This document presents a methodology for mapping multidimensional transforms onto reconfigurable architectures like FPGAs. The methodology uses tensor product decompositions and permutation matrices to express transforms recursively in terms of lower-order blocks. This allows large transforms to be computed by combining many parallel, smaller transform blocks. Specific examples are given for mapping one-dimensional linear convolution and discrete cosine transforms. The overall goal is to provide a unified framework and design process for implementing multidimensional transforms in a modular, parallel architecture.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
IJCER (www.ijceronline.com) International Journal of computational Engineerin...ijceronline
The document discusses efficient VLSI implementations of image encryption using minimal operations. It proposes using discrete cosine transform (DCT) for image compression and encryption simultaneously. For encryption, a linear feedback shift register generates random numbers added to some DCT outputs. The DCT algorithm and arithmetic operators are optimized to reduce operations and increase throughput. Simulation results show encryption in the frequency domain at 656 million samples per second on an 82 MHz clock.
3 - A critical review on the usual DCT Implementations (presented in a Malays...Youness Lahdili
1) The document reviews and compares various implementations of the Discrete Cosine Transform (DCT), which is widely used in video and image compression standards.
2) It finds that the Loeffler algorithm achieves the theoretical minimum of 11 multiplications for an 8-point DCT, while the Arai algorithm requires only 5 multiplications and 29 additions.
3) The document concludes that while the DCT has been very successful, other transforms like the Discrete Wavelet Transform used in the Daala standard may provide alternatives worth further research to reduce blocking artifacts at high compression.
This document summarizes a research paper that proposes a novel architecture for implementing a 1D lifting integer wavelet transform (IWT) using residue number system (RNS). The key aspects covered are:
1) RNS offers advantages over binary representations for digital signal processing by avoiding carry propagation. A ROM-based approach is proposed for RNS division.
2) The lifting scheme for discrete wavelet transforms is summarized, including split, predict, and update stages.
3) A novel RNS-based architecture is proposed using three main blocks - split, predict, and update - that repeat at each decomposition level. Pipelined implementations of the predict and update blocks are detailed.
This document summarizes an article from the International Journal of Electronics and Communication Engineering & Technology. The article proposes techniques for image steganography based on chaos theory and the contourlet transform. It describes using a modified Arnold cat map to scramble secret data for increased security before embedding it in the contourlet domain of an image. Experimental results showed the proposed method provides high embedding capacity while maintaining good stego image quality compared to wavelet-based approaches. It embeds data in subbands with lower energy to minimize distortion, and evaluates quality using PSNR, SNR, and correlation metrics.
Determining the Efficient Subband Coefficients of Biorthogonal Wavelet for Gr...CSCJournals
In this paper, we propose an invisible blind watermarking scheme for the gray-level images. The cover image is decomposed using the Discrete Wavelet Transform with Biorthogonal wavelet filters and the watermark is embedded into significant coefficients of the transformation. The Biorthogonal wavelet is used because it has the property of perfect reconstruction and smoothness. The proposed scheme embeds a monochrome watermark into a gray-level image. In the embedding process, we use a localized decomposition, means that the second level decomposition is performed on the detail sub-band resulting from the first level decomposition. The image is decomposed into first level and for second level decomposition we consider Horizontal, vertical and diagonal subband separately. From this second level decomposition we take the respective Horizontal, vertical and diagonal coefficients for embedding the watermark. The robustness of the scheme is tested by considering the different types of image processing attacks like blurring, cropping, sharpening, Gaussian filtering and salt and pepper noise effect. The experimental result shows that the embedding watermark into diagonal subband coefficients is robust against different types of attacks.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
IOSR Journal of Electronics and Communication Engineering(IOSR-JECE) is an open access international journal that provides rapid publication (within a month) of articles in all areas of electronics and communication engineering and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in electronics and communication engineering. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
IJCER (www.ijceronline.com) International Journal of computational Engineerin...ijceronline
The document discusses efficient VLSI implementations of image encryption using minimal operations. It proposes using discrete cosine transform (DCT) for image compression and encryption simultaneously. For encryption, a linear feedback shift register generates random numbers added to some DCT outputs. The DCT algorithm and arithmetic operators are optimized to reduce operations and increase throughput. Simulation results show encryption in the frequency domain at 656 million samples per second on an 82 MHz clock.
