Abstract Several neural network approaches for solving differential equations employ trial solutions with a feedforward neural network. There are different means to incorporate the trial solution in the construction, for instance, one may include them directly in the cost function. Used within the corresponding neural network, the trial solutions define the so-called neural form. Such neural forms represent general, flexible tools by which one may solve various differential equations. In this article, we consider time-dependent initial value problems, which require to set up the neural form framework adequately. The neural forms presented up to now in the literature for such a setting can be considered as first-order polynomials. In this work, we propose to extend the polynomial order of the neural forms. The novel collocation-type construction includes several feedforward neural networks, one for each order. Additionally, we propose the fragmentation of the computational domain into subdomains. The neural forms are solved on each subdomain, whereas the interfacing grid points overlap in order to provide initial values over the whole fragmentation. We illustrate in experiments that the combination of collocation neural forms of higher order and the domain fragmentation allows to solve initial value problems over large domains with high accuracy and reliability.

Abstract Feedforward neural networks offer a possible approach for solving differential equations. However, the reliability and accuracy of the approximation still represent delicate issues that are not fully resolved in the current literature. Computational approaches are in general highly dependent on a variety of computational parameters as well as on the choice of optimisation methods, a point that has to be seen together with the structure of the cost function. The intention of this paper is to make a step towards resolving these open issues. To this end, we study here the solution of a simple but fundamental stiff ordinary differential equation modelling a damped system. We consider two computational approaches for solving differential equations by neural forms. These are the classic but still actual method of trial solutions defining the cost function, and a recent direct construction of the cost function related to the trial solution method. Let us note that the settings we study can easily be applied more generally, including solution of partial differential equations. By a very detailed computational study, we show that it is possible to identify preferable choices to be made for parameters and methods. We also illuminate some interesting effects that are observable in the neural network simulations. Overall we extend the current literature in the field by showing what can be done in order to obtain useful and accurate results by the neural network approach. By doing this we illustrate the importance of a careful choice of the computational setup.

Photometric stereo is a fundamental technique in computer vision known to produce 3D shape with high accuracy. It uses several input images of a static scene taken from one and the same camera position but under varying illumination. The vast majority of studies in this 3D reconstruction method assume orthographic projection for the camera model. In addition, they mainly use the Lambertian reflectance model as the way that light scatters at surfaces. Thus, providing reliable photometric stereo results from real world objects still remains a challenging task. We address 3D reconstruction by use of a more realistic set of assumptions, combining for the first time the complete Blinn–Phong reflectance model and perspective projection. Furthermore, we compare two different methods of incorporating the perspective projection into our model. Experiments are performed on both synthetic and real world images; the latter do not benefit from laboratory conditions. The results show the high potential of our method even for complex real world applications such as medical endoscopy images which may include many specular highlights.

The integration of surface normals for the purpose of computing the shape of a surface in 3D space is a classic problem in computer vision. However, even nowadays it is still a challenging task to devise a method that is flexible enough to work on non-trivial computational domains with high accuracy, robustness, and computational efficiency. By uniting a classic approach for surface normal integration with modern computational techniques, we construct a solver that fulfils these requirements. Building upon the Poisson integration model, we use an iterative Krylov subspace solver as a core step in tackling the task. While such a method can be very efficient, it may only show its full potential when combined with suitable numerical preconditioning and problem-specific initialisation. We perform a thorough numerical study in order to identify an appropriate preconditioner for this purpose. To provide suitable initialisation, we compute this initial state using a recently developed fast marching integrator. Detailed numerical experiments illustrate the benefits of this novel combination. In addition, we show on real-world photometric stereo datasets that the developed numerical framework is flexible enough to tackle modern computer vision applications.

In spite of significant advances in Shape from Shading (SfS) over the last years, it is still a challenging task to design SfS approaches that are flexible enough to handle a wide range of input scenes. In this paper, we address this lack of flexibility by proposing a novel model that extends the range of possible applications. To this end, we consider the class of modern perspective SfS models formulated via partial differential equations (PDEs). By combining a recent spherical surface parametrisation with the advanced non-Lambertian Oren-Nayar reflectance model, we obtain a robust approach that allows to deal with an arbitrary position of the light source while being able to handle rough surfaces and thus more realistic objects at the same time. To our knowledge, the resulting model is currently the most advanced and most flexible approach in the literature on PDE-based perspective SfS. Apart from deriving our model, we also show how the corresponding set of sophisticated Hamilton-Jacobi equations can be efficiently solved by a specifically tailored fast marching scheme. Experiments with medical real-world data demonstrate that our model works in practice and that is offers the desired flexibility.

The integration of surface normals is a classic and fundamental task in computer vi-sion. 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 for-mulation 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 ex-tension of DEFM that allows to integrate surface normals over non-trivial domains, e.g. featuring holes. Numerical results confirm desirable qualities of this method. 1

Due to their improved capability to handle realistic illumination scenarios, nonLambertian reflectance models are becoming increasingly more popular in the Shape from Shading (SfS) community. One of these advanced models is the Oren-Nayar model which is particularly suited to handle rough surfaces. However, not only the proper selection of the model is important, also the validation of stable and efficient algorithms plays a fundamental role when it comes to the practical applicability. While there are many works dealing with such algorithms in the case of Lambertian SfS, no such analysis has been performed so far for the Oren-Nayar model. In our paper we address this problem and present an in-depth study for such an advanced SfS model. To this end, we investigate under which conditions, i.e. model parameters, the Fast Marching (FM) method can be applied – a method that is known to be one of the most efficient algorithms for solving the underlying partial differential equations of Hamilton-Jacobi type. In this context, we do not only perform a general investigation of the model using Osher’s criterion for verifying the suitability of the FM method. We also conduct a parameter dependent analysis that shows, that FM can safely be used for the model for a wide range of settings relevant for practical applications. Thus, for the first time, it becomes possible to theoretically justify the use of the FM method as solver for the Oren-Nayar model which has been applied so far on a purely empirical basis only. Numerical experiments demonstrate the validity of our theoretical analysis. They show a stable behaviour of the FM method for the predicted range of model parameters.

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.

Recently, non-Lambertian models for perspective shape from shading (PSFS) have received attention in the literature. The Phong-based PSFS model combining Lambertian and specular reflection has been shown to give good results for objects in real-world images. In this paper, we present at hand of this model the first analysis of a non-Lambertian PSFS model in the literature. We show mathematically and experimentally how crucial the specular part of the model is for the reconstruction. Moreover, we give a detailed analysis of the Hamiltonian defining the model. While the non-Lambertian reflectance is generally assumed to lead to numerical difficulties, this investigation shows that an efficient fast marching scheme can be applied successfully without losing depth information. Our work represents a first step towards the thorough understanding of non-Lambertian PSFS models.

Shape from shading (SFS) is one of the most fundamental problems in Computer Vision: reconstruction of the three-dimensional (3D) shape of photographed objects given a single input image. Image formation is then modelled based on assumptions on illumination and light reflectance. Solving the 3D shape from the model is useful for many applications. SFS has been extended, for example, to several input images of the same scene taken under different lighting conditions, known as photometric stereo. The aim of the session is to introduce state-of-the-art models and algorithms in the field, and to discuss future directions of research.