Oliver Vogel

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.

Morphological Amoebas Are Self-snakes

This paper is concerned with amoeba median filtering, a structure-adaptive morphological image filter. It has been introduced by Lerallut et al. in a discrete formulation. Experimental evidence shows that iterated amoeba median filtering leads to segmentation-like results that are similar to those obtained by self-snakes, an image filter based on a partial differential equation. We establish this correspondence by analysing a space-continuous formulation of iterated amoeba median filtering. We prove that in the limit of vanishing radius of the structuring elements, iterated amoeba median filtering indeed approximates the partial differential equation of self-snakes. This result holds true under very general assumptions on the metric used to construct the amoebas. We present experiments with discrete iterated amoeba median filtering that confirm qualitative and quantitative predictions of our analysis.

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.

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.

Linear Osmosis Models for Visual Computing

Osmosis is a transport phenomenon that is omnipresent in nature. It differs from diffusion by the fact that it allows nonconstant steady states. In our paper we lay the foundations of osmosis filtering for visual computing applications. We model filters with osmotic properties by means of linear drift-diffusion processes. They preserve the average grey value and the nonnegativity of the initial image. Most interestingly, however, we show how the nonconstant steady state of an osmosis evolution can be steered by its drift vector field. We interpret this behaviour as a data integration mechanism. In the integrable case, we characterise the steady state as a minimiser of a suitable energy functional. In the nonintegrable case, we can exploit osmosis as a framework to fuse incompatible data in a visually convincing way. Osmotic data fusion differs from gradient domain methods by its intrinsic invariance under multiplicative grey scale changes. The osmosis framework constitutes a novel class of methods that can be taylored to solve various problems in image processing, computer vision, and computer graphics. We demonstrate its versatility by proposing osmosis models for compact image respresentation, shadow removal, and seamless image cloning.

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.

Direct shape-from-shading with adaptive higher order regularisation

Although variational methods are popular techniques in the context of shape-from-shading, they are in general restricted to indirect approaches that only estimate the gradient of the surface depth. Such methods suffer from two drawbacks: (i) They need additional constraints to enforce the integrability of the solution. (ii) They require the application of depth-from-gradient algorithms to obtain the actual surface. In this paper we present three novel approaches that avoid the aforementioned drawbacks by construction: (i) First, we present a method that is based on homogeneous higher order regularisation. Thus it becomes possible to estimate the surface depth directly by solving a single partial differential equation. (ii) Secondly, we develop a refined technique that adapts this higher order regularisation to semantically important structures in the original image. This addresses another drawback of existing variational methods: the blurring of the results due to the regularisation. (iii) Thirdly, we present an even further improved approach, in which the smoothness process is steered directly by the evolving depth map. This in turn allows to tackle the well-known problem of spontaneous concave-convex switches in the solution. In our experimental section both qualitative and quantitative experiments on standard shape-from-shading data sets are performed. A comparison to the popular variational method of Frankot and Chellappa shows the superiority of all three approaches.

A Lattice Boltzmann Model for Rotationally Invariant Dithering

In this paper, we present a novel algorithm for dithering of gray-scale images. Our algorithm is based on the lattice Boltzmann method, a well-established and powerful concept known from computational physics. We describe the method and show the consistency of the new scheme to a partial differential equation. In contrast to widely-used error diffusion methods our lattice Boltzmann model is rotationally invariant by construction. In several experiments on real and synthetic images, we show that our algorithm produces clearly superior results to these methods.