For the recently developed quasi-reversible BCH codes with long lengths and high error-correcting capability, this paper is aimed at proposing a new and faster decoding procedure. It consists of four steps: 1) compute the consecutive syndromes; 2) calculate the syndrome functions by the forward and backward recursions; 3) solve a linear subsystem together with one matrix multiplication in order to find an error-locator polynomial; 4) determine the errors from the obtained polynomial by using the root-finding algorithm. This procedure, especially in Steps 2 and 3, differs greatly from the conventional procedures, which determine an error-locator polynomial directly from solving a linear system with the aid of the consecutive syndromes. The key idea behind this decoding technique is that the computational complexity of such a small subsystem instead of an originally large linear system can be significantly reduced, although there are additional forward and backward syndrome calculations with low complexity increasing. Finally, the illustrative examples and numerical simulations can be helpful to demonstrate the accuracy and efficacy of the presented decoding technique at different error-correcting capabilities.

It is well-known that the decoding of algebraic codes with error locators have been extensively conceived in the literature over a half century. The radical locator, which is an ω th root of error locator, has recently been discovered. This paper focuses on the two classes of radical locators. The first is complete radical locators, where all the locators are assigned to radical locators. The second is partial radical locators in that there are only a few locators being radical locators. In the former case based on complete radical locators, a new square matrix whose determinant is a univariate radical-locator polynomial is proposed. In particular, this matrix is modified to allow a large square matrix. It can be transformed by Gaussian elimination to the matrix in a row-echelon form in which zeros appearing in the diagonal entries of the current matrix are able to determine errors implicitly. Furthermore, this work further extends the previous results from the univariate case to multivariate cases. The matrix methods found herein enable one to decode a large class of cyclic codes efficiently. In the latter case, the cyclotomic cosets are a practical approach to select a small subset of positive integers. They are employed to develop partial radical locators. Finally, in contrast with the complete radical locators, the algebraic decoding methods make a natural use of partial radical locators more widely and flexibly to arbitrary cyclic codes.

The problem of blind image recovery using multiple blurry images of the same scene is addressed in this paper. To perform blind deconvolution, which is also called blind image recovery, the blur kernel and image are represented by groups of sparse domains to exploit the local and nonlocal information such that a novel joint deblurring approach is conceived. In the proposed approach, the group sparse regularization on both the blur kernel and image is provided, where the sparse solution is promoted by -norm. In addition, the reweighted data fidelity is developed to further improve the recovery performance, where the weight is determined by the estimation error. Moreover, to reduce the undesirable noise effects in group sparse representation, distance measures are studied in the block matching process to find similar patches. In such a joint deblurring approach, a more sophisticated two-step interactive process is needed in which each step is solved by means of the well-known split Bregman iteration algorithm, which is generally used to efficiently solve the proposed joint deblurring problem. Finally, numerical studies, including synthetic and real images, demonstrate that the performance of this joint estimation algorithm is superior to the previous state-of-the-art algorithms in terms of both objective and subjective evaluation standards. The recovery results of real captured images using unmanned aerial vehicles are also provided to further validate the effectiveness of the proposed method.

A class of cyclic codes has recently been decoded by using the weak-locator polynomials instead of the conventional error-locator polynomials. In this paper, a generalization of weak-locator polynomials to the multivariate cases, called the multivariate weak-locator polynomials, is defined, and a new matrix whose determinant can be expressed as a multivariate weak-locator polynomial is developed for cyclic codes. Moreover, a modified matrix along with Gaussian elimination found herein enables one to determine error positions precisely. The presented matrices result in a reduction of the number of syndromes when compared with the previously known matrices. The simulation shows that, for example, with the capability of correcting more errors, a considerably sophistical scheme of decoding the (97, 49, 15) binary cyclic code based on the proposed matrices is around 115 times faster than other existing decoders. Therefore, in general, it is developed to facilitate faster decoding of a large class of cyclic codes and is naturally suitable for the software implementation.

