A persistence lens is a hierarchy of disjoint upper and lower level sets of a continuous luminance image's Reeb graph. The boundary components of a persistence lens's interior components are Jordan curves that serve as a hierarchical segmentation of the image, and may be rendered as vector graphics. A persistence lens determines a varilet basis for the luminance image, in which image simplification is a realized by subspace projection. Image scale space, and image fractal analysis, result from applying a scale measure to each basis function.
Varilet scale space different than these references, but shares their basic intent: a sequential removal of detail, where at each stage the removed detail has smaller scale than what remains. Varilet scale space fully embraces the semantics of causality'' @cite_5 , using simplification's monotone quotient map to track identity (section ).
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This paper describes a generalized axiomatic scale-space theory that makes it possible to derive the notions of linear scale-space, affine Gaussian scale-space and linear spatio-temporal scale-space using a similar set of assumptions (scale-space axioms). The notion of non-enhancement of local extrema is generalized from previous application over discrete and rotationally symmetric kernels to continuous and more general non-isotropic kernels over both spatial and spatio-temporal image domains. It is shown how a complete classification can be given of the linear (Gaussian) scale-space concepts that satisfy these conditions on isotropic spatial, non-isotropic spatial and spatio-temporal domains, which results in a general taxonomy of Gaussian scale-spaces for continuous image data. The resulting theory allows filter shapes to be tuned from specific context information and provides a theoretical foundation for the recently exploited mechanisms of shape adaptation and velocity adaptation, with highly useful applications in computer vision. It is also shown how time-causal spatio-temporal scale-spaces can be derived from similar assumptions. The mathematical structure of these scale-spaces is analyzed in detail concerning transformation properties over space and time, the temporal cascade structure they satisfy over time as well as properties of the resulting multi-scale spatio-temporal derivative operators. It is also shown how temporal derivatives with respect to transformed time can be defined, leading to the formulation of a novel analogue of scale normalized derivatives for time-causal scale-spaces. The kernels generated from these two types of theories have interesting relations to biological vision. We show how filter kernels generated from the Gaussian spatio-temporal scale-space as well as the time-causal spatio-temporal scale-space relate to spatio-temporal receptive field profiles registered from mammalian vision. Specifically, we show that there are close analogies to space-time separable cells in the LGN as well as to both space-time separable and non-separable cells in the striate cortex. We do also present a set of plausible models for complex cells using extended quasi-quadrature measures expressed in terms of scale normalized spatio-temporal derivatives. The theories presented as well as their relations to biological vision show that it is possible to describe a general set of Gaussian and or time-causal scale-spaces using a unified framework, which generalizes and complements previously presented scale-space formulations in this area. We propose a novel and fast multiscale feature detection and description approach that exploits the benefits of nonlinear scale spaces. Previous attempts to detect and describe features in nonlinear scale spaces such as KAZE [1] and BFSIFT [6] are highly time consuming due to the computational burden of creating the nonlinear scale space. In this paper we propose to use recent numerical schemes called Fast Explicit Diffusion (FED) [3, 4] embedded in a pyramidal framework to dramatically speed-up feature detection in nonlinear scale spaces. In addition, we introduce a Modified-Local Difference Binary (M-LDB) descriptor that is highly efficient, exploits gradient information from the nonlinear scale space, is scale and rotation invariant and has low storage requirements. Our features are called Accelerated-KAZE (A-KAZE) due to the dramatic speed-up introduced by FED schemes embedded in a pyramidal framework. These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions. We consider monotone embeddings of a finite metric space into low-dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on n points can be embedded into l2n, while (in a sense to be made precise later), for almost every n-point metric space, every monotone map must be into a space of dimension Ω(n) (Lemma 3).It becomes natural, then, to seek explicit constructions of metric spaces that cannot be monotonically embedded into spaces of sublinear dimension. To this end, we employ known results on sphericity of graphs, which suggest one example of such a metric space--that is defined by a complete bipartite graph. We prove that an δn-regular graph of order n, with bounded diameter has sphericity Ω(n (λ2+1)), where λ2 is the second largest eigenvalue of the adjacency matrix of the graph, and 0<δ≤½ is constant (Theorem 4). We also show that while random graphs have linear sphericity, there are quais-random graphs of logarithmic sphericity (Lemma 7).For the above bound to be linear, λ2 must be constant. We show that if the second eigenvalue of an n 2-regular graph is bounded by a constant, then the graph is close to being complete bipartite. Namely, its adjacency matrix differs from that of a complete bipartite graph in only o(n2) entries (Theorem 5). Furthermore, for any 0<δ<½, and λ2, there are only finitely many δn-regular graphs with second eigenvalue at most λ2 (Corollary 4).