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

Automatic construction of robust spherical harmonic subspaces

In this paper we propose a method to automatically recover a class specific low dimensional spherical harmonic basis from a set of in-the-wild facial images. We combine existing techniques for uncalibrated photometric stereo and low rank matrix decompositions in order to robustly recover a combined model of shape and identity. We build this basis without aid from a 3D model and show how it can be combined with recent efficient sparse facial feature localisation techniques to recover dense 3D facial shape. Unlike previous works in the area, our method is very efficient and is an order of magnitude faster to train, taking only a few minutes to build a model with over 2000 images. Furthermore, it can be used for real-time recovery of facial shape

Automatic construction of robust spherical harmonic subspaces

In this paper we propose a method to automatically recover a class specific low dimensional spherical harmonic basis from a set of in-the-wild facial images. We combine existing techniques for uncalibrated photometric stereo and low rank matrix decompositions in order to robustly recover a combined model of shape and identity. We build this basis without aid from a 3D model and show how it can be combined with recent efficient sparse facial feature localisation techniques to recover dense 3D facial shape. Unlike previous works in the area, our method is very efficient and is an order of magnitude faster to train, taking only a few minutes to build a model with over 2000 images. Furthermore, it can be used for real-time recovery of facial shape

Phenomenological modeling of image irradiance for non-Lambertian surfaces under natural illumination.

Various vision tasks are usually confronted by appearance variations due to changes of illumination. For instance, in a recognition system, it has been shown that the variability in human face appearance is owed to changes to lighting conditions rather than person\u27s identity. Theoretically, due to the arbitrariness of the lighting function, the space of all possible images of a fixed-pose object under all possible illumination conditions is infinite dimensional. Nonetheless, it has been proven that the set of images of a convex Lambertian surface under distant illumination lies near a low dimensional linear subspace. This result was also extended to include non-Lambertian objects with non-convex geometry. As such, vision applications, concerned with the recovery of illumination, reflectance or surface geometry from images, would benefit from a low-dimensional generative model which captures appearance variations w.r.t. illumination conditions and surface reflectance properties. This enables the formulation of such inverse problems as parameter estimation. Typically, subspace construction boils to performing a dimensionality reduction scheme, e.g. Principal Component Analysis (PCA), on a large set of (real/synthesized) images of object(s) of interest with fixed pose but different illumination conditions. However, this approach has two major problems. First, the acquired/rendered image ensemble should be statistically significant vis-a-vis capturing the full behavior of the sources of variations that is of interest, in particular illumination and reflectance. Second, the curse of dimensionality hinders numerical methods such as Singular Value Decomposition (SVD) which becomes intractable especially with large number of large-sized realizations in the image ensemble. One way to bypass the need of large image ensemble is to construct appearance subspaces using phenomenological models which capture appearance variations through mathematical abstraction of the reflection process. In particular, the harmonic expansion of the image irradiance equation can be used to derive an analytic subspace to represent images under fixed pose but different illumination conditions where the image irradiance equation has been formulated in a convolution framework. Due to their low-frequency nature, irradiance signals can be represented using low-order basis functions, where Spherical Harmonics (SH) has been extensively adopted. Typically, an ideal solution to the image irradiance (appearance) modeling problem should be able to incorporate complex illumination, cast shadows as well as realistic surface reflectance properties, while moving away from the simplifying assumptions of Lambertian reflectance and single-source distant illumination. By handling arbitrary complex illumination and non-Lambertian reflectance, the appearance model proposed in this dissertation moves the state of the art closer to the ideal solution. This work primarily addresses the geometrical compliance of the hemispherical basis for representing surface reflectance while presenting a compact, yet accurate representation for arbitrary materials. To maintain the plausibility of the resulting appearance, the proposed basis is constructed in a manner that satisfies the Helmholtz reciprocity property while avoiding high computational complexity. It is believed that having the illumination and surface reflectance represented in the spherical and hemispherical domains respectively, while complying with the physical properties of the surface reflectance would provide better approximation accuracy of image irradiance when compared to the representation in the spherical domain. Discounting subsurface scattering and surface emittance, this work proposes a surface reflectance basis, based on hemispherical harmonics (HSH), defined on the Cartesian product of the incoming and outgoing local hemispheres (i.e. w.r.t. surface points). This basis obeys physical properties of surface reflectance involving reciprocity and energy conservation. The basis functions are validated using analytical reflectance models as well as scattered reflectance measurements which might violate the Helmholtz reciprocity property (this can be filtered out through the process of projecting them on the subspace spanned by the proposed basis, where the reciprocity property is preserved in the least-squares sense). The image formation process of isotropic surfaces under arbitrary distant illumination is also formulated in the frequency space where the orthogonality relation between illumination and reflectance bases is encoded in what is termed as irradiance harmonics. Such harmonics decouple the effect of illumination and reflectance from the underlying pose and geometry. Further, a bilinear approach to analytically construct irradiance subspace is proposed in order to tackle the inherent problem of small-sample-size and curse of dimensionality. The process of finding the analytic subspace is posed as establishing a relation between its principal components and that of the irradiance harmonics basis functions. It is also shown how to incorporate prior information about natural illumination and real-world surface reflectance characteristics in order to capture the full behavior of complex illumination and non-Lambertian reflectance. The use of the presented theoretical framework to develop practical algorithms for shape recovery is further presented where the hitherto assumed Lambertian assumption is relaxed. With a single image of unknown general illumination, the underlying geometrical structure can be recovered while accounting explicitly for object reflectance characteristics (e.g. human skin types for facial images and teeth reflectance for human jaw reconstruction) as well as complex illumination conditions. Experiments on synthetic and real images illustrate the robustness of the proposed appearance model vis-a-vis illumination variation. Keywords: computer vision, computer graphics, shading, illumination modeling, reflectance representation, image irradiance, frequency space representations, {hemi)spherical harmonics, analytic bilinear PCA, model-based bilinear PCA, 3D shape reconstruction, statistical shape from shading

