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Marching Cubes: A High Resolution 3D Surface Construction Algorithm
Abstract
We present a new algorithm, called marching cubes, that creates triangle models of constant density surfaces from 3D medical data. Using a divide-and-conquer approach to generate inter-slice connectivity, we create a case table that defines triangle topology. The algorithm processes the 3D medical data in scan-line order and calculates triangle vertices using linear interpolation. We find the gradient of the original data, normalize it, and use it as a basis for shading the models. The detail in images produced from the generated surface models is the result of maintaining the inter-slice connectivity, surface data, and gradient information present in the original 3D data. Results from computed tomography (CT), magnetic resonance (MR), and single-photon emission computed tomography (SPECT) illustrate the quality and functionality of marching cubes. We also discuss improvements that decrease processing time and add solid modeling capabilities.
... In the implementation of volume integral averaging, we utilize an auxiliary grid to ensure that we only compute numerical integration on the cells crossed by the interfaces, significantly improving computational efficiency. For computing the angle at which the interface intersects the cell in the TTI equivalent medium parameterization method, we utilize a pre-built lookup table based on the Marching Cube method (Lorensen & Cline 1987;Newman & Yi 2006). This lookup table helps to identify which edges of the cell the interface intersects with, preventing incorrect topological relationships and reducing the computational workload required for locating intersection points. ...
... In the computation of volume averaging, we have already acquired auxiliary grid points labeled with the medium layer. To swiftly determine which edges contain intersection using these labeled points, we employ the method of pre-constructing lookup tables, which is widely used in the Marching Cubes algorithm (Lorensen & Cline 1987;Newman & Yi 2006). table values of cases 3 , 4, 6, 7, 10, 11, 12, and 13 to zero. ...
... The principle of constructing lookup tables is to enumerate the 256 relationships between the cell states (e.g., configurations of the vertex states) and the edges containing intersection points. Fortunately, due to the equivalence of many states through rotations and mirror reflections, these 256 states of grid cells can be reduced to 15 basic cases(Lorensen & Cline 1987;Newman & Yi 2006). ...
Grid point discretization of the model has a significant impact on the accuracy of finite-difference seismic waveform simulations. Discretizing the discontinuous velocity model using local point medium parameters can lead to artifact diffraction caused by the stair-step representation and inaccuracies in calculated waveforms due to interface errors, particularly evident when employing coarse grids. To accurately represent model interfaces and reduce interface errors in finite-difference calculations, various equivalent medium parametrization methods have been developed in recent years. Most of these methods require volume-integrated averaging calculations of the medium parameter values within grid cells. The simplest way to achieve this volume averaging is to apply numerical integration averaging to all grid cells. However, this approach demands considerable computational time. To address this computational challenge, we propose employing a set of auxiliary grids to identify which grid cells intersected by the welded interface and perform volume averaging only on these specific cells, thereby reducing unnecessary computational overhead. Additionally, we present a three-dimensional tilted transversely isotropic equivalent medium parameterization method, which effectively suppresses interface errors and artefact diffraction under the application of coarse grids. We also provide an approach for computing the normal direction of the interface, which is essential for the tilted transversely isotropic equivalent medium parameterization. Numerical tests validate the accuracy of the tilted transversely isotropic equivalent medium parameterization method and demonstrate the practicality of the implementation proposed in this paper for complex models.
... We demonstrate the robustness of our method using the original FAUST dataset [5] as the source and the mc dataset [53] as the target shape. The shape from the mc dataset is generated by the marching-cubes algorithm [29], which includes many low-quality triangular grids Robustness to Discretizations. The ideal matching techniques should be insensitive to shape discretization since real objects often have complex and varied shapes and geometric features that result in incompatible meshing or representations, In particular, when real objects are discretized into high-resolution models containing a large amount of detail, they usually need to be processed using mesh simplification techniques [20,58]. ...
