Estimation of Facial Angular Information Using a Complex-Number-Based Statistical Model

This paper describes a graph-spectral method for shape-from-shading. The algorithm takes as its input a field of initial surface normal estimates computed using the Lambertian irradiance cone and the Canny edge gradient. We illustrate how to refine this field of surface normals using a graph-spectral method. From the initial field of surface normals we make curvature estimates. The curvature estimates are in turn used to compute a transition probability matrix. From the theory of random walks on graphs, the leading eigenvector of this matrix is a curvature minimising path through the field of surface normals. We impose curvature consistency on the initially noisy field of surface normals by rotating them about their irradiance cones so that they follow the path defined by the leading eigenvector. This method is applied to a variety of real-world images and is shown to lead to improved surface reconstruction.

We make two contributions to the problem of shape-from-shading. First, we develop a new method for surface normal recovery. We pose the problem as that of solving the steady state heat equation subject to the hard constraint that Lambert's law is satisfied. According to this picture, the surface normals are found by taking the gradient of a scalar field. The heat equation for the scalar field can be solved using simple finite difference methods and leads to an iterative procedure for surface normal estimation. The second contribution is to show how surface height recovery from the field of surface normals can be posed as one of low dimensional embedding. We experiment with the resulting method on a variety of real-world image data, where it produces qualitatively good reconstructed surfaces.

Recovering high quality surfaces from noisy triangulated surfaces is a fundamental important problem in geometry processing. Sharp features including edges and corners can not be well preserved in most existing denoising methods except the recent total variation (TV) and $\ell_0$ regularization methods. However, these two methods have suffered producing staircase artifacts in smooth regions. In this paper, we first introduce a second order regularization method for restoring a surface normal vector field, and then propose a new vertex updating scheme to recover the desired surface according to the restored surface normal field. The proposed model can preserve sharp features and simultaneously suppress the staircase effects in smooth regions which overcomes the drawback of the first order models. In addition, the new vertex updating scheme can prevent ambiguities introduced in existing vertex updating methods. Numerically, the proposed high order model is solved by the augmented Lagrangian method with a dynamic weighting strategy. Intensive numerical experiments on a variety of surfaces demonstrate the superiority of our method by visually and quantitatively.

We study Isometric Non-Rigid Shape-from-Motion (Iso-NRSfM): given multiple intrinsically calibrated monocular images, we want to reconstruct the time-varying 3D shape of an object undergoing isometric deformations. We show that Iso-NRSfM is solvable from the warps (the inter-image geometric transformations). We propose a new theoretical framework based on Riemmanian manifolds to represent the unknown 3D surfaces, as embeddings of the camera's retinal planes. This allows us to use the manifolds' metric tensor and Christoffel Symbol fields, which we prove are related across images by simple rules depending only on the warps. This forms a set of important theoretical results. Using the infinitesimal planarity formulation, it then allows us to derive a system of two quartics in two variables for each image pair. The sum-of-squares of these polynomials is independent of the number of images and can be solved globally, forming a well-posed problem for N ≥ 3 images, whose solution directly leads to the surface's normal field. The proposed method outperforms existing work in terms of accuracy and computation cost on synthetic and real datasets.

The aim in this paper is to show how to discriminate gender using a parameterized representation of fields of facial surface normals (needle-maps) which can be extracted from 2D intensity images using shape-from-shading (SFS). We makes use of principle geodesic analysis (PGA) to parameterize the facial needle-maps. Using feature selection, we determine which of the components of the resulting parameter vector are the most significant in distinguishing gender. Using the EM algorithm we distinguish gender by fitting a two component mixture model to the vectors of selected features. Results on real-world data reveal that the method gives gender discrimination results that are comparable to human observers.KeywordsFeature SelectionMixture ModelGaussian Mixture ModelGender DiscriminationFeature ComponentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper presents a novel method for 3D surface reconstruction that uses polarization and shading information from two views. The method relies on polarization data acquired using a standard digital camera and a linear polarizer. Fresnel theory is used to process the raw images and to obtain initial estimates of surface normals, assuming that the reflection type is diffuse. Based on this idea, the paper presents two novel contributions to the problem of surface reconstruction. The first is a technique to enhance the surface normal estimates by incorporating shading information into the method. This is done using robust statistics to estimate how the measured pixel brightnesses depend on the surface orientation. This gives an estimate of the object material reflectance function, which is used to refine the estimates of the surface normals. The second contribution is to use the refined estimates to establish correspondence between two views of an object. To do this, a set of patches are extracted from each view and are aligned by minimizing an energy functional based on the surface normal estimates and local topographic properties. The optimum alignment parameters for different patch pairs are then used to establish stereo correspondence. This process results in an unambiguous field of surface normals, which can be integrated to recover the surface depth. Our technique is most suited to smooth, non-metallic surfaces. It complements existing stereo algorithms since it does not require salient surface features to obtain correspondences. An extensive set of experiments, yielding reconstructed objects and reflectance functions, are presented and compared to ground truth.

This paper deals with the detection of small magnetization using a thin film magneto-impedance sensor with subjecting to strong normal field. The sensor was made by soft magnetic amorphous thin-film with uniaxial magnetic anisotropy in the width direction of the element. It was reported that the sensor has very high sensitivity, such as pico-tesla order, when it is driven by hundreds of MHz. In this paper, a sensitive measurement method aiming for detection of a small particle or a cluster of nano-particles, having low-remanence, is proposed. The point is the application of strong normal field in the measurement area including sensor element and particle. The normal strong field is applied in the normal direction of the sensor plane in the value almost hundreds of mT. Instead of such strong normal field, the sensor keeps high sensitivity, because of the demagnetizing force in the thickness direction. A theoretical estimation for clarifying an efficiency of the method, experimental results of sensor property and sensitivity with subjecting to the normal field, and also a confirmation of detection of a small particle using the proposed method is reported. As a special mention, detection fundamentals when a applied surface normal field has a distribution and also a particle would run through in the vicinity of sensor is discussed.

