Boundedness of weakly singular integral operators on domains

The monograph dissertation deals with kernel integral operators and their mapping properties on Euclidean domains. The associated kernels are weakly singular and examples of such are given by Green functions of certain elliptic partial differential equations. It is well known that mapping properties of the corresponding Green operators can be used to deduce a priori estimates for the solutions of these equations. In the dissertation, natural size- and cancellation conditions are quantified for kernels defined in domains. These kernels induce integral operators which are then composed with any partial differential operator of prescribed order, depending on the size of the kernel. The main object of study in this dissertation being the boundedness properties of such compositions, the main result is the characterization of their Lp-boundedness on suitably regular domains. In case the aforementioned kernels are defined in the whole Euclidean space, their partial derivatives of prescribed order turn out to be so called standard kernels that arise in connection with singular integral operators. The Lp-boundedness of singular integrals is characterized by the T1 theorem, which is originally due to David and Journé and was published in 1984 (Ann. of Math. 120). The main result in the dissertation can be interpreted as a T1 theorem for weakly singular integral operators. The dissertation deals also with special convolution type weakly singular integral operators that are defined on Euclidean spaces.

Automating Crop Damage Assessments with Unmanned Aerial Systems and Machine Learning

Agricultural pests are responsible for millions of dollars in crop losses and management costs every year. In order to implement optimal site-specific treatments and reduce control costs, new methods to accurately monitor and assess pest damage need to be investigated. In this paper we explore the combination of unmanned aerial vehicles (UAV), remote sensing and machine learning techniques as a promising methodology to address this challenge. The deployment of UAVs as a sensor platform is a rapidly growing field of study for biosecurity and precision agriculture applications. In this experiment, a data collection campaign is performed over a sorghum crop severely damaged by white grubs (Coleoptera: Scarabaeidae). The larvae of these scarab beetles feed on the roots of plants, which in turn impairs root exploration of the soil profile. In the field, crop health status could be classified according to three levels: bare soil where plants were decimated, transition zones of reduced plant density and healthy canopy areas. In this study, we describe the UAV platform deployed to collect high-resolution RGB imagery as well as the image processing pipeline implemented to create an orthoimage. An unsupervised machine learning approach is formulated in order to create a meaningful partition of the image into each of the crop levels. The aim of this approach is to simplify the image analysis step by minimizing user input requirements and avoiding the manual data labelling necessary in supervised learning approaches. The implemented algorithm is based on the K-means clustering algorithm. In order to control high-frequency components present in the feature space, a neighbourhood-oriented parameter is introduced by applying Gaussian convolution kernels prior to K-means clustering. The results show the algorithm delivers consistent decision boundaries that classify the field into three clusters, one for each crop health level as shown in Figure 1. The methodology presented in this paper represents a venue for further esearch towards automated crop damage assessments and biosecurity surveillance.

Stationary processes for object detection and non-rigid structure-from-motion

Stationary processes are random variables whose value is a signal and whose distribution is invariant to translation in the domain of the signal. They are intimately connected to convolution, and therefore to the Fourier transform, since the covariance matrix of a stationary process is a Toeplitz matrix, and Toeplitz matrices are the expression of convolution as a linear operator. This thesis utilises this connection in the study of i) efficient training algorithms for object detection and ii) trajectory-based non-rigid structure-from-motion.

Assessment of crop insect damage using unmanned aerial systems: A machine learning approach

Agricultural pests are responsible for millions of dollars in crop losses and management costs every year. In order to implement optimal site-specific treatments and reduce control costs, new methods to accurately monitor and assess pest damage need to be investigated. In this paper we explore the combination of unmanned aerial vehicles (UAV), remote sensing and machine learning techniques as a promising technology to address this challenge. The deployment of UAVs as a sensor platform is a rapidly growing field of study for biosecurity and precision agriculture applications. In this experiment, a data collection campaign is performed over a sorghum crop severely damaged by white grubs (Coleoptera: Scarabaeidae). The larvae of these scarab beetles feed on the roots of plants, which in turn impairs root exploration of the soil profile. In the field, crop health status could be classified according to three levels: bare soil where plants were decimated, transition zones of reduced plant density and healthy canopy areas. In this study, we describe the UAV platform deployed to collect high-resolution RGB imagery as well as the image processing pipeline implemented to create an orthoimage. An unsupervised machine learning approach is formulated in order to create a meaningful partition of the image into each of the crop levels. The aim of the approach is to simplify the image analysis step by minimizing user input requirements and avoiding the manual data labeling necessary in supervised learning approaches. The implemented algorithm is based on the K-means clustering algorithm. In order to control high-frequency components present in the feature space, a neighbourhood-oriented parameter is introduced by applying Gaussian convolution kernels prior to K-means. The outcome of this approach is a soft K-means algorithm similar to the EM algorithm for Gaussian mixture models. The results show the algorithm delivers decision boundaries that consistently classify the field into three clusters, one for each crop health level. The methodology presented in this paper represents a venue for further research towards automated crop damage assessments and biosecurity surveillance.

