Invertible Orientation Scores of 3D Images

The enhancement and detection of elongated structures in noisy image data is relevant for many biomedical applications. To handle complex crossing structures in 2D images, 2D orientation scores were introduced, which already showed their use in a variety of applications. Here we extend this work to 3D orientation scores. First, we construct the orientation score from a given dataset, which is achieved by an invertible coherent state type of transform. For this transformation we introduce 3D versions of the 2D cake-wavelets, which are complex wavelets that can simultaneously detect oriented structures and oriented edges. For efficient implementation of the different steps in the wavelet creation we use a spherical harmonic transform. Finally, we show some first results of practical applications of 3D orientation scores.Comment: ssvm 2015 published version in LNCS contains a mistake (a switch notation spherical angles) that is corrected in this arxiv versio

Left-invariant evolutions of wavelet transforms on the Similitude Group

Enhancement of multiple-scale elongated structures in noisy image data is relevant for many biomedical applications but commonly used PDE-based enhancement techniques often fail at crossings in an image. To get an overview of how an image is composed of local multiple-scale elongated structures we construct a multiple scale orientation score, which is a continuous wavelet transform on the similitude group, SIM(2). Our unitary transform maps the space of images onto a reproducing kernel space defined on SIM(2), allowing us to robustly relate Euclidean (and scaling) invariant operators on images to left-invariant operators on the corresponding continuous wavelet transform. Rather than often used wavelet (soft-)thresholding techniques, we employ the group structure in the wavelet domain to arrive at left-invariant evolutions and flows (diffusion), for contextual crossing preserving enhancement of multiple scale elongated structures in noisy images. We present experiments that display benefits of our work compared to recent PDE techniques acting directly on the images and to our previous work on left-invariant diffusions on orientation scores defined on Euclidean motion group.Comment: 40 page

Invertible Orientation Scores as an Application of Generalized Wavelet Theory,”

Abstract -Inspired by the visual system of many mammals, we consider the construction of-and reconstruction from-an orientation score of an image, via a wavelet transform corresponding to the left-regular representation of the Euclidean motion group in ‫ތ‬ 2 ( ‫ޒ‬ 2 ) and oriented wavelet ψ ∈ ‫ތ‬ 2 ( ‫ޒ‬ 2 ). Because this representation is reducible, the general wavelet reconstruction theorem does not apply. By means of reproducing kernel theory, we formulate a new and more general wavelet theory, which is applied to our specific case. As a result we can quantify the well-posedness of the reconstruction given the wavelet ψ and deal with the question of which oriented wavelet ψ is practically desirable in the sense that it both allows a stable reconstruction and a proper detection of local elongated structures. This enables image enhancement by means of left-invariant operators on orientation scores

Left-invariant Stochastic Evolution Equations on SE(2) and its Applications to Contour Enhancement and Contour Completion via Invertible Orientation Scores

We provide the explicit solutions of linear, left-invariant, (convection)-diffusion equations and the corresponding resolvent equations on the 2D-Euclidean motion group SE(2). These diffusion equations are forward Kolmogorov equations for stochastic processes for contour enhancement and completion. The solutions are group-convolutions with the corresponding Green's function, which we derive in explicit form. We mainly focus on the Kolmogorov equations for contour enhancement processes which, in contrast to the Kolmogorov equations for contour completion, do not include convection. The Green's functions of these left-invariant partial differential equations coincide with the heat-kernels on SE(2), which we explicitly derive. Then we compute completion distributions on SE(2) which are the product of a forward and a backward resolvent evolved from resp. source and sink distribution on SE(2). On the one hand, the modes of Mumford's direction process for contour completion coincide with elastica curves minimizing $\int \kappa^{2} + \epsilon ds$, related to zero-crossings of 2 left-invariant derivatives of the completion distribution. On the other hand, the completion measure for the contour enhancement concentrates on geodesics minimizing $\int \sqrt{\kappa^{2} + \epsilon} ds$. This motivates a comparison between geodesics and elastica, which are quite similar. However, we derive more practical analytic solutions for the geodesics. The theory is motivated by medical image analysis applications where enhancement of elongated structures in noisy images is required. We use left-invariant (non)-linear evolution processes for automated contour enhancement on invertible orientation scores, obtained from an image by means of a special type of unitary wavelet transform