The functional maps framework has achieved remarkable success in non-rigid shape matching. However, the traditional functional map representations do not explicitly encode surface orientation, which can easily lead to orientation-reversing correspondence. The complex functional map addresses this issue by linking oriented tangent bundles to favor orientation-preserving correspondence. Nevertheless, the absence of effective restrictions on the complex functional maps hinders them from obtaining high-quality correspondences. To this end, we introduce novel and powerful constraints to determine complex functional maps by incorporating multiple complex spectral filter operator preservation constraints with a rigorous theoretical guarantee. Such constraints encode the surface orientation information and enforce the isometric property of the map. Based on these constraints, we propose a novel and efficient method to obtain orientation-preserving and accurate correspondences across shapes by alternatively updating the functional maps, complex functional maps, and pointwise maps. Extensive experiments demonstrate our significant improvements in correspondence quality and computing efficiency. In addition, our constraints can be easily adapted to other functional maps-based methods to enhance their performance.
... For the simulations, we employed the FEM-based Isaac Gym simulator [61] for its advanced capabilities in realistically simulating deformable objects [62]. To facilitate the simulation of deformable objects, we apply the Marching Cubes algorithm [63] to the generated density fields to derive the object meshes. Subsequently, we utilize fTetWild [64] for the tetrahedralization of these meshes. ...
This paper studies the problem of estimating physical properties (system identification) through visual observations. To facilitate geometry-aware guidance in physical property estimation, we introduce a novel hybrid framework that leverages 3D Gaussian representation to not only capture explicit shapes but also enable the simulated continuum to deduce implicit shapes during training. We propose a new dynamic 3D Gaussian framework based on motion factorization to recover the object as 3D Gaussian point sets across different time states. Furthermore, we develop a coarse-to-fine filling strategy to generate the density fields of the object from the Gaussian reconstruction, allowing for the extraction of object continuums along with their surfaces and the integration of Gaussian attributes into these continuums. In addition to the extracted object surfaces, the Gaussian-informed continuum also enables the rendering of object masks during simulations, serving as implicit shape guidance for physical property estimation. Extensive experimental evaluations demonstrate that our pipeline achieves state-of-the-art performance across multiple benchmarks and metrics. Additionally, we illustrate the effectiveness of the proposed method through real-world demonstrations, showcasing its practical utility. Our project page is at https://jukgei.github.io/project/gic.
... Finally, with the trained MLP on our provided scene, we utilize the Marching Cubes (MC) algorithm to convert the implicit representation into an explicit representation with voxels to obtain the 3D mesh model of human head [18]. The 3D mesh model is presented in Figure 2. ...
As the significance of simulation in medical care and intervention continues to grow, it is anticipated that a simplified and low-cost platform can be set up to execute personalized diagnoses and treatments. 3D Slicer can not only perform medical image analysis and visualization but can also provide surgical navigation and surgical planning functions. In this paper, we have chosen 3D Slicer as our base platform and monocular cameras are used as sensors. Then, We used the neural radiance fields (NeRF) algorithm to complete the 3D model reconstruction of the human head. We compared the accuracy of the NeRF algorithm in generating 3D human head scenes and utilized the MarchingCube algorithm to generate corresponding 3D mesh models. The individual's head pose, obtained through single-camera vision, is transmitted in real-time to the scene created within 3D Slicer. The demonstrations presented in this paper include real-time synchronization of transformations between the human head model in the 3D Slicer scene and the detected head posture. Additionally, we tested a scene where a tool, marked with an ArUco Maker tracked by a single camera, synchronously points to the real-time transformation of the head posture. These demos indicate that our methodology can provide a feasible real-time simulation platform for nasopharyngeal swab collection or intubation.
... Marching Cubes are used to create 3D geometries from image segmentation (Lorensen and Cline, 1987). Semi-automated segmentation and geometry creation tools are available in many proprietary and open-source software packages (Mandolini et al., 2022;Virzì et al., 2020), and fully automatic tools using various algorithms (Mandolini et al., 2022;Virzì et al., 2020) and machine learning are a robust area of research (Burton et al., 2020;Seo et al., 2020). ...
Modern medicine has dramatically improved the lives of many. In orthopaedics, robotic surgery has given clinicians superior accuracy when performing interventions over conventional methods. Nevertheless, while these and many other methods are available to ensure treatments are performed successfully, far fewer methods exist to predict the proper treatment option for a given person. Clinicians are forced to categorize individuals, choosing the best treatment on “average.” However, many individuals differ significantly from the “average” person, for which many of these treatments are designed. Going forward, a method of testing, evaluating, and predicting different treatment options' short- and long-term effects on an individual is needed. Digital Twins have been proposed as one such method.
