Algorithm for generating new orthonormal transforms

2009 International Workshop on Local and Non-Local Approximation in Image Processing, 2009

In this work a new complete system of trigonometric functions for representation of finite continuous functions is defined. New discrete orthogonal transforms are synthesized by discretization and orthonormalization of the proposed system of functions. An image compression algorithm using the synthesized transform is developed. The proposed algorithm illustrates high compression ratio and good reconstruction quality as confirmed by a numerous of experiments.

Signal Processing, 1991

This paper considers the generation of integer transforms based on the principle of dyadic symmetry. A set of new order-16 integer transforms is found and the kernel components of the new integer transforms can be represented by one-byte integers. Among the I 19 new integer transforms obtained, 12 of them have better performance, in terms of transform efficiency, than the order-16 integer cosine transforms and other well-known transforms like the slant transform and the Walsh transform. They also have similar transform efficiency and basis restriction mean-square-error compared to the discrete cosine transform (DCT). Zusammenfassung. In dieser Arbeit wird die Erzeugung ganzzahliger Transformationen basierend auf dem Prinzip dyadischer Symmetric betrachlet. Ein Satz neuer Transformationen der Ordnung 16 wird angegeben, die Komponenten des Kernes der neuen Transformation krnnen durch Ein-Byte-Integer dargestellet werden. Unter den 119 neuen Transformationen sind 12 effizienter als die entsprechenden Integer Kosinus-Transformationen und andere bekannte Transformationen wie die Slantoder Walsh-Transformation. Sie besitzen auch eine /ihnliche Effizienz und grundsfitzliche Einschriinkung beim Mittleren Quadratischen Fehler wie die Diskrete Kosinustransformation. R~um~. Cet article traite de la crration de transform~es en nombre basres sur le principle de symmrtrie dyadique. Un ensemble de nouvelles transformres en nombre entier d'ordre 16 est proposre et les composants des noyeaux peuvent 6tre reprrsentrs par des entiers d'un octet. Parmi les 119 nouvelles transformres obtenues, 12 ont de meilleures performances, en termes d'efficacit+, que la transformre en cosinus en nombre entier d'ordre 16 et que d'autres transformres crlrbres comme celle en pente et celle de Walsh. Elles ont 6galement la m~me 6fficacit+ et la m~me restriction de base en erreur quadratique moyenne que la transformre en cosinus discret.

Applications of Digital Image Processing XXXII, 2009

We study factorization techniques and performance of Discrete Cosine Transforms of various sizes (including nondyadic and odd numbers). In our construction we utilize an array of known techniques (such as Heideman's mapping between odd-sized DCT and DFT, Winograd fast DFT algorithms, prime-factoring, etc), and also propose a new decimation strategy for construction of even-sized scaled transforms. We then analyze complexity and coding gain of such transforms with sizes 2-64 and identify ones that show best complexity/performance tradeoffs.

1994

A simplified version of the integer cosine transform (ICT) is described. For practical reasons, the transform is considered jointly with the quantization of its coefficients. It differs from conventional ICT algorithms in that the combined factors for normalization and quantization are approximated by powers of two. In conventional algorithms, the normalization/quantization stage typically requires as many integer divisions as the number of transform coefficients. By restricting the factors to powers of two, these divisions can be performed by variable shifts in the binary representation of the coefficients, with speed and cost advantages to the hardware implementation of the algorithm. The error introduced by the factor approximations is compensated for in the inverse ICT operation, executed with floating point precision. The simplified ICT algorithm has potential applications in image-compression systems with disparate cost and speed requirements in the encoder and decoder ends. F...

IEEE Transactions on Signal Processing, 1991

It is possible to replace the real-numbered elements of a discrete cosine transform (DCT) matrix by integers and still maintain the structure, i.e., relative magnitudes and orthogonality, among the matrix elements. The result is an integer cosine transform (ICT). Thirteen ICT's have been found and some of them have performance comparable to the DCT.

IEEE Transactions on Circuits and Systems, 2008

2009

We study factorization techniques and performance of Discrete Cosine Transforms of various sizes (including nondyadic and odd numbers). In our construction we utilize an array of known techniques (such as Heideman's mapping between odd-sized DCT and DFT, Winograd fast DFT algorithms, prime-factoring, etc), and also propose a new decimation strategy for construction of even-sized scaled transforms. We then analyze complexity and coding gain of such transforms with sizes 2-64 and identify ones that show best complexity/performance tradeoffs.

