Voxel-based Objects Research Articles

A Measure of Tortuosity for Enclosing Surfaces of Voxel-Based Objects

An easy measure of tortuosity for enclosing surfaces of 3D objects composed of face-connect voxels is presented. We consider that any 3D object composed of face-connected voxels has three implicit surface areas: the enclosing surface area, that corresponds to the sum of the areas of the external plane faces of the voxels which form the visible faces of the solid; the contact surface area that corresponds to the sum of the areas of the contact surfaces which are common to two voxels; and total surface area that corresponds to the sum of all the surface areas of the faces of the all voxels of the solid. The relation among these different surface areas allowed us to define the proposed measure of tortuosity for enclosing surfaces of voxel-based objects. The measure proposed here of tortuosity is invariant under translation, rotation, scaling, and mirror and rotational symmetries. This measure of tortuosity is preserved at different resolutions and is valid for objects holding holes and tunnels. Also, the proposed measure of tortuosity is normalized into a continuous range from 0 to 1 which allows us to improve the object classification. Thus, the minimum and maximum values of tortuosity for enclosing surfaces of voxel-based objects are described. Different families of particular objects are generated based on the above-mentioned concepts. Finally, we present the computation of tortuosity of different kind of voxel-based objects from the real world including examples of brain structures.

Multifingered Grasp Planning via Inference in Deep Neural Networks: Outperforming Sampling by Learning Differentiable Models

We propose a novel approach to multifingered grasp planning that leverages learned deep neural network (DNN) models. We trained a voxel-based 3D convolutional neural network (CNN) to predict grasp-success probability as a function of both visual information of an object and grasp configuration. From this, we formulated grasp planning as inferring the grasp configuration that maximizes the probability of grasp success. In addition, we learned a prior over grasp configurations as a mixture-density network (MDN) conditioned on our voxel-based object representation. We show that this object-conditional prior improves grasp inference when used with the learned grasp success-prediction network compared to a learned, objectagnostic prior or an uninformed uniform prior. Our work is the first to directly plan high-quality multifingered grasps in configuration space using a DNN without the need of an external planner. We validated our inference method by performing multifinger grasping on a physical robot. Our experimental results show that our planning method outperforms existing grasp-planning methods for neural networks (NNs).

Chain coding representation of voxel-based objects with enclosing, edging and intersecting trees

Thrifty methods to represent and store three dimensional objects are important. Two different methods for describing voxel-based objects (VBOs) by means of edging (ETs) and intersecting (ITs) trees are demonstrated. Each tree comes from a different kind of border of the underlying VBO, and both trees are one dimensional alternative descriptors to skeletons for VBOs representation. Vertices in the trees correspond to the vertices of the VBO enclosing surface where some surface vertices have been conveniently suppressed. These descriptors are computed using a base-five digit chain code (combined with parentheses) and has been used to illustrate three dimensional curves and enclosing trees. The descriptors are invariant under rotation and translation, and preserve the VBO shape. Using either descriptor, the description of the mirror image of a VBO is easily obtained. The proposed descriptor notation is a good tool for storing VBOs, and intersecting trees providing further storage savings. Enclosing trees (EcTs) are briefly reviewed as a preamble to introduce ETs and ITs.

Efficient simulation of voxelized phantom in GATE with embedded SimSET multiple photon history generator

GEANT4 Application for Tomographic Emission (GATE) is a powerful Monte Carlo simulator that combines the advantages of the general-purpose GEANT4 simulation code and the specific software tool implementations dedicated to emission tomography. However, the detailed physical modelling of GEANT4 is highly computationally demanding, especially when tracking particles through voxelized phantoms. To circumvent the relatively slow simulation of voxelized phantoms in GATE, another efficient Monte Carlo code can be used to simulate photon interactions and transport inside a voxelized phantom. The simulation system for emission tomography (SimSET), a dedicated Monte Carlo code for PET/SPECT systems, is well-known for its efficiency in simulation of voxel-based objects. An efficient Monte Carlo workflow integrating GATE and SimSET for simulating pinhole SPECT has been proposed to improve voxelized phantom simulation. Although the workflow achieves a desirable increase in speed, it sacrifices the ability to simulate decaying radioactive sources such as non-pure positron emitters or multiple emission isotopes with complex decay schemes and lacks the modelling of time-dependent processes due to the inherent limitations of the SimSET photon history generator (PHG). Moreover, a large volume of disk storage is needed to store the huge temporal photon history file produced by SimSET that must be transported to GATE. In this work, we developed a multiple photon emission history generator (MPHG) based on SimSET/PHG to support a majority of the medically important positron emitters. We incorporated the new generator codes inside GATE to improve the simulation efficiency of voxelized phantoms in GATE, while eliminating the need for the temporal photon history file. The validation of this new code based on a MicroPET R4 system was conducted for 124I and 18F with mouse-like and rat-like phantoms. Comparison of GATE/MPHG with GATE/GEANT4 indicated there is a slight difference in energy spectra for energy below 50 keV due to the lack of x-ray simulation from 124I decay in the new code. The spatial resolution, scatter fraction and count rate performance are in good agreement between the two codes. For the case studies of 18F-NaF (124I-IAZG) using MOBY phantom with 1 × 1 × 1 mm3 voxel sizes, the results show that GATE/MPHG can achieve acceleration factors of approximately 3.1 × (4.5 ×), 6.5 × (10.7 ×) and 9.5 × (31.0 ×) compared with GATE using the regular navigation method, the compressed voxel method and the parameterized tracking technique, respectively. In conclusion, the implementation of MPHG in GATE allows for improved efficiency of voxelized phantom simulations and is suitable for studying clinical and preclinical imaging.