3 - A critical review on the usual DCT Implementations (presented in a Malays...Youness Lahdili
1) The document reviews and compares various implementations of the Discrete Cosine Transform (DCT), which is widely used in video and image compression standards.
2) It finds that the Loeffler algorithm achieves the theoretical minimum of 11 multiplications for an 8-point DCT, while the Arai algorithm requires only 5 multiplications and 29 additions.
3) The document concludes that while the DCT has been very successful, other transforms like the Discrete Wavelet Transform used in the Daala standard may provide alternatives worth further research to reduce blocking artifacts at high compression.
This document summarizes a research paper that proposes a novel architecture for implementing a 1D lifting integer wavelet transform (IWT) using residue number system (RNS). The key aspects covered are:
1) RNS offers advantages over binary representations for digital signal processing by avoiding carry propagation. A ROM-based approach is proposed for RNS division.
2) The lifting scheme for discrete wavelet transforms is summarized, including split, predict, and update stages.
3) A novel RNS-based architecture is proposed using three main blocks - split, predict, and update - that repeat at each decomposition level. Pipelined implementations of the predict and update blocks are detailed.
This document summarizes an article from the International Journal of Electronics and Communication Engineering & Technology. The article proposes techniques for image steganography based on chaos theory and the contourlet transform. It describes using a modified Arnold cat map to scramble secret data for increased security before embedding it in the contourlet domain of an image. Experimental results showed the proposed method provides high embedding capacity while maintaining good stego image quality compared to wavelet-based approaches. It embeds data in subbands with lower energy to minimize distortion, and evaluates quality using PSNR, SNR, and correlation metrics.
Determining the Efficient Subband Coefficients of Biorthogonal Wavelet for Gr...CSCJournals
In this paper, we propose an invisible blind watermarking scheme for the gray-level images. The cover image is decomposed using the Discrete Wavelet Transform with Biorthogonal wavelet filters and the watermark is embedded into significant coefficients of the transformation. The Biorthogonal wavelet is used because it has the property of perfect reconstruction and smoothness. The proposed scheme embeds a monochrome watermark into a gray-level image. In the embedding process, we use a localized decomposition, means that the second level decomposition is performed on the detail sub-band resulting from the first level decomposition. The image is decomposed into first level and for second level decomposition we consider Horizontal, vertical and diagonal subband separately. From this second level decomposition we take the respective Horizontal, vertical and diagonal coefficients for embedding the watermark. The robustness of the scheme is tested by considering the different types of image processing attacks like blurring, cropping, sharpening, Gaussian filtering and salt and pepper noise effect. The experimental result shows that the embedding watermark into diagonal subband coefficients is robust against different types of attacks.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
IOSR Journal of Electronics and Communication Engineering(IOSR-JECE) is an open access international journal that provides rapid publication (within a month) of articles in all areas of electronics and communication engineering and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in electronics and communication engineering. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
IOSR Journal of Electronics and Communication Engineering(IOSR-JECE) is an open access international journal that provides rapid publication (within a month) of articles in all areas of electronics and communication engineering and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in electronics and communication engineering. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
This document provides an overview of multi-dimensional RNNs and some architectural issues and recent results related to them. It begins with an introduction to RNNs compared to feedforward neural networks, and solutions like LSTM and GRU to address the vanishing gradient problem. It then discusses several generalizations of the simple RNN architecture, including directionality with BRNN/BLSTM, dimensionality with MDRNN/MDLSTM, and directionality + dimensionality with MDMDRNN. It also covers hierarchical subsampling with HSRNN. The document concludes by summarizing some recent examples that apply these ideas, such as 2D LSTM for scene labeling, as well as new ideas like ReNet, PyraMiD-LSTM, and Grid LSTM.