While Digital Twins have grown in popularity in recent years in many fields to understand and predict a range of phenomena, healthcare has been slow to adopt and create Digital Twins, as until recently, few methods existed to make measurements on living individuals to determine individual geometry and material properties accurately.
This work aims to determine what kinds of data should be captured in living individuals and how best to use this data to build computer models able to replicate their joint behavior. A set of template models and FEA manipulation algorithms are presented to improve the accuracy and reduce the time required to build models. Then, a set of works investigating the reproducibility and accuracy associated with different model
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calibration methods and data sources on model performance is presented. Lastly, ongoing work is introduced that uses lessons from all other sections to develop and validate Digital Twin models of two living individuals.
Overall, this work shows that currently available in vivo techniques for laxity measurement are sufficient to create a model. Yet, the accuracy of models can often hinge on the accuracy of the geometries, particularly ligament attachments. This work introduced methods to improve the speed and accuracy used to create geometries through the morphing of template geometries and using subject-specific kinematics to inform better predictions of ligament attachment sites. The processes and lessons learned can be used in future work to better model living individuals.
... signed distance field (SDF) or unsigned distance field (UDF), to the radiance field. In this way, we can optimize them together so that we can easily extract the mesh directly from learned signed implicit field by matching cube [23] to get watertight meshes. ...
Our objective is to leverage a differentiable radiance field \eg NeRF to reconstruct detailed 3D surfaces in addition to producing the standard novel view renderings. There have been related methods that perform such tasks, usually by utilizing a signed distance field (SDF). However, the state-of-the-art approaches still fail to correctly reconstruct the small-scale details, such as the leaves, ropes, and textile surfaces. Considering that different methods formulate and optimize the projection from SDF to radiance field with a globally constant Eikonal regularization, we improve with a ray-wise weighting factor to prioritize the rendering and zero-crossing surface fitting on top of establishing a perfect SDF. We propose to adaptively adjust the regularization on the signed distance field so that unsatisfying rendering rays won't enforce strong Eikonal regularization which is ineffective, and allow the gradients from regions with well-learned radiance to effectively back-propagated to the SDF. Consequently, balancing the two objectives in order to generate accurate and detailed surfaces. Additionally, concerning whether there is a geometric bias between the zero-crossing surface in SDF and rendering points in the radiance field, the projection becomes adjustable as well depending on different 3D locations during optimization. Our proposed \textit{RaNeuS} are extensively evaluated on both synthetic and real datasets, achieving state-of-the-art results on both novel view synthesis and geometric reconstruction.
... Specifically, we first apply the Marching Cube algorithm [21] to obtain a discrete mesh M t = (V t , E t ) from the implicit surface s θ (x, t) = 0 (See Figure 3a (Left)). Each vertex v ti ∈ V t has a position v ti and a color c ti , and they depend on the network parameters θ. ...
This paper presents a novel approach 4DRecons that takes a single camera RGB-D sequence of a dynamic subject as input and outputs a complete textured deforming 3D model over time. 4DRecons encodes the output as a 4D neural implicit surface and presents an optimization procedure that combines a data term and two regularization terms. The data term fits the 4D implicit surface to the input partial observations. We address fundamental challenges in fitting a complete implicit surface to partial observations. The first regularization term enforces that the deformation among adjacent frames is as rigid as possible (ARAP). To this end, we introduce a novel approach to compute correspondences between adjacent textured implicit surfaces, which are used to define the ARAP regularization term. The second regularization term enforces that the topology of the underlying object remains fixed over time. This regularization is critical for avoiding self-intersections that are typical in implicit-based reconstructions. We have evaluated the performance of 4DRecons on a variety of datasets. Experimental results show that 4DRecons can handle large deformations and complex inter-part interactions and outperform state-of-the-art approaches considerably.
