Aperture filters: Theory, application, and multiresolution analysis
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We have discussed how it is possible to treat automatic W-operator design for gray-level images in the context of finite discrete lattices. This imposes new restrictions to the space of operators and the trade-off is the potential suboptimality of the designed operator. Aperture operators have been characterized, their representation presented, their design analyzed and two representations, via lattice intervals or decision trees, have been given. Unfortunately, both representations are computationally expensive for most useful operators and the more refined is the representation, the more memory is necessary. Hybrid design has been presented and its usability characterized. Its error due to the restrictions has been analyzed and it has been shown that a considerable improvement can be achieved if good human constraints are designed.Multiresolution design has been presented and analyzed. The representation of this class of operators has been characterized and generalization is inherent in the representation. Hundreds of experiments have been done (some shown here or in other publications) and they show the superiority of aperture operators over optimal restricted window linear operators for signals and images, using both the MAE and MSE criteria. For some problems, the number of training images has not been enough to make the aperture operator better than the linear filter for MSE, only for MAE. Nevertheless, the visual result is better for aperture operators because these are less error prone on image edges. Moreover, the graphics show that the aperture operators would eventually be better than the linear operators for MSE if more examples were given for training the aperture. Decision trees have been initially chosen because they are fast to implement and they can represent the full class of aperture operators. However, there are two main disadvantages: (1) since the representation of the operator is done by a partition of the space of possible configurations and the morphological representation is done by maximal intervals from the operator's kernel, it is difficult to integrate the decision-tree representation with the software; (2) the representation by decision trees is not adequate to impose algebraic restrictions to the design of the operator. The superiority of nonlinear operators in relation to linear ones is not difficult to understand when one thinks in terms of lattices. Besides the inherent suboptimality of linearity relative to the most general class of operators, when a linear operator is applied to an image, the result usually cannot be represented by a point in the output lattice. What is usually done is that one makes the result discrete in order to have a representation in the lattice. Therefore, the discretization in practice transforms linear operators into nonlinear ones. This is true from electronic acquisition devices to the theory of optimal linear filters in discrete spaces. In the latter case, the discretization makes the principal hypothesis of the optimal linear theory (that one is finding the inverse transform of a linear operator) void. There are several problems still to be addressed in the area of automatic design of operators: strategies to choose the aperture (i.e., W, K, and M), strategies/ algorithms to speed up the search of the operator, and compact ways to represent the operator, to name a few. 2006 Hindawi Publishing Corporation.
