A fast sparse QR factorization for solving linear least squares problems in graphics

Nuclear magnetic resonance imaging (MRI) has become the gold standard for many imaging applications. It is routinely used to detect diseases and tissue pathologies due to its superior soft tissue contrast and high spatial resolution. Unlike other imaging techniques such as X-rays, computed tomography, it does not harm patients with ionising radiation.However, MRIs are expensive and only a small portion of patients who would benefit from them can access them. The pursuit of higher sensitivity has generated a trend towards increasing the already large static magnetic field. This has improved imaging quality but has made this technology more expensive and further hindered MRI accessibility. The main cost driver of MRI machines is the strong magnet.The collection of techniques being developed to overcome the intrinsic low NMR sensitivity has fostered interest in low field (LF) MRI to improve cost-effectiveness of MRI. First, the purchase and operational cost of LF systems is relatively low. Second, they are more compatible with other instrumentation which enables versatile concepts such as application-specific open systems and intraoperative imaging. LF systems can offer a wide range of configurations by using permanent magnets. Third, the small fringe field lowers the risk to patients and workers and makes the magnet easier to site. Fourth, artefacts which rely on the strength of field are reduced, as well as the specific absorption rate (SAR) in tissues.Ultra-low field (ULF) MRI systems employ field strengths lower than 10 mT. These systems can complement the advantages of LF systems mentioned above with additional potential benefits. The requirement of homogeneity in ULF is relaxed extensively. It is possible to adjust the strength and direction of the measurement field, allowing flexibility in pulse sequences. Moreover, ULF systems operate in the low kHz frequency range, avoiding the need for radio frequency (RF) noise shielding and enhancing T1-weighted contrast.Most ULF MRI research concentrates on using expensive and fragile sensors, produces the magnetic fields through resistive coils and employs linear gradient fields for encoding. Although ULF systems are important low-cost and portable MRI candidates, they still need considerable space to generate linear fields and need expensive and bulky high permeability shielding to suppress ambient noise.This PhD project develops two major solutions towards a portable and low-cost ULF MRI system: a software gradiometer based on machine learning for ambient noise cancellation and a fast iterative algorithm with total variation (TV) regularisation for image reconstruction. In addition, a concept for a dynamical permanent magnet array (PMA) ULF MRI system is proposed with nonlinear 3D spatial encoding.First, I introduce an experimental setup as a platform for evaluating the MR signal in the ULF regime using a Helmholtz coil as a framework. It is composed of a measurement field, an NMR console, a transmit and receive coil, and a shielding box. The NMR signal is successfully acquired and analysed in partial shielding and full shielding. The results suggest that partial shielding of ULF MRI instruments has the potential to generate a strong signal. This system is built for a subsequent study on noise cancellation presented in following chapter.The second study is a convolutional neural network (CNN) based software gradiometer for background noise cancellation. It includes three ambient noise monitoring coils which are placed away from the NMR sensing coil. The noise induced in the sensing coils is estimated using machine learning. The results show that the software gradiometer reduces ambient noise and can be used to relax shielding requirements, thereby reducing the size and weight of the shielding box.The concept of using small magnets for 3D-spatial encoding in dynamical PMA based ULF MRI is described. It uses small rectangular permanent magnets to generate a 3D-spatial encoding field. The encoding magnets move along an optimised trajectory to achieve an encoding matrix with a low condition number. A prototype of the 3D-spatial encoding has been built and its feasibility tested. The described encoding system can reduce the volume and energy requirements and improve the portability of ULF MRI systems.Finally, iterative imaging reconstruction algorithms for ULF MR are studied. The current state-of-the-art methods are evaluated, which includes Simultaneous Iterative Reconstruction Technique (ART/SIRT), Conjugate Gradient Least Squares (CGLS), and the Chambolle-Pock (CP) algorithms. A dynamic weight Column Action Reconstruction Technique (dwCART) method is proposed and applied to image reconstruction with TV regularisation in the setting of undersampling. In the absence of having a finalised prototype to acquire real ULF MRI imaging data, the methods are tested on CT models (using both simulated and real CT data). The dwCART algorithm performed best on the tested CT problems. It paves the way for fast and accurate imaging with the PMA-based ULF MRI system.

We present algorithms to determine the number of nonzeros in each row and column of the factors of a sparse matrix, for both the QR factorization and the LU factorization with partial pivoting. The algorithms use only the nonzero structure of the input matrix, and run in time nearly linear in the number of nonzeros in that matrix. They may be used to set up data structures or schedule parallel operations in advance of the numerical factorization. The row and column counts we compute are upper bounds on the actual counts. If the input matrix is strong Hall and there is no coincidental numerical cancellation, the counts are exact for QR factorization and are the tightest bounds possible for LU factorization. These algorithms are based on our earlier work on computing row and column counts for sparse Cholesky factorization, plus an efficient method to compute the column elimination tree of a sparse matrix without explicitly forming the product of the matrix and its transpose.

