Application of Computer Algebra to Photometric Stereo with Two Light Sources (original) (raw)
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Programming and Computer Software, 2017
The problem of reconstructing a Lambertian surface from its two photometric stereo images is discussed. Previously, the solution to this problem was only obtained for a special choice of two light source directions. In this paper, using the computer algebra system Mathematica, the necessary and sufficient conditions for the unique reconstruction of the surface from its two images is analyzed in a more general setting. Photometric images of various surfaces are simulated, and the validity of the theoretical results is demonstrated.
Application of Computer Algebra to the Reconstruction of Surface from Its Photometric Images
Programming and Computer Software, 2018
This paper addresses the problem of reconstructing the shape of an unknown Lambertian surface given in the 3D space by a continuously differentiable function. The surface is reconstructed from its photometric images obtained by its successive illumination with three different remote light sources. Using computer algebra methods, we show that the unique solution of the problem, which exists in the domain of all three images, can be continued beyond this domain based on the solutions obtained for any pair of the three images. To disambiguate the reconstruction of the surface from its two images, we compute the corresponding value of the parameter at the boundary of the domain. Soundness of the theoretical results is confirmed by simulating photometric images of various surfaces.
Second-Order Algebraic Surfaces and Two Image Photometric Stereo
2018
This paper discusses the special case of reconstructing the unknown Lambertian surface from two-image photometric stereo. Both images are assumed here to be formed by a genuine second-order algebraic surface. The corresponding uniqueness issue is discussed for different pairs of image irradiance equations under various illumination settings. Illustrative examples supplement presented analysis and demonstrate its main gist.
Orthogonal Illuminations in Two Light-Source Photometric Stereo
Lecture Notes in Computer Science, 2016
In this paper we investigate the case of ambiguous shape reconstruction from two light-source photometric stereo based on illuminating the unknown Lambertian surface. So-far this problem is merely well-understood for two linearly independent light-source directions with one illumination assumed as overhead. As already established, a necessary and sufficient condition to disambiguate the entire shape reconstruction process is controlled by the satisfaction of the corresponding secondorder linear PDE with constant coefficients in two independent variables. This work extends the latter to an arbitrary pair of light-source directions transforming the above constraint into a special nonlinear PDE. In addition, a similar ambiguity analysis is also performed for a special configuration of two light-source directions assumed this time as orthogonal and contained in the vertical plane. Finally, this work is supplemented by illustrative examples exploiting symbolic computation used within a framework of continuous reflectance map model (i.e. an image irradiance equation) and applied to a genuine Lambertian surfaces.
Photometric stereo: Lambertian reflectance and light sources with unknown direction and strength
This paper reconsiders the familiar case of photometric stereo under the assumption of Lambertian surface re ectance and three distant point sources of illumination. Here, it is assumed that the directions to and the relative strengths of the three light sources are not known a priori. Rather, estimation of these parameters becomes part of the problem formulation. Each light source is represented by a 3-D vector that points in the direction of the light source and has magnitude proportional to the strength of the light source. Thus, nine parameters are required to characterize the three light sources. It is shown that, regardless of object shape, triples of measured intensity values are constrained to lie on a quadratic surface having six degrees of freedom. Estimation of the six parameters of the quadratic surface allows the determination of the nine parameters of the light sources up to an unknown rotation. This is su cient to determine object shape, although attitude with respect to the world-based or the camera-based coordinate system can not be simultaneously recovered without additional information.
An Application of the Photometric Stereo Method
1979
7' The orientation of patches on the surface of an object can be determined from multiple images taken with different illumination, but from the Sante viewing position. This method. refer red to as photometric stereo, can be implemented using table lookup based on nunwrica l inversion of experimentally determined reflectance maps. Here we concentrate on
Photometric stereo through an adapted alternation approach
2008 15th IEEE International Conference on Image Processing, 2008
Photometric stereo aims at finding the surface normal and reflectance at every point of an object from a set of images obtained under different lighting conditions. The obtained intensity image data are stacked into a matrix that can be approximated by a low-dimensional linear subspace, under the Lambertian model. The current paper proposes to use an adaptation of the Alternation technique to tackle this problem when the images contain missing data, which correspond to pixels in shadow and saturated regions. Experimental results considering both synthetic and real images show the good performance of the proposed Alternation-based strategy.
Photometric Stereo Using Constrained Bivariate Regression for General Isotropic Surfaces
2014 IEEE Conference on Computer Vision and Pattern Recognition, 2014
This paper presents a photometric stereo method that is purely pixelwise and handles general isotropic surfaces in a stable manner. Following the recently proposed sumof-lobes representation of the isotropic reflectance function, we constructed a constrained bivariate regression problem where the regression function is approximated by smooth, bivariate Bernstein polynomials. The unknown normal vector was separated from the unknown reflectance function by considering the inverse representation of the image formation process, and then we could accurately compute the unknown surface normals by solving a simple and efficient quadratic programming problem. Extensive evaluations that showed the state-of-the-art performance using both synthetic and real-world images were performed.
Photometric Stereo with General, Unknown Lighting
International Journal of Computer Vision, 2006
Work on photometric stereo has shown how to recover the shape and reflectance properties of an object using multiple images taken with a fixed viewpoint and variable lighting conditions. This work has primarily relied on the presence of a single point source of light in each image. In this paper we show how to perform photometric stereo assuming that all lights in a scene are isotropic and distant from the object but otherwise unconstrained. Lighting in each image may be an unknown and arbitrary combination of diffuse, point and extended sources. Our work is based on recent results showing that for Lambertian objects, general lighting conditions can be represented using low order spherical harmonics. Using this representation we can recover shape by performing a simple optimization in a low-dimensional space. We also analyze the shape ambiguities that arise in such a representation.
Unambigous Determination of Shape from Photometric Stereo with Unknown Light Sources
2001
Photometric stereo with uncalibrated lights determines surface orientations ambiguously up to any regular transformation. If the surface reflectance model is separable with respect to the illumination and viewing directions, its inherent symmetries enable to design two previously unrecognized constraints on normals that reduce this ambiguity. The two constraints represent projections of normals onto planes perpendicular to the viewing and illumination directions, respectively. We identify the classes of transformations that leave each constraint invariant. We construct the constraints using polarization measurement under the assumption of separable reflectance model for smooth dielectrics. We verify that applying the first constraint together with the integrability constraint results in bas-relief ambiguity, while application of the second constraint on integrable normals reduces the ambiguity to convex/concave ambiguity. Importantly, the latter result is also obtained when the first and second constraints alone are combined.