Fractal Techniques for Face Recognition

Fractals are popular because of their ability to create complex images using only several simple codes. This is possible by capturing image redundancy and presenting the image in compressed form using the self similarity feature. For many years fractals were used for image compression. In the last few years they have also been used for face recognition. In this research we present new fractal methods for recognition, especially human face recognition. This research introduces 3 new methods for using fractals for face recognition, the use of fractal codes directly as features, Fractal image-set coding and Subfractals. In the first part, the mathematical principle behind the application of fractal image codes for recognition is investigated. An image Xf can be represented as Xf = A × Xf + B which A and B are fractal parameters of image Xf . Different fractal codes can be presented for any arbitrary image. With the definition of a fractal transformation, T (X) = A(X − Xf ) + Xf , we can define the relationship between any image produced in the fractal decoding process starting with any arbitrary image X0 as Xn = Tn(X) = An(X − Xf ) + Xf . We show that some choices for A or B lead to faster convergence to the final image. Fractal image-set coding is based on the fact that a fractal code of an arbitrary gray-scale image can be divided in two parts – geometrical parameters and luminance parameters. Because the fractal codes for an image are not unique, we can change the set of fractal parameters without significant change in the quality of the reconstructed image. Fractal image-set coding keeps geometrical parameters ii the same for all images in the database. Differences between images are captured in the non-geometrical or luminance parameters which are faster to compute. For recognition purposes, the fractal code of a query image is applied to all the images in the training set for one iteration. The distance between an image and the result after one iteration is used to define a similarity measure between this image and the query image. The fractal code of an image is a set of contractive mappings each of which transfer a domain block to its corresponding range block. The distribution of selected domain blocks for range blocks in an image depends on the content of image and the fractal encoding algorithm used for coding. A small variation in a part of the input image may change the contents of the range and domain blocks in the fractal encoding process, resulting in a change in the transformation parameters in the same part or even other parts of the image. A subfractal is a set of fractal codes related to range blocks of a part of the image. These codes are calculated to be independent of other codes of the other parts of the same image. In this case the domain blocks nominated for each range block must be located in the same part of the image which the range blocks come from. The proposed fractal techniques were applied to face recognition using the MIT and XM2VTS face databases. Accuracies of 95% were obtained with up to 156 images.