How to reduce 3-D gray-scale mathematical morphology to 2-D

A planar encoding of three-dimensional (3-D) images, which is commutative with respect to set and gray-scale morphological operations, is proposed. Such an encoding indicates a way to reduce 3-D set and mathematical morphology operations to two-dimensional (2-D) ones     

3-D/2-D registration by integrating 2-D information in 3-D

In image-guided therapy, high-quality preoperative images serve for planning and simulation, and intraoperatively as "background", onto which models of surgical instruments or radiation beams are projected. The link between a preoperative image and intraoperative physical space of the patient is established by image-to-patient registration. In this paper, we present a novel 3-D/2-D registration method. First, a 3-D image is reconstructed from a few 2-D X-ray images and next, the preoperative 3-D image is brought into the best possible spatial correspondence with the reconstructed image by optimizing a similarity measure (SM). Because the quality of the reconstructed image is generally low, we introduce a novel SM, which is able to cope with low image quality as well as with different imaging modalities. The novel 3-D/2-D registration method has been evaluated and compared to the gradient-based method (GBM) using standardized evaluation methodology and publicly available 3-D computed tomography (CT), 3-D rotational X-ray (3DRX), and magnetic resonance (MR) and 2-D X-ray images of two spine phantoms, for which gold standard registrations were known. For each of the 3DRX, CT, or MR images and each set of X-ray images, 1600 registrations were performed from starting positions, defined as the mean target registration error (mTRE), randomly generated and uniformly distributed in the interval of 0-20 mm around the gold standard. The capture range was defined as the distance from gold standard for which the final TRE was less than 2 mm in at least 95% of all cases. In terms of success rate, as the function of initial misalignment and capture range the proposed method outperformed the GBM. TREs of the novel method and the GBM were approximately the same. For the registration of 3DRX and CT images to X-ray images as few as 2-3 X-ray views were sufficient to obtain approximately 0.4 mm TREs, 7-9 mm capture range, and 80%-90% of successful registrations. To obtain similar results for MR to X-ray registrations, an image, reconstructed from at least 11 X-ray images was required. Reconstructions from more than 11 images had no effect on the registration results.     

Finite-element method modeling of superconductors: from 2-D to 3-D

A three-dimensional (3-D) numerical modeling technique for solving problems involving superconducting materials is presented. The model is implemented in finite-element method software and is based on a recently developed 3-D formulation for general electromagnetic problems with solid conductors. It has been adapted for modeling of superconductors with nonlinear resistivity in 3-D, characterized by a power-law E-J relation. It has first been compared with an existing and verified two-dimensional (2-D) model: Compared are the current density distribution inside the conductors and the self-field ac losses for different applied transport currents. Second, the model has been tested for computing the current distribution with typical 3-D geometries, such as corner-shaped and twisted superconductors. Finally, it has been used with two superconducting filaments in the presence of external magnetic field for verifying the existence of coupling currents. This effect deals with the finite length of the conductors and cannot be taken into account by 2-D models.     

An order-recursive algorithm to solve the 3-D Yule-Walker equationsof causal 3-D AR models

An order-recursive algorithm is proposed to solve the 3-D Yule-Walker equations of causal 3-D AR models. It is computationally efficient and can be easily transformed into a computer program. Moreover, it can be utilized to determine the orders of a causal 3-D AR process     

Atlas-based indexing of brain sections via 2-D to 3-D image registration

A 2-D to 3-D nonlinear intensity-based registration method is proposed in which the alignment of histological brain sections with a volumetric brain atlas is performed. First, sparsely cut brain sections were linearly matched with an oblique slice automatically extracted from the atlas. Second, a planar-to-curved surface alignment was employed in order to match each section with its corresponding image overlaid on a curved-surface within the atlas. For the latter, a PDE-based registration technique was developed that is driven by a local normalized-mutual-information similarity measure. We demonstrate the method and evaluate its performance with simulated and real data experiments. An atlas-guided segmentation of mouse brains' hippocampal complex, retrieved from the Mouse Brain Library (MBL) database, is demonstrated with the proposed algorithm.     