In QR factorization of a sparse m{times}n matrix A (m {ge} n) the orthogonal factor Q is often stored implicitly as a lower trapezoidal matrix H known as the Householder matrix. This paper presents a simple characterization of the row structure of Q, which could be used as the basis for a sparse data structure that can store Q explicitly. The new characterization is a simple extension of a well known row-oriented characterization of the structure of H. Hare, Johnson, Olesky, and van den Driessche have recently provided a complete sparsity analysis of the QR factorization. Let U be the matrix consisting of the first n columns of Q. Using results from, we show that the data structures for H and U resulting from our characterizations are tight when A is a strong Hall matrix. We also show that H and the lower trapezoidal part of U have the same sparsity characterization when A is strong Hall. We then show that this characterization can be extended to any weak Hall matrix that has been permuted into block upper triangular form. Finally, we show that permuting to block triangular form never increases the fill incurred during the factorization.

Parallel solution of sparse linear least squares problems on distributed-memory multiprocessors

We develop two different techniques to study volume mapping problem in Computer Graphics and Medical Imaging fields.The first one is to find a harmonic map from a 3 manifold to a 3D solid sphere and the second is a sphere carving algorithm which calculates the simplicial decomposition of volume adapted to surfaces.We derive the 3D harmonic energy equation and it can be easily extended to higher dimensions.We use a textrehedral mesh to represent the volume data.We demonstrate our method on various solid 3D models.We suggest that 3D harmonic mapping of volume can provide a canonical coordinate system for feature identification and registration for computer animation and medical imaging.

High-resolution (0.3-1 m) digital-elevation data is widely available from commercial sources. Whereas the production of two-dimensional (2D) mapping products from such data is standard practice, the visualization of such three-dimensional (3D) data has been problematic. The basis for this problem is the same as that for the large-model problem in computer graphics-- large amounts of geometry are difficult for current rendering algorithms and hardware. This paper describes a cost-effective solution to this problem that has two parts. First is the employment of the latest in cost-effective 3D chips and video boards that have recently emerged. The second part is the employment of quad-tree data structures for efficient data storage and retrieval during rendering. The result is the capability for real-time display of large (over tens of millions of samples) digital elevation models on modest PC-based systems. This paper shows several demonstrations of this approach using airborne lidar data. The implication of this work is a paradigm shift for geo-spatial information systems--3D data can now be as easy to use as 2D data.© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

It has been observed that it is possible to represent the solution to certain interior scattering problems, such as simple cavities, in terms of solutions to the exterior problem and the pseudo inverse of the plane wave transform (PWT) operator. This may provide a pathway to sparse direct representations for interior problems provided that sparse representations of the exterior solution operator and the pseudo inverse of the PWT are available. This paper considers the latter possibility. The approach used consists of first compressing the PWT via spatially localized current modes that radiate far fields only to specific angular regions. It is shown that the resulting operator has a sparse QR factorization, which permits the efficient direct implementation of the desired pseudo inverse operation.

There is often a trade-off between preserving sparsity and numerical stability in sparse matrix factorizations. In applications like the direct solution of Equality Constrained Least Squares problem, the accurate detection of the rank of a large and sparse constraint matrix is a key issue. Column pivoting is not suitable for distributed memory machines because it forces the program into a lock-step mode, preventing any overlapping of computations. So factorization algorithms on such machines need to use a reliable, yet inexpensive incremental condition estimator to decide on which columns to be included. We describe an incremental condition estimator that can be used during a sparse QR factorization. We show that it is quite reliable and is well suited for use on parallel machines. We supply experimental results to support its effectiveness as well as suitability for parallel architectures.

We propose an FPGA design for soft-output data detection in orthogonal frequency-division multiplexing (OFDM)-based large-scale (multi-user) MIMO systems. To reduce the high computational complexity of data detection, our design uses a modified version of the conjugate gradient least square (CGLS) algorithm. In contrast to existing linear detection algorithms for massive MIMO systems, our method avoids two of the most complex tasks, namely Gram-matrix computation and matrix inversion, while still being able to compute soft-outputs. Our architecture uses an array of reconfigurable processing elements to compute the CGLS algorithm in a hardware-efficient manner. Implementation results on Xilinx Virtex-7 FPGA for a 128 antenna, 8 user large-scale MIMO system show that our design only uses 70% of the area-delay product of the competitive method, while exhibiting superior error-rate performance.