Measurement of the top quark mass with dilepton events selected using neuroevolution at CDF

We report a measurement of the top quark mass $M_t$ in the dilepton decay channel $t\bar{t}\to b\ell'^{+}\nu'_\ell\bar{b}\ell^{-}\bar{\nu}_{\ell}$. Events are selected with a neural network which has been directly optimized for statistical precision in top quark mass using neuroevolution, a technique modeled on biological evolution. The top quark mass is extracted from per-event probability densities that are formed by the convolution of leading order matrix elements and detector resolution functions. The joint probability is the product of the probability densities from 344 candidate events in 2.0 fb$^{-1}$ of $p\bar{p}$ collisions collected with the CDF II detector, yielding a measurement of $M_t= 171.2\pm 2.7(\textrm{stat.})\pm 2.9(\textrm{syst.})\mathrm{GeV}/c^2$.

Measurement of the Top-Quark Mass with Dilepton Events Selected Using Neuroevolution at CDF

We report a measurement of the top quark mass $M_t$ in the dilepton decay channel $t\bar{t}\to b\ell'^{+}\nu'_\ell\bar{b}\ell^{-}\bar{\nu}_{\ell}$. Events are selected with a neural network which has been directly optimized for statistical precision in top quark mass using neuroevolution, a technique modeled on biological evolution. The top quark mass is extracted from per-event probability densities that are formed by the convolution of leading order matrix elements and detector resolution functions. The joint probability is the product of the probability densities from 344 candidate events in 2.0 fb$^{-1}$ of $p\bar{p}$ collisions collected with the CDF II detector, yielding a measurement of $M_t= 171.2\pm 2.7(\textrm{stat.})\pm 2.9(\textrm{syst.})\mathrm{GeV}/c^2$.

PIV studies of large scale structures in the near field of small aspect ratio elliptic jets

The near flow field of small aspect ratio elliptic turbulent free jets (issuing from nozzle and orifice) was experimentally studied using a 2D PIV. Two point velocity correlations in these jets revealed the extent and orientation of the large scale structures in the major and minor planes. The spatial filtering of the instantaneous velocity field using Gaussian convolution kernel shows that while a single large vortex ring circumscribing the jet seems to be present at the exit of nozzle, the orifice jet exhibited a number of smaller vortex ring pairs close to jet exit. The smaller length scale observed in the case of the orifice jet is representative of the smaller azimuthal vortex rings that generate axial vortex field as they are convected. This results in the axis-switching in the case of orifice jet and may have a mechanism different from the self induction process as observed in the case of contoured nozzle jet flow.

Support theorems on R-n and non-compact symmetric spaces

We consider convolution equations of the type f * T = g, where f, g is an element of L-P (R-n) and T is a compactly supported distribution. Under natural assumptions on the zero set of the Fourier transform of T, we show that f is compactly supported, provided g is. Similar results are proved for non-compact symmetric spaces as well. (C) 2010 Elsevier Inc. All rights reserved.

Detection of edges from projections

In a number of applications of computerized tomography, the ultimate goal is to detect and characterize objects within a cross section. Detection of edges of different contrast regions yields the required information. The problem of detecting edges from projection data is addressed. It is shown that the class of linear edge detection operators used on images can be used for detection of edges directly from projection data. This not only reduces the computational burden but also avoids the difficulties of postprocessing a reconstructed image. This is accomplished by a convolution backprojection operation. For example, with the Marr-Hildreth edge detection operator, the filtering function that is to be used on the projection data is the Radon transform of the Laplacian of the 2-D Gaussian function which is combined with the reconstruction filter. Simulation results showing the efficacy of the proposed method and a comparison with edges detected from the reconstructed image are presented

Dual-DSP system for signal and image processing

The design of a dual-DSP microprocessor system and its application for parallel FFT and two-dimensional convolution are explained. The system is based on a master-salve configuration. Two ADSP-2101s are configured as slave processors and a PC/AT serves as the master. The master serves as a control processor to transfer the program code and data to the DSPs. The system architecture and the algorithms for the two applications, viz. FFT and two-dimensional convolutions, are discussed.