3-D/2-D registration of CT and MR to X-ray images

A crucial part of image-guided therapy is registration of preoperative and intraoperative images, by which the precise position and orientation of the patient's anatomy is determined in three dimensions. This paper presents a novel approach to register three-dimensional (3-D) computed tomography (CT) or magnetic resonance (MR) images to one or more two-dimensional (2-D) X-ray images. The registration is based solely on the information present in 2-D and 3-D images. It does not require fiducial markers, intraoperative X-ray image segmentation, or timely construction of digitally reconstructed radiographs. The originality of the approach is in using normals to bone surfaces, preoperatively defined in 3-D MR or CT data, and gradients of intraoperative X-ray images at locations defined by the X-ray source and 3-D surface points. The registration is concerned with finding the rigid transformation of a CT or MR volume, which provides the best match between surface normals and back projected gradients, considering their amplitudes and orientations. We have thoroughly validated our registration method by using MR, CT, and X-ray images of a cadaveric lumbar spine phantom for which "gold standard" registration was established by means of fiducial markers, and its accuracy assessed by target registration error. Volumes of interest, containing single vertebrae L1-L5, were registered to different pairs of X-ray images from different starting positions, chosen randomly and uniformly around the "gold standard" position. CT/X-ray (MR/ X-ray) registration, which is fast, was successful in more than 91% (82% except for L1) of trials if started from the "gold standard" translated or rotated for less than 6 mm or 17 degrees (3 mm or 8.6 degrees), respectively. Root-mean-square target registration errors were below 0.5 mm for the CT to X-ray registration and below 1.4 mm for MR to X-ray registration.     

Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation

An efficient Gauss-Newton iterative imaging technique utilizing a three-dimensional (3-D) field solution coupled to a two-dimensional (2-D) parameter estimation scheme (3-D/2-D) is presented for microwave tomographic imaging in medical applications. While electromagnetic wave propagation is described fully by a 3-D vector field, a 3-D scalar model has been applied to improve the efficiency of the iterative reconstruction process with apparently limited reduction in accuracy. In addition, the image recovery has been restricted to 2-D but is generalizable to three dimensions. Image artifacts related primarily to 3-D effects are reduced when compared with results from an entirely two-dimensional inversion (2-D/2-D). Important advances in terms of improving algorithmic efficiency include use of a block solver for computing the field solutions and application of the dual mesh scheme and adjoint approach for Jacobian construction. Methods which enhance the image quality such as the log-magnitude/unwrapped phase minimization were also applied. Results obtained from synthetic measurement data show that the new 3-D/2-D algorithm consistently outperforms its 2-D/2-D counterpart in terms of reducing the effective imaging slice thickness in both permittivity and conductivity images over a range of inclusion sizes and background medium contrasts.     

Gradient-based 2-D/3-D rigid registration of fluoroscopic X-ray to CT

We present a gradient-based method for rigid registration of a patient preoperative computed tomography (CT) to its intraoperative situation with a few fluoroscopic X-ray images obtained with a tracked C-arm. The method is noninvasive, anatomy-based, requires simple user interaction, and includes validation. It is generic and easily customizable for a variety of routine clinical uses in orthopaedic surgery. Gradient-based registration consists of three steps: 1) initial pose estimation; 2) coarse geometry-based registration on bone contours, and; 3) fine gradient projection registration (GPR) on edge pixels. It optimizes speed, accuracy, and robustness. Its novelty resides in using volume gradients to eliminate outliers and foreign objects in the fluoroscopic X-ray images, in speeding up computation, and in achieving higher accuracy. It overcomes the drawbacks of intensity-based methods, which are slow and have a limited convergence range, and of geometry-based methods, which depend on the image segmentation quality. Our simulated, in vitro, and cadaver experiments on a human pelvis CT, dry vertebra, dry femur, fresh lamb hip, and human pelvis under realistic conditions show a mean 0.5-1.7 mm (0.5-2.6 mm maximum) target registration accuracy.     