The cubic‐spline interpolation (CSI) scheme is known to be designed to resample the discrete image data based on the least‐square method in conjunction with the cubic convolution interpolation (CCI) function. In this CSI scheme, the improved quality of resampling can be achieved as the interpolation kernel size increases. However, the improvement of the performance gets less and less. This means that the performance of the CSI scheme has not been significantly improved and converges toward a constant value when the interpolation kernel size exceeds a certain value. A proof of this result is given in this study and has never been seen in the literature to the authors' knowledge. Moreover, this study analyses the relationship between the performance and computational complexity of the CSI schemes with different interpolation kernel sizes and compares them from a structural point of view. Simulation results indicate that it is in agreement with the theoretic derivations. Since the arithmetic operations required are increasing linearly with the increment of the interpolation kernel size, selecting an interpolation kernel size gives the best trade‐off between the performance and computational complexity in practical applications. However, the optimum choice of the interpolation kernel size depends crucially on effective demand.

In this study, a new approach is developed to facilitate faster decoding of a binary systematic (71, 36, 11) quadratic residue (QR) code. In this decoder, it simplifies the step of calculating the condition and avoids calculating the unknown syndrome, thereby yielding a fast algebraic decoder for correcting four possible errors. Moreover, while using the proposed algorithm, if uses the channel measurement information proposed by Chase to sequentially invert the bits of the received word until one of the errors is cancelled for the five‐error case and apply the new algebraic decoding algorithm mentioned above to correct the remaining four errors, the algorithm has been verified through a software simulation in C‐language. The simulation shows that the decoding scheme developed here is more efficient than the previous decoding algorithm developed for the (71, 36, 11) QR code and it is naturally suitable for software implementation.

Based on an error-flattened, non-uniform-region linear-approximation algorithm, this brief proposes a low-error and a cost-efficient design procedure for realizing an optimized shift-and-add Logarithmic Unit (LU), which uses minimum hardware to meet the desired error constraint for embedded graphics systems. Mathematically, this brief first derives two solutions of the error-flattened algorithm. Subsequently, for an error constraint, the minimum number of approximation regions, n, the corresponding i th interpolation coefficients (ai, bi), and the regional endpoints (xi-1, xi), 1 ≤ i ≤ n, are obtained accordingly. Using the unique properties of the logarithmic function, spacing of xi is non-uniform to make errors in each region consistent. Carefully examining the cost of the applied add/sub network, a low-cost candidate for a shift-and-add LU is determined. Next, a cost exploration process, which gradually increases n, is performed. A large number of regions results in a more accurate conversion algorithm that might tolerate more implementation errors by using simple hardware whose cost is lower than that of the inferior candidate. After exploring the cost based on error tolerance, the proposed design procedure finally generates a hardware to meet the desired error constraint. Slightly modifying the regional endpoint xi increases hardware efficiency at the cost of increased error. Proposed circuits were synthesized in UMC 65RVT CMOS technology. Compared to state-of-the-art shift-and-add logarithmic converters, simulation results reveal that the proposed design saves approximately 12.7-51.1 percent area of polynomial approximation and improves approximately 1-14-dB SNR gain while achieving a tighter error constraint.

The algebraic decoding of a p-ary cyclic code consists of four steps: 1) computation of the known syndromes using the received word; 2) computation of the unknown syndromes from the known syndromes; 3) computation of the error positions by a use of the Berlekamp-Massey (BM) algorithm and Chien's search; and 4) computation of the error values by solving a linear system. This paper addresses two problems of determining the error positions and computing the unknown syndromes. To solve the first problem, a new matrix, together with Gaussian elimination instead of the BM algorithm and Chien's search, is proposed. In this new simplified decoder, finding an error-locator polynomial is completely avoided. A main advantage of the presented decoding method is when the Bose-Chaudhuri-Hocquenghem bound is unequal to the minimum distance of the code. Some cyclic codes, which do not have 2t consecutive known syndromes, can be decoded up to their actual minimum distance by using the presented matrix once. To solve the second problem, two algorithms for different square matrices reported recently by Lee et al. are also provided to calculate the value of an unknown syndrome. Finally, an algebraic decoding of the (47,23,15) ternary cyclic code is given.