Single-view depth estimation refers to the ability to derive three-dimensional information per pixel from a single two-dimensional image. Single-view depth estimation is an ill-posed problem because there are multiple depth solutions that explain 3D geometry from a single view. While deep neural networks have been shown to be effective at capturing depth from a single view, the majority of current methodologies are deterministic in nature. Accounting for uncertainty in the predictions can avoid disastrous consequences when applied to fields such as autonomous driving or medical robotics. We have addressed this problem by quantifying the uncertainty of supervised single-view depth for Bayesian deep neural networks. There are scenarios, especially in medicine in the case of endoscopic images, where such annotated data is not available. To alleviate the lack of data, we present a method that improves the transition from synthetic to real domain methods. We introduce an uncertainty-aware teacher-student architecture that is trained in a self-supervised manner, taking into account the teacher uncertainty. Given the vast amount of unannotated data and the challenges associated with capturing annotated depth in medical minimally invasive procedures, we advocate a fully self-supervised approach that only requires RGB images and the geometric and photometric calibration of the endoscope. In endoscopic imaging, the camera and light sources are co-located at a small distance from the target surfaces. This setup indicates that brighter areas of the image are nearer to the camera, while darker areas are further away. Building on this observation, we exploit the fact that for any given albedo and surface orientation, pixel brightness is inversely proportional to the square of the distance. We propose the use of illumination as a strong single-view self-supervisory signal for deep neural networks.
With the rise of open data, identifiability of individuals based on 3D renderings obtained from routine structural magnetic resonance imaging (MRI) scans of the head has become a growing privacy concern. To protect subject privacy, several algorithms have been developed to de‐identify imaging data using blurring, defacing or refacing. Completely removing facial structures provides the best re‐identification protection but can significantly impact post‐processing steps, like brain morphometry. As an alternative, refacing methods that replace individual facial structures with generic templates have a lower effect on the geometry and intensity distribution of original scans, and are able to provide more consistent post‐processing results by the price of higher re‐identification risk and computational complexity. In the current study, we propose a novel method for anonymized face generation for defaced 3D T1‐weighted scans based on a 3D conditional generative adversarial network. To evaluate the performance of the proposed de‐identification tool, a comparative study was conducted between several existing defacing and refacing tools, with two different segmentation algorithms (FAST and Morphobox). The aim was to evaluate (i) impact on brain morphometry reproducibility, (ii) re‐identification risk, (iii) balance between (i) and (ii), and (iv) the processing time. The proposed method takes 9 s for face generation and is suitable for recovering consistent post‐processing results after defacing.
We present a new learning-based framework S-3D-RCNN that can recover accurate object orientation in SO(3) and simultaneously predict implicit rigid shapes from stereo RGB images. For orientation estimation, in contrast to previous studies that map local appearance to observation angles, we propose a progressive approach by extracting meaningful Intermediate Geometrical Representations (IGRs). This approach features a deep model that transforms perceived intensities from one or two views to object part coordinates to achieve direct egocentric object orientation estimation in the camera coordinate system. To further achieve finer description inside 3D bounding boxes, we investigate the implicit shape estimation problem from stereo images. We model visible object surfaces by designing a point-based representation, augmenting IGRs to explicitly address the unseen surface hallucination problem. Extensive experiments validate the effectiveness of the proposed IGRs, and S-3D-RCNN achieves superior 3D scene understanding performance. We also designed new metrics on the KITTI benchmark for our evaluation of implicit shape estimation.
A geometric modeling technique called Octree Encoding is presented. Arbitrary 3-D objects can be represented to any specified resolution in a hierarchical 8-ary tree structure or “octree” Objects may be concave or convex, have holes (including interior holes), consist of disjoint parts, and possess sculptured (i.e., “free-form”) surfaces. The memory required for representation and manipulation is on the order of the surface area of the object. A complexity metric is proposed based on the number of nodes in an object's tree representation. Efficient (linear time) algorithms have been developed for the Boolean operations (union, intersection and difference), geometric operations (translation, scaling and rotation), N-dimensional interference detection, and display from any point in space with hidden surfaces removed. The algorithms require neither floating-point operations, integer multiplications, nor integer divisions. In addition, many independent sets of very simple calculations are typically generated, allowing implementation over many inexpensive high-bandwidth processors operating in parallel. Real time analysis and manipulation of highly complex situations thus becomes possible.
Two computer graphics systems for the presentation of biomedical information for diagnosis and treatment planning are described. Both systems presented utilize computer tomographic (CT) data as input. One of the systems produces three-dimensional surface representations of organs and anatomical features found within the body. The other system is a radiation treatment planning aid which uses tomographic data in its computations.