FAST AND EFFICIENT IMAGE COMPRESSION BASED ON PARALLEL COMPUTING USING MATLABJournal For Research
Image compression technique is used in many applications for example, satellite imaging, medical imaging, video where the size of the iamge requires more space to store, in such application image compression effectively can be used. There are two types in image compression techniques Lossy and Lossless comression. Both these techniques are used for compression of images, but these techniques are not fast. The image compression techniques both lossy and lossless image compression techniques are not fast, they take more time for compression and decompression. For fast and efficient image compression a parallel computing technique is used in matlab. Matlab is used in this project for parallel computing of images. In this paper we will discuss Regular image compression technique, three alternatives of parallel computing using matlab, comparison of image compression with and without parallel computing.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Efficient Implementation of Low Power 2-D DCT ArchitectureIJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This summary provides the key details from the document in 3 sentences:
The document proposes a new method for encrypting two images into a single encrypted image using generalized weighted fractional Fourier transform (GWFRFT) with double random phase encoding. The encryption process involves applying pixel scrambling, phase encoding, and two rounds of GWFRFT with random phase masks on the combined image signal. This technique is shown to provide comparable security to the Advanced Encryption Standard (AES) with a 232-bit key size through a high number of possible permutations in the GWFRFT parameters and orders.
An Efficient Multiplierless Transform algorithm for Video CodingCSCJournals
This paper presents an efficient algorithm to accelerate software video encoders/decoders by reducing the number of arithmetic operations for Discrete Cosine Transform (DCT). A multiplierless Ramanujan Ordered Number DCT (RDCT) is presented which computes the coefficients using shifts and addition operations only. The reduction in computational complexity has improved the performance of the video codec by almost 58% compared with the commonly used integer DCT. The results show that significant computation reduction can be achieved with negligible average peak signal-to-noise ratio (PSNR) degradation. The average structural similarity index matrix (SSIM) also ensures that the degradation due to the approximation is minimal.
This document discusses matrix inversion techniques for MIMO wireless communication systems. It begins by introducing how matrix inversion is used in algorithms for MIMO systems and standards like 802.11n. Existing matrix inversion approaches cannot achieve the performance needed for real-time 802.11n systems. The document then presents a new matrix inversion algorithm based on modified squared Givens rotations (MSGR) that enables real-time implementation with high throughput and low latency. This algorithm overcomes limitations of other QR decomposition techniques. Finally, the document evaluates this algorithm integrated into a MIMO receiver and demonstrates it can support the requirements of modern wireless standards like 802.11n.
This document summarizes a study of modified noise-shaper architectures for oversampled sigma-delta digital-to-analog converters (ΣΔDACs). Two hybrid architectures, A1 and A2, are investigated to trade off noise-shaper and digital-to-analog converter (DAC) complexity while maintaining signal-to-noise ratio (SNR). Simulation results show that architecture A1 achieves fairly good SNR by reducing the number of bits to the noise shaper, while architecture A2 further reduces DAC complexity at the cost of doubling the number of DACs. The number of required DAC unit elements is computed and compared for different architectures and parameter values, illustrating the complexity tradeoffs between noise shaping
❶ Capture total energy of relevant mode (Mechanical, Electrostatic, Dissipation)
❷ Krylov/Arnoldi methods to generate Lagrangian formulation
❸ Create Compact model for system modeling
MATHEON D-Day: Numerical simulation of integrated circuits for future chip ge...Dagmar Monett
The document discusses numerical simulation methods for integrated circuits. It notes that current circuit models contain many parameters that cannot be fully interpreted physically. New modeling concepts and rigorous mathematical analysis are needed to develop simulation methods for future chip generations. The document outlines an algorithm that uses automatic differentiation via ADOL-C to compute the tractability index of differential algebraic equations, which is important for index determination and consistent initialization. This helps analyze complex systems described by high-order derivatives.
A Novel Algorithm for Watermarking and Image Encryption cscpconf
Digital watermarking is a method of copyright protection of audio, images, video and text. We
propose a new robust watermarking technique based on contourlet transform and singular value
decomposition. The paper also proposes a novel encryption algorithm to store a signed double
matrix as an RGB image. The entropy of the watermarked image and correlation coefficient of
extracted watermark image is very close to ideal values, proving the correctness of proposed
algorithm. Also experimental results show resiliency of the scheme against large blurring attack
like mean and gaussian filtering, linear filtering (high pass and low pass filtering) , non-linear
filtering (median filtering), addition of a constant offset to the pixel values and local exchange of pixels .Thus proving the security, effectiveness and robustness of the proposed watermarking algorithm.
