A supervised classification scheme using positive Boolean function

A new fuzzy filter, called fuzzy stack filter (FSF), is proposed to extend the filtering capability of conventional stack filter (SF), which is based on the positive Boolean function (PBF) as its window operator. We fuzzify the onset and off-set of a given PBF to obtain two types of fuzzy PBFs. Then, we adopt the architecture of threshold decomposition to develop this new fuzzy filter with a fuzzy PBF as its window operator. Each fuzzy PBF is associated with a set of control parameters. Therefore, the original PBF can be estimated from above and below by two fuzzy PBFs with appropriate control parameters. Furthermore, we can apply the fuzzy modifiers to modify the fuzzy PBFs such that the PBFs can be completely estimated by the fuzzy PBFs. Hence, the stack filter is a special case of fuzzy stack filter. Since some control parameters are added in this new filter, the neural learning algorithms can be easily developed under the flexibility of the given control parameters. We first propose the fuzzy (m,n) rank-order filter to test our proposed learning algorithm. In this simple learning algorithm, we can remove the noise-corrupted images very well in contrast to the filtering behavior of rank-order filters. We believe that the results presented will lead to more fruitful research on more advanced and powerful learning algorithms dedicated to the appropriate applications.

The signed binary number representation (SBNR) is used to reduce the number of PBF calculation stages in stack filters. The approach is based on the possibility of minimization of signed power-of-two terms in the SBNR representation of input data. A coder and a decoder for the mutual transformation of binary-weighted code and the minimal SBNR are proposed. An efficient algorithm and a processor structure for stack filtering based on the minimal SBNR are presented. The time complexity of the proposed filter is of order kT/sub PBF/, where k is the number of bits in the input signals and T/sub PBF/ is the delay in the incorporated circuit for the positive Boolean function (PBF). A stack filtering algorithm for the generalized nonadjacent form (GNAF) of signed digit number representation (SDNR) is also presented.

A hand-based personal authentication using a coarse-to-fine strategy

A novel approach to personal verification using the thermal images of palm-dorsa vein patterns is presented in this paper. The characteristics of the proposed method are that no prior knowledge about the objects is necessary and the parameters can be set automatically. In our work, an infrared (IR) camera is adopted as the input device to capture the thermal images of the palm-dorsa. In the proposed approach, two of the finger webs are automatically selected as the datum points to define the region of interest (ROI) on the thermal images. Within each ROI, feature points of the vein patterns (FPVPs) are extracted by modifying the basic tool of watershed transformation based on the properties of thermal images. According to the heat conduction law (the Fourier law), multiple features can be extracted from each FPVP for verification. Multiresolution representations of images with FPVPs are obtained using multiple multiresolution filters (MRFs) that extract the dominant points by filtering miscellaneous features for each FPVP. A hierarchical integrating function is then applied to integrate multiple features and multiresolution representations. The former is integrated by an inter-to-intra personal variation ratio and the latter is integrated by a positive Boolean function. We also introduce a logical and reasonable method to select a trained threshold for verification. Experiments were conducted using the thermal images of palm-dorsas and the results are satisfactory with an acceptable accuracy rate (FRR:2.3% and FAR:2.3%). The experimental results demonstrate that our proposed approach is valid and effective for vein-pattern verification.

This paper presents a new supervised classification technique for hyperspectral imagery, which consists of two algorithms, referred to as greedy modular eigenspace (GME) and positive Boolean function (PBF). The GME method is designed to extract features by a simple and efficient GME feature module. The GME makes use of the data correlation matrix to reorder spectral bands from which a group of feature eigenspaces can be generated to reduce dimensionality. It can be implemented as a feature extractor to generate a particular feature eigenspace for each of the material classes present in hyperspectral data. The residual reconstruction errors (RRE) are then calculated by projecting the samples into different individual GME-generated modular eigenspaces. The PBF is further developed for classification. It is a stack filter built by using the binary RRE as classifier parameters for supervised training. It implements the minimum classification error (MCE) as a criterion so as to improve classification performance. It utilizes the positive and negative sample learning ability of the MCE criteria to improve classification accuracy. The performance of the proposed method is evaluated by MODIS/ASTER airborne simulator (MASTER) images for land cover classification during the Pacrim II campaign. Experimental results demonstrate that the GME feature extractor suits the nonlinear PBF-based multi-class classifier well for classification preprocessing. The proposed approach is not only an effective method for land cover classification in earth remote sensing but also dramatically improves the eigen-decomposition computational complexity compared to the conventional principal components analysis (PCA).

Weighted medians — positive Boolean functions conversion algorithms

Selection probabilities, viz. rank and sample selection probabilities, quantify the impulse suppression, robustness, lowpass filtering ability, and detail preserving ability of stack filters. Closed-form expressions exist for computing selection probabilities from positive Boolean function describing the stack filter. We first present an analysis framework that interprets a Boolean vector to denote membership in a partition set. Using this framework a new algorithm to compute selection probabilities from positive Boolean functions is derived. Next a method to synthesize a stack filter from specified selection probabilities is presented. An underdetermined system of linear Diophantine equations is derived in the unknown elements of the C-matrix and the desired selection probabilities. This system of equations is solved to obtain the candidate solutions for the C-matrix. For each candidate C-matrix, an algorithm based on partition sets is used to derive one or more positive Boolean functions of the stack filter or show that none exists.

