A Semi-Supervised Reduced-Space Method for Hyperspectral Imaging Segmentation

The development of the hyperspectral remote sensor technology allows the acquisition of images with a very detailed spectral information for each pixel. Because of this, hyperspectral images (HSI) potentially possess larger capabilities in solving many scientific and practical problems in agriculture, biomedical, ecological, geological, hydrological studies. However, their analysis requires developing specialized and fast algorithms for data processing, due the high dimensionality of the data. In this work, we propose a new semi-supervised method for multilabel segmentation of HSI that combines a suitable linear discriminant analysis, a similarity index to compare different spectra, and a random walk based model with a direct label assignment. The user-marked regions are used for the projection of the original high-dimensional feature space to a lower dimensional space, such that the class separation is maximized. This allows to retain in an automatic way the most informative features, lightening the successive computational burden. The part of the random walk is related to a combinatorial Dirichlet problem involving a weighted graph, where the nodes are the projected pixel of the original HSI, and the positive weights depend on the distances between these nodes. We then assign to each pixel of the original image a probability quantifying the likelihood that the pixel (node) belongs to some subregion. The computation of the spectral distance involves both the coordinates in a features space of a pixel and of its neighbors. The final segmentation process is therefore reduced to a suitable optimization problem coupling the probabilities from the random walker computation, and the similarity with respect the initially labeled pixels. We discuss the properties of the new method with experimental results carried on benchmark images.

Record statistics for random walks and Lévy flights with resetting

We compute exactly the mean number of records $\langle R_N \rangle$ for a time-series of size $N$ whose entries represent the positions of a discrete time random walker on the line with resetting. At each time step, the walker jumps by a length $\eta$ drawn independently from a symmetric and continuous distribution $f(\eta)$ with probability $1-r$ (with $0\leq r < 1$) and with the complementary probability $r$ it resets to its starting point $x=0$. This is an exactly solvable example of a weakly correlated time-series that interpolates between a strongly correlated random walk series (for $r=0$) and an uncorrelated time-series (for $(1-r) \ll 1$). Remarkably, we found that for every fixed $r \in [0,1[$ and any $N$, the mean number of records $\langle R_N \rangle$ is completely universal, i.e., independent of the jump distribution $f(\eta)$. In particular, for large $N$, we show that $\langle R_N \rangle$ grows very slowly with increasing $N$ as $\langle R_N \rangle \approx (1/\sqrt{r})\, \ln N$ for $0<r <1$. We also computed the exact universal crossover scaling functions for $\langle R_N \rangle$ in the two limits $r \to 0$ and $r \to 1$. Our analytical predictions are in excellent agreement with numerical simulations.

"Spectrally gapped" random walks on networks: a Mean First Passage Time formula

We derive an approximate but explicit formula for the Mean First Passage Time of a random walker between a source and a target node of a directed and weighted network. The formula does not require any matrix inversion, and it takes as only input the transition probabilities into the target node. It is derived from the calculation of the average resolvent of a deformed ensemble of random sub-stochastic matrices H=\langle H\rangle +\delta HH=⟨H⟩+δH, with \langle H\rangle⟨H⟩ rank-11 and non-negative. The accuracy of the formula depends on the spectral gap of the reduced transition matrix, and it is tested numerically on several instances of (weighted) networks away from the high sparsity regime, with an excellent agreement.

A Semiautomatic Multi-Label Color Image Segmentation Coupling Dirichlet Problem and Colour Distances

Image segmentation is an essential but critical component in low level vision, image analysis, pattern recognition, and now in robotic systems. In addition, it is one of the most challenging tasks in image processing and determines the quality of the final results of the image analysis. Colour based segmentation could hence offer more significant extraction of information as compared to intensity or texture based segmentation. In this work, we propose a new local or global method for multi-label segmentation that combines a random walk based model with a direct label assignment computed using a suitable colour distance. Our approach is a semi-automatic image segmentation technique, since it requires user interaction for the initialisation of the segmentation process. The random walk part involves a combinatorial Dirichlet problem for a weighted graph, where the nodes are the pixel of the image, and the positive weights are related to the distances between pixels: in this work we propose a novel colour distance for computing such weights. In the random walker model we assign to each pixel of the image a probability quantifying the likelihood that the node belongs to some subregion. The computation of the colour distance is pursued by employing the coordinates in a colour space (e.g., RGB, XYZ, YCbCr) of a pixel and of the ones in its neighbourhood (e.g., in a 8–neighbourhood). The segmentation process is, therefore, reduced to an optimisation problem coupling the probabilities from the random walker approach, and the similarity with respect the labelled pixels. A further investigation involves an adaptive preprocess strategy using a regression tree for learning suitable weights to be used in the computation of the colour distance. We discuss the properties of the new method also by comparing with standard random walk and k−means approaches. The experimental results carried on the White Blood Cell (WBC) dataset and GrabCut datasets show the remarkable performance of the proposed method in comparison with state-of-the-art methods, such as normalised random walk and normalised lazy random walk, with respect to segmentation quality and computational time. Moreover, it reveals to be very robust with respect to the presence of noise and to the choice of the colourspace.

A synthetic tubular molecular transport system

Creating artificial macromolecular transport systems that can support the movement of molecules along defined routes is a key goal of nanotechnology. Here, we report the bottom-up construction of a macromolecular transport system in which molecular pistons diffusively move through micrometer-long, hollow filaments. The pistons can cover micrometer distances in fractions of seconds. We build the system using multi-layer DNA origami and analyze the structures of the components using transmission electron microscopy. We study the motion of the pistons along the tubes using single-molecule fluorescence microscopy and perform Langevin simulations to reveal details of the free energy surface that directs the motions of the pistons. The tubular transport system achieves diffusivities and displacement ranges known from natural molecular motors and realizes mobility improvements over five orders of magnitude compared to previous artificial random walker designs. Electric fields can also be employed to actively pull the pistons along the filaments, thereby realizing a nanoscale electric rail system. Our system presents a platform for artificial motors that move autonomously driven by chemical fuels and for performing nanotribology studies, and it could form a basis for future molecular transportation networks.

Image segmentation through the scale-space random walker

This thesis proposes an extension to the Random Walks assisted segmentation algorithm that allows it to operate on a scale-space. Scale-space is a multi-resolution signal analysis method that retains all of the structures in an image through progressive blurring with a Gaussian kernel. The input of the algorithm is setup so that Random Walks will operate on the scale-space, rather than the image itself. The result is that the finer scales retain the detail in the image and the coarser scales filter out the noise. This augmented algorithm is referred to as "Scale-Space Random Walks" (SSRW) and it is shown in both artifical and natural images to be superior to Random Walks when an image has been corrupted by noise. It is also shown that SSRW can impove the segmentation when texture, such as the artifical edges created by JPEG compression, has made the segmentation boundary less accurate. This thesis also presents a practical application of the SSRW in an assisted rotoscoping tool. The tool is implemented as a plugin for a popular commerical compositing application that leverages the power of a Graphics Processing Unit (GPU) to improve the algorithm's performance so that it is near-realtime. Issues such as memory handling, user input and performing vector-matrix algebra are addressed.