Fast calculation of computer-generated holograms based on 3-D Fourier spectrum for omnidirectional diffraction from a 3-D voxel-based object

We have derived the basic spectral relation between a 3-D object and its 2-D diffracted wavefront by interpreting the diffraction calculation in the 3-D Fourier domain. Information on the 3-D object, which is inherent in the diffracted wavefront, becomes clear by using this relation. After the derivation, a method for obtaining the Fourier spectrum that is required to synthesize a hologram with a realistic sampling number for visible light is described. Finally, to verify the validity and the practicality of the above-mentioned spectral relation, fast calculation of a series of wavefronts radially diffracted from a 3-D voxel-based object is demonstrated.

Enclosing trees

A method is described for representing voxel-based objects by means of enclosing trees. An enclosing tree is a tree which totally covers a voxel-based object, the vertices of the enclosing tree correspond to the vertices of the enclosing surface of the analyzed voxel-based object. An enclosing tree is represented by a chain of base-five digit strings suitably combined by means of parentheses. The enclosing-tree notation is invariant under rotation and translation. Furthermore, using this notation it is possible to obtain the mirror image of any voxel-based object with ease. The enclosing-tree notation preserves the shape of voxel-based objects, allowing us to know some of their topological and geometrical properties. Also, the proposed enclosing-tree notation is a good tool for storing of voxel-based objects.

A "Group Marching Cube" (GMC) Algorithm for Speeding up the Marching Cube Algorithm

Although the Marching Cube (MC) algorithm is very popular for displaying images of voxel-based objects, its slow surface extraction process is usually considered to be one of its major disadvantages. It was pointed out that for the original MC algorithm, we can limit vertex calculations to once per vertex to speed up the surface extraction process, however, it did not mention how this process could be done efficiently. Neither was the reuse of these MC vertices looked into seriously in the literature. In this paper, we propose a “Group Marching Cube” (GMC) algorithm, to reduce the time needed for the vertex identification process, which is part of the surface extraction process. Since most of the triangle-vertices of an iso-surface are shared by many MC triangles, the vertex identification process can avoid the duplication of the vertices in the vertex array of the resultant triangle data. The MC algorithm is usually done through a hash table mechanism proposed in the literature and used by many software systems. Our proposed GMC algorithm considers a group of voxels simultaneously for the application of the MC algorithm to explore interesting features of the original MC algorithm that have not been discussed in the literature. Based on our experiments, for an object with more than 1 million vertices, the GMC algorithm is 3 to more than 10 times faster than the algorithm using a hash table. Another significant advantage of GMC is its compatibility with other algorithms that accelerate the MC algorithm. Together, the overall performance of the original MC algorithm is promoted even further.

Monte Carlo simulation and scatter correction of the GE Advance PET scanner with SimSET and Geant4

For Monte Carlo simulations to be used as an alternative solution to perform scatter correction, accurate modelling of the scanner as well as speed is paramount. General-purpose Monte Carlo packages (Geant4, EGS, MCNP) allow a detailed description of the scanner but are not efficient at simulating voxel-based geometries (patient images). On the other hand, dedicated codes (SimSET, PETSIM) will perform well for voxel-based objects but will be poor in their capacity of simulating complex geometries such as a PET scanner. The approach adopted in this work was to couple a dedicated code (SimSET) with a general-purpose package (Geant4) to have the efficiency of the former and the capabilities of the latter. The combined SimSET+Geant4 code (SimG4) was assessed on the GE Advance PET scanner and compared to the use of SimSET only. A better description of the resolution and sensitivity of the scanner and of the scatter fraction was obtained with SimG4. The accuracy of scatter correction performed with SimG4 and SimSET was also assessed from data acquired with the 20 cm NEMA phantom. SimG4 was found to outperform SimSET and to give slightly better results than the GE scatter correction methods installed on the Advance scanner (curve fitting and scatter modelling for the 300–650 keV and 375–650 keV energy windows, respectively). In the presence of a hot source close to the edge of the field of view (as found in oxygen scans), the GE curve-fitting method was found to fail whereas SimG4 maintained its performance.