Frequent itemsets are useful for discovering interesting associations hidden in large databases. Many mining algorithms use data with binary attributes to represent the occurrence of items and find frequent itemsets. However, many real-world applications provide a richer source of transactions with quantitative values. The fuzzy frequent pattern tree algorithm was thus proposed for extracting fuzzy frequent itemsets from the quantitative transactions. In this paper, a tree structure called the upper-bound multiple fuzzy frequent-pattern (UBMFFP)-tree is designed for improving the pruning effect in the mining process. A two-phase fuzzy mining approach based on the tree structure is also proposed to obtain the complete fuzzy frequent itemsets from a quantitative database. The proposed fuzzy mining approach recursively and efficiently finds the upper-bound fuzzy counts of itemsets with the aid of the tree structure. It prunes unpromising itemsets in the first phase, and then finds the actual fuzzy frequent itemsets in the second phase. Experimental results indicate that the proposed UBMFFP-tree algorithm has good performance in terms of execution time and number of tree nodes.

Recently, mining useful information and knowledge from transactions is evolving into an important research issue. Many algorithms have thus been proposed for mining association rules based on items with binary values. Transactions with quantitative values are, however, also commonly seen in real‐world applications. The fuzzy frequent‐pattern tree (FP tree) algorithm has been proposed for extracting fuzzy frequent itemsets from quantitative transactions. Only the term with the maximum cardinality in later processes is used, making the number of fuzzy regions processed equal to the number of original items, which reduces the processing time. In real world applications, however, multiple fuzzy regions of an item produce better fuzzy association rules than those obtained using a single region. In this paper, the multiple fuzzy FP tree (MFFP tree) algorithm is proposed for mining fuzzy frequent itemsets from transactions with quantitative values. When extending the FP‐tree structure to handle fuzzy data, the processing becomes much more complex than that for the original FP‐tree structure since the fuzzy intersection in each transaction has to be handled. The MFFP‐tree construction algorithm is designed and the MFFP‐growth mining approach is proposed for mining the fuzzy frequent itemsets from the tree structure. Experiments were conducted to evaluate the performance of the proposed approach.

In this paper, Three decoding methods of the (89, 45, 17) binary quadratic residue (QR) code to be presented are hard, soft and linear programming decoding algorithms. Firstly, a new hybrid algebraic decoding algorithm for the (89, 45, 17) QR code is proposed. It uses the Laplace formula to obtain the primary unknown syndromes, as done in Lin et al.'s algorithm when the number of errors v is less than or equal to 5, whereas Gaussian elimination is adopted to compute the unknown syndromes when v ≥ 6. Secondly, an appropriate modification to the algorithm developed by Chase is also given in this paper. Therefore, combining the proposed algebraic decoding algorithm with the modified Chase-II algorithm, called a new soft-decision decoding algorithm, becomes a complete soft decoding of QR codes. Thirdly, in order to further improve the error-correcting performance of the code, linear programming (LP) is utilized to decode the (89, 45, 17) QR code. Simulation results show that the proposed algebraic decoding algorithm reduces the decoding time when compared with Lin et al.'s hard decoding algorithm, and thus significantly reduces the decoding complexity of soft decoding while maintaining the same bit error rate (BER) performance. Moreover, the LP-based decoding improves the error-rate performance almost without increasing the decoding complexity, when compared with the new soft-decision decoding algorithm. It provides a coding gain of 0.2 dB at BER = 2 × 10-6.

The cubic-spline interpolation (CSI) scheme can be utilized to obtain a better quality reconstructed image. It is based on the least-squares method with cubic convolution interpolation (CCI) function. Within the parametric CSI scheme, it is difficult to determine the optimal parameter for various target images. In this paper, a novel method involving the concept of opportunity costs is proposed to identify the most suitable parameter for the CCI function needed in the CSI scheme. It is shown that such an optimal four-point CCI function in conjunction with the least-squares method can achieve a better performance with the same arithmetic operations in comparison with the existing CSI algorithm. In addition, experimental results show that the optimal six-point CSI scheme together with cross-zonal filter is superior in performance to the optimal four-point CSI scheme without increasing the computational complexity.

