Shape-from-shading using the heat equation

This paper offers two new directions to shape-from-shading, namely the use of the heat equation to smooth the field of surface normals and the recovery of surface height using a low-dimensional embedding. Turning our attention to the first of these contributions, we pose the problem of surface normal recovery as that of solving the steady state heat equation subject to the hard constraint that Lambert's law is satisfied. We perform our analysis on a plane perpendicular to the light source direction, where the z component of the surface normal is equal to the normalized image brightness. The x - y or azimuthal component of the surface normal is found by computing the gradient of a scalar field that evolves with time subject to the heat equation. We solve the heat equation for the scalar potential and, hence, recover the azimuthal component of the surface normal from the average image brightness, making use of a simple finite difference method. The second contribution is to pose the problem of recovering the surface height function as that of embedding the field of surface normals on a manifold so as to preserve the pattern of surface height differences and the lattice footprint of the surface normals. We experiment with the resulting method on a variety of real-world image data, where it produces qualitatively good reconstructed surfaces

Recovering facial shape using a statistical model of surface normal direction

In this paper, we show how a statistical model of facial shape can be embedded within a shape-from-shading algorithm. We describe how facial shape can be captured using a statistical model of variations in surface normal direction. To construct this model, we make use of the azimuthal equidistant projection to map the distribution of surface normals from the polar representation on a unit sphere to Cartesian points on a local tangent plane. The distribution of surface normal directions is captured using the covariance matrix for the projected point positions. The eigenvectors of the covariance matrix define the modes of shape-variation in the fields of transformed surface normals. We show how this model can be trained using surface normal data acquired from range images and how to fit the model to intensity images of faces using constraints on the surface normal direction provided by Lambert's law. We demonstrate that the combination of a global statistical constraint and local irradiance constraint yields an efficient and accurate approach to facial shape recovery and is capable of recovering fine local surface details. We assess the accuracy of the technique on a variety of images with ground truth and real-world images

Image Degradation Due To Surface Scattering In The Presence Of Aberrations

This dissertation focuses on the scattering phenomena by well-polished optical mirror surfaces. Specifically, predicting image degradation by surface scatter from rough mirror surfaces for a two-mirror telescope operating at extremely short wavelengths (9nm~30nm) is performed. To evaluate image quality, surface scatter is predicted from the surface metrology data and the point spread function in the presence of both surface scatter and aberrations is calculated. For predicting the scattering intensity distribution, both numerical and analytic methods are considered. Among the numerous analytic methods, the small perturbation method (classical Rayleigh-Rice surface scatter theory), the Kirchhoff approximation method (classical BeckmanKirchhoff surface scatter theory), and the generalized Harvey-Shack surface scatter theory are adopted. As a numerical method, the integral equation method (method of moments) known as a rigorous solution is discussed. Since the numerical method is computationally too intensive to obtain the scattering prediction directly for the two mirror telescope, it is used for validating the three analytic approximate methods in special cases. In our numerical comparison work, among the three approximate methods, the generalized Harvey-Shack model shows excellent agreement to the rigorous solution and it is used to predict surface scattering from the mirror surfaces. Regarding image degradation due to surface scatter in the presence of aberrations, it is shown that the composite point spread function is obtained in explicit form in terms of convolutions of the geometrical point spread function and scaled bidirectional scattering distribution functions of the individual surfaces of the imaging system. The approximations and assumptions in this iv formulation are discussed. The result is compared to the irradiance distribution obtained using commercial non-sequential ray tracing software for the case of a two-mirror telescope operating at the extreme ultra-violet wavelengths and the two results are virtually identical. Finally, the image degradation due to the surface scatter from the mirror surfaces and the aberration of the telescope is evaluated in terms of the fractional ensquared energy (for different wavelengths and field angles) which is commonly used as an image quality requirement on many NASA astronomy programs

Estimating the surface radiance function from single images

This paper describes a simple method for estimating the surface radiance function from single images of smooth surfaces made of materials whose reflectance function is isotropic and monotonic. The method makes implicit use of the Gauss map between the surface and a unit sphere. We assume that the material brightness is monotonic with respect to the angle between the illuminant direction and the surface normal. Under conditions in which the light source and the viewer directions are identical, we show how a tabular representation of the surface radiance function can be estimated using the cumulative distribution of image gradients. Using this tabular representation of the radiance function, surfaces may be rendered under varying light source direction by rotating the corresponding reflectance map on the Gauss sphere about the specular spike direction. We present a sensitivity study on synthetic and real-world imagery. We also present two applications which make use of the estimated radiance function. The first of these illustrates how the radiance function estimates can be used to render objects when the light and viewer directions are no longer coincident. The second application involves applying corrected Lambertian radiance to rough and shiny surfaces

Estimating the surface radiance function from single images

This paper describes a simple method for estimating the surface radiance function from single images of smooth surfaces made of materials whose reflectance function is isotropic and monotonic. The method makes implicit use of the Gauss map between the surface and a unit sphere. We assume that the material brightness is monotonic with respect to the angle between the illuminant direction and the surface normal. Under conditions in which the light source and the viewer directions are identical, we show how a tabular representation of the surface radiance function can be estimated using the cumulative distribution of image gradients. Using this tabular representation of the radiance function, surfaces may be rendered under varying light source direction by rotating the corresponding reflectance map on the Gauss sphere about the specular spike direction. We present a sensitivity study on synthetic and real-world imagery. We also present two applications which make use of the estimated radiance function. The first of these illustrates how the radiance function estimates can be used to render objects when the light and viewer directions are no longer coincident. The second application involves applying corrected Lambertian radiance to rough and shiny surfaces.