Abstract

(ProQuest: ... denotes formulae omitted.)1 IntroductionPeople have always had the ability to recognize and distinguish two different faces, while computers have begun to show the same ability not very long ago. According to [1] by the mid-1960s, researchers began the work on the recognition of human faces using the computer.Lately robust facial recognition systems are becoming more useful, being helpful in various fields such as terrorism and crime fighting and different user authentication in real or virtual spaces for better security. But despite the amazing innovations and developments in which all take part, the extent of a person from a photograph by comparing it with other images previously saved in a database it is still problematic and a very complex topic. This is mainly due to the variety of human faces depending on the area and the different conditions in which they can be, such as camera performance, facial expressions, lighting, makeup, glasses and more.Most times, a very simple variation of these conditions affect the accuracy of the systems through which the facial recognition is performed, mainly when using highly populated databases. This is the main cause for facial recognition systems are not yet widely used.Facial recognition is usually used in security systems and can be compared to other biometric techniques such as fingerprint or iris recognition systems [2]. Also, facial and form recognition could help marketers in having a better impact on their clients because the visual aspect is one of the classic three channels: visual, auditory and kinaesthetic [3]. Some facial recognition algorithms identify faces by extracting landmarks or features of an image with the subject's face, which is nothing more than a set of pixels ordered by a particular pattern [4].Each human face has many distinctive landmarks, different deck peaks that make up facial features. Each face has approximately 80 nodal points of which the following are used for facial recognition [5]:* Distance between the eyes;* Width of the nose;* Depth of the eye sockets;* The shape of the cheekbones;* The length of the jaw line.Before the actual process of recognition is necessary to create image galleries. From the perspective of facial recognition, the gallery is a set of models serving as a reference for biometric matching process [6].The following are the stages of facial recognition [7]:1. The image capture is performed using a video or photo camera. The camera that is used to capture the image is more powerful so the recognition will be much more accurate.2. The face detection is usually a fairly complex process because an image has almost always a background or other faces. Thus, the system will try the standardization of the photo so it has the same characteristics as previously stored images in the gallery.3. The extraction of features and the generation of a model is a mathematical representation named biometric reference that will be the recognition substance.4. Comparing the models is the process by which the biometric reference is compared to the other models of familiar faces in the gallery.5. Declaring identity is establishing the kinship between two references, but which is often carried out by the human factor.2 Mathematical Theories and Techniques for Facial RecognitionTo achieve facial recognition there are some complex mathematical, geometry and photometry algorithms that support the processing and calculations that are made on a photo. First algorithms that were used to process facial recognition were based only on geometry, with which we only identified similarities between the main features as the position of the head, eyes, nose or mouth.A Cartesian plan is a plan made by perpendicular lines that are called lines. If we have two points with coordinates A (xA, yA) and B with coordinates (xB, yB) the calculation of the distance between these two points is. …

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