Analysis and approximation of some Shape-from-Shading models for non-Lambertian surfaces

The reconstruction of a 3D object or a scene is a classical inverse problem in Computer Vision. In the case of a single image this is called the Shape-from-Shading (SfS) problem and it is known to be ill-posed even in a simplified version like the vertical light source case. A huge number of works deals with the orthographic SfS problem based on the Lambertian reflectance model, the most common and simplest model which leads to an eikonal type equation when the light source is on the vertical axis. In this paper we want to study non-Lambertian models since they are more realistic and suitable whenever one has to deal with different kind of surfaces, rough or specular. We will present a unified mathematical formulation of some popular orthographic non-Lambertian models, considering vertical and oblique light directions as well as different viewer positions. These models lead to more complex stationary nonlinear partial differential equations of Hamilton-Jacobi type which can be regarded as the generalization of the classical eikonal equation corresponding to the Lambertian case. However, all the equations corresponding to the models considered here (Oren-Nayar and Phong) have a similar structure so we can look for weak solutions to this class in the viscosity solution framework. Via this unified approach, we are able to develop a semi-Lagrangian approximation scheme for the Oren-Nayar and the Phong model and to prove a general convergence result. Numerical simulations on synthetic and real images will illustrate the effectiveness of this approach and the main features of the scheme, also comparing the results with previous results in the literature.Comment: Accepted version to Journal of Mathematical Imaging and Vision, 57 page

An overview of some mathematical techniques and problems linking 3D vision to 3D printing

Computer Vision and 3D printing have rapidly evolved in the last 10 years but interactions among them have been very limited so far, despite the fact that they share several mathematical techniques. We try to fill the gap presenting an overview of some techniques for Shape-from-Shading problems as well as for 3D printing with an emphasis on the approaches based on nonlinear partial differential equations and optimization. We also sketch possible couplings to complete the process of object manufacturing starting from one or more images of the object and ending with its final 3D print. We will give some practical examples of this procedure

A unified approach to the well-posedness of some non-Lambertian models in Shape-from-Shading theory

In this paper we show that the introduction of an attenuation factor in the %image irradiance brightness equations relative to various perspective Shape from Shading models allows to make the corresponding differential problems well-posed. We propose a unified approach based on the theory of viscosity solution and we show that the brightness equations with the attenuation term admit a unique viscosity solution. We also discuss in detail the possible boundary conditions that we can use for the Hamilton-Jacobi equations associated to these models

A PDE approach to Shape from Shading via Photometric Stereo

We present a new analytic and numerical approach to the shape from shading using photometric stereo technique. That is, we solve the problem to find the 3D surface of an object starting from its several 2D pictures taken from the same point of view, but changing, for every image, the direction of the light source

3D surface reconstruction for lower limb prosthetic model using modified radon transform

Computer vision has received increased attention for the research and innovation on three-dimensional surface reconstruction with aim to obtain accurate results. Although many researchers have come up with various novel solutions and feasibility of the findings, most require the use of sophisticated devices which is computationally expensive. Thus, a proper countermeasure is needed to resolve the reconstruction constraints and create an algorithm that is able to do considerably fast reconstruction by giving attention to devices equipped with appropriate specification, performance and practical affordability. This thesis describes the idea to realize three-dimensional surface of the residual limb models by adopting the technique of tomographic imaging coupled with the strategy based on multiple-views from a digital camera and a turntable. The surface of an object is reconstructed from uncalibrated two-dimensional image sequences of thirty-six different projections with the aid of Radon transform algorithm and shape-from-silhouette. The results show that the main objective to reconstruct three-dimensional surface of lower limb model has been successfully achieved with reasonable accuracy as the starting point to reconstruct three-dimensional surface and extract digital reading of an amputated lower limb model where the maximum percent error obtained from the computation is approximately 3.3 % for the height whilst 7.4%, 7.9% and 8.1% for the diameters at three specific heights of the objects. It can be concluded that the reconstruction of three-dimensional surface for the developed method is particularly dependent to the effects the silhouette generated where high contrast two-dimensional images contribute to higher accuracy of the silhouette extraction. The advantage of the concept presented in this thesis is that it can be done with simple experimental setup and the reconstruction of three-dimensional model neither involves expensive equipment nor require any service by an expert to handle sophisticated mechanical scanning system

A Semi-Lagrangian Approximation of the Oren-Nayar PDE for the Orthographic Shape–from–Shading Problem

Several advances have been made in the last ten years to improve the Shape–from–Shading model in order to allow its use on real images. The classic Lambertian model, suitable to reconstruct 3D surfaces with uniform reflection properties has shown to be unsuitable for other types of surfaces, for example for rough objects consisting of materials such as clay. Other models have been proposed but it is still unclear what would be the best model. For this reason, we start our analysis for non-Lambertian surfaces. The goal being to find a unique model which should be flexible enough to deal with many kinds of real images. As a starting point for this big project, we consider the non-Lambertian Oren–Nayar reflectance model. In this paper we construct a semi-Lagrangian approximation scheme for its nonlinear partial differential equation and we compare its performances with the classical model in terms of some error indicators on series of benchmarks images