O. Alshibami

Radix-2 /spl times/ 2 /spl times/ 2 algorithm for the 3-D discrete Hartley transform

The discrete Hartley transform (DHT) has proved to be a valuable tool in digital signal/image processing and communications and has also attracted research interests in many multidimensional applications. Although many fast algorithms have been developed for the calculation of one- and two-dimensional (1-D and 2-D) DHT, the development of multidimensional algorithms in three and more dimensions is still unexplored and has not been given similar attention; hence, the multidimensional Hartley transform is usually calculated through the row-column approach. However, proper multidimensional algorithms can be more efficient than the row-column method and need to be developed. Therefore, it is the aim of this paper to introduce the concept and derivation of the three-dimensional (3-D) radix-2 /spl times/ 2 /spl times/ 2 algorithm for fast calculation of the 3-D discrete Hartley transform. The proposed algorithm is based on the principles of the divide-and-conquer approach applied directly in 3-D. It has a simple butterfly structure and has been found to offer significant savings in arithmetic operations compared with the row-column approach based on similar algorithms.

Fast algorithm for the 3D DCT

The three-dimensional discrete cosine transform (3D DCT) has been used in many 3D applications such as video coding and compression. Many fast algorithms have been developed for the calculation of the 1D DCT. These algorithms are then used for the calculation of the 3D DCT using the row-column approach. However, 3D algorithms involve fewer arithmetic operations and can be faster. The 3D decimation-in-frequency vector-radix algorithm (3D DIF VR), for the 3D DCT-II, is developed and its arithmetic complexity analysed and compared to similar algorithms. In comparison with the familiar row-column approach, the 3D vector-radix reduces the number of multiplications significantly while keeping the number of additions the same and hence can be used for fast 3D image and video coding and compression.

Fast algorithm for the 3-D discrete Hartley transform

The application of multidimensional fast transforms to solve problems in image processing, motion analysis and multidimensional signal processing is growing. The discrete Hartley transform (DHT) is one of the new tools used in many applications including signal and image processing, digital filters, communication etc. This transform is closely related to the discrete Fourier transform, but it is a real-to-real transform and it has the same inverse. Many fast algorithms have been developed for the calculation of one-dimensional DHT. These algorithms are then used for the calculation of multidimensional Hartley transform through an intermediate transform using the row-column approach. However proper multidimensional algorithms can be more efficient and need to be developed. It is the aim of this paper to derive the 3-D vector radix for the 3-D discrete Hartley transform. The arithmetic operations of this algorithm are compared to similar algorithms using the row-column approach.

Fast 3-D decimation-in-frequency algorithm for 3-D Hartley transform

The three-dimensional discrete Hartley transform (3-D DHT) has been applied in a wide range of 3-D applications such as 3-D power spectrum analysis, 3-D filtering, and medical applications, etc. In this paper, a three-dimensional algorithm for fast computation of the three-dimensional discrete Hartley transform is developed. The mathematical concept and derivation is presented and the arithmetic complexity is analysed and compared to the familiar row-column approach. It is found that this algorithm offers substantial savings in both the number of multiplications and additions.

Fast algorithm for the 2-D new Mersenne number transform

Abstract This paper introduces the development and derivations of the 2-D vector-radix algorithm for the calculation of the 2-D new Mersenne number transform. The algorithm is implemented, its arithmetic complexity is analysed and compared to the row–column approach. The vector-radix algorithm is found to be more efficient and involves fewer arithmetic operations than the row–column approach. Using random data, an example is given showing the validity of the developed algorithm and the exact nature of the transform.

Radix-4 algorithm for the new Mersenne number transform

The one-dimensional new Mersenne number transform (NMNT) was proposed for the calculation of error free convolutions and correlations for signal processing purposes. The aim of this paper is to develop the radix-4 decimation-in-time algorithm for fast calculation of the NMNT with a sequence length equal to a power of four. The arithmetic complexity of this algorithm is analysed and the number of multiplications and additions is calculated. An example is given to prove the validity of the algorithm and the exact nature of this transform.

Radix-4 decimation-in-frequency algorithm for the new Mersenne number transform

The development of efficient algorithms for fast calculation of discrete transforms has led to a wide spread use of these transforms in a large number of applications. Consequently, fast algorithms for the new Mersenne number transform (NMNT) need to be developed to make it suitable for efficient and fast implementation of error-free convolutions/correlations and related applications. In this paper the radix-4 decimation-in-frequency algorithm is developed for fast calculation of the 1-D NMNT. The mathematical development of this algorithm is presented, its arithmetic complexity is analyzed and the numbers of multiplications and additions are calculated. An example for the calculation of the auto-correlation using the 1-D NMNT and the developed algorithm is given.

Decimation-in-frequency vector radix algorithm for fast calculation of the 2-D NMNT

For fast calculation of the two-dimensional New Mersenne Number Transform (2-D NMNT), the decimation-in-frequency vector-radix algorithm is developed. The arithmetic complexity of the developed algorithm has been analysed and compared to the row-column approach using multiple butterflies. The results show that the 2-D vector radix algorithm reduces both the number of multiplications and additions.

Vector radix-4 × 4 for fast calculation of the 2-D new Mersenne number transform

This paper describes the vector-radix-4 × 4 decimation-in-time (VR-4 × 4 DIT) for fast calculation of the two-dimensional new Mersenne number transform (2-D NMNT). The new 2-D algorithm is developed and implemented. Its arithmetic complexity is also analysed. Since the 2-D NMNT can be used for error-free calculation of 2-D convolution and auto/cross-correlation functions, the use of this algorithm will improve the efficiency of calculating these functions using the 2-D NMNT leading to a better performance of related applications. The new VR-4 × 4 and existing 2-D NMNT algorithms are compared with respect to their arithmetic operations and computer runtimes. Finally, an example is given to prove the validity of the developed algorithm and the error-free nature of the 2-D NMNT.

Fast 3-D algorithm for the 3-D IDCT

We introduce a fast three-dimensional algorithm for the calculation of the three-dimensional inverse discrete cosine transform (3-D IDCT). The derivation of the algorithm is presented and its arithmetic complexity is analysed and compared to that of the familiar row-column-frame (RCF) method. The proposed algorithm is found to reduce the number of multiplications by about 41%, whilst keeping the number of additions the same. Also, based on computer run-time, it is found to reduce the time involved in calculating the 3-D IDCT significantly. This makes the developed algorithm more suitable for 3-D image and video compression decoders involving the 3-D IDCTs.