Visualizing curved spacetime

I present a way to visualize the concept of curved spacetime. The result is a curved surface with local coordinate systems (Minkowski Systems) living on it, giving the local directions of space and time. Relative to these systems, special relativity holds. The method can be used to visualize gravitational time dilation, the horizon of black holes, and cosmological models. The idea underlying the illustrations is first to specify a field of timelike four-velocities. Then, at every point, one performs a coordinate transformation to a local Minkowski system comoving with the given four-velocity. In the local system, the sign of the spatial part of the metric is flipped to create a new metric of Euclidean signature. The new positive definite metric, called the absolute metric, can be covariantly related to the original Lorentzian metric. For the special case of a 2-dimensional original metric, the absolute metric may be embedded in 3-dimensional Euclidean space as a curved surface.Comment: 15 pages, 20 figure

Reflectance Hashing for Material Recognition

We introduce a novel method for using reflectance to identify materials. Reflectance offers a unique signature of the material but is challenging to measure and use for recognizing materials due to its high-dimensionality. In this work, one-shot reflectance is captured using a unique optical camera measuring {\it reflectance disks} where the pixel coordinates correspond to surface viewing angles. The reflectance has class-specific stucture and angular gradients computed in this reflectance space reveal the material class. These reflectance disks encode discriminative information for efficient and accurate material recognition. We introduce a framework called reflectance hashing that models the reflectance disks with dictionary learning and binary hashing. We demonstrate the effectiveness of reflectance hashing for material recognition with a number of real-world materials

Embedding nonrelativistic physics inside a gravitational wave

Gravitational waves with parallel rays are known to have remarkable properties: Their orbit space of null rays possesses the structure of a non-relativistic spacetime of codimension-one. Their geodesics are in one-to-one correspondence with dynamical trajectories of a non-relativistic system. Similarly, the null dimensional reduction of Klein-Gordon's equation on this class of gravitational waves leads to a Schroedinger equation on curved space. These properties are generalized to the class of gravitational waves with a null Killing vector field, of which we propose a new geometric definition, as conformally equivalent to the previous class and such that the Killing vector field is preserved. This definition is instrumental for performing this generalization, as well as various applications. In particular, results on geodesic completeness are extended in a similar way. Moreover, the classification of the subclass with constant scalar invariants is investigated.Comment: 56 pages, 9 figures, v3:Minor correction

A Visualization Tool Used to Develop New Photon Mapping Techniques

We present a visualisation tool aimed specifically at the development and optimisation of photon map denoising methods. Our tool allows the rapid testing of hypotheses and algorithms through the use of parallel coordinates, domain-specific scripting, color mapping and point plots. Interaction is carried out by brushing, adjusting parameters and focus-plus-context, and yields interactive visual feedback and debugging information. We demonstrate the use of the tool to explore high-dimensional photon map data, facilitating the discovery of novel parameter spaces which can be used to dissociate complex caustic illumination. We then show how these new parameterisations may be used to improve upon pre-existing noise removal methods in the context of the photon relaxation framework