In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in ... more In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in the spirit of the Hughes dynamic continuum model. The pedestrians, represented by a density function, move on the graph choosing a route to minimize the instantaneous travel cost to the destination. The density is governed by a conservation law while the minimization principle is described by a graph eikonal equation. We show that the model is well posed and we implement some numerical examples to demonstrate the validity of the proposed model.
In this paper we show that the introduction of an attenuation factor in the brightness equations ... more 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 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.
IEEE transactions on pattern analysis and machine intelligence, Jan 6, 2018
We present a method for estimating surface height directly from a single polarisation image simpl... more 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 estim...
In this paper we show that the introduction of an attenuation factor in the brightness equations ... more 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 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.
Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference
Thermal insulation represents one of the major challenges for energy efficiency. Problems related... more Thermal insulation represents one of the major challenges for energy efficiency. Problems related to insulation are well-known and widely studied in mathematical physics. Neverthless, mathematics involved is still very tricky especially when one looks at shape optimization issues [1, 2], and sometimes the answers are not so intuitive [3].In this talk we will consider two domains: an internal (fixed) ball of radius r and an external domain whose geometry varies. Physically, we are considering a domain of given temperature, thermally insulated by surrounding it with a constant thickness of thermal insulator. Our question is related to the best (or worst) shape for the external domain, in terms of heat dispersion (of course, under prescribed geometrical constraints). Mathematically, our problem is composed by an elliptic PDE with Robin-Dirichlet boundary conditions. This work is still in progress and we want to share someremarks and open questions, in addition to the results obtained ...
In this paper we propose a high-order accurate scheme for image segmentation based on the level-s... more 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 1D. 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 to detect the regularity of the approximate solution adapting the scheme in an automatic way. Some numerical tests on synthetic and r...
We consider a class of "filtered" schemes for first order time dependent Hamilton-Jacob... more We consider a class of "filtered" schemes for first order time dependent Hamilton-Jacobi equations and prove a general convergence result for this class of schemes. A typical filtered scheme is obtained mixing a high-order scheme and a monotone scheme according to a filter function F which decides where the scheme has to switch from one scheme to the other. A crucial role for this switch is played by a parameter ε=ε(Δ t,Δ x)>0 which goes to 0 as the time and space steps (Δ t,Δ x) are going to 0 and does not depend on the time t_n, for each iteration n. The tuning of this parameter in the code is rather delicate and has an influence on the global accuracy of the filtered scheme. Here we introduce an adaptive and automatic choice of ε=ε ^n (Δ t, Δ x) at every iteration modifying the classical set up. The adaptivity is controlled by a smoothness indicator which selects the regions where we modify the regularity threshold ε^n. A convergence result and some error estimates f...
Photometric 3D-reconstruction techniques aim at inferring the geometry of a scene from one or sev... more Photometric 3D-reconstruction techniques aim at inferring the geometry of a scene from one or several images, by inverting a physical model describing the image formation. This chapter presents an introductory overview of the main pho-tometric 3D-reconstruction techniques which are shape-from-shading, photometric stereo and shape-from-polarisation.
Lecture Notes in Computational Science and Engineering, 2019
In this paper we consider a class of “filtered” schemes for some first order time dependent Hamil... more In this paper we consider a class of “filtered” schemes for some first order time dependent Hamilton-Jacobi equations. A typical feature of a filtered scheme is that at the node xj the scheme is obtained as a mixture of a high-order scheme and a monotone scheme according to a filter function F. The mixture is usually governed by F and by a fixed parameter e = e(Δt, Δx) > 0 which goes to 0 as (Δt, Δx) is going to 0 and does not depend on n. Here we improve the standard filtered scheme introducing an adaptive and automatic choice of the parameter e = en(Δt, Δx) at every iteration. To this end, we use a smoothness indicator in order to select the regions where we can compute the regularity threshold en. The numerical tests presented confirms the effectiveness of the adaptive scheme.
A Comparison of Non-Lambertian Models for the Shape-from-Shading Problem
In this paper we present in a unified approach Shape-from-Shading models under orthographic proje... more In this paper we present in a unified approach Shape-from-Shading models under orthographic projection for non-Lambertian surfaces and compare them with the classical Lambertian model. Those non-Lambertian models have been proposed in the literature by various authors in order to take into account more realistic surfaces such as rough and specular surfaces. The advantage of our unified mathematical model is the possibility to easily modify a single differential model to various situations just changing some control parameters. Moreover, the numerical approximation we propose is valid for that general model and can be easily adapted to the real situation. Finally, we compare the models on some benchmarks including real and synthetic images.
