Andrea L. Bertozzi

Navier-stokes, fluid dynamics, and image and video inpainting

Image inpainting involves filling in part of an image or video using information from the surrounding area. Applications include the restoration of damaged photographs and movies and the removal of selected objects. We introduce a class of automated methods for digital inpainting. The approach uses ideas from classical fluid dynamics to propagate isophote lines continuously from the exterior into the region to be inpainted. The main idea is to think of the image intensity as a stream function for a two-dimensional incompressible flow. The Laplacian of the image intensity plays the role of the vorticity of the fluid; it is transported into the region to be inpainted by a vector field defined by the stream function. The resulting algorithm is designed to continue isophotes while matching gradient vectors at the boundary of the inpainting region. The method is directly based on the Navier-Stokes equations for fluid dynamics, which has the immediate advantage of well-developed theoretical and numerical results. This is a new approach for introducing ideas from computational fluid dynamics into problems in computer vision and image analysis.

Swarming patterns in a two-dimensional kinematic model for biological groups

We construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern-forming behavior. The model takes the form of a conservation law in two spatial dimensions. The social interactions are modeled in the velocity term, which is nonlocal in the population density and includes a parameter that controls the interaction length scale. The dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion. For the purely incompressible case, the model resembles one for fluid dynamical vortex patches. There exist solutions which have constant population density and compact support for all time. Numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills. The sign of the social interaction term determines the direction of the rotation, and the interaction length scale affects the degree o...

Image Recovery via Nonlocal Operators

This paper considers two nonlocal regularizations for image recovery, which exploit the spatial interactions in images. We get superior results using preprocessed data as input for the weighted functionals. Applications discussed include image deconvolution and tomographic reconstruction. The numerical results show our method outperforms some previous ones.

The lubrication approximation for thin viscous films: Regularity and long-time behavior of weak solutions

We consider the fourth-order degenerate diffusion equation, in one space dimension. This equation, derived from a lubrication approximation, models the surface-tension-dominated motion of thin viscous films and spreading droplets [15]. The equation with f(h) = |h| also models a thin neck of fluid in the Hele-Shaw cell [10], [11], [23]. In such problems h(x,t) is the local thickness of the the film or neck. This paper considers the properties of weak solutions that are more relevant to the droplet problem than to Hele-Shaw. For simplicity we consider periodic boundary conditions with the interpretation of modeling a periodic array of droplets. We consider two problems: The first has initial data h0 ≥ 0 and f(h) = |h|n, 0 < n < 3. We show that there exists a weak nonnegative solution for all time. Also, we show that this solution becomes a strong positive solution after some finite time T*, and asymptotically approaches its means as t ∞. The weak solution is in the classical sense of distributions for 3/8 < n < 3 and in a weaker sense introduced in [1] for the remaining 0 < n ≤ 3/8. Furthermore, the solutions have high enough regularity to just include the unique source-type solutions [2] with zero slope at the edge of the support. They do not include any of the less regular solutions with positive slope at the edge of the support. Second, we consider strictly positive initial data h0 ≥ m > 0 and f(h) = |h|n, 0 < n < ∞. For this problem we show that even if a finite-time singularity of the form h 0 does occur, there exists a weak nonnegative solution for all time t. This weak solution becomes strong and positive again after some critical time T*. As in the first problem, we show that the solution approaches its mean as t ∞. The main technical idea is to introduce new classes of dissipative entropies to prove existence and higher regularity. We show that these entropies are related to norms of the difference between the solution and its mean to prove the relaxation result.

A STATISTICAL MODEL OF CRIMINAL BEHAVIOR

Motivated by empirical observations of spatio-temporal clusters of crime across a wide variety of urban settings, we present a model to study the emergence, dynamics, and steady-state properties of crime hotspots. We focus on a two-dimensional lattice model for residential burglary, where each site is characterized by a dynamic attractiveness variable, and where each criminal is represented as a random walker. The dynamics of criminals and of the attractiveness field are coupled to each other via specific biasing and feedback mechanisms. Depending on parameter choices, we observe and describe several regimes of aggregation, including hotspots of high criminal activity. On the basis of the discrete system, we also derive a continuum model; the two are in good quantitative agreement for large system sizes. By means of a linear stability analysis we are able to determine the parameter values that will lead to the creation of stable hotspots. We discuss our model and results in the context of established crim...

Linear stability and transient growth in driven contact lines

Fluid flowing down an inclined plane commonly exhibits a fingering instability in which the contact line corrugates. We show that below a critical inclination angle the base state before the instability is linearly stable. Several recent experiments explore inclination angles below this critical angle, yet all clearly show the fingering instability. We explain this paradox by showing that regardless of the long time linear stability of the front, microscopic scale perturbations at the contact line grow on a transient time scale to a size comparable with the macroscopic structure of the front. This amplification is sufficient to excite nonlinearities and thus initiate finger formation. The amplification is a result of the well-known singular dependence of the macroscopic profiles on the microscopic length scale near the contact line. Implications for other types of forced contact lines are discussed.

Inpainting of Binary Images Using the Cahn–Hilliard Equation

Image inpainting is the filling in of missing or damaged regions of images using information from surrounding areas. We outline here the use of a model for binary inpainting based on the Cahn-Hilliard equation, which allows for fast, efficient inpainting of degraded text, as well as super-resolution of high contrast images

Blow-up in multidimensional aggregation equations with mildly singular interaction kernels*

We consider the multidimensional aggregation equation ut − ∇ (u∇K * u) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better). In the case of bounded initial data, finite time singularity has been proved for kernels with a Lipschitz point at the origin (Bertozzi and Laurent 2007 Commun. Math. Sci. 274 717–35), whereas for C2 kernels there is no finite-time blow-up. We prove, under mild monotonicity assumptions on the kernel K, that the Osgood condition for well-posedness of the ODE characteristics determines global in time well-posedness of the PDE with compactly supported bounded nonnegative initial data. When the Osgood condition is violated, we present a new proof of finite time blow-up that extends previous results, requiring radially symmetric data, to general bounded, compactly supported nonnegative initial data without symmetry. We also present a new analysis of radially symmetric solutions under less strict monotonicity conditions. Finally, we conclude with a discussion of similarity solutions for the case K(x) = |x| and some open problems.

Long-Wave Instabilities and Saturation in Thin Film Equations

Hocherman and Rosenau conjectured that long-wave unstable Cahn-Hilliard type interface models develop nite-time singularities when the nonlinearity in the destabilizing term grows faster at large amplitudes than the nonlinearity in the stabilizing term (Phys. D 67:113-125, 1993). We consider this conjecture for a class of equations, often used to model thin lms in a lubrication context, by showing that if the solutions are uniformly bounded above or below (e.g. are nonnegative) then the destabilizing term can be stronger than previously conjectured yet the solution still remains globally bounded. For example, they conjecture that for the long-wave unstable equation ht = (h n hxxx)x (h m hx)x; m>n leads to blow-up. Using a conservation of volume constraint for the case m> n> 0, we conjecture a dierent critical exponent for possible singularities of nonnegative solutions. We prove that nonlinearities with exponents below the conjectured critical exponent yield globally bounded solutions. Specically, for the above equation, solutions are bounded if m<n + 2. The bound is proved using energy methods and is then used to prove the existence of non-negative weak solutions in the sense of distributions. We present preliminary numerical evidence suggesting that m n +2 can allow blow-up.

Global regularity for vortex patches

We present a proof of Chemins [4] result which states that the boundary of a vortex patch remains smooth for all time if it is initially smooth.