3D facial shape estimation from a single image under arbitrary pose and illumination.

Humans have the uncanny ability to perceive the world in three dimensions (3D), otherwise known as depth perception. The amazing thing about this ability to determine distances is that it depends only on a simple two-dimensional (2D) image in the retina. It is an interesting problem to explain and mimic this phenomenon of getting a three-dimensional perception of a scene from a flat 2D image of the retina. The main objective of this dissertation is the computational aspect of this human ability to reconstruct the world in 3D using only 2D images from the retina. Specifically, the goal of this work is to recover 3D facial shape information from a single image of unknown pose and illumination. Prior shape and texture models from real data, which are metric in nature, are incorporated into the 3D shape recovery framework. The output recovered shape, likewise, is metric, unlike previous shape-from-shading (SFS) approaches that only provide relative shape. This work starts first with the simpler case of general illumination and fixed frontal pose. Three optimization approaches were developed to solve this 3D shape recovery problem, starting from a brute-force iterative approach to a computationally efficient regression method (Method II-PCR), where the classical shape-from-shading equation is cast as a regression framework. Results show that the output of the regression-like approach is faster in timing and similar in error metrics when compared to its iterative counterpart. The best of the three algorithms above, Method II-PCR, is compared to its two predecessors, namely: (a) Castelan et al. [1] and (b) Ahmed et al. [2]. Experimental results show that the proposed method (Method II-PCR) is superior in all aspects compared to the previous state-of-the-art. Robust statistics was also incorporated into the shape recovery framework to deal with noise and occlusion. Using multiple-view geometry concepts [3], the fixed frontal pose was relaxed to arbitrary pose. The best of the three algorithms above, Method II-PCR, once again is used as the primary 3D shape recovery method. Results show that the pose-invariant 3D shape recovery version (for input with pose) has similar error values compared to the frontal-pose version (for input with frontal pose), for input images of the same subject. Sensitivity experiments indicate that the proposed method is, indeed, invariant to pose, at least for the pan angle range of (-50° to 50°). The next major part of this work is the development of 3D facial shape recovery methods, given only the input 2D shape information, instead of both texture and 2D shape. The simpler case of output 3D sparse shapes was dealt with, initially. The proposed method, which also use a regression-based optimization approach, was compared with state-of-the art algorithms, showing decent performance. There were five conclusions that drawn from the sparse experiments, namely, the proposed approach: (a) is competitive due to its linear and non-iterative nature, (b) does not need explicit training, as opposed to [4], (c) has comparable results to [4], at a shorter computational time, (d) better in all aspects than Zhang and Samaras [5], and (e) has the limitation, together with [4] and [5], in terms of the need to manually annotate the input 2D feature points. The proposed method was then extended to output 3D dense shapes simply by replacing the sparse model with its dense equivalent, in the regression framework inside the 3D face recovery approach. The numerical values of the mean height and surface orientation error indicate that even if shading information is unavailable, a decent 3D dense reconstruction is still possible

04131 Abstracts Collection -- Geometric Properties from Incomplete Data

From 21.03.04 to 26.03.04, the Dagstuhl Seminar 04131 ``Geometric Properties from Incomplete Data\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available