IEEE Transactions on Pattern Analysis and Machine Intelligence
In this paper we present methods for estimating shape from polarisation and shading information, ... more 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.
We consider a class of "filtered" schemes for first order time dependent Hamilton-Jacobi equation... more We consider a class of "filtered" schemes for first order time dependent Hamilton-Jacobi equations and prove a general convergence result for this class of schemes. A typical filtered scheme is obtained mixing a high-order scheme and a monotone scheme according to a filter function F which decides where the scheme has to switch from one scheme to the other. A crucial role for this switch is played by a parameter ε = ε(∆t, ∆x) > 0 which goes to 0 as the time and space steps (∆t, ∆x) are going to 0 and does not depend on the time t n , for each iteration n. The tuning of this parameter in the code is rather delicate and has an influence on the global accuracy of the filtered scheme. Here we introduce an adaptive and automatic choice of ε = ε n (∆t, ∆x) at every iteration modifying the classical set up. The adaptivity is controlled by a smoothness indicator which selects the regions where we modify the regularity threshold ε n. A convergence result and some error estimates for the new adaptive filtered scheme are proved, this analysis relies on the properties of the scheme and of the smoothness indicators. Finally, we present some numerical tests to compare the adaptive filtered scheme with other methods.
Astronomical images are of crucial importance for astronomers since they contain a lot of informa... more Astronomical images are of crucial importance for astronomers since they contain a lot of information about celestial bodies that can not be directly accessible. Most of the information available for the analysis of these objects starts with sky explorations via telescopes and satellites. Unfortunately, the quality of astronomical images is usually very low with respect to other real images and this is due to technical and physical features related to their acquisition process. This increases the percentage of noise and makes more difficult to use directly standard segmentation methods on the original image. In this work we will describe how to process astronomical images in two steps: in the first step we improve the image quality by a rescaling of light intensity whereas in the second step we apply level-set methods to identify the objects. Several experiments will show the effectiveness of this procedure and the results obtained via various discretization techniques for level-set equations.
2017 IEEE International Conference on Computer Vision (ICCV), Oct 1, 2017
In this paper we present a differential approach to photopolarimetric shape estimation. We propos... more In this paper we present a differential approach to photopolarimetric 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.
We present a method for estimating surface height directly from a single polarisation image simpl... more 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 depth. The ambiguity in the surface normal azimuth angle is resolved globally when the optimal surface height is reconstructed. Our method is applicable to objects with uniform albedo exhibiting diffuse and specular reflectance. We extend it to an uncalibrated scenario by demonstrating that the illumination (point source or first/second order spherical harmonics) can be estimated from the polarisation image, up to a binary convex/concave ambiguity. We believe that our method is the first monocular, passive shape-from-x technique that enables well-posed depth estimation with only a single, uncalibrated illumination condition. We present results on glossy objects, including in uncontrolled, outdoor illumination.
In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in ... more In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in the spirit of the Hughes dynamic continuum model. The pedestrians, represented by a density function, move on the graph choosing a route to minimize the instantaneous travel cost to the destination. The density is governed by a conservation law while the minimization principle is described by a graph eikonal equation. We show that the model is well posed and we implement some numerical examples to demonstrate the validity of the proposed model.
Recovering the 3D shape of an object from shading is a challenging problem due to the complexity ... more Recovering the 3D shape of an object from shading is a challenging problem due to the complexity of modeling light propagation and surface reflections. Photometric Stereo (PS) is broadly considered a suitable approach for high-resolution shape recovery, but its functionality is restricted to a limited set of object surfaces and controlled lighting setup. In particular, PS models generally consider reflection from objects as purely diffuse, with specularities being regarded as a nuisance that breaks down shape reconstruction. This is a serious drawback for implementing PS approaches since most common materials have prominent specular components. In this paper, we propose a PS model that solves the problem for both diffuse and specular components aimed at shape recovery of generic objects with the approach being independent of the albedo values
In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in ... more In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in the spirit of the Hughes dynamic continuum model. The pedestrians, represented by a density function, move on the graph choosing a route to minimize the instantaneous travel cost to the destination. The density is governed by a conservation law while the minimization principle is described by a graph eikonal equation. We show that the model is well posed and we implement some numerical examples to demonstrate the validity of the proposed model.