FPGA Implementation of 2-D DCT & DWT Engines for Vision Based Tracking of Dyn...IJERA Editor
Real time motion estimation for tracking is a challenging task. Several techniques can transform an image into frequency domain, such as DCT, DFT and wavelet transform. Direct implementation of 2-D DCT takes N^4 multiplications for an N x N image which is impractical. The proposed architecture for implementation of 2-D DCT uses look up tables. They are used to store pre-computed vector products that completely eliminate the multiplier. This makes the architecture highly time efficient, and the routing delay and power consumption is also reduced significantly. Another approach, 2-D discrete wavelet transform based motion estimation (DWT-ME) provides substantial improvements in quality and area. The proposed architecture uses Haar wavelet transform for motion estimation. In this paper, we present the comparison of the performance of discrete cosine transform, discrete wavelet transform for implementation in motion estimation.
Low complexity design of non binary ldpc decoder using extended min-sum algor...eSAT Journals
This document summarizes a research paper on reducing the computational complexity of non-binary LDPC decoders using an extended min-sum algorithm. It introduces low-density parity check codes and non-binary LDPC codes. It then describes an extended min-sum decoding algorithm and proposes two modifications to the parity check matrix - a lower diagonal matrix and a doubly diagonal matrix - to reduce complexity while maintaining performance. Simulation results on code lengths of 504 and 648 bits show the doubly diagonal matrix achieves the best bit error rate. Analysis finds the lower diagonal matrix has the lowest computational complexity of the approaches.
Analytical Delay Model for Distributed On-Chip RLCG Global Interconnects for ...IDES Editor
Fast delay estimation methods, as compared to
simulation techniques, are needed for incremental
performance-driven layout synthesis. On-chip inductive and
conductive effects are becoming predominant in deep
submicron (DSM) interconnects due to increasing clock
speeds; circuit complexity and interconnect lengths.
Inductance causes noise in the signal waveforms, which can
adversely affect the performance of the circuit and signal
integrity. Elmore delay-based estimation methods, although
efficient, fails to accurately estimate the delay for RLCG
interconnect lines. This paper presents an analytical delay
model, based on first and second moments of RLCG
interconnection lines, that considers the effect of inductance
and conductance for the estimation of delay in interconnection
lines. Simulation results justify the efficacy of the proposed
delay modelling approach.
Este documento analiza el ciclo urbano del agua en la cuenca del río Guadalquivir. Presenta datos sobre usos del suelo, población, demandas actuales y futuras de agua por uso, presiones como la contaminación, el estado de las masas de agua, y la recuperación de costes. Predice que para el año 2015 la demanda de agua corregida será de 393 Hm3 para abastecimiento y 3.101 Hm3 para uso agrícola, lo que representa una disminución del 15% y 9% respectivamente. También
El documento presenta un recurso interactivo para que estudiantes entre 10 y 13 años tomen decisiones que contribuyan a la paz y armonía social. El recurso ofrece escenarios comunes con situaciones que requieren una decisión, y evalúa las consecuencias de las opciones para promover valores como la solidaridad y justicia.
O documento discute a doação de órgãos no Brasil e em Portugal, incluindo legislação sobre doação viva e cadáver, contraindicações para doação, identificação de doadores e órgãos transplantados.
El documento describe el proceso de instalación de osCommerce, una solución de código abierto para tiendas de comercio electrónico. Explica los requisitos previos como PHP 4.1 o superior, MySQL y Apache, y los pasos de instalación que incluyen importar el catálogo de muestra, completar la configuración del servidor, eliminar la carpeta install, proteger la carpeta admin y establecer permisos en los archivos.
El poema Cuerpo del amanecer de Miguel Hernández describe una flor, posiblemente un girasol, que se abre y cierra en el transcurso de un día y eventualmente muere, pero cuya semilla permanece bajo tierra para dar vida a nuevas flores. El poema sugiere que aunque mueran, los seres humanos dejan descendencia que continúa la cadena de la vida, como las semillas bajo tierra que germinan nuevas plantas.
