A Matlab Implementation of a Flat Norm Motivated Polygonal Edge Matching Method using a Decomposition of Boundary into Four 1-Dimensional Currents
We describe and provide code and examples for a polygonal edge matching method.
cs.CV↗
arXiv subjects
Kevin R. Vixie
Publications and source records attributed to Kevin R. Vixie.
Some properties of minimizers for the Chan-Esedoglu L1TV functional
We present two results characterizing minimizers of the Chan-Esedoglu L1TV functional $F(u) \equiv \int |\nabla u | dx + λ\int |u - f| dx $; $u,f:\Bbb{R}^n \to \Bbb{R}$. If we restrict to $u = χ_Σ$ and $f = χ_Ω$, $Σ, Ω\in \Bbb{R}^n$, the $L^1$TV functional reduces to $E(Σ) = \Per(Σ) + λ|Σ\vartriangle Ω|$. We show that there is a minimizer $Σ$ such that its boundary $\partialΣ$ lies between the union of all balls of radius $\frac{n}λ$ contained in $Ω$ and the corresponding union of $\frac{n}λ$-balls in $Ω^c$. We also show that if a ball of radius $\frac{n}λ + ε$ is almost contained in $Ω$, a slightly smaller concentric ball can be added to $Σ$ to get another minimizer. Finally, we comment on recent results Allard has obtained on $L^1$TV minimizers and how these relate to our results.
math.OC↗
L1TV computes the flat norm for boundaries
We show that the recently introduced L1TV functional can be used to explicitly compute the flat norm for co-dimension one boundaries. While this observation alone is very useful, other important implications for image analysis and shape statistics include a method for denoising sets which are not boundaries or which have higher co-dimension and the fact that using the flat norm to compute distances not only gives a distance, but also an informative decomposition of the distance. This decomposition is made to depend on scale using the "flat norm with scale" which we define in direct analogy to the L1TV functional. We illustrate the results and implications with examples and figures.
math.DG↗
Reconstruction from projections using dynamics: Non-Stochastic Case
The problem of determining three-dimensional density fields from single two-dimensional projections is hopelessly underdetermined without additional assumptions. While parameterized inversions are typically used to solve this problem, we present theoretical results along a different route to the elimination of indeterminacy. Suppose that we have a series of N radiographs measuring X-ray or proton attenuation through an evolving object at a sequence of times. Suppose also that we know the deterministic dynamical laws governing the evolution of the object we are probing with radiography. The these laws can be used to combine the radiographs into one "super"-measurement which can then be inverted to the object sequence. Now suppose the objects are points in $R^n$ and the radiographs are points in $R^d$. It would be expected that the best we could do would be to get invertability when N = $\lceil n/d \rceil$. The worst case is that we never get invertability. We look at both cases carefully for linear dynamics. We show that dynamical laws giving this optimally short invertibility time are generic. We then illustrate with numerical examples and present a conjecture for the nonlinear case.
math.DS↗
The Bispectral Aliasing Test: A Clarification and Some Key Examples
Controversy regarding the correctness of a test for aliasing proposed by Hinich and Wolinsky has been surprisingly long-lived. Two factors have prolonged this controversy. One factor is the presence of deep-seated intuitions that such a test is fundamentally incoherent. Perhaps the most compelling objection is that, given a set of discrete-time samples, one can construct an unaliased continuous-time series which exactly fits those samples. Therefore, the samples alone can not show that the original time series was aliased. The second factor prolonging the debate has been an inability of its proponents to unseat those objections. In fact, as is shown here, all objections can be met and the test as stated is correct. In particular, the role of stationarity as knowledge in addition to the sample values turns out to be crucial. Under certain conditions, including those addressed by the bispectral aliasing test, the continuous-time signals reconstructed from aliased samples are non-stationary. Therefore detecting aliasing in (at least some) stationary continuous-time processes both makes sense and can be done. The merits of the bispectral test for practical use are briefly addressed, but our primary concern here is its theoretical soundness.
chao-dyn↗
Detection of Aliasing in Persistent Signals
We explain why aliasing can be detected in a generic temporally-sampled stationary signal process. We then define a concept of stationarity that makes sense for single waveforms. (This is done without assuming that the waveform is a sample path of some underlying stochastic process.) We show how to use this concept to detect aliasing in sampled waveforms. The constraint that must be satisfied to make detection of aliasing possible is shown to be fairly unrestrictive. We use simple harmonic signals to elucidate the method. We then demonstrate that the method works for continuous-spectrum signals---specifically, for time series from the Lorenz and Rossler systems. Finally we explain how the method might permit the recovery of additional information about Fourier components outside the Nyquist band.
chao-dyn↗