Extended kac-akhiezer formulae and the fredholm determinant of finite section hilbert-schmidt kernels

This paper deals with some results (known as Kac-Akhiezer formulae) on generalized Fredholm determinants for Hilbert-Schmidt operators on L2-spaces, available in the literature for convolution kernels on intervals. The Kac-Akhiezer formulae have been obtained for kernels which are not necessarily of convolution nature and for domains in R(n).

Image reconstruction from incomplete projections data: A multiresolution modeling approach

Computerized tomography is an imaging technique which produces cross sectional map of an object from its line integrals. Image reconstruction algorithms require collection of line integrals covering the whole measurement range. However, in many practical situations part of projection data is inaccurately measured or not measured at all. In such incomplete projection data situations, conventional image reconstruction algorithms like the convolution back projection algorithm (CBP) and the Fourier reconstruction algorithm, assuming the projection data to be complete, produce degraded images. In this paper, a multiresolution multiscale modeling using the wavelet transform coefficients of projections is proposed for projection completion. The missing coefficients are then predicted based on these models at each scale followed by inverse wavelet transform to obtain the estimated projection data.

A nonlinear spectral finite element model for analysis of wave propagation in solid with internal friction and dissipation

A geometrically non-linear Spectral Finite Flement Model (SFEM) including hysteresis, internal friction and viscous dissipation in the material is developed and is used to study non-linear dissipative wave propagation in elementary rod under high amplitude pulse loading. The solution to non-linear dispersive dissipative equation constitutes one of the most difficult problems in contemporary mathematical physics. Although intensive research towards analytical developments are on, a general purpose cumputational discretization technique for complex applications, such as finite element, but with all the features of travelling wave (TW) solutions is not available. The present effort is aimed towards development of such computational framework. Fast Fourier Transform (FFT) is used for transformation between temporal and frequency domain. SFEM for the associated linear system is used as initial state for vector iteration. General purpose procedure involving matrix computation and frequency domain convolution operators are used and implemented in a finite element code. Convergnence of the spectral residual force vector ensures the solution accuracy. Important conclusions are drawn from the numerical simulations. Future course of developments are highlighted.

Linear Filtering in MDCT domain

In this paper, expressions for convolution multiplication properties of MDCT are derived starting from the equivalent DFT representations. Using these expressions, methods for implementing linear filtering through block convolution in the MDCT domain are presented. The implementation is exact for symmetric filters and approximate for non-symmetric filters in the case of rectangular window based MDCT. For a general MDCT window function, the filtering is done on the windowed segments and hence the convolution is approximate for symmetric as well as non-symmetric filters. This approximation error is shown to be perceptually insignificant for symmetric impulse response filters. Moreover, the inherent $50 \%$ overlap between adjacent frames used in MDCT computation does reduce this approximation error similar to smoothing of other block processing errors. The presented techniques are useful for compressed domain processing of audio signals.

3D Image Reconstruction From Truncated Helical Cone Beam Projection Data - A Linear Prediction Approach

With the introduction of 2D flat-panel X-ray detectors, 3D image reconstruction using helical cone-beam tomography is fast replacing the conventional 2D reconstruction techniques. In 3D image reconstruction, the source orbit or scanning geometry should satisfy the data sufficiency or completeness condition for exact reconstruction. The helical scan geometry satisfies this condition and hence can give exact reconstruction. The theoretically exact helical cone-beam reconstruction algorithm proposed by Katsevich is a breakthrough and has attracted interest in the 3D reconstruction using helical cone-beam Computed Tomography.In many practical situations, the available projection data is incomplete. One such case is where the detector plane does not completely cover the full extent of the object being imaged in lateral direction resulting in truncated projections. This result in artifacts that mask small features near to the periphery of the ROI when reconstructed using the convolution back projection (CBP) method assuming that the projection data is complete. A number of techniques exist which deal with completion of missing data followed by the CBP reconstruction. In 2D, linear prediction (LP)extrapolation has been shown to be efficient for data completion, involving minimal assumptions on the nature of the data, producing smooth extensions of the missing projection data.In this paper, we propose to extend the LP approach for extrapolating helical cone beam truncated data. The projection on the multi row flat panel detectors has missing columns towards either ends in the lateral direction in truncated data situation. The available data from each detector row is modeled using a linear predictor. The available data is extrapolated and this completed projection data is backprojected using the Katsevich algorithm. Simulation results show the efficacy of the proposed method.