IOSR Journal of Electronics and Communication Engineering(IOSR-JECE) is an open access international journal that provides rapid publication (within a month) of articles in all areas of electronics and communication engineering and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in electronics and communication engineering. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
This document provides an overview of multi-dimensional RNNs and some architectural issues and recent results related to them. It begins with an introduction to RNNs compared to feedforward neural networks, and solutions like LSTM and GRU to address the vanishing gradient problem. It then discusses several generalizations of the simple RNN architecture, including directionality with BRNN/BLSTM, dimensionality with MDRNN/MDLSTM, and directionality + dimensionality with MDMDRNN. It also covers hierarchical subsampling with HSRNN. The document concludes by summarizing some recent examples that apply these ideas, such as 2D LSTM for scene labeling, as well as new ideas like ReNet, PyraMiD-LSTM, and Grid LSTM.
FAST AND EFFICIENT IMAGE COMPRESSION BASED ON PARALLEL COMPUTING USING MATLABJournal For Research
Image compression technique is used in many applications for example, satellite imaging, medical imaging, video where the size of the iamge requires more space to store, in such application image compression effectively can be used. There are two types in image compression techniques Lossy and Lossless comression. Both these techniques are used for compression of images, but these techniques are not fast. The image compression techniques both lossy and lossless image compression techniques are not fast, they take more time for compression and decompression. For fast and efficient image compression a parallel computing technique is used in matlab. Matlab is used in this project for parallel computing of images. In this paper we will discuss Regular image compression technique, three alternatives of parallel computing using matlab, comparison of image compression with and without parallel computing.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Efficient Implementation of Low Power 2-D DCT ArchitectureIJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This summary provides the key details from the document in 3 sentences:
The document proposes a new method for encrypting two images into a single encrypted image using generalized weighted fractional Fourier transform (GWFRFT) with double random phase encoding. The encryption process involves applying pixel scrambling, phase encoding, and two rounds of GWFRFT with random phase masks on the combined image signal. This technique is shown to provide comparable security to the Advanced Encryption Standard (AES) with a 232-bit key size through a high number of possible permutations in the GWFRFT parameters and orders.
An Efficient Multiplierless Transform algorithm for Video CodingCSCJournals
This paper presents an efficient algorithm to accelerate software video encoders/decoders by reducing the number of arithmetic operations for Discrete Cosine Transform (DCT). A multiplierless Ramanujan Ordered Number DCT (RDCT) is presented which computes the coefficients using shifts and addition operations only. The reduction in computational complexity has improved the performance of the video codec by almost 58% compared with the commonly used integer DCT. The results show that significant computation reduction can be achieved with negligible average peak signal-to-noise ratio (PSNR) degradation. The average structural similarity index matrix (SSIM) also ensures that the degradation due to the approximation is minimal.
This document discusses matrix inversion techniques for MIMO wireless communication systems. It begins by introducing how matrix inversion is used in algorithms for MIMO systems and standards like 802.11n. Existing matrix inversion approaches cannot achieve the performance needed for real-time 802.11n systems. The document then presents a new matrix inversion algorithm based on modified squared Givens rotations (MSGR) that enables real-time implementation with high throughput and low latency. This algorithm overcomes limitations of other QR decomposition techniques. Finally, the document evaluates this algorithm integrated into a MIMO receiver and demonstrates it can support the requirements of modern wireless standards like 802.11n.
