arXiv subjects

Gregery T. Buzzard

Publications and source records attributed to Gregery T. Buzzard.

50 records · Page 3Linked to original sources

arXiv2018

Plug-and-Play Unplugged: Optimization Free Reconstruction using Consensus Equilibrium

Regularized inversion methods for image reconstruction are used widely due to their tractability and ability to combine complex physical sensor models with useful regularity criteria. Such methods motivated the recently developed Plug-and-Play prior method, which provides a framework to use advanced denoising algorithms as regularizers in inversion. However, the need to formulate regularized inversion as the solution to an optimization problem limits the possible regularity conditions and physical sensor models. In this paper, we introduce Consensus Equilibrium (CE), which generalizes regularized inversion to include a much wider variety of both forward components and prior components without the need for either to be expressed with a cost function. CE is based on the solution of a set of equilibrium equations that balance data fit and regularity. In this framework, the problem of MAP estimation in regularized inversion is replaced by the problem of solving these equilibrium equations, which can be approached in multiple ways. The key contribution of CE is to provide a novel framework for fusing multiple heterogeneous models of physical sensors or models learned from data. We describe the derivation of the CE equations and prove that the solution of the CE equations generalizes the standard MAP estimate under appropriate circumstances. We also discuss algorithms for solving the CE equations, including ADMM with a novel form of preconditioning and Newton's method. We give examples to illustrate consensus equilibrium and the convergence properties of these algorithms and demonstrate this method on some toy problems and on a denoising example in which we use an array of convolutional neural network denoisers, none of which is tuned to match the noise level in a noisy image but which in consensus can achieve a better result than any of them individually.

cs.CV↗

arXiv2017

A Framework for Dynamic Image Sampling Based on Supervised Learning (SLADS)

Sparse sampling schemes have the potential to dramatically reduce image acquisition time while simultaneously reducing radiation damage to samples. However, for a sparse sampling scheme to be useful it is important that we are able to reconstruct the underlying object with sufficient clarity using the sparse measurements. In dynamic sampling, each new measurement location is selected based on information obtained from previous measurements. Therefore, dynamic sampling schemes have the potential to dramatically reduce the number of measurements needed for high fidelity reconstructions. However, most existing dynamic sampling methods for point-wise measurement acquisition tend to be computationally expensive and are therefore too slow for practical applications. In this paper, we present a framework for dynamic sampling based on machine learning techniques, which we call a supervised learning approach for dynamic sampling (SLADS). In each step of SLADS, the objective is to find the pixel that maximizes the expected reduction in distortion (ERD) given previous measurements. SLADS is fast because we use a simple regression function to compute the ERD, and it is accurate because the regression function is trained using data sets that are representative of the specific application. In addition, we introduce a method to terminate dynamic sampling at a desired level of distortion, and we extended the SLADS methodology to sample groups of pixels at each step. Finally, we present results on computationally-generated synthetic data and experimentally-collected data to demonstrate a dramatic improvement over state-of-the-art static sampling methods.

cs.CV↗

arXiv2015

Simultaneous state and exogenous input estimation for nonlinear systems using boundary-layer sliding mode observers

While sliding mode observers (SMOs) using discontinuous relays are widely analyzed, most SMOs are implemented computationally using a continuous approximation of the discontinuous relays. This approximation results in the formation of a boundary layer in a neighborhood of the sliding manifold in the observer error space. Therefore, it becomes necessary to develop methods for attenuating the effect of the boundary layer and guaranteeing performance bounds on the resulting state estimation error. In this paper, a method is proposed for constructing boundary-layer SMOs (BL-SMOs) with prescribed state estimation error bounds. The BL-SMO formulation is then extended to simultaneously estimate exogenous inputs (disturbance signals in the state and output vector fields), along with the system state. Two numerical examples are presented to illustrate the effectiveness of the proposed approach.