Reconstruction of 3-D horizons from 3-D seismic datasets

We propose a method for extracting automatically and simultaneously the quasi-horizontal surfaces in three-dimensional (3-D) seismic data. The proposed algorithm identifies connected sets of points which form surfaces in 3-D space. To improve reliability, this algorithm takes into consideration the relative positions of all horizons, and uses globally self-consistent connectivity criteria which respect the temporal order of horizon creation. The first stage of the algorithm consists of the preliminary estimation of the local direction of each horizon at each point of the 3-D space. The second stage consists of smoothing the signal along the detected layer structure to reduce noise. The last stage consists of the simultaneous building of all 3-D horizons. The output of the processing is a set of 3-D horizons represented by a series of triangulated surfaces.     

3-D object recognition using 2-D views

We consider the problem of recognizing 3-D objects from 2-D images using geometric models and assuming different viewing angles and positions. Our goal is to recognize and localize instances of specific objects (i.e., model-based) in a scene. This is in contrast to category-based object recognition methods where the goal is to search for instances of objects that belong to a certain visual category (e.g., faces or cars). The key contribution of our work is improving 3-D object recognition by integrating Algebraic Functions of Views (AFoVs), a powerful framework for predicting the geometric appearance of an object due to viewpoint changes, with indexing and learning. During training, we compute the space of views that groups of object features can produce under the assumption of 3-D linear transformations, by combining a small number of reference views that contain the object features using AFoVs. Unrealistic views (e.g., due to the assumption of 3-D linear transformations) are eliminated by imposing a pair of rigidity constraints based on knowledge of the transformation between the reference views of the object. To represent the space of views that an object can produce compactly while allowing efficient hypothesis generation during recognition, we propose combining indexing with learning in two stages. In the first stage, we sample the space of views of an object sparsely and represent information about the samples using indexing. In the second stage, we build probabilistic models of shape appearance by sampling the space of views of the object densely and learning the manifold formed by the samples. Learning employs the Expectation-Maximization (EM) algorithm and takes place in a "universal," lower-dimensional, space computed through Random Projection (RP). During recognition, we extract groups of point features from the scene and we use indexing to retrieve the most feasible model groups that might have produced them (i.e., hypothesis generation). The likelihood of each hypothesis is then computed using the probabilistic models of shape appearance. Only hypotheses ranked high enough are considered for further verification with the most likely hypotheses verified first. The proposed approach has been evaluated using both artificial and real data, illustrating promising performance. We also present preliminary results illustrating extensions of the AFoVs framework to predict the intensity appearance of an object. In this context, we have built a hybrid recognition framework that exploits geometric knowledge to hypothesize the location of an object in the scene and both geometrical and intesnity information to verify the hypotheses.     

Least-Square NUFFT Methods Applied to 2-D and 3-D Radially Encoded MR Image Reconstruction

Radially encoded MRI has gained increasing attention due to its motion insensitivity and reduced artifacts. However, because its samples are collected nonuniformly in the $k$-space, multidimensional (especially 3-D) radially sampled MRI image reconstruction is challenging. The objective of this paper is to develop a reconstruction technique in high dimensions with on-the-fly kernel calculation. It implements general multidimensional nonuniform fast Fourier transform (NUFFT) algorithms and incorporates them into a $k$-space image reconstruction framework. The method is then applied to reconstruct from the radially encoded $k$-space data, although the method is applicable to any non-Cartesian patterns. Performance comparisons are made against the conventional Kaiser–Bessel (KB) gridding method for 2-D and 3-D radially encoded computer-simulated phantoms and physically scanned phantoms. The results show that the NUFFT reconstruction method has better accuracy–efficiency tradeoff than the KB gridding method when the kernel weights are calculated on the fly. It is found that for a particular conventional kernel function, using its corresponding deapodization function as a scaling factor in the NUFFT framework has the potential to improve accuracy. In particular, when a cosine scaling factor is used, the NUFFT method is faster than KB gridding method since a closed-form solution is available and is less computationally expensive than the KB kernel (KB griding requires computation of Bessel functions). The NUFFT method has been successfully applied to 2-D and 3-D in vivo studies on small animals.      

