This paper presents a method to extend PDE surfaces to high dimensional spaces. We review a common existing analytic solution, and show how it can be used straightforwardly to increase the dimension of the space the surface is embedded within. We then further develop a numerical scheme suitable for increasing the number of variables that parametrise the surface, and investigate some of the properties of this solution with a view to future work.

The prevalence of micromoulded components has steadily increased over recent years. The production of such components is extremely sensitive to a number of variables that may potentially lead to significant changes in the surface geometry, often regarded as a crucial determinant of the product¿s functionality and quality. So far, traditional large-scale quality assessment techniques have been used in micromoulding. However, these techniques are not entirely suitable for small scales . Techniques such as Atomic Force Mi- croscopy (AFM) or White Light Interferometry (WLI) have been used for obtaining full three-dimensional profiles of micromoulded components, pro- ducing large data sets that are very difficult to manage. This work presents a method of characterizing surface features of micro and nano scale based on the use of the Biharmonic equation as means of describing surface profiles whilst guaranteeing tangential (C1) continuity. Thus, the problem of rep- resenting surface features of micromoulded components from massive point clouds is transformed into a boundary-value problem, reducing the amount of data required to describe any given surface feature.The boundary condi- tions needed for finding a particular solution to the Biharmonic equation are extracted from the data set and the coefficients associated with a suitable analytic solution are used to describe key design parameters or geometric properties of a surface feature. Moreover, the expressions found for describ- ing key design parameters in terms of the analytic solution to the Biharmonic equation may lead to a more suitable quality assessment technique for micromoulding than the criteria currently used. In summary this technique provides a means for compressing point clouds representing surface features whilst providing an analytic description of such features. The work is applicable to many other instances where surface topography is in need of efficient representation.

With its numerous applications, automatic facial emotion recognition has recently become a very active area of research. Yet there has been little detailed study of the dynamic components of facial expressions. This article reviews research that shows gender is encoded in the dynamics of a smile, and how it may be possible to use the dynamic components of facial expressions as a form of biometric.

In this paper, we study Bézier surfaces in View the MathML source three-dimensional Minkowski space. In particular, we focus on timelike and spacelike cases for Bézier surfaces. We also deal with the Plateau¿Bézier problem in View the MathML source, obtaining conditions over the control net to be extremal of the Dirichlet function for both timelike and spacelike Bézier surfaces. Moreover, we provide interesting examples showing the behavior of the Plateau¿Bézier problem in View the MathML source and illustrating the relationship between it and the corresponding Plateau¿Bézier problem in the Euclidean space R3.

An alternative technique for shape morphing using a surface generating method using partial differential equations is outlined throughout this work. The boundaryvalue nature that is inherent to this surface generation technique together with its mathematical properties are hereby exploited for creating intermediate shapes between an initial shape and a final one. Four alternative shape morphing techniques are proposed here. The first one is based on the use of a linear combination of the boundary conditions associated with the initial and final surfaces, the second one consists of varying the Fourier mode for which the PDE is solved whilst the third results from a combination of the first two. The fourth of these alternatives is based on the manipulation of the spine of the surfaces, which is computed as a by-product of the solution. Results of morphing sequences between two topologically nonequivalent surfaces are presented. Thus, it is shown that the PDE based approach for morphing is capable of obtaining smooth intermediate surfaces automatically in most of the methodologies presented in this work and the spine has been revealed as a powerful tool for morphing surfaces arising from the method proposed here.