In the past, the multiple fuzzy frequent pattern tree (MFFP tree) was proposed for extracting multiple fuzzy frequent itemsets from quantitative transactions. It kept the multiple transformed fuzzy regions of an item to form the multiple fuzzy frequent itemsets. In this paper, an incremental algorithm is proposed for efficiently mining multiple fuzzy frequent itemsets based on the FUP concepts and the MFFP-tree structure. Experimental results show that the proposed incremental algorithm runs faster than the batch one.

In this paper, a novel two-phase fuzzy mining approach based on the designed upper-bound multiple fuzzy frequent-pattern (UBMFFP) tree is proposed to obtain all fuzzy frequent itemsets from a quantitative database. It prunes unpromising itemsets in the first phase, and then finds the actual fuzzy frequent itemsets in the second phase. Experimental results indicate that the proposed approach has better performance than some previous ones.

It is well-known that the Euclidean algorithm can be used o find the systematic errata-locator polynomial and the errata-evaluator polynomial simultaneously in Berlekamp's key equation that is needed to decode a Reed-Solomon (RS) codes. In this paper, a simplified decoding algorithm to correct both errors and erasures is used in conjunction with the Euclidean algorithm for efficiently decoding nonsystematic RS codes. In fact, this decoding algorithm is an appropriate modification to the algorithm developed by Shiozaki and Gao. Based on the ideas presented above, a fast algorithm described from Blahut's classic book is derivated and proved in this paper to correct erasures as well as errors by replacing the Euclidean algorithm by the Berlekamp-Massey (BM) algorithm. These facts lead to significantly reduce the decoding complexity of the proposed RS decoder. In addition, computer simulations show that this simple and fast decoding technique reduces the decoding time when compared with existing efficient algorithms including the new Euclidean-algorithm-based decoding approach proposed in this paper.

In this paper, we attempt to extend the fuzzy FP-tree mining process to mine all fuzzy frequent itemsets, instead of only the representative linguistic terms, from a set of quantitative transactions. A multiple fuzzy-term FP (MFFP) tree with the consideration of fuzzy operations is proposed to help the execution of the fuzzy mining process. The corresponding MFFP-growth mining algorithm is also designed to derive all fuzzy frequent itemsets from the tree structure by fuzzy operations. Experimental results also show the performance of the proposed approach.

It has been shown that the 2-D cubic spline interpolation (CSI) proposed by Truong et al. is one of the best algorithms for image resampling or compression. Such a CSI algorithm together with the image coding standard, e.g., JPEG, can be used to obtain a modified image codec while still maintaining a good quality of the reconstructed image for higher compression ratios. In this paper, a fast direct computation algorithm is developed to improve the computational efficiency of the original FFT-based 2-D CSI methods. In fact, this algorithm computes the 2-D CSI directly without explicitly calculating the complex division usually needed in the FFT or Winograd discrete Fourier transform (WDFT) algorithm. In addition, this paper describes a novel way to derivate the 2-D CSI from the 1-D CSI by using the row-column method. This new fast 2-D CSI provides a regular and simple structure based upon linear correlations. Therefore, it can be implemented by the use of a modification of Kung’s pipeline structure and is naturally suitable for VLSI implementations. Experimental results show that the proposed new fast 2-D CSI algorithm can achieve almost the same CSI performance with much fewer arithmetic operations in comparison with existing efficient algorithms.

In this paper, a new soft-decision decoder of the (24, 12, 8) binary extended Golay code up to six errors is proposed. First, by using the error pattern obtained from hard decoder, the method of determining possible error patterns is developed. The emblematic probability value of each error pattern is then defined as the product of the individual bit-error probabilities corresponding to the locations of the possible error patterns. The most likely one among these error patterns is obtained by choosing the maximum of the emblematic probability values of possible error patterns. Finally, simulation results in additive white Gaussian noise (AWGN) show that this decoder reduce the decoding complexity although it performs a slight loss of coding gain than the modified Chase's II algorithm proposed by Hackett.