eess.SY↗

arXiv2013

Experimental Design for Dynamics Identification of Cellular Processes

We address the problem of using nonlinear models to design experiments to characterize the dynamics of cellular processes by using the approach of the Maximally Informative Next Experiment (MINE), which was introduced in [W. Dong, et al. Systems biology of the clock in neurospora crassa. {\em PLoS ONE}, page e3105, 2008] and independently in [M. M. Donahue, et al. Experiment design through dynamical characterization of non-linear systems biology models utilising sparse grids. {\em IET System Biology}, 4:249--262, 2010]. In this approach, existing data is used to define a probability distribution on the parameters; the next measurement point is the one that yields the largest model output variance with this distribution. Building upon this approach, we introduce the Expected Dynamics Estimator (EDE), which is the expected value using this distribution of the output as a function of time. We prove the consistency of this estimator (uniform convergence to true dynamics) even when the chosen experiments cluster in a finite set of points. We extend this proof of consistency to various practical assumptions on noisy data and moderate levels of model mismatch. Through the derivation and proof, we develop a relaxed version of MINE that is more computationally tractable and robust than the original formulation. The results are illustrated with numerical examples on two nonlinear ordinary differential equation models of biomolecular and cellular processes.

q-bio.QM↗

arXiv2003

Maps conjugating holomorphic maps in C^n

If f is a bijection from C^n onto a complex manifold M, which conjugates every holomorphic map in C^n to an endomorphism in M, then we prove that f is necessarily biholomorphic or antibiholomorphic. This extends a result of A. Hinkkanen to higher dimensions. As a corollary, we prove that if there is an epimorphism from the semigroup of all holomorphic endomorphisms of C^n to the semigroup of holomorphic endomorphisms in M, or an epimorphism in the opposite direction for a doubly-transitive M, then it is given by conjugation by some biholomorphic or antibiholomorphic map. We show also that there are two unbounded domains in C^n with isomorphic endomorphism semigroups but which are neither biholomorphically nor antibiholomorphically equivalent.

math.CV↗

arXiv2000

Double Sections and Dominating Maps

As is well-known, there exist nonconstant holomorphic maps from the plane into the Riemann sphere $\PP^1$ minus two points, the simplest example of which is an explicit realization of the uniformization map given by applying the exponential map and then composing with a Mobius transformation taking 0 and $\infty$ to the two given punctures. Likewise, we can map the plane into the sphere minus one point by simply applying directly a Mobius transformation taking $\infty$ to this puncture. In this paper we prove two parametrized versions of this uniformization result using different methods. We first present a slightly easier version using complex analysis, which allows us to construct the uniformizing maps more or less explicitly in terms of the initial coordinates given. Then we give the more general and coordinate invariant version, which we prove by extending Kodaira's theory of the Jacobian fibration to a family of singular curves constructed via algebraic geometry. Finally, using the results obtained with the Jacobian fibration, we obtain two equivalent conditions for an complex surface nonhyperbolically fibered over a complex curve to be holomorphically dominable by $\CC^2$. In theorem \ref{thm:surf} we show that this dominability is equivalent to the apparently weaker condition of the existence of a Zariski dense image of $\CC$ and equivalent to the quasiprojectivity of the base curve together with the nonnegativity of the orbifold Euler characteristic.

math.CV↗

arXiv2000

Algebraic Surfaces Holomorphically Dominable by C2

An n-dimensional complex manifold M is said to be (holomorphically) dominable by $\CC^n$ if there is a map $F:\CC^n \ra M$ which is holomorphic such that the Jacobian determinant $\det(DF)$ is not identically zero. Such a map F is called a dominating map. In this paper, we attempt to classify algebraic surfaces X which are dominable by $\CC^2$ using a combination of techniques from algebraic topology, complex geometry and analysis. One of the key tools in the study of algebraic surfaces is the notion of Kodaira dimension (defined in section 2). By Kodaira's pioneering work and its extensions, an algebraic surface which is dominable by $\CC^2$ must have Kodaira dimension less than two. Using the Kodaira dimension and the fundamental group of X, we succeed in classifying algebraic surfaces which are dominable by $\CC^2$ except for certain cases in which X is an algebraic surface of Kodaira dimension zero and the case when X is rational without any logarithmic 1-form. More specifically, in the case when X is compact (namely projective), we need to exclude only the case when X is birationally equivalent to a K3 surface (a simply connected compact complex surface which admits a globally non-vanishing holomorphic 2-form) that is neither elliptic nor Kummer (see sections 3 and 4 for the definition of these types of surfaces).