Perturbation method for magnetic field calculations of nonconductive objects.

Inhomogeneous magnetic fields produce artifacts in MR images including signal dropout and spatial distortion. A novel perturbative method for calculating the magnetic field to first order (error is second order) within and around nonconducting objects is presented. The perturbation parameter is the susceptibility difference between the object and its surroundings (for example, approximately 10 ppm in the case of brain tissue and air). This method is advantageous as it is sufficiently accurate for most purposes, can be implemented as a simple convolution with a voxel-based object model, and is linear. Furthermore, the method is simple to use and can quickly calculate the field for any orientation of an object using a set of precalculated basis images.

Volumetric virtual environments

Driven by fast development of both virtual reality and volume visualization, we discuss some critical techniques towards building a volumetric VR system, specifically the modeling, rendering, and manipulations of a volumetric scene. Techniques such as voxel-based object simplification, accelerated volume rendering, fast stereo volume rendering, and volumetric “collision detection” are introduced and improved, with the idea of demonstrating the possibilities and potential benefits of incorporating volumetric models into VR systems.

3D line voxelization and connectivity control

Voxelization algorithms that convert a 3D continuous line representation into a discrete line representation have a dual role in graphics. First, these algorithms synthesize voxel-based objects in volume graphics. The 3D line itself is a fundamental primitive, also used as a building block for voxelizing more complex objects. For example, sweeping a 3D voxelized line along a 3D voxelized circle generates a voxelized cylinder. The second application of 3D line voxelization algorithms is for ray traversal in voxel space. Rendering techniques that cast rays through a volume of voxels are based on algorithms that generate the set of voxels visited (or pierced) by the continuous ray. Discrete ray algorithms have been developed for traversing a 3D space partition or a 3D array of sampled or computed data. These algorithms produce one discrete point per step, in contrast to ray casting algorithms for volume rendering, which track a continuous ray at constant intervals, and to voxelization algorithms that generate nonbinary voxel values (for example, partial occupancies). Before considering algorithms for generating discrete lines, we introduce the topology and geometry of discrete lines.

Function representation in geometric modeling: concepts, implementation and applications

Concepts of functionally based geometric modeling including sets of objects, operations, and relations are discussed. Transformations of a defining real function are described for set-theoretic operations, blending, offsetting, bijective mapping, projection, cartesian products, and metamorphosis. Inclusion, point membership, and intersection relations are also described. We use a high-level geometric language that can extend the interactive modeling system by input symbolic descriptions of primitives, operations, and predicates. This approach supports combinations of representational styles, including constructive geometry, sweeping, soft objects, voxel-based objects, deformable and other animated objects. Application examples of aesthetic design, collisions simulation, NC machining, range data processing, and 3D texture generation are given.

Normal estimation in 3 D discrete space

Three-dimensional voxel-based objects are inherently discrete and do not maintain any notion of a continuous surface or normal values, which are crucial for the simulation of light behavior. Thus, in volume rendering, the normal vector of the displayed surfaces must be estimated prior to rendering. We survey several methods for normal estimation and analyze their performance. One unique method, the context-sensitive approach, employs segmentation and segment-bounded operators that are based on object and slope discontinuities in order to achieve high fidelity normal estimation for rendering volumetric objects.

Dynamic elastic interpolation for 3D medical image reconstruction from serial cross sections

An interpolation method is proposed for generating the intermediate contours between a start contour and a goal contour. Coupled with the display method for voxel-based objects, it provides a powerful tool for reconstructing the 3D object from serial cross sections. The method tries to fill in the lost information between two slices, assuming that there is smooth change between them. This is a reasonable assumption provided that the sampling is at least twice the Nyquist rate, in which case the result of the interpolation is expected to be very close to reality. One of the major advantages of this approach is its ability to handle the branching problem. Another major advantage is that after each intermediate contour is generated and sent to display device, there is no need to keep it in the memory unless the solid model will be used for further processing. Thus, the space complexity of this algorithm is relatively low.

Viewing Transformations of Voxel-Based Objects Via Linear Octrees

This article gives three viewing-transformation algorithms for displaying on a screen 3D pictures represented by linear octrees. All the procedures take advantage of the recursive labeling used to identify the successive decomposition of an object into octants. The first algorithm performs transformations directly on the linear octree, while the second and third algorithms determine the 3D border of the given object first and then project onto the screen the surface voxels thus found. All the algorithms perform the viewing transformations in O(RN) time, where R is the resolution of the picture and N is the number of elements in the linear octree. One of the algorithms provides views of the object at different layers of gray level, while another allows internal views.