This document summarizes a study of modified noise-shaper architectures for oversampled sigma-delta digital-to-analog converters (ΣΔDACs). Two hybrid architectures, A1 and A2, are investigated to trade off noise-shaper and digital-to-analog converter (DAC) complexity while maintaining signal-to-noise ratio (SNR). Simulation results show that architecture A1 achieves fairly good SNR by reducing the number of bits to the noise shaper, while architecture A2 further reduces DAC complexity at the cost of doubling the number of DACs. The number of required DAC unit elements is computed and compared for different architectures and parameter values, illustrating the complexity tradeoffs between noise shaping
❶ Capture total energy of relevant mode (Mechanical, Electrostatic, Dissipation)
❷ Krylov/Arnoldi methods to generate Lagrangian formulation
❸ Create Compact model for system modeling
MATHEON D-Day: Numerical simulation of integrated circuits for future chip ge...Dagmar Monett
The document discusses numerical simulation methods for integrated circuits. It notes that current circuit models contain many parameters that cannot be fully interpreted physically. New modeling concepts and rigorous mathematical analysis are needed to develop simulation methods for future chip generations. The document outlines an algorithm that uses automatic differentiation via ADOL-C to compute the tractability index of differential algebraic equations, which is important for index determination and consistent initialization. This helps analyze complex systems described by high-order derivatives.
A Novel Algorithm for Watermarking and Image Encryption cscpconf
Digital watermarking is a method of copyright protection of audio, images, video and text. We
propose a new robust watermarking technique based on contourlet transform and singular value
decomposition. The paper also proposes a novel encryption algorithm to store a signed double
matrix as an RGB image. The entropy of the watermarked image and correlation coefficient of
extracted watermark image is very close to ideal values, proving the correctness of proposed
algorithm. Also experimental results show resiliency of the scheme against large blurring attack
like mean and gaussian filtering, linear filtering (high pass and low pass filtering) , non-linear
filtering (median filtering), addition of a constant offset to the pixel values and local exchange of pixels .Thus proving the security, effectiveness and robustness of the proposed watermarking algorithm.
FPGA Implementation of 2-D DCT & DWT Engines for Vision Based Tracking of Dyn...IJERA Editor
Real time motion estimation for tracking is a challenging task. Several techniques can transform an image into frequency domain, such as DCT, DFT and wavelet transform. Direct implementation of 2-D DCT takes N^4 multiplications for an N x N image which is impractical. The proposed architecture for implementation of 2-D DCT uses look up tables. They are used to store pre-computed vector products that completely eliminate the multiplier. This makes the architecture highly time efficient, and the routing delay and power consumption is also reduced significantly. Another approach, 2-D discrete wavelet transform based motion estimation (DWT-ME) provides substantial improvements in quality and area. The proposed architecture uses Haar wavelet transform for motion estimation. In this paper, we present the comparison of the performance of discrete cosine transform, discrete wavelet transform for implementation in motion estimation.
Low complexity design of non binary ldpc decoder using extended min-sum algor...eSAT Journals
This document summarizes a research paper on reducing the computational complexity of non-binary LDPC decoders using an extended min-sum algorithm. It introduces low-density parity check codes and non-binary LDPC codes. It then describes an extended min-sum decoding algorithm and proposes two modifications to the parity check matrix - a lower diagonal matrix and a doubly diagonal matrix - to reduce complexity while maintaining performance. Simulation results on code lengths of 504 and 648 bits show the doubly diagonal matrix achieves the best bit error rate. Analysis finds the lower diagonal matrix has the lowest computational complexity of the approaches.
Analytical Delay Model for Distributed On-Chip RLCG Global Interconnects for ...IDES Editor
Fast delay estimation methods, as compared to
simulation techniques, are needed for incremental
performance-driven layout synthesis. On-chip inductive and
conductive effects are becoming predominant in deep
submicron (DSM) interconnects due to increasing clock
speeds; circuit complexity and interconnect lengths.
Inductance causes noise in the signal waveforms, which can
adversely affect the performance of the circuit and signal
integrity. Elmore delay-based estimation methods, although
efficient, fails to accurately estimate the delay for RLCG
interconnect lines. This paper presents an analytical delay
model, based on first and second moments of RLCG
interconnection lines, that considers the effect of inductance
and conductance for the estimation of delay in interconnection
lines. Simulation results justify the efficacy of the proposed
delay modelling approach.