Specular flow is the motion field induced on the image plane by the movement of points reflected by a curved, mirror-like surface. This flow provides information about surface shape, and when the camera and surface move as a fixed pair, shape can be recovered by solving linear differential equations along integral curves of flow. Previous analysis has shown that two distinct motions (i.e., two flow fields) are generally sufficient to guarantee a unique solution without externally-provided initial conditions. In this work, we show that we can often succeed with only one flow. The key idea is to exploit the fact that smooth surfaces induce integrability constraints on the surface normal field. We show that this induces a new differential equation that facilitates the propagation of shape information between integral curves of flow, and that combining this equation with known methods often permits the recovery of unique shape from a single specular flow given only a single seed point.

We introduce a new approach for recovering shape-from-shading (SFS) from synthetic aperture radar (SAR) images of the terrain. Three contributions are proposed: 1) we show how the direction of surface normals is constrained by the geometry of the radar reflectivity cone; 2) we show how topographic features can be used as boundary constraints on the recovered surface normals; and 3) the resulting field of surface normals is smoothed using robust statistics.

Knowledge of the structural impedance is significantly important in the computation of acoustic scattering and radiation because it contains all the information about the structure which is necessary for the evaluation of the scattered or radiated field in any fluid medium. The inverse problem of determining, or reconstructing, the structural impedance from scattering measurements is treated. An algorithm is developed for the reconstruction of the structural impedance from knowledge of scattered field for a set of incident plane wave directions. An intermediate step includes the reconstruction of the surface pressure and normal velocity fields using acoustical holography. The resolution limit of the reconstruction is found in the case of the infinite cylinder and the sphere. The case of the sphere is generalizable to other finite shapes. Problems caused by noise in the inversion of the data are discussed. Alternative reconstructions of wet impedance and wet admittance which can be performed for any scattering data are defined.

The shape of the optimal rod determined in the work meets the condition of mass conservation in relation to the reference rod. At the same time, this rod shows a significant increase in resistance to axial force. In the examples presented, this increase was <i>80</i>% and <i>117</i>%, respectively, for rods with slenderness of 125 and 175. A practical benefit from the use of compression rods of the proposed shapes is clearly visible. The example presented in this publication shows how great the utility in the structural mechanics can be, resulting from the applications of complex analysis (complex numbers). This approach to many problems can find its solutions, while they are lacking in the real numbers domains. What is more, although these are operations on complex numbers, these solutions have often their real representations, as the numerical example shows. There are too few applications of complex numbers in the technique and science, therefore it is obvious that the use of complex analysis should have an increasing range. One of the first people to use complex numbers was Girolamo Cardano. Cardano, using complex numbers, was solving cubic equations, unsolvable to his times – as the famous Franciscan and professor of mathematics Luca Pacioli put it in his paper <i>Summa de arithmetica, geometria, proportioni et proportionalita</i> (1494). It is worth mentioning that history has given Cardano priority in the use of complex numbers, but most probably they were discovered by another professor of mathematics – Scipione del Ferro (cf. [1]). We can see, that already then, they were definitely important (complex numbers).

Wave-particle duality is one of the basic features of quantum mechanics, giving rise to the use of complex numbers in describing states of quantum systems and their dynamics and interaction. Since the inception of quantum theory, it has been debated whether complex numbers are essential or whether an alternative consistent formulation is possible using real numbers only. Here, we attack this long-standing problem theoretically and experimentally, using the powerful tools of quantum resource theories. We show that, under reasonable assumptions, quantum states are easier to create and manipulate if they only have real elements. This gives an operational meaning to the resource theory of imaginarity. We identify and answer several important questions, which include the state-conversion problem for all qubit states and all pure states of any dimension and the approximate imaginarity distillation for all quantum states. As an application, we show that imaginarity plays a crucial role in state discrimination, that is, there exist real quantum states that can be perfectly distinguished via local operations and classical communication but that cannot be distinguished with any nonzero probability if one of the parties has no access to imaginarity. We confirm this phenomenon experimentally with linear optics, discriminating different two-photon quantum states by local projective measurements. Our results prove that complex numbers are an indispensable part of quantum mechanics.

Inevitably, when the topic of rotation is discussed, the use of complex numbers comes up. Since rotational physics is central to magnetic resonance spectroscopy it is necessary to have a firm understanding of the fundamental properties of complex numbers. The historical origin of numbers in general and complex numbers in particular is briefly discussed and some of the more useful properties of these objects is presented. Much use is made of these properties in later chapters and the reader who is unfamiliar or “rusty” with complex numbers is urged to read this chapter.

The framework for playing quantum games in an Einstein-Podolsky-Rosen (EPR) type setting is investigated using the mathematical formalism of geometric algebra (GA). The main advantage of this framework is that the players' strategy sets remain identical to the ones in the classical mixed-strategy version of the game, and hence the quantum game becomes a proper extension of the classical game, avoiding a criticism of other quantum game frameworks. We produce a general solution for two-player games, and as examples, we analyze the games of Prisoners' Dilemma and Stag Hunt in the EPR setting. The use of GA allows a quantum-mechanical analysis without the use of complex numbers or the Dirac Bra-ket notation, and hence is more accessible to the non-physicist.

On the Use of Complex Numbers