Weitzenböck Derivations and Classical Invariant Theory II: The Symbolic Method

2000 Mathematics Subject Classification: 13N15, 13A50, 13F20.

The Lindelöf number greater than continuum is u-invariant

2000 Mathematics Subject Classification: 54C35, 54D20, 54C60.

High performance shift invariant motion estimation and compensation in wavelet domain video compression

The contributions of this dissertation are in the development of two new interrelated approaches to video data compression: (1) A level-refined motion estimation and subband compensation method for the effective motion estimation and motion compensation. (2) A shift-invariant sub-decimation decomposition method in order to overcome the deficiency of the decimation process in estimating motion due to its shift-invariant property of wavelet transform. ^ The enormous data generated by digital videos call for an intense need of efficient video compression techniques to conserve storage space and minimize bandwidth utilization. The main idea of video compression is to reduce the interpixel redundancies inside and between the video frames by applying motion estimation and motion compensation (MEMO) in combination with spatial transform coding. To locate the global minimum of the matching criterion function reasonably, hierarchical motion estimation by coarse to fine resolution refinements using discrete wavelet transform is applied due to its intrinsic multiresolution and scalability natures. ^ Due to the fact that most of the energies are concentrated in the low resolution subbands while decreased in the high resolution subbands, a new approach called level-refined motion estimation and subband compensation (LRSC) method is proposed. It realizes the possible intrablocks in the subbands for lower entropy coding while keeping the low computational loads of motion estimation as the level-refined method, thus to achieve both temporal compression quality and computational simplicity. ^ Since circular convolution is applied in wavelet transform to obtain the decomposed subframes without coefficient expansion, symmetric-extended wavelet transform is designed on the finite length frame signals for more accurate motion estimation without discontinuous boundary distortions. ^ Although wavelet transformed coefficients still contain spatial domain information, motion estimation in wavelet domain is not as straightforward as in spatial domain due to the shift variance property of the decimation process of the wavelet transform. A new approach called sub-decimation decomposition method is proposed, which maintains the motion consistency between the original frame and the decomposed subframes, improving as a consequence the wavelet domain video compressions by shift invariant motion estimation and compensation. ^

Rota’s universal operators and invariant subspaces in Hilbert spaces

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a quasi-nilpotent injective compact operator with dense range. In articular, this new universal operator invites an approach to the Invariant Subspace Problem that uses properties of operators that commute with the universal operator.

Some examples of special Lagrangian tori

A short paper giving some examples of smooth hypersurfaces M of degree n+1 in complex projective n-space that are defined by real polynomial equations and whose real slice contains a component diffeomorphic to an n-1 torus, which is then special Lagrangian with respect to the Calabi-Yau metric on M.

Analytic Torsion, the Eta Invariant, and Closed Differential Forms on Spaces of Metrics

The central idea of this dissertation is to interpret certain invariants constructed from Laplace spectral data on a compact Riemannian manifold as regularized integrals of closed differential forms on the space of Riemannian metrics, or more generally on a space of metrics on a vector bundle. We apply this idea to both the Ray-Singer analytic torsion

and the eta invariant, explaining their dependence on the metric used to define them with a Stokes' theorem argument. We also introduce analytic multi-torsion, a generalization of analytic torsion, in the context of certain manifolds with local product structure; we prove that it is metric independent in a suitable sense.

A solution to the three disjoint path problem on honeycomb tori

In a previous paper we solved an open problem named as the three disjoint path problem on honeycomb meshes. In this paper we extend the technique used to solve the related problem on honeycomb tori. The result gives the minimum possible length of the longest of any three disjoint paths between two given nodes in a torus. The problem has practical benefits in the fault tolerant aspects of interconnection topologies.

Automatically Evolving Rotation-invariant Texture Image Descriptors by Genetic Programming

In computer vision, training a model that performs classification effectively is highly dependent on the extracted features, and the number of training instances. Conventionally, feature detection and extraction are performed by a domain-expert who, in many cases, is expensive to employ and hard to find. Therefore, image descriptors have emerged to automate these tasks. However, designing an image descriptor still requires domain-expert intervention. Moreover, the majority of machine learning algorithms require a large number of training examples to perform well. However, labelled data is not always available or easy to acquire, and dealing with a large dataset can dramatically slow down the training process. In this paper, we propose a novel Genetic Programming based method that automatically synthesises a descriptor using only two training instances per class. The proposed method combines arithmetic operators to evolve a model that takes an image and generates a feature vector. The performance of the proposed method is assessed using six datasets for texture classification with different degrees of rotation, and is compared with seven domain-expert designed descriptors. The results show that the proposed method is robust to rotation, and has significantly outperformed, or achieved a comparable performance to, the baseline methods.

Rota's universal operators and invariant subspaces in Hilbert spaces

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a quasi-nilpotent injective compact operator with dense range. In particular, this new universal operator invites an approach to the Invariant Subspace Problem that uses properties of operators that commute with the universal operator.

Rover autonomous integrity monitoring of GNSS RTK positioning solutions with multi-constellations

This paper studies receiver autonomous integrity monitoring (RAIM) algorithms and performance benefits of RTK solutions with multiple-constellations. The proposed method is generally known as Multi-constellation RAIM -McRAIM. The McRAIM algorithms take advantage of the ambiguity invariant character to assist fast identification of multiple satellite faults in the context of multiple constellations, and then detect faulty satellites in the follow-up ambiguity search and position estimation processes. The concept of Virtual Galileo Constellation (VGC) is used to generate useful data sets of dual-constellations for performance analysis. Experimental results from a 24-h data set demonstrate that with GPS&VGC constellations, McRAIM can significantly enhance the detection and exclusion probabilities of two simultaneous faulty satellites in RTK solutions.

Robust outdoor visual localization using a three-dimensional-edge map

Visual localization systems that are practical for autonomous vehicles in outdoor industrial applications must perform reliably in a wide range of conditions. Changing outdoor conditions cause difficulty by drastically altering the information available in the camera images. To confront the problem, we have developed a visual localization system that uses a surveyed three-dimensional (3D)-edge map of permanent structures in the environment. The map has the invariant properties necessary to achieve long-term robust operation. Previous 3D-edge map localization systems usually maintain a single pose hypothesis, making it difficult to initialize without an accurate prior pose estimate and also making them susceptible to misalignment with unmapped edges detected in the camera image. A multihypothesis particle filter is employed here to perform the initialization procedure with significant uncertainty in the vehicle's initial pose. A novel observation function for the particle filter is developed and evaluated against two existing functions. The new function is shown to further improve the abilities of the particle filter to converge given a very coarse estimate of the vehicle's initial pose. An intelligent exposure control algorithm is also developed that improves the quality of the pertinent information in the image. Results gathered over an entire sunny day and also during rainy weather illustrate that the localization system can operate in a wide range of outdoor conditions. The conclusion is that an invariant map, a robust multihypothesis localization algorithm, and an intelligent exposure control algorithm all combine to enable reliable visual localization through challenging outdoor conditions.