Este documento analiza el ciclo urbano del agua en la cuenca del río Guadalquivir. Presenta datos sobre usos del suelo, población, demandas actuales y futuras de agua por uso, presiones como la contaminación, el estado de las masas de agua, y la recuperación de costes. Predice que para el año 2015 la demanda de agua corregida será de 393 Hm3 para abastecimiento y 3.101 Hm3 para uso agrícola, lo que representa una disminución del 15% y 9% respectivamente. También
El documento presenta un recurso interactivo para que estudiantes entre 10 y 13 años tomen decisiones que contribuyan a la paz y armonía social. El recurso ofrece escenarios comunes con situaciones que requieren una decisión, y evalúa las consecuencias de las opciones para promover valores como la solidaridad y justicia.
O documento discute a doação de órgãos no Brasil e em Portugal, incluindo legislação sobre doação viva e cadáver, contraindicações para doação, identificação de doadores e órgãos transplantados.
El documento describe el proceso de instalación de osCommerce, una solución de código abierto para tiendas de comercio electrónico. Explica los requisitos previos como PHP 4.1 o superior, MySQL y Apache, y los pasos de instalación que incluyen importar el catálogo de muestra, completar la configuración del servidor, eliminar la carpeta install, proteger la carpeta admin y establecer permisos en los archivos.
El poema Cuerpo del amanecer de Miguel Hernández describe una flor, posiblemente un girasol, que se abre y cierra en el transcurso de un día y eventualmente muere, pero cuya semilla permanece bajo tierra para dar vida a nuevas flores. El poema sugiere que aunque mueran, los seres humanos dejan descendencia que continúa la cadena de la vida, como las semillas bajo tierra que germinan nuevas plantas.
Este documento describe la composición, tipos, funciones y clasificación de los huesos del esqueleto humano. Los huesos están compuestos principalmente de minerales como fosfato y carbonato de calcio, así como colágeno y otras proteínas. Cumplen funciones como dar soporte al cuerpo, permitir el movimiento, proteger órganos internos y almacenar minerales y energía. El esqueleto se divide en esqueleto axial, que incluye los huesos de la columna vertebral y cráneo, y esquele
Modelling Quantum Transport in Nanostructuresiosrjce
IOSR Journal of Electronics and Communication Engineering(IOSR-JECE) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of electronics and communication engineering and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in electronics and communication engineering. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
This document summarizes three methods for modeling quantum transport in nanostructures:
1) The non-equilibrium Green's function (NEGF) method provides a rigorous description of quantum transport by solving Poisson's equation and the quantum transport solver based on NEGF formalism self-consistently.
2) The recursive Green's function method computes the Green's function recursively without full matrix inversion, reducing computational efforts.
3) The Gauss estimation method computes spectral coefficients representing the Green's function to estimate current at discrete longitudinal field values rather than integrating over the entire field.
DESPECKLING OF SAR IMAGES BY OPTIMIZING AVERAGED POWER SPECTRAL VALUE IN CURV...ijistjournal
The document describes a novel algorithm for despeckling synthetic aperture radar (SAR) images using particle swarm optimization (PSO) in the curvelet domain. The algorithm first identifies homogeneous regions in the speckled image using variance calculations. It then uses PSO to optimize the thresholding of curvelet coefficients, with the objective of minimizing the average power spectral value. This provides an optimized threshold to apply curvelet-based despeckling. The proposed method is tested on standard images and shown to outperform conventional filters like median and Lee filters in reducing speckle noise.
DESPECKLING OF SAR IMAGES BY OPTIMIZING AVERAGED POWER SPECTRAL VALUE IN CURV...ijistjournal
Synthetic Aperture Radar (SAR) images are inherently affected by multiplicative speckle noise, due to the coherent nature of scattering phenomena. In this paper, a novel algorithm capable of suppressing speckle noise using Particle Swarm Optimization (PSO) technique is presented. The algorithm initially identifies homogenous region from the corrupted image and uses PSO to optimize the Thresholding of curvelet coefficients to recover the original image. Average Power Spectrum Value (APSV) has been used as objective function of PSO. The Proposed algorithm removes Speckle noise effectively and the performance of the algorithm is tested and compared with Mean filter, Median filter, Lee filter, Statistic Lee filter, Kuan filter, frost filter and gamma filter., outperforming conventional filtering methods.