In many three-dimensional imaging applications the three-dimensional scene is represented by a three-dimensional array of volume elements, or voxels for short. A subset Q of the voxels is specified by some property. The objects in the scene are then defined as subsets of Q formed by voxels which are “connected” in some appropriate sense. It is often of interest to detect and display the surface of an object in the scene, specified say by one of the voxels in it.
In this paper, the problem of surface detection is translated into a problem of traversal of a directed graph, G. The nodes of G correspond to faces separating voxels in Q from voxels not in Q. It has been proven that connected subgraphs of G correspond to surfaces of connected components of Q (i.e., of objects in the scene). Further properties of the directed graph have been proven, which allow us to keep the number of marked nodes (needed to avoid loops in the graph traversal) to a small fraction of the total number of visited nodes.
This boundary detection algorithm has been implemented. We discuss the interaction between the underlying mathematical theory and the design of the working software. We illustrate the software on some clinical studies in which the input is computed tomographic (CT) data and the output is dynamically rotating three-dimensional displays of isolated organs. Even though the medical application leads to very large scale problems, our theory and design allows us to use our method routinely on the minicomputer of a CT scanner.
An algorithm is described for obtaining an optimal approximation, using triangulation, of a three-dimensional surface defined by randomly distributed points along contour lines. The combinatorial problem of finding the best arrangement of triangles is treated by assuming an adequate objective function. The optimal triangulation is found using classical methods of graph theory. An illustrative example gives the procedure for triangulation of contour lines of a human head for use in radiation therapy planning.
A simple algorithm is presented for processing complex contour arrangements to produce polygonal element mosaics which are suitable for line drawing and continuous tone display. The program proceeds by mapping adjacent contours onto the same unit square and, subject to ordering limitations, connecting nodes of one contour to their nearest neighbors in the other contour. While the mapping procedure provides a basis for branching decisions, highly ambiguous situations are resolved by user interaction. The program was designed to interface a contour definition of the components of a human brain. These brain data are a most complex definition and, as such, serve to illustrate both the capabilities and limitations of the procedures.
In many scientific and technical endeavors, a three-dimensional solid must be reconstructed from serial sections, either to aid in the comprehension of the object's structure or to facilitate its automatic manipulation and analysis. This paper presents a general solution to the problem of constructing a surface over a set of cross-sectional contours. This surface, to be composed of triangular tiles, is constructed by separately determining an optimal surface between each pair of consecutive contours. Determining such a surface is reduced to the problem of finding certain minimum cost cycles in a directed toroidal graph. A new fast algorithm for finding such cycles is utilized. Also developed is a closed-form expression, in terms of the number of contour points, for an upper bound on the number of operations required to execute the algorithm. An illustrated example which involves the construction of a minimum area surface describing a human head is included.
Modern scanning techniques, such as computed tomography, have begun to produce true three-dimensional imagery of internal structures. The first stage in finding structure in these images, like that for standard two-dimensional images, is to evaluate a local edge operator over the image. If an edge segment in two dimensions is modeled as an oriented unit line segment that separates unit squares (i.e., pixels) of different intensities, then a three-dimensional edge segment is an oriented unit plane that separates unit volumes (i.e., voxels) of different intensities. In this correspondence we derive an operator that finds the best oriented plane at each point in the image. This operator, which is based directly on the 3-D problem, complements other approaches that are either interactive or heuristic extensions of 2-D techniques.
In visualizing objects in 3-D digital images based on shaded-surface displays, segmentation of the 3-D regions is a basic processing operation. When object and nonobject regions touch or overlap and have identical features, automatic segmentation methods fail. We present an interactive technique which permits isolation of arbitrary subregions of the 3-D image by a region filling procedure. We illustrate the usefulness of this capability, through computerized tomography data, in preoperative surgical planning and for producing arbitrary fragmentations of objects in 3-D images. The techniques are applicable in other fields as well, where independent display of overlapping structures is important. When the object regions are specified through a set of contours in a sequence of slices, we show that the boundary surfaces of the object are produceable in an efficient way using the cuberille-based discrete representation of the object surfaces. The algorithms we present for forming boundary surfaces do not impose restrictions on the shape and pattern of distribution of contours in the slices as do the surface tiling schemes reported in the literature. We assume that a contour is made up of at least three points.