math.AG↗

arXiv1999

Double sections and dominating maps

As is well-known, given the complex sphere P^1 minus two points, there exist nonconstant holomorphic maps from the plane into this set, the simplest example of which is given by applying the exponential map and then composing with a Möbius transformation taking 0 and 1 to the two given punctures. Likewise, given the sphere minus one point, we can map the plane into this set by simply applying directly a Möbius transformation taking 1 to this puncture. In this paper we prove a parametrized version of this result.

math.CV↗

arXiv1999

Algebraic surfaces holomorphically dominable by C^2

Using the Kodaira dimension and the fundamental group of X, we succeed in classifying algebraic surfaces which are dominable by C^2 except for certain cases in which X is an algebraic surface of Kodaira dimension zero and the case when X is rational without any logarithmic 1-form. More specifically, in the case when X is compact (namely projective), we need to exclude only the case when X is birationally equivalent to a K3 surface (a simply connected compact complex surface which admits a globally non-vanishing holomorphic 2-form) that is neither elliptic nor Kummer. With the exceptions noted above, we show that for any algebraic surface of Kodaira dimension less than 2, dominability by C^2 is equivalent to the apparently weaker requirement of the existence of a holomorphic image of C which is Zariski dense in the surface. With the same exceptions, we will also show the very interesting and revealing fact that dominability by C^2 is preserved even if a sufficiently small neighborhood of any finite set of points is removed from the surface. In fact, we will provide a complete classification in the more general category of (not necessarily algebraic) compact complex surfaces before tackling the problem in the case of non-compact algebraic surfaces.

math.CV↗

arXiv1996

A Carleman type theorem for proper holomorphic embeddings

In 1927, Carleman showed that a continuous, complex-valued function on the real line can be approximated in the Whitney topology by an entire function restricted to the real line. In this paper, we prove a similar result for proper holomorphic embeddings. Namely, we show that a proper $\cC^r$ embedding of the real line into $\C^n$ can be approximated in the strong $\cC^r$ topology by a proper holomorphic embedding of $\C$ into $\C^n$.

math.CV↗

arXiv1996

An interpolation theorem for holomorphic automorphisms of {\bf C}$^n$

We construct automorphisms of $\C^n$ which map certain discrete sequences one onto another with prescribed finite jet at each point, thus solving a general Mittag-Leffler interpolation problem for automorphisms. Under certain circumstances, this can be done while also approximating a given automorphism on a compact set.

math.CV↗

arXiv1996

Complex dynamics in several variables

These notes are the outgrowth of a series of lectures given at MSRI in January 1995 at the beginning of the special semester in complex dynamics and hyperbolic geometry. In these notes, the primary aim is to motivate the study of complex dynamics in two variables and to introduce the major ideas in the field. Hence the treatment of the subject is mostly expository.

math.DS↗

arXiv1995

An embedding of {\bf C} in {\bf C}$^2$ with hyperbolic complement

Let $X$ be a closed, $1$-dimensional, complex subvariety of $\CC^2$ and let $\ol{\BB}$ be a closed ball in $\CC^2 - X$. Then there exists a Fatou-Bieberbach domain $Ω$ with $X \subseteq Ω\subseteq \CC^2 - \ol{\BB}$ and a biholomorphic map $Φ: Ω\ra \CC^2$ such that $\CC^2 - Φ(X)$ is Kobayashi hyperbolic. As corollaries, there is an embedding of the plane in $\CC^2$ whose complement is hyperbolic, and there is a nontrivial Fatou-Bieberbach domain containing any finite collection of complex lines.

math.DS↗

arXiv1995

Compositional roots of Hénon maps

Let $H$ denote a composition of complex Hénon maps in {\bf C}$^2$. In this paper we show that the only possible compositional roots of $H$ are also compositions of Hénon maps, and that $H$ can have compositional roots of only finitely many distinct orders.

math.DS↗