This document describes two techniques for designing optical XNOR and NAND logic gates. The first technique uses a 2D array of coupled optical cavities with Kerr nonlinearity. Discrete cavity solitons are numerically simulated and used to demonstrate optical XNOR and NAND gates by controlling soliton interactions with a Gaussian beam. The second technique uses multi-mode interference waveguides to convert the phase of binary-phase-shift keying input signals to amplitude at the output, implementing optical XNOR and NAND logic. Numerical simulations using the finite element method show contrast ratios of 21.5 dB for the XNOR gate and 22.3 dB for the NAND gate.
This document summarizes an article from the International Journal of Electronics and Communication Engineering & Technology. The article proposes techniques for image steganography based on chaos theory and the contourlet transform. It describes using a modified Arnold cat map to scramble secret data for increased security before embedding it in the contourlet domain of an image. Experimental results showed this approach provides high embedding capacity while maintaining good stego image quality as measured by PSNR, SNR, and correlation metrics. It was found to perform better than similar wavelet-based steganography methods.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
Investigation of repeated blasts at Aitik mine using waveform cross correlationIvan Kitov
We present results of signal detection from repeated events at the Aitik and Kiruna mines in Sweden as based on waveform cross correlation. Several advanced methods based on tensor Singular Value Decomposition is applied to waveforms measured at seismic array ARCES, which consists of three-component sensors.
This document summarizes a research paper on using wavelet neural networks (WNNs) for adaptive equalization in digital communication systems. The paper proposes using WNNs structured with wavelet basis functions as the activation functions. The orthogonal least squares (OLS) algorithm is then used to update the weighting matrix and select the most important wavelet basis units, reducing redundancy. The experimental results showed that a WNN equalizer using OLS outperformed conventional neural network equalizers in terms of signal-to-noise ratio and ability to handle non-linear channels.
This document summarizes a research paper on using wavelet neural networks (WNNs) for adaptive equalization in digital communication systems. The paper proposes using WNNs structured with wavelet basis functions as the activation functions. The orthogonal least squares (OLS) algorithm is then used to update the weighting matrix and select the most important wavelet basis units, reducing redundancy. The experimental results showed that a WNN equalizer using OLS outperformed conventional neural network equalizers in terms of signal-to-noise ratio and ability to handle non-linear channels.
FINGERPRINTS IMAGE COMPRESSION BY WAVE ATOMScsandit
The document presents a study comparing fingerprint image compression using wavelets and wave atoms transforms. It finds that wave atoms transforms provide better performance than current wavelet-based standards like WSQ. Specifically:
- Wave atoms achieved higher PSNR values and compression ratios than wavelets when reconstructing images from a reduced number of coefficients.
- An algorithm was proposed using wave atom decomposition, non-uniform quantization, and entropy coding that achieved a compression ratio of 18 with a PSNR of 35.04 dB, outperforming the WSQ standard.
- Minutiae detection on original and reconstructed images showed wave atoms better preserved local fingerprint structures. Therefore, wave atoms are concluded to be more suitable than wavelets
FINGERPRINTS IMAGE COMPRESSION BY WAVE ATOMScsandit
The fingerprint images compression based on geometric transformed presents important
research topic, these last year’s many transforms have been proposed to give the best
representation to a particular type of image “fingerprint image”, like classics wavelets and
wave atoms. In this paper we shall present a comparative study between this transforms, in
order to use them in compression. The results show that for fingerprint images, the wave atom
offers better performance than the current transform based compression standard. The wave
atoms transformation brings a considerable contribution on the compression of fingerprints
images by achieving high values of ratios compression and PSNR, with a reduced number of
coefficients. In addition, the proposed method is verified with objective and subjective testing.
This document discusses a proposed architecture for a higher Nyquist-range digital-to-analog converter (DAC) that employs sinusoidal interpolation.
[1] Conventional DACs operate within the Nyquist range, but the proposed architecture aims to utilize higher Nyquist ranges by approximating an oscillating signal from an RF DAC concept using sinusoidal interpolation in the time domain.
[2] The proposed architecture quantizes both the input signal and pulse amplitude modulation waveform and combines them digitally, replacing analog oscillatory circuits with a digital data stream. This