The 3-D Computer

The 3-D Computer [1]–[4] is a unique implementation of a cellular array processor. We have developed two radically new technologies which enable massive numbers of communication channels both between silicon wafers and through them. A parallel processor (single instruction-multiple data stream cellular array processor) has been designed and built to demonstrate the potential of this technological approach. While the 3-D Computer which has been built and operated in a small scale implementation relative to the long-term aims of this technology, it is nevertheless an extremely powerful computer. The current feasibility demonstration 3-D Computer is a 32×32 array of processors partitioned over five wafers stacked one on top of another. The throughput of this current machine is >600 million operations per second (MOPS) with a 10 MHz clock, while the projected throughput of a full scale machine is >100 billion operations per second (BOPS), again with a 10 MHz clock. The extension of the level of circuit integration beyond that of VLSI and WSI, which is made possible by the 3-D technologies of wafer feedthroughs and microbridges, enable us to achieve these enormous throughputs in a very compact form and at very low power. The small size and low power attributes of the 3-D Computer result from the elimination of the chip level and board level packaging and the intraboard wiring required by conventional levels of circuit integration.     

Geodesic morphometry with applications to 3-D morpho-functional anatomy

The quantitative analysis of three-dimensional (3-D) shapes in terms of morphology and functionality is one of the most challenging problems in medical image analysis. This paper proposes a general methodology that aims at solving part of this problem. It introduces a nonparametric hierarchical partitioning approach that operates on any arbitrary 3-D shape described as a triangle mesh. It first extends the concept of basin districts to the case of curved spaces through a partitioning process on a valuation representing the main curvatures over a polyhedral support. A hierarchical construction of basin districts is obtained from a watershed transform. The speed of the front propagation on the polyhedral surface is controlled by the local characteristics of the surface geometry. As a prerequisite, a set of co-processing tools has been developed that operates directly on a triangulated domain. This includes classical signal processing tasks (e.g., re-sampling, filtering) on a polyhedral support performing a trade-off between accuracy and efficiency. The ability to provide an intrinsic shape partition from any triangular mesh is useful in a wide range of applications from accurate geometric modeling, and hierarchical shape dissection to robust mesh compression. Examples are presented in the paper to illustrate the principles and methodology.     

3-D vector radix algorithm for the 3-D new Mersenne numbertransform

Three-dimensional convolutions and correlations are used for three-dimensional image-processing applications. Their calculation involves extensive computation, which makes the use of fast transforms very advantageous. As the number of arithmetic operations is very large, the accumulation of rounding or truncation errors arising in the use of the fast Fourier and Hartley transforms tends to increase. Number theoretic transforms are calculated modulo an integer and hence they are not subject to these errors. Previously, a one-dimensional transform called the new Mersenne number transform (NMNT) was introduced and applied successfully to the calculation of 1-D convolutions/correlations. Unlike other Mersenne number transforms, the NMNT can handle long data sequences and has fast algorithms. In the paper, the 1-D definitions are first extended to the 3-D case in detail for use in 3-D image processing applications. The concept and derivation of the 3-D vector radix algorithm is then introduced for the fast calculation of the 3-D NMNT. The proposed algorithm is found to offer substantial savings over the row-column approach in terms of arithmetic operations. Examples are given showing the validity of both transform and algorithm     