In this paper we show that the introduction of an attenuation factor in the brightness equations ... more 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 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.
IEEE transactions on pattern analysis and machine intelligence, Jan 6, 2018
We present a method for estimating surface height directly from a single polarisation image simpl... more 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 estim...
In this paper we show that the introduction of an attenuation factor in the brightness equations ... more 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 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.
Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference
Thermal insulation represents one of the major challenges for energy efficiency. Problems related... more Thermal insulation represents one of the major challenges for energy efficiency. Problems related to insulation are well-known and widely studied in mathematical physics. Neverthless, mathematics involved is still very tricky especially when one looks at shape optimization issues [1, 2], and sometimes the answers are not so intuitive [3].In this talk we will consider two domains: an internal (fixed) ball of radius r and an external domain whose geometry varies. Physically, we are considering a domain of given temperature, thermally insulated by surrounding it with a constant thickness of thermal insulator. Our question is related to the best (or worst) shape for the external domain, in terms of heat dispersion (of course, under prescribed geometrical constraints). Mathematically, our problem is composed by an elliptic PDE with Robin-Dirichlet boundary conditions. This work is still in progress and we want to share someremarks and open questions, in addition to the results obtained ...
In this paper we propose a high-order accurate scheme for image segmentation based on the level-s... more 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 1D. 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 to detect the regularity of the approximate solution adapting the scheme in an automatic way. Some numerical tests on synthetic and r...
We consider a class of "filtered" schemes for first order time dependent Hamilton-Jacob... more We consider a class of "filtered" schemes for first order time dependent Hamilton-Jacobi equations and prove a general convergence result for this class of schemes. A typical filtered scheme is obtained mixing a high-order scheme and a monotone scheme according to a filter function F which decides where the scheme has to switch from one scheme to the other. A crucial role for this switch is played by a parameter ε=ε(Δ t,Δ x)>0 which goes to 0 as the time and space steps (Δ t,Δ x) are going to 0 and does not depend on the time t_n, for each iteration n. The tuning of this parameter in the code is rather delicate and has an influence on the global accuracy of the filtered scheme. Here we introduce an adaptive and automatic choice of ε=ε ^n (Δ t, Δ x) at every iteration modifying the classical set up. The adaptivity is controlled by a smoothness indicator which selects the regions where we modify the regularity threshold ε^n. A convergence result and some error estimates f...
Photometric 3D-reconstruction techniques aim at inferring the geometry of a scene from one or sev... more Photometric 3D-reconstruction techniques aim at inferring the geometry of a scene from one or several images, by inverting a physical model describing the image formation. This chapter presents an introductory overview of the main pho-tometric 3D-reconstruction techniques which are shape-from-shading, photometric stereo and shape-from-polarisation.
Lecture Notes in Computational Science and Engineering, 2019
In this paper we consider a class of “filtered” schemes for some first order time dependent Hamil... more In this paper we consider a class of “filtered” schemes for some first order time dependent Hamilton-Jacobi equations. A typical feature of a filtered scheme is that at the node xj the scheme is obtained as a mixture of a high-order scheme and a monotone scheme according to a filter function F. The mixture is usually governed by F and by a fixed parameter e = e(Δt, Δx) > 0 which goes to 0 as (Δt, Δx) is going to 0 and does not depend on n. Here we improve the standard filtered scheme introducing an adaptive and automatic choice of the parameter e = en(Δt, Δx) at every iteration. To this end, we use a smoothness indicator in order to select the regions where we can compute the regularity threshold en. The numerical tests presented confirms the effectiveness of the adaptive scheme.
A Comparison of Non-Lambertian Models for the Shape-from-Shading Problem
In this paper we present in a unified approach Shape-from-Shading models under orthographic proje... more In this paper we present in a unified approach Shape-from-Shading models under orthographic projection for non-Lambertian surfaces and compare them with the classical Lambertian model. Those non-Lambertian models have been proposed in the literature by various authors in order to take into account more realistic surfaces such as rough and specular surfaces. The advantage of our unified mathematical model is the possibility to easily modify a single differential model to various situations just changing some control parameters. Moreover, the numerical approximation we propose is valid for that general model and can be easily adapted to the real situation. Finally, we compare the models on some benchmarks including real and synthetic images.