Now that technology allows us to present photorealistic animations of scenically lit objects acting in realtime, the problem of computer graphics has changed fi om making displays recognisable, to ensuring that users notice what they are intended to see, without being distracted by irrelevant information. Worse than that, the use of veridical displays that are intended to be lifelike runs the risk of introducing unpredictable sources of information, that can lead users to infer all sorts of unwanted details. Traditional visual theory, based upon bottom-up models of feature extraction from the retinal image, cannot inform us about these aspects of perception. Broader based cognitive theories are required that integrate visual perception with attention, memory, emotion and inference. Theories such as Barnard's Interacting Cognitive Subsystems enable phenomena such as change blindness and the craft principles of film editing to be interpreted within a common framework, supporting extrapolation to computer graphics.

The modeling of plants is an active area of research in computer graphics and the geometrical modeling of veins is essential for obtaining plant images with a realistic appearance. The venation patterns determine the anisotropy of plant tissues, and affect the shadowing and masking of light incident on these tissues, hence the images. The current trend in the area of plant image generation is to aim for realism using biologically-based and predictable algorithms. Such algorithms do not depend on ad hoc parameters that have to be tuned whenever a new image is being generated. A number of plant image generation algorithms are now available in the literature that have achieved an impressive level of sophistication in many aspects, generating realistic looking images. One notable exception is the geometrical modeling of veins. This remains as an open problem in computer graphics. In this paper we review the state of art of the realistic simulation of plant venation systems from a geometric modeling perspective and propose a specified set of requirements for evaluating possible solutions for this problem.

In this paper, we propose a triangulation method for a set of points in the plane. The method is based on the idea of constructing convex layers by Graham’s scan. It allows to develop an algorithm with the optimal complexity of O(N logN) (in case of constant number of layers) and an easy implementation. Firstly, convex hulls are constructed for the set S of N points, forming k layers. Then, each layer is triangulated in one scan of the adjacent convex hulls. Algorithm is easily parallelized: each layer can be triangulated independently. The main feature of the proposed algorithm is that it has a very simple implementation and the elements (triangles) of the resulting triangulation are presented in the form of simple and at the same time fast data structures: concatenable triangle queue or triangle tree. This makes the algorithm convenient for solving a wide range of applied problems of computational geometry and computer graphics, including simulation in science and engineering, rendering and morphing.

Realistic facial synthesis is one of the most fundamental problems in computer graphics and one of the most difficult. Human lips synchronization of the faces is very important part of this. In this paper it solves the issue of how to make human lips synchronization in one of the most popular 3D modelling software - Autodesk Maya. It is described an automatic way to analyse voice, using Adobe after effects and how the result interacts to Maya, but this way is still in progress, so voice is analyzed and classified manually. Using programming language of Maya - MEL (Maya Embedded Language) the recognized phonemes are associated with mouth positions to provide visemes for computer animation of speech. Different mouth positions are created using blend shape deformers, that let you to deform a surface into the shapes of other surfaces. Lip animation is facilitated by activating facial muscles and the jaw on the given facial model created specially for animation. Highspeed natural-looking synchronized lips animation is achieved. It is planed to model the expressive visual features of expressive speech. Informal user testing suggests that the addition of detailed internal mouth structures, such as the tongue, would improve the phrase recognition.

Collision detection is a fundamental problem in computer graphics. In this paper, temporal coherence is studied and an algorithm exploiting it for bounding volume hierarchies, is presented. We show that maintaining some of the intersection tests computed in the previous frame, along with certain information, is able to speedup the intersection tests considerably. The algorithm is able to accelerate the collision detection for small motions and works as fast as the regular algorithm for large motions, where temporal coherence does not exist. The algorithm framework can be implemented for any type of bounding volume hierarchy. To demonstrate this, it was implemented for the OBB and the AABB data structures and tested on several benchmark scenarios.

Texture synthesis is a fundamental problem in computer graphics that would benefit various applications. Existing methods are effective in handling 2D image textures. In contrast, many real-world textures contain meso-structure in the 3D geometry space, such as grass, leaves, and fabrics, which cannot be effectively modeled using only 2D image textures. We propose a novel texture synthesis method with Neural Radiance Fields (NeRF) to capture and synthesize textures from given multi-view images. In the proposed NeRF texture representation, a scene with fine geometric details is disentangled into the meso-structure textures and the underlying base shape. This allows textures with meso-structure to be effectively learned as latent features situated on the base shape, which are fed into a NeRF decoder trained simultaneously to represent the rich view-dependent appearance. Using this implicit representation, we can synthesize NeRF-based textures through patch matching of latent features. However, inconsistencies between the metrics of the reconstructed content space and the latent feature space may compromise the synthesis quality. To enhance matching performance, we further regularize the distribution of latent features by incorporating a clustering constraint. In addition to generating NeRF textures over a planar domain, our method can also synthesize NeRF textures over curved surfaces, which are practically useful. Experimental results and evaluations demonstrate the effectiveness of our approach.