractured Reservoirs," by Mishal Al-Harbi, SPE, Hao Cheng, SPE, Zhong He, and Akhil Datta-Gupta, SPE, Texas A&amp;M U., prepared for the 2004 SPE Annual Technical Conference and Exhibition, Houston, 26-29 September. Streamline-based models have shown great potential in reconciling high-resolution geologic models with production data. Use of streamline-based production-data integration has been extended to naturally fractured reservoirs. A dual-porosity streamline model was used for fracture-flow simulation by treating the fracture and matrix as separate continua connected through a transfer function. The sensitivities that define the relationship between reservoir properties and the production response in fractured reservoirs were computed analytically. The production-data integration was carried out with a generalized travel-time inversion. Introduction Natural fractures play a significant role in subsurface flow and transport of fluids. Seismic imaging and horizontal drilling have revealed the extent of fractures in many reservoirs and enabled operators to use novel ways to use fracture connectivity to enhance recovery. The number of reservoirs that are considered naturally fractured has risen significantly, increasing the need for robust fracture-characterization methods that integrate both static and dynamic data. Discrete-fracture-network (DFN) techniques map fracture planes in 3D space by use of statistical properties of fracture swarms, fracture-network geometry, and flow characteristics. DFN models can incorporate complex-fracture patterns by use of field-data sources such as cores, well logs, borehole images, seismics, and geomechanics. Although the DFN models can reproduce very realistic fracture geometry, these models must be conditiond to dynamic data (such as well-test, tracer, and production data) to reproduce the flow behavior in the reservoir. Such conditioning is particularly important for fractured reservoirs because only a few of the fractures in the DFN model might carry most of the fluid flow. Streamline models have shown great potential in integrating dynamic data into high-resolution geologic models. Streamline models can determine the sensitivity of the production data to reservoir parameters such as porosity and permeability. These sensitivities are partial derivatives that quantify how the production response will be affected by changes in reservoir properties. Typically, integrating dynamic data into reservoir models involves solving an inverse problem in which sensitivities play a key role. Streamline-based sensitivities were used in conjunction with a generalized travel-time-inversion method to integrate production data into geologic models.

Theory of Acyclic Chemical Networks and Enumeration of Polyenoids via Two-Dimensional Chirality

Procedure for constructing acyclic chemical networks is delineated through the example of polyenoids, CNHN+2. Chirality in two-dimensional space is introduced to enumerate polyenoids. Cis−trans dichotomies, each arising from a 2D chiral center, are stacked to form stereoisomers. Redundancy caused by rotating the molecular plane in 3D space is eliminated to obtain the net count of isomers.

Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits

B.D. Smith (ibid., vol.MI-4, p.15-25, 1985; Opt. Eng., vol.29, p.524-34, 1990) and P. Grangeat (These de doctorat, 1987; Lecture Notes in Mathematics 1497, p.66-97, 1991) derived a cone-beam inversion formula that can be applied when a nonplanar orbit satisfying the completeness condition is used. Although Grangeat's inversion formula is mathematically different from Smith's one, they have similar overall structures to each other. The contribution of the present paper is two-fold. First, based on the derivation of Smith, the authors point out that Grangeat's inversion formula and Smith's one can be conveniently described using a single formula (the Smith-Grangeat inversion formula) that is in the form of space-variant filtering followed by cone-beam back projection. Furthermore, the resulting formula is reformulated for data acquisition systems with a planar detector to obtain a new reconstruction algorithm. Second, the authors make two significant modifications to the new algorithm to reduce artifacts and numerical errors encountered in direct implementation of the new algorithm. As for exactness of the new algorithm, the following fact can be stated. The algorithm based on Grangeat's intermediate function is exact for any complete orbit, whereas that based on Smith's intermediate function should be considered as an approximate inverse excepting the special case where almost every plane in 3D space meets the orbit. The validity of the new algorithm is demonstrated by simulation studies.