The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shape-from-shading and for recent dynamic theories of shape. Its numerical simulation can be delicate, owing to the formation of singularities in the evolving front and is typically based on level set methods. However, there are more classical approaches rooted in Hamiltonian physics which have yet to be widely used by the computer vision community. In this paper we review the Hamiltonian formulation, which offers specific advantages when it comes to the detection of singularities or shocks. We specialize to the case of Blum's grass fire flow and measure the average outward ux of the vector field that underlies the Hamiltonian system. This measure has very different limiting behaviors depending upon whether the region over which it is computed shrinks to a singular point or a non-singular one. Hence, it is an effective way to distinguish between these two cases. We combine the ux measurement with a homotopy preserving thinning process applied in a discrete lattice. This leads to a robust and accurate algorithm for computing skeletons in 2D as well as 3D, which has low computational complexity. We illustrate the approach with several computational examples.

The class of geometric deformable models, so-called level sets, has brought tremendous impact to medical imagery due to its capability to preserve topology and fast shape recovery. In an effort to facilitate a clear and full understanding of these powerful state-of-the-art applied mathematical tools, this paper is an attempt to explore these geometric methods, their implementations and integration of regularization terms to improve the robustness of these topologically independent propagating curves/surfaces. This paper first presents the origination of the level sets, followed by the taxonomy tree of level sets. We then derive the fundamental equation of curve/surface evolution and zero-level curves/surfaces. The paper then focuses on the first core class of level sets, the so-called level sets "without regularizers". The next section is devoted on a second kind, so-called level sets "with regularizers". In this class, we present four kinds of systems on the design of the regularizers. Next, the paper presents a third kind of level sets, so-called the "bubble-based" techniques. An entire section is dedicated to optimization and quantification techniques for shape recovery when used with the level sets. Finally, the paper concludes with 22 general merits and four demerits on level sets and the future of level sets in medical image segmentation. We present the applications of level sets to complex shapes likethehuman cortex acquired via MRI for neurological image analysis.

We propose a variational algorithm to jointly estimate the shape, albedo, and light configuration of a Lambertian scene from a collection of images taken from different vantage points. Our work can be thought of as extending classical multi-view stereo to cases where point correspondence cannot be established, or extending classical shape from shading to the case of multiple views with unknown light sources. We show that a first naive formalization of this problem yields algorithms that are numerically unstable, no matter how close the initialization is to the true geometry. We then propose a computational scheme to overcome this problem, resulting in provably stable algorithms that converge to (local) minima of the cost functional. Although we restrict our attention to Lambertian objects with uniform albedo, extensions of our framework are conceivable. 1

The eikonal equation and variants of it are of significant interest for problems in computer vision and image processing. It is the basis for continuous versions of mathematical morphology, stereo, shape-from-shading and for recent dynamic theories of shape. Its numerical simulation can be delicate, owing to the formation of singularities in the evolving front, and is typically based on level set methods introduced by Osher and Sethian. However, there are more classical approaches rooted in Hamiltonian physics, which have received little consideration in the computer vision literature. Here the front is interpreted as minimizing a particular action functional. In this context, we introduce a new algorithm for simulating the eikonal equation, which offers a number of computational advantages over the earlier methods. In particular, the locus of shocks is computed in a robust and efficient manner. We illustrate the approach with several numerical examples.

Abstract—In this paper, we explore how graph-spectral methods can be used to develop a new shape-from-shading algorithm. We characterize the field of surface normals using a weight matrix whose elements are computed from the sectional curvature between different image locations and penalize large changes in surface normal direction. Modeling the blocks of the weight matrix as distinct surface patches, we use a graph seriation method to find a surface integration path that maximizes the sum of curvature-dependent weights and that can be used for the purposes of height reconstruction. To smooth the reconstructed surface, we fit quadrics to the height data for each patch. The smoothed surface normal directions are updated ensuring compliance with Lambert’s law. The processes of height recovery and surface normal adjustment are interleaved and iterated until a stable surface is obtained. We provide results on synthetic and real-world imagery. Index Terms—Graph seriation, graph-spectral methods, shapefrom-shading. I.

Numerical analysis of conservation laws plays an important role in the implementation of curve evolution equations. This paper reviews the relevant concepts in numerical analysis and the relation between curve evolution, Hamilton-Jacobi partial differential equations, and differential conservation laws. This close relation enables us to introduce finite difference approximations, based on the theory of conservation laws, into curve evolution. It is shown how curve evolution serves as a powerful tool for image analysis, and how these mathematical relations enable us to construct efficient and accurate numerical schemes. Some examples demonstrate the importance of the CFL condition as a necessary condition for the stability of the numerical schemes. 1 Introduction Recently, researchers in the field of image processing and computer vision started to pay attention to new ways of analyzing and representing two-dimensional, stationary or moving images, via planar curve evolution. In fact, a...

In this paper we propose a solution of the Lambertian Shape-From-Shading (SFS) problem by designing a new mathematical framework based on the notion of viscosity solution. The power of our approach is twofolds: 1) it defines a notion of weak solutions (in the viscosity sense) which does not necessarily require boundary data. Moreover, it allows to characterize the viscosity solutions by their “minimums”; 2) it unifies the works of Rouy et al. [18, 25], Falcone et al. [12], Prados et al. [21, 24], based on the notion of viscosity solutions and the work of Dupuis and Oliensis [11] dealing with classical solutions.

This paper describes a graph-spectral method for 3D surface integration. The algorithm takes as its input a 2D field of surface normal estimates,delivered,for instance,by a shape-from-shading or shape-from-texture procedure. We commence by using the surface normals to obtain an affinity weight matrix whose elements are related to the surface curvature. The weight matrix is used to compute a row-normalized transition probability matrix,and we pose the recovery of the integration path as that of finding the steady-state random walk for the Markov chain defined by this matrix. The steady-state random walk is given by the leading eigenvector of the original affinity weight matrix. By threading the surface normals together along the path specified by the magnitude order of the components of the leading eigenvector we perform surface integration. The height increments along the path are simply related to the traversed path length and the slope of the local tangent plane. The method is evaluated on needle-maps delivered by a shape-from-shading algorithm applied to real-world data and also on synthetic data. The method is compared with the local geometric height reconstruction method of Bors,Hancock and Wilson, and the global methods of Horn and Brooks and Frankot and Chellappa.

We address the problem of integrating shading and multi-frame stereo cues within the framework of optimization in the innite-dimensional space of piecewise smooth surfaces. Cue integration then reduces to the determination of regions where prior assumptions on the reectance of the surfaces can be enforced, and results in a novel re-formulation of the correspondence problem as a \segmentation" or \grouping" task. In general, our formulation combines stereo and shading, and therefore allows dening a well-posed problem even when reconstruction from each cue in isolation would be ill-posed. For a simplied model we prove the necessary conditions for optimality, and propose an iterative optimization algorithm. We implement the algorithm using ultra-narrowband level set methods, and perform some preliminary experiments on two-dimensional scenes seen from one-dimensional images. We use a deterministic formulation of the problem, although a statistical interpretation is possible. 1 Introduc...

This paper describes two new methods for the reconstruction of discrete surfaces from shading images. Both approaches are based on the reconstruction of a discrete surface by mixing photometric and geometric techniques. The processing of photometric information is based on reflectance maps, which are classic tools of Shape from Shading. The geometric features are extracted from the discrete surface and propagated along the surface. The propagation is based in one case on equal height discrete contour propagation and in the other case on region propagation. Both methods allow photometric stereo. Results of reconstruction from synthetic and real images are presented. q 2004 Elsevier B.V. All rights reserved.