Median filtering with programmable window length is needed for environments with impulse noises of different lengths. The paper proposes a new architecture for an A/D convertor with median filtering of programmable window length. Both digital and analogue medians are available at the output. Previous architecture was implemented using a positive boolean function (PBF) circuit which requires a large area for large window length. In this architecture, the majority circuit, which has a smaller area and faster speed, is used to replace the PBF circuit. The throughput is independent of the window size and the hardware complexity is linearly dependent on the window size. The silicon area of this architecture is small and it is therefore suitable for monolithic implementation.

The nonlinear digital filtering methods which are based on threshold decomposition and positive Boolean functions are becoming widely used. They are in use especially in image processing and biomedical signal processing, both of which usually involve large amounts of data. It is thus important to develop fast algorithms to realize weighted median weighted order statistics and more general stack filtering operations. The amounts of data are so large that fast algorithms are needed no matter how powerful the computers are that are available. In the following paper we introduce new algorithms for some of these problems and discuss the use of logical matrix algorithms for the solution of nonlinear digital filtering problems.

Given a positive Boolean function fand a subset δ of its variables, we give a combinatorial condition characterizing the existence of a prime implicant Dˆof the Boolean dual f d of f having the property that every variable in δ appears in Dˆ We show that the recognition of this property is an NP-complete problem, suggesting an inherent computational difficulty of Boolean dualization, independently of the size of the dual function. Finally it is shown that if the cardinality of δ is bounded by a constant, then the above recognition problem is polynomial. In particular, it follows that the co-ocurrence graph of the dual of a positive Boolean function can be always generated in time polynomial in the size of the function.

Worst-case groundness analysis using positive Boolean functions

Stack filters are a class of nonlinear filters for removing noise that is uncorrelated with signal. Their design is formulated as a high nonlinear optimization problem. An improved immune memory clonal selection algorithm (IMCSA) is suitable for designing stack filters. But as many algorithms for designing stack filters have to face that evaluation of each candidate solution is still takes the most computation complexity. By representing the positive Boolean functions (PBF) in objective function, a hybrid computation approach (software and hardware) for the calculation tasks has been proposed. Compared with available methods, the proposed method can improve the efficiencies of design stack filters and the design time is significantly reduced. The procedure of design and implementation for stack filters can be reached at the same time by the method

We consider to identify an unknown Boolean function f by asking an oracle the functional values f(a) for a selected set of test vectors a ε {0,1}n. If f is known to be a positive function of n variables, the algorithm by Gainanov can achieve the goal by issuing O(mn) queries, where m = ¦minT(f)¦ + ¦maxF(f)¦ and minT(f) (resp. maxF(f)) denotes the set of minimal true vectors (resp. maximal false vectors) of f. However, it is not known whether this whole task including the generation of test vectors can be carried out in polynomial time in n and m or not. To partially answer this question, we propose here two algorithms that, given an unknown positive function f of n variables, decide whether f is 2-monotonic or not, and if f is 2-monotonic, output sets min T(f) and max F(f). The first algorithm uses O(nm 2+n 2 m) time and O(nm) queries while the second one uses O(n3m) time and O(n 3 m) queries.

Feature extraction using weighted training (FEWT) samples is proposed in this letter. Different spectral bands (features) play different roles in identification of land-cover classes. In the FEWT, the relative importance of each feature of a training sample in predicting the class label of that sample is obtained and considered as a weight for that feature. Then, the weighted training samples can be used in each arbitrary feature extraction method. In this letter, we use the weighted training samples in supervised locality preserving projection. The experimental results on three popular hyperspectral images show that FEWT has better performance and more speed than some state-of-the-art supervised feature extraction methods using limited number of available training samples.

In this work, we present tensor-based linear and nonlinear models for\nhyperspectral data classification and analysis. By exploiting principles of\ntensor algebra, we introduce new classification architectures, the weight\nparameters of which satisfies the {\\it rank}-1 canonical decomposition\nproperty. Then, we introduce learning algorithms to train both the linear and\nthe non-linear classifier in a way to i) to minimize the error over the\ntraining samples and ii) the weight coefficients satisfies the {\\it rank}-1\ncanonical decomposition property. The advantages of the proposed classification\nmodel is that i) it reduces the number of parameters required and thus reduces\nthe respective number of training samples required to properly train the model,\nii) it provides a physical interpretation regarding the model coefficients on\nthe classification output and iii) it retains the spatial and spectral\ncoherency of the input samples. To address issues related with linear\nclassification, characterizing by low capacity, since it can produce rules that\nare linear in the input space, we introduce non-linear classification models\nbased on a modification of a feedforward neural network. We call the proposed\narchitecture {\\it rank}-1 Feedfoward Neural Network (FNN), since their weights\nsatisfy the {\\it rank}-1 caconical decomposition property. Appropriate learning\nalgorithms are also proposed to train the network. Experimental results and\ncomparisons with state of the art classification methods, either linear (e.g.,\nSVM) and non-linear (e.g., deep learning) indicates the outperformance of the\nproposed scheme, especially in cases where a small number of training samples\nare available. Furthermore, the proposed tensor-based classfiers are evaluated\nagainst their capabilities in dimensionality reduction.\n