IEEE Transactions on Pattern Analysis and Machine Intelligence
In this paper we present methods for estimating shape from polarisation and shading information, ... more 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.
We consider a class of "filtered" schemes for first order time dependent Hamilton-Jacobi equation... more We consider a class of "filtered" schemes for first order time dependent Hamilton-Jacobi equations and prove a general convergence result for this class of schemes. A typical filtered scheme is obtained mixing a high-order scheme and a monotone scheme according to a filter function F which decides where the scheme has to switch from one scheme to the other. A crucial role for this switch is played by a parameter ε = ε(∆t, ∆x) > 0 which goes to 0 as the time and space steps (∆t, ∆x) are going to 0 and does not depend on the time t n , for each iteration n. The tuning of this parameter in the code is rather delicate and has an influence on the global accuracy of the filtered scheme. Here we introduce an adaptive and automatic choice of ε = ε n (∆t, ∆x) at every iteration modifying the classical set up. The adaptivity is controlled by a smoothness indicator which selects the regions where we modify the regularity threshold ε n. A convergence result and some error estimates for the new adaptive filtered scheme are proved, this analysis relies on the properties of the scheme and of the smoothness indicators. Finally, we present some numerical tests to compare the adaptive filtered scheme with other methods.
Astronomical images are of crucial importance for astronomers since they contain a lot of informa... more Astronomical images are of crucial importance for astronomers since they contain a lot of information about celestial bodies that can not be directly accessible. Most of the information available for the analysis of these objects starts with sky explorations via telescopes and satellites. Unfortunately, the quality of astronomical images is usually very low with respect to other real images and this is due to technical and physical features related to their acquisition process. This increases the percentage of noise and makes more difficult to use directly standard segmentation methods on the original image. In this work we will describe how to process astronomical images in two steps: in the first step we improve the image quality by a rescaling of light intensity whereas in the second step we apply level-set methods to identify the objects. Several experiments will show the effectiveness of this procedure and the results obtained via various discretization techniques for level-set equations.
2017 IEEE International Conference on Computer Vision (ICCV), Oct 1, 2017
In this paper we present a differential approach to photopolarimetric shape estimation. We propos... more In this paper we present a differential approach to photopolarimetric 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.
We present a method for estimating surface height directly from a single polarisation image simpl... more 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 depth. The ambiguity in the surface normal azimuth angle is resolved globally when the optimal surface height is reconstructed. Our method is applicable to objects with uniform albedo exhibiting diffuse and specular reflectance. We extend it to an uncalibrated scenario by demonstrating that the illumination (point source or first/second order spherical harmonics) can be estimated from the polarisation image, up to a binary convex/concave ambiguity. We believe that our method is the first monocular, passive shape-from-x technique that enables well-posed depth estimation with only a single, uncalibrated illumination condition. We present results on glossy objects, including in uncontrolled, outdoor illumination.
In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in ... more In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in the spirit of the Hughes dynamic continuum model. The pedestrians, represented by a density function, move on the graph choosing a route to minimize the instantaneous travel cost to the destination. The density is governed by a conservation law while the minimization principle is described by a graph eikonal equation. We show that the model is well posed and we implement some numerical examples to demonstrate the validity of the proposed model.
Recovering the 3D shape of an object from shading is a challenging problem due to the complexity ... more Recovering the 3D shape of an object from shading is a challenging problem due to the complexity of modeling light propagation and surface reflections. Photometric Stereo (PS) is broadly considered a suitable approach for high-resolution shape recovery, but its functionality is restricted to a limited set of object surfaces and controlled lighting setup. In particular, PS models generally consider reflection from objects as purely diffuse, with specularities being regarded as a nuisance that breaks down shape reconstruction. This is a serious drawback for implementing PS approaches since most common materials have prominent specular components. In this paper, we propose a PS model that solves the problem for both diffuse and specular components aimed at shape recovery of generic objects with the approach being independent of the albedo values
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Papers by Silvia Tozza