Robust Gradient-Based 3-D/2-D Registration of CT and MR to X-Ray Images

One of the most important technical challenges in image-guided intervention is to obtain a precise transformation between the intrainterventional patient's anatomy and corresponding preinterventional 3-D image on which the intervention was planned. This goal can be achieved by acquiring intrainterventional 2-D images and matching them to the preinterventional 3-D image via 3-D/2-D image registration. A novel 3-D/2-D registration method is proposed in this paper. The method is based on robustly matching 3-D preinterventional image gradients and coarsely reconstructed 3-D gradients from the intrainterventional 2-D images. To improve the robustness of finding the correspondences between the two sets of gradients, hypothetical correspondences are searched for along normals to anatomical structures in 3-D images, while the final correspondences are established in an iterative process, combining the robust random sample consensus algorithm (RANSAC) and a special gradient matching criterion function. The proposed method was evaluated using the publicly available standardized evaluation methodology for 3-D/2-D registration, consisting of 3-D rotational X-ray, computed tomography, magnetic resonance (MR), and 2-D X-ray images of two spine segments, and standardized evaluation criteria. In this way, the proposed method could be objectively compared to the intensity, gradient, and reconstruction-based registration methods. The obtained results indicate that the proposed method performs favorably both in terms of registration accuracy and robustness. The method is especially superior when just a few X-ray images and when MR preinterventional images are used for registration, which are important advantages for many clinical applications.      

The 3-D Computer

The 3-D Computer [1]–[4] is a unique implementation of a cellular array processor. We have developed two radically new technologies which enable massive numbers of communication channels both between silicon wafers and through them. A parallel processor (single instruction-multiple data stream cellular array processor) has been designed and built to demonstrate the potential of this technological approach. While the 3-D Computer which has been built and operated in a small scale implementation relative to the long-term aims of this technology, it is nevertheless an extremely powerful computer. The current feasibility demonstration 3-D Computer is a 32×32 array of processors partitioned over five wafers stacked one on top of another. The throughput of this current machine is >600 million operations per second (MOPS) with a 10 MHz clock, while the projected throughput of a full scale machine is >100 billion operations per second (BOPS), again with a 10 MHz clock. The extension of the level of circuit integration beyond that of VLSI and WSI, which is made possible by the 3-D technologies of wafer feedthroughs and microbridges, enable us to achieve these enormous throughputs in a very compact form and at very low power. The small size and low power attributes of the 3-D Computer result from the elimination of the chip level and board level packaging and the intraboard wiring required by conventional levels of circuit integration.     

Nonlinear filtering approach to 3-D gray-scale image interpolation

Three-dimensional (3-D) images are now common in radiology. A 3-D image is formed by stacking a contiguous sequence of two-dimensional cross-sectional images, or slices. Typically, the spacing between known slices is greater than the spacing between known points on a slice. Many visualization and image-analysis tasks, however, require the 3-D image to have equal sample spacing in all directions. To meet this requirement, one applies an interpolation technique to the known 3-D image to generate a new uniformly sampled 3-D image. The authors propose a nonlinear-filter-based approach to gray-scale interpolation of 3-D images. The method, referred to as column-fitting interpolation, is reminiscent of the maximum-homogeneity filter used for image enhancement. The authors also draw upon the paradigm of relaxation labeling to devise an improved column-fitting interpolator. Both methods are typically more effective than traditional gray-scale interpolation techniques.     

Vessels as 4-D curves: global minimal 4-D paths to extract 3-D tubular surfaces and centerlines

In this paper, we propose an innovative approach to the segmentation of tubular structures. This approach combines all of the benefits of minimal path techniques such as global minimizers, fast computation, and powerful incorporation of user input, while also having the capability to represent and detect vessel surfaces directly which so far has been a feature restricted to active contour and surface techniques. The key is to represent the trajectory of a tubular structure not as a 3-D curve but to go up a dimension and represent the entire structure as a 4-D curve. Then we are able to fully exploit minimal path techniques to obtain global minimizing trajectories between two user supplied endpoints in order to reconstruct tubular structures from noisy or low contrast 3-D data without the sensitivity to local minima inherent in most active surface techniques. In contrast to standard purely spatial 3-D minimal path techniques, however, we are able to represent a full tubular surface rather than just a curve which runs through its interior. Our representation also yields a natural notion of a tube's "central curve." We demonstrate and validate the utility of this approach on magnetic resonance (MR) angiography and computed tomography (CT) images of coronary arteries.