One of the challenging problems in geometric modeling and computer graphics is the construction of realistic human facial geometry. Such geometry are essential for a wide range of applications, such as 3D face recognition, virtual reality applications, facial expression simulation and computer based plastic surgery application. This paper addresses a method for the construction of 3D geometry of human faces based on the use of Elliptic Partial Differential Equations (PDE). Here the geometry corresponding to a human face is treated as a set of surface patches, whereby each surface patch is represented using four boundary curves in the 3-space that formulate the appropriate boundary conditions for the chosen PDE. These boundary curves are extracted automatically using 3D data of human faces obtained using a 3D scanner. The solution of the PDE generates a continuous single surface patch describing the geometry of the original scanned data. In this study, through a number of experimental verifications we have shown the efficiency of the PDE based method for 3D facial surface reconstruction using scan data. In addition to this, we also show that our approach provides an efficient way of facial representation using a small set of parameters that could be utilized for efficient facial data storage and verification purposes.

Bezier curves and surfaces are two very useful tools in Geometric Modeling, with many applications. In this paper, we will offer a new method to provide approximations of regular curves and surfaces by Bezier ones, with the corresponding examples.

In this paper, a new approach to modelling 3D faces based on 2D images is introduced. Here 3D faces are created using two photographs from which we extract facial features based on image manipulation techniques. Through the image manipulation techniques we extract the crucial feature lines of the face in two views. These are then used in modifying a template base mesh which is created in 3D. This base mesh, which has been designed by keeping facial animation in mind, is then subdivided to provide the level of detail required. The methodology, as it stands, is semi-automatic whereby our goal is to automate this process in order to provide an inexpensive and expedient way of producing realistic face models intended for animation purposes. Thus, we show how image manipulation techniques can be used to create binary images which can in turn be used in manipulating a base mesh that can be adapted to a given facial geometry. In order to explain our approach more clearly we discuss a series of examples where we create 3D facial geometry of individuals given the corresponding image data.

Today, computer based face recognition is a mature and reliable mechanism which is being practically utilised for many access control scenarios. As such, face recognition or authentication is predominantly performed using ‘perfect’ data of full frontal facial images. Though that may be the case, in reality, there are numerous situations where full frontal faces may not be available — the imperfect face images that often come from CCTV cameras do demonstrate the case in point. Hence, the problem of computer based face recognition using partial facial data as probes is still largely an unexplored area of research. Given that humans and computers perform face recognition and authentication inherently differently, it must be interesting as well as intriguing to understand how a computer favours various parts of the face when presented to the challenges of face recognition. In this work, we explore the question that surrounds the idea of face recognition using partial facial data. We explore it by applying novel experiments to test the performance of machine learning using partial faces and other manipulations on face images such as rotation and zooming, which we use as training and recognition cues. In particular, we study the rate of recognition subject to the various parts of the face such as the eyes, mouth, nose and the cheek. We also study the effect of face recognition subject to facial rotation as well as the effect of recognition subject to zooming out of the facial images. Our experiments are based on using the state of the art convolutional neural network based architecture along with the pre-trained VGG-Face model through which we extract features for machine learning. We then use two classifiers namely the cosine similarity and the linear support vector machines to test the recognition rates. We ran our experiments on two publicly available datasets namely, the controlled Brazilian FEI and the uncontrolled LFW dataset. Our results show that individual parts of the face such as the eyes, nose and the cheeks have low recognition rates though the rate of recognition quickly goes up when individual parts of the face in combined form are presented as probes.

This paper presents a method for interactive design by means of extending the PDE based approach for surface generation. The governing partial differential equation is generalized to arbitrary order allowing complex shapes to be designed as single patch PDE surfaces. Using this technique a designer has the flexibility of creating and manipulating the geometry of shape that satisfying an arbitrary set of boundary conditions. Both the boundary conditions which are defined as curves in 3-space and the spine of the corresponding PDE are utilized as interactive design tools for creating and manipulating geometry intuitively. In order to facilitate interactive design in real time, a compact analytic solution for the chosen arbitrary order PDE is formulated. This solution scheme even in the case of general boundary conditions satisfies exactly the boundary conditions where the resulting surface has an closed form representation allowing real time shape manipulation. In order to enable users to appreciate the powerful shape design and manipulation capability of the method, we present a set of practical examples.