Michael Breuß

Anisotropic Continuous-Scale Morphology

We describe a new approach to incorporate adaptivity into the partial differential equations (PDEs) of morphological dilation and erosion. By multiplication of the image gradient with a space-variant matrix, the speed of the evolution is locally adapted to the data. This is used to create anisotropic morphological evolutions that enhance coherent, flow-like image structures. We show that our adaptive method can be implemented by means of a simple modification of the classical Rouy-Tourin finite difference scheme. Numerical experiments confirm that the proposed dilations and erosions are capable of real anisotropic behaviour that can be used for closing interrupted lines.

Perspective Shape from Shading: Ambiguity Analysis and Numerical Approximations

In this paper we study a perspective model for shape from shading and its numerical approximation. We show that an ambiguity still persists, although the model with light attenuation factor has previously been shown to be well-posed under appropriate assumptions. Analytical results revealing the ambiguity are complemented by various numerical tests. Moreover, we present convergence results for two iterative approximation schemes. The first is based on a finite difference discretization, whereas the second is based on a semi-Lagrangian discretization. The convergence results are obtained in the general framework of viscosity solutions of the underlying partial differential equation. In addition to these theoretical and numerical results, we propose an algorithm for reconstructing discontinuous surfaces, making it possible to obtain results of reasonable quality even for complex scenes. To this end, we solve the constituting equation on a previously segmented input image, using state constraint boundary conditions at the segment borders.

Perspective Shape from Shading with Non-Lambertian Reflectance

In this work, we extend the applicability of perspective Shape from Shading to images incorporating non-Lambertian surfaces. To this end, we derive a new model inspired by the perspective model for Lambertian surfaces recently studied by Prados et al. and the Phong reflection model incorporating ambient, diffuse and specular components. Besides the detailed description of the modeling process, we propose an efficient and stable semi-implicit numerical realisation of the resulting Hamilton-Jacobi equation. Numerical experiments on both synthetic and simple real-world images show the benefits of our new model: While computational times stay modest, a large qualitative gain can be achieved.

A Shock-Capturing Algorithm for the Differential Equations of Dilation and Erosion

Dilation and erosion are the fundamental operations in morphological image processing. Algorithms that exploit the formulation of these processes in terms of partial differential equations offer advantages for non-digitally scalable structuring elements and allow sub-pixel accuracy. However, the widely-used schemes from the literature suffer from significant blurring at discontinuities. We address this problem by developing a novel, flux corrected transport (FCT) type algorithm for morphological dilation/erosion with a flat disc. It uses the viscosity form of an upwind scheme in order to quantify the undesired diffusive effects. In a subsequent corrector step we compensate for these artifacts by means of a stabilised inverse diffusion process that requires a specific nonlinear multidimensional formulation. We prove a discrete maximum–minimum principle in this multidimensional framework. Our experiments show that the method gives a very sharp resolution of moving fronts, and it approximates rotation invariance very well.

Making Shape from Shading Work for Real-World Images

Although shape from shading (SfS) has been studied for almost four decades, the performance of most methods applied to real-world images is still unsatisfactory: This is often caused by oversimplified reflectance and projection models as well as by ignoring light attenuation and nonconstant albedo behavior. We address this problem by proposing a novel approach that combines three powerful concepts: (i) By means of a Chan-Vese segmentation step, we partition the image into regions with homogeneous reflectance properties. (ii) This homogeneity is further improved by an adaptive thresholding that singles out unreliable details which cause fluctuating albedos. Using an inpainting method based on edge-enhancing anisotropic diffusion, structures are filled in such that the albedo does no longer suffer from fluctuations. (iii) Finally a sophisticated SfS method is used that features a perspective projection model, considers physical light attenuation and models specular highlights. In our experiments we demonstrate that each of these ingredients improves the reconstruction quality significantly. Their combination within a single method gives favorable perfomance also for images that are taken under real-world conditions where simpler approaches fail.

An adaptive domain-decomposition technique for parallelization of the fast marching method

Abstract The fast marching method (FMM) is an efficient technique to solve numerically the Eikonal equation. The parallelization of the FMM is not easy because of its intrinsic sequential nature. In this paper we propose a novel approach to parallelize the FMM. It leads to an equation-dependent domain decomposition and it turns out to be particularly suitable for machines with two or four cores that are in common use today. Compared to other techniques in the field, the proposed method is much simpler to implement and it gives a slightly better computational speed-up. In order to test the new method on a real-world application, we solve the shape-from-shading problem based on a Hamilton–Jacobi equation. On a standard four-core machine, the method confirms the good properties. It shows a reasonable speedup factor of about 2.5, and it reveals its potential to good performance if the arithmetic density of the problem is high.

Numerical algorithms for perspective shape from shading

The Shape-From-Shading (SFS) problem is a fundamental and classic problem in computer vision. It amounts to compute the 3-D depth of objects in a single given 2-D image. This is done by exploiting information about the illumination and the image brightness. We deal with a recent model for Perspective SFS (PSFS) for Lambertian surfaces. It is defined by a Hamilton-Jacobi equation and complemented by state constraints boundary conditions. In this paper we investigate and compare three state-of-the-art numerical approaches. We begin with a presentation of the methods. Then we discuss the use of some acceleration techniques, including cascading multigrid, for all the tested algorithms. The main goal of our paper is to analyze and compare recent solvers for the PSFS problem proposed in the literature.

Fast and Robust Surface Normal Integration by a Discrete Eikonal Equation.

The integration of surface normals is a classic and fundamental task in computer vision. In this paper we deal with a highly efficient fast marching (FM) method to perform the integration. In doing this we build upon a previous work of Ho and his coauthors. Their FM scheme is based on an analytic model that incorporates the eikonal equation. Our method is also built upon this equation, but it makes use of a complete discrete formulation for constructing the FM integrator (DEFM). We not only provide a theoretical justification of the proposed method, but also illustrate at hand of a simple example that our approach is much better suited to the task. Several more sophisticated tests confirm the robustness and higher accuracy of the DEFM model. Moreover, we present an extension of DEFM that allows to integrate surface normals over non-trivial domains, e.g. featuring holes. Numerical results confirm desirable qualities of this method.

Fast Shape from Shading for Phong-Type Surfaces

Shape from Shading (SfS) is one of the oldest problems in image analysis that is modelled by partial differential equations (PDEs). The goal of SfS is to compute from a single 2-D image a reconstruction of the depicted 3-D scene. To this end, the brightness variation in the image and the knowledge of illumination conditions are used. While the quality of models has reached maturity, there is still the need for efficient numerical methods that enable to compute sophisticated SfS processes for large images in reasonable time. In this paper we address this problem. We consider a so-called Fast Marching (FM) scheme,which is one of the most efficient numerical approaches available. However, the FM scheme is not trivial to use for modern non-linear SfS models. We show how this is done for a recent SfS model incorporating the non-Lambertian reflectance model of Phong. Numerical experiments demonstrate that --- without compromising quality --- our FM scheme is two orders of magnitude faster than standard methods.

Numerical aspects of TV flow

The singular diffusion equation called total variation (TV) flow plays an important role in image processing and appears to be suitable for reducing oscillations in other types of data. Due to its singularity for zero gradients, numerical discretizations have to be chosen with care. We discuss different ways to implement TV flow numerically, and we show that a number of discrete versions of this equation may introduce oscillations such that the scheme is in general not TV-decreasing. On the other hand, we show that TV flow may act self-stabilising: even if the total variation increases by the filtering process, the resulting oscillations remain bounded by a constant that is proportional to the ratio of mesh widths. For our analysis we restrict ourselves to the one-dimensional setting.