2013

Dong, Jie; Ngan, King Ngi; Fong, Chi Keung; Cham, Wai Kuen; A Universal Approach to Developing Fast Algorithm for Simpli?ed Order-16 ICT, IEEE International Symposium on Circuits and Sys tems, 2007, May 27-30, 2007, pp. 281-284, IEEE, US. Cham, W.K.; Chan,Y.T., An order16 integer cosine transform, IEEE Transactions on Signal Processing, May 1991 pp. 1205-1208, vol. 39, Issue 5, IEEE, US. Ma, SiWei; Jao, Kuo, C.-C. J., High-de?nition video coding With super-macroblocks, ProceedingsiSPIE The International Society for Optical Engineering, p. 41, vol. 6508, part 1, International Society for Optical Engineering, US, Jan. 2007.

Signal Processing, 2007

We provide different transformation formulae between the different discrete cosine transform (DCT) types of the same size. The transformations use only diagonal and special lower/upper triangular matrices that minimize the overhead of transformation. These transformations provide a tool for using any of the DCT types as a core module for computing all other types. r

2013 European Conference on Circuit Theory and Design (ECCTD), 2013

In image data compression, integer cosine transforms (ICTs) have been preferred to discrete cosine transforms (DCTs) due to their similar transform efficiency and lower implementation cost. However, there exist many alternative ICTs with different performance measures and implementation costs. In this work, we explore all possible ICTs, compute their performance measures, and find their implementation costs in terms of the number of adders/subtractors, where a state-of-art technique is used to realize ICTs under a shift-adds architecture. We also investigate the tradeoff between performance and implementation cost, present the pareto-optimal points of this tradeoff, and introduce promising ICTs that were not considered before.

Signal Processing, 2020

Discrete transforms play an important role in many signal processing applications, and low-complexity alternatives for classical transforms became popular in recent years. Particularly, the discrete cosine transform (DCT) has proven to be convenient for data compression, being employed in well-known image and video coding standards such as JPEG, H.264, and the recent high efficiency video coding (HEVC). In this paper, we introduce a new class of low-complexity 8-point DCT approximations based on a series of works published by Bouguezel, Ahmed and Swamy. Also, a multiparametric fast algorithm that encompasses both known and novel transforms is derived. We select the best-performing DCT approximations after solving a multicriteria optimization problem, and submit them to a scaling method for obtaining larger size transforms. We assess these DCT approximations in both JPEG-like image compression and video coding experiments. We show that the optimal DCT approximations present compelling results in terms of coding efficiency and image quality metrics, and require only few addition or bit-shifting operations, being suitable for low-complexity and low-power systems.

2011

We propose fast algorithms for computing Discrete Sine and Discrete Cosine Transforms (DCT and DST) of types VI and VII. Particular attention is paid to derivation of fast algorithms for computing DST-VII of lengths 4 and 8, which are currently under consideration for inclusion in ISO/IEC/ITU-T High Efficiency Video Coding (HEVC) standard.

IJSRD, 2014

In this paper, Integer cosine transform (ICT) is introduced in MPEG-4 instead of Discrete cosine transform (DCT. Integer cosine transform (ICT) is adopted by H.264/AVC for its bit-exact implementation and significant complexity reduction. As adoption of the newly standardized H.264 becomes increasingly more widespread, efforts must be made to transcode video from earlier standards, such as MPEG-4, to the H.264 format. The block sizes of most appropriate are 8 or 16 for the transform coding of the image data. Therefore, implementation of the order 8 and 16 DCTs has fast computing time and costeffectiveness for realization of a transform coding. However, the components of the basis vectors of the DCT exist irrational numbers then cannot be reduced to integers by simple scaling. Therefore, it is hard to implement and using floating point arithmetic is complex and expensive, so integer cosine transforms (ICTs) are proposed to implement the DCT by using simple integer arithmetic. By investigating the structure of the transform kernels, two efficient schemes are introduced to convert an 8 × 8 DCT block into its corresponding four 4 × 4 integer cosine transform blocks. This technique can be used to improve the result of MPEG-4 video compression standard.

Multidimensional Systems and Signal Processing, 1990

This paper presents vector and parallel algorithms and implementations of one-and two-dimensional orthogonal transforms. The speed performances are evaluated on Cray X-MP/48 vector computer. The sinusoidal orthogonal transforms are computed using fast real Fourier transform (FFT) kernel. The non-sinusoidal orthogonal transform algorithms are derived by using direct factorizations of transform matrices. Concurrent processing is achieved by using the multitasking capability of Cray X-MP/48 to transform long dam vectors and two-dimensional data vectors. The discrete orthogonal transforms discussed in this paper include: Fourier transform (DFT), cosine transform (DCT), sine transform (DST), Hartley transform (DHT), Walsh transform (DWHT) and Hadamard transform (DHDT). The factors affecting the speedup of vector and parallel processing of these transforms are considered. The vectorization techniques are illustrated by an FFT example.