Semi-analytic modeling of stacked metasurfaces

Ziel dieser Arbeit war die Entwicklung eines semi-analytischen Models mehrschichtiger nano- strukturierter Oberflächen. Einzelne Schichten werden hierbei in Forschungsgemeinschaft als “Metasurface” bezeichnet. In Folge nennt man Schichtsysteme aus Metasurfaces “Metasurface Stacks” oder “Stacked Metasurfaces”. Das besondere an Metasurfaces liegt an einer speziellen Art der Licht-Materie-Wechselwirkung. Im Gegensatz zu herkömmlichen, natürlich vorkommenden optischen Materialien, welche im Wesentlichen durch ihre atom- und molekularphysikalischen Eigenschaften wechselwirken, besitzen Metasurfaces mesoskopische Strukturen. Diese haben Größen, die der von Lichtwellen entsprechen. Dadurch entstehen zum einen Streuphänomene die komplexe Feldwechselwirkungen erzeugen. Darüberhinaus sorgen evaneszente Felder, die auf der Oberfläche der Nano-Strukturen angeregt werden können, für ein geändertes Resonanzverhalten, welches sich durch verschiedene Reflektions- und Absorptionseigenschaften auszeichnet. Sind die Strukturen einer Metasurface periodisch angeordnet lassen sich die dort angeregten Felder durch sogenannte Bloch-Moden beschreiben. Diese sind periodische Feldlösungen der Maxwell-Gleichungen. Betrachtet man nun die Gesamtheit aller Bloch-Moden der Metasurface, kann man eine dominante Mode mit, im Vergleich zu allen anderen, maximalem Energietransport in das Fernfeld identifizieren. Diese nennt man in der Literatur Fundamentalmode. Ist die Metasurface so beschaffen, dass bei Wechselwirkung mit Licht einer bestimmten Wellenlänge diese Fundamentalmode signifikant alle anderen Moden dominiert und letztere stark dämpfen, das heißt evaneszent abfallen, so kann das betreffende Medium als homogen gedeutet werden. Darauf basierend wurde in der vorliegenden Arbeit ein semi-analytisches Model von Stacked Metasurfaces entwickelt, welches verschiedenen experimentellen Tests stand hielt. Ein besonderer Erfolg liegt in der Erweiterung des Modells zur Untersuchung von Feynman-Pfaden

Evaluation of sensor, environment and operational factors impacting the use of multiple sensor constellations for long term resource monitoring

Moderate resolution remote sensing data offers the potential to monitor the long and short term trends in the condition of the Earth’s resources at finer spatial scales and over longer time periods. While improved calibration (radiometric and geometric), free access (Landsat, Sentinel, CBERS), and higher level products in reflectance units have made it easier for the science community to derive the biophysical parameters from these remotely sensed data, a number of issues still affect the analysis of multi-temporal datasets. These are primarily due to sources that are inherent in the process of imaging from single or multiple sensors. Some of these undesired or uncompensated sources of variation include variation in the view angles, illumination angles, atmospheric effects, and sensor effects such as Relative Spectral Response (RSR) variation between different sensors. The complex interaction of these sources of variation would make their study extremely difficult if not impossible with real data, and therefore, a simulated analysis approach is used in this study. A synthetic forest canopy is produced using the Digital Imaging and Remote Sensing Image Generation (DIRSIG) model and its measured BRDFs are modeled using the RossLi canopy BRDF model. The simulated BRDF matches the real data to within 2% of the reflectance in the red and the NIR spectral bands studied. The BRDF modeling process is extended to model and characterize the defoliation of a forest, which is used in factor sensitivity studies to estimate the effect of each factor for varying environment and sensor conditions. Finally, a factorial experiment is designed to understand the significance of the sources of variation, and regression based analysis are performed to understand the relative importance of the factors. The design of experiment and the sensitivity analysis conclude that the atmospheric attenuation and variations due to the illumination angles are the dominant sources impacting the at-sensor radiance

