Analysis of scalar fields over point cloud data. A sampling theory for compact sets in Euclidean space. On the local behavior of spaces of natural images. Bulletin of the American Mathematical Society, 46(2):255–308, 2009. American Mathematical Society, Providence, RI, 2001. A Course in Metric Geometry, volume 33 of Graduate Studies in Mathematics. Computational Geometry: Theory and Applications, 46(4):448–465, 2012. Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes. These techniques were developed to provide our theoretical analysis of the signal-to-noise ratio of Rips-like zigzags, but they are of independent interest as they apply to zigzag modules generally.ĭ. In particular, we give methods for reversing arrows and removing spaces from a zigzag while controlling the changes occurring in its barcode. Along the way, we develop new techniques for manipulating and comparing persistence barcodes from zigzag modules. Thus, Rips zigzags can offer improvements in both size complexity and signal-to-noise ratio. Interestingly, we show that some species of Rips zigzags will exhibit less noise than the (nonzigzag) Rips filtration itself. Some of these Rips-like zigzags have been available as part of the Dionysus library for several years while others are new. We present several species of Rips-like zigzags and compare them with respect to the signal-to-noise ratio, a measure of how well the underlying homology is represented in the persistence barcode relative to the noise in the barcode at the relevant scales. The Rips filtration is prohibitively large however, zigzag persistence can be used to keep the size linear in \(n\), with a constant factor depending only (exponentially) on the intrinsic dimension of \(X\). I am intrigued by the symbiotic relationship of body to jewelry, both on and off the body.For \(n\) points sampled near a compact set \(X\), the persistence barcode of the Rips filtration built from the sample contains information about the homology of \(X\) as long as \(X\) satisfies some geometric assumptions. The body is a dwelling for jewelry, and how we move affects its life. It is a collection of movements that live parallel to the interaction that happens with a piece of jewelry. Walking is a way of knowing the body: the subtle gestures it makes while progressing through space, how it moves – how it interacts with its surrounding landscape. I want to know the space of the body as it moves through specific environments and connect these to my research and inquiries. “As we move, we begin to understand the spaces we encounter. Through jewelry James sees a connection to landscapes that are traversed by a body, conforming to existing pathways, and landscapes that are created by wearing, tracing the curves of our physical existence. By physically embedding her movements into the surfaces James evidences a tangible relationship to the ephemeral. Engraving a sheet of metal is a process of ritual much like the act of walking. James observes the patterns and pathways that a body takes in its everyday routines and finds parallels within her studio practice. Valerie James’ work conveys her interest in movement, mapping, and mark making.
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