Binary quadratic residue (QR) codes, which have code rates greater than or equal to 1/2 and generally have large minimum distances, are among the best known codes. This paper considers a modified algebraic decoding algorithm for the (89,45,17) binary QR code that utilizes the Berlekamp-Massey algorithm. It identifies the primary unknown syndromes and provides methods to determine these on a case-by-case basis for any number of correctable errors. Numerical evaluation shows that the proposed algorithm significantly reduces at least 52% of decoding time for two or more errors.

A new decoder is proposed to decode the (24, 12, 8) binary extended Golay code up to four errors. It consists of the conventional hard decoder for correcting up to three errors, the detection algorithm for four errors and the soft decoding for four errors. For a weight-4 error in a received 24-bit word, Method 1 or 2 is developed to determine all six possible error patterns. The emblematic probability value of each error pattern is then defined as the product of four individual bit-error probabilities corresponding to the locations of the four errors. The most likely one among these six error patterns is obtained by choosing the maximum of the emblematic probability values of all possible error patterns. Finally, simulation results of this decoder in additive white Gaussian noise show that at least 93% and 99% of weight-4 error patterns that occur are corrected if the two Eb/N0 ratios are greater than 2 and 5 dB, respectively. Consequently, the proposed method can achieve a better percentage of successful decoding for four errors at variable signal-to-noise ratios than Lu et al.'s algorithm in software. However, the speed of the method is slower than Lu et al.'s algorithm.

Gao's algorithm similar to Shiozaki's algorithm for decoding nonsystematic Reed-Solomon (RS) codes which is a subclass of the redundant residue polynomial codes is considered here. This algorithm computes the message polynomial directly without explicitly finding the error-locator polynomial and errorevaluator polynomial. In this letter, a simplified decoding algorithm to correct both errors and erasures is used in conjunction with Gao's algorithm for efficiently decoding RS codes. In addition, we show that the extended Gao algorithm similar to the Shiozaki-Truong-Cheung-Reed algorithm significantly reduces the decoding complexity.

This paper proposes a novel scheme which consists of a weight-counting algorithm, the combinatorial designs of the Assmus-Mattson theorem, and the weight polynomial of Gleason’s theorem to determine the weight distributions of binary extended quadratic residue codes. As a consequence, the weight distribution of binary (168, 84, 24) extended quadratic residue code is given.

In this paper, Fedorenko and Trifonov's procedure is applied to evaluate the syndrome of the received word in time-domain Reed-Solomon decoders. This application leads to a substantial reduction of the computational complexity of the syndrome polynomial for correcting both errors and erasures. Moreover, simulation results for this new syndrome method are given.

In this paper, there are 17 types of wavelet coefficients obtained from the Matlab software and an Aurora-2 database used to evaluate which wavelet type has a better accuracy in speaker recognition. We first determine the frequency cepstral coefficient (FCC) level to form a 114-dimentional feature vector by the use of Daubechies-4 wavelet and support vector machines (SVMs) with pre-selected exponential radial basis kernel function (ERBF) and under some additional conditions. Then, average, for each wavelet, the accuracy of 42 possible combinations about the gender of speakers considered in seven kinds of experiments corresponding to two to eight speakers. The experimental results show that the best accuracy in average will be achieved by using the reverse biorthogonal-3.5 or reverse biorthogonal-3.7 wavelet. The reverse biorthogonal-3.5 wavelet is then chosen to be the proposed wavelet function for speaker recognition in terms of shorter filter length

This paper proposes an improved DCT-based image codec embedded cubic spline interpolation (CSI) with optimal quantization. It is shown in literature that CSI is superior in performance to the other interpolation functions. One of main applications of the CSI is to cooperate with standard DCT-based JPEG to obtain the modified JPEG codec and still obtain a better quality of the reconstructed image for higher compression ratios. This paper shows that the compression performance of such a modified JPEG codec can be further improved by the use of optimal quantization. The optimal quantization makes use of the rate distortion optimization algorithm to measure DCT coefficient statistic for an image generated from the CSI scheme and then construct a DCT quantization table for the given rate/distortion specification. Experimental results show that the proposed optimal quantization can achieve a better PSNR than that of the modified JPEG codec with a default quantization table