Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.

Trihalomethanes (THM), especially trichloromethane, play an important role in photochemical processes of the lower atmosphere, but the current knowledge of the known sources and sinks of trichloromethane is still incomplete. The trichloromethane flux through the environment is estimated at approximately 660 kt year(-1) and 90% of the emissions are of natural origin. Next to offshore seawater contributing approximately 360 kt year(-1) unknown soil processes are the most prominent source (approximately 220 kt year(-1)). This paper describes a new abiotic source of trichloromethane from the terrestrial environment induced by the oxidation of organic matter by iron(III) and hydrogen peroxide in the presence of chloride. Different organic-rich soils and a series of organic substances regarded as monomeric constituents of humus were investigated for their release of trichloromethene. The influence of iron(III), hydrogen peroxide, halide, and pH on its formation was assayed. The optimal reaction turn over for the representative compound catechol was 58.4 ng of CHCl3 from 1.8 mg of carbon applying chloride and 1.55 microg of CHBr3 from 1.8 mg of carbon applying bromide; resorcin and hydroquinone displayed similar numbers. Results presented in this paper pinpoint 1,2,4,5-tetrahydroxybenzene as playing a key role as intermediate in the formation pathway of the trihalomethanes. The highest THM yields were obtained when applying the oxidized form of 1,2,4,5-tetrahydroxybenzene as THM precursor. These findings are consistent with the well-known degradation pathway starting from resorcin-like dihydroxylated compounds proceeding via further hydroxylation and after halogenation finally ending up in trihalomethanes. In conclusion, Fenton-like reaction conditions (iron(III) and hydrogen peroxide), elevated halide content and an extended reaction time can be seen as the most important parameters required for an optimal THM formation.

No abstract