The deep structure of Gaussian scale space images

In order to be able to deal with the discrete nature of images in a continuous way, one can use results of the mathematical field of 'distribution theory'. Under almost trivial assumptions, like 'we know nothing', one ends up with convolving the image with a Gaussian filter. In this manner scale is introduced by means of the filter's width. The ensemble of the image and its convolved versions at al scales is called a 'Gaussian scale space image'. The filter's main property is that the scale derivative equals the Laplacean of the spatial variables: it is the Greens function of the so-called Heat, or Diffusion, Equation. The investigation of the image all scales simultaneously is called 'deep structure'. In this thesis I focus on the behaviour of the elementary topological items 'spatial critical points' and 'iso-intensity manifolds'. The spatial critical points are traced over scale. Generically they are annihilated and sometimes created pair wise, involving extrema and saddles. The locations of these so-called 'catastrophe events' are calculated with sub-pixel precision. Regarded in the scale space image, these spatial critical points form one-dimensional manifolds, the so-called critical curves. A second type of critical points is formed by the scale space saddles. They are the only possible critical points in the scale space image. At these points the iso-intensity manifolds exhibit special behaviour: they consist of two touching parts, of which one intersects an extremum that is part of the critical curve containing the scale space saddle. This part of the manifold uniquely assigns an area in scale space to this extremum. The remaining part uniquely assigns it to 'other structure'. Since this can be repeated, automatically an algorithm is obtained that reveals the 'hidden' structure present in the scale space image. This topological structure can be hierarchically presented as a binary tree, enabling one to (de-)select parts of it, sweeping out parts, simplify, etc. This structure can easily be projected to the initial image resulting in an uncommitted 'pre-segmentation': a segmentation of the image based on the topological properties without any user-defined parameters or whatsoever. Investigation of non-generic catastrophes shows that symmetries can easily be dealt with. Furthermore, the appearance of creations is shown to be nothing but (instable) protuberances at critical curves. There is also biological inspiration for using a Gaussian scale space, since the visual system seems to use Gaussian-like filters: we are able of seeing and interpreting multi-scale

Data security in photonic information systems using quantum based approaches

The last two decades has seen a revolution in how information is stored and transmitted across the world. In this digital age, it is vital for banking systems, governments and businesses that this information can be transmitted to authorised receivers quickly and efficiently. Current classical cryptosystems rely on the computational difficulty of calculating certain mathematical functions but with the advent of quantum computers, implementing efficient quantum algorithms, these systems could be rendered insecure overnight. Quantum mechanics thankfully also provides the solution, in which information is transmitted on single-photons called qubits and any attempt by an adversary to gain information on these qubits is limited by the laws of quantum mechanics. This thesis looks at three distinct different quantum information experiments. Two of the systems describe the implementation of distributing quantum keys, in which the presence of an eavesdropper introduces unavoidable errors by the laws of quantum mechanics. The first scheme used a quantum dot in a micropillar cavity as a singlephoton source. A polarisation encoding scheme was used for implementing the BB84, quantum cryptographic protocol, which operated at a wavelength of 905 nm and a clock frequency of 40 MHz. A second system implemented phase encoding using asymmetric unbalanced Mach-Zehnder interferometers, with a weak coherent source, operating at a wavelength of 850 nm and pulsed at a clock rate of 1 GHz. The system used depolarised light propagating in the fibre quantum channel. This helps to eliminate the random evolution of the state of polarisation of photons, as a result of stress induced changes in the intrinsic birefringence of the fibre. The system operated completely autonomously, using custom software to compensate for path length fluctuations in the arms of the interferometer and used a variety of different single-photon detector technologies. The final quantum information scheme looked at quantum digital signatures, which allows a sender, Alice, to distribute quantum signatures to two parties, Bob and Charlie, such that they are able to authenticate that the message originated from Alice and that the message was not altered in transmission