In this paper we present methods for estimating shape from polarisation and shading information, i.e. photo-polarimetric shape estimation, under varying, but unknown, illumination, i.e. in an uncalibrated scenario. We propose several alternative photo-polarimetric constraints that depend upon the partial derivatives of the surface and show how to express them in a unified system of partial differential equations of which previous work is a special case. By careful combination and manipulation of the constraints, we show how to eliminate non-linearities such that a discrete version of the problem can be solved using linear least squares. We derive a minimal, combinatorial approach for two source illumination estimation which we use with RANSAC for robust light direction and intensity estimation. We also introduce a new method for estimating a polarisation image from multichannel data and provide methods for estimating albedo and refractive index. We evaluate lighting, shape, albedo and refractive index estimation methods on both synthetic and real-world data showing improvements over existing state-of-the-art.

In this paper, we propose a high-order accurate scheme for image segmentation based on the level-set method. In this approach, the curve evolution is described as the 0-level set of a representation function, but we modify the velocity that drives the curve to the boundary of the object in order to obtain a new velocity with additional properties that are extremely useful to develop a more stable high-order approximation with a small additional cost. The approximation scheme proposed here is the first 2D version of an adaptive “filtered" scheme recently introduced and analyzed by the authors in one dimension. This approach is interesting since the implementation of the filtered scheme is rather efficient and easy. The scheme combines two building blocks (a monotone scheme and a high-order scheme) via a filter function and smoothness indicators that allow one to detect the regularity of the approximate solution adapting the scheme in an automatic way. Some numerical tests on synthetic and real images confirm the accuracy of the proposed method and the advantages given by the new velocity.

We present a method for estimating surface height directly from a single polarisation image simply by solving a large, sparse system of linear equations. To do so, we show how to express polarisation constraints as equations that are linear in the unknown height. The local ambiguity in the surface normal azimuth angle is resolved globally when the optimal surface height is reconstructed. Our method is applicable to dielectric objects exhibiting diffuse and specular reflectance, though lighting and albedo must be known. We relax this requirement by showing that either spatially varying albedo or illumination can be estimated from the polarisation image alone using nonlinear methods. In the case of illumination, the estimate can only be made up to a binary ambiguity which we show is a generalised Bas-relief transformation corresponding to the convex/concave ambiguity. We believe that our method is the first passive, monocular shape-from-x technique that enables well-posed height estimation with only a single, uncalibrated illumination condition. We present results on real world data, including in uncontrolled, outdoor illumination.

In this paper we present a differential approach to photo-polarimetric shape estimation. We propose several alternative differential constraints based on polarisation and photometric shading information and show how to express them in a unified partial differential system. Our method uses the image ratios technique to combine shading and polarisation information in order to directly reconstruct surface height, without first computing surface normal vectors. Moreover, we are able to remove the non-linearities so that the problem reduces to solving a linear differential problem. We also introduce a new method for estimating a polarisation image from multichannel data and, finally, we show it is possible to estimate the illumination directions in a two source setup, extending the method into an uncalibrated scenario. From a numerical point of view, we use a least-squares formulation of the discrete version of the problem. To the best of our knowledge, this is the first work to consider a unified differential approach to solve photo-polarimetric shape estimation directly for height. Numerical results on synthetic and real-world data confirm the effectiveness of our proposed method.

In this paper we show that the introduction of an attenuation factor in the brightness equations relative to various perspective shape-from-shading models allows us to make the corresponding differential problems well-posed. We propose a unified approach based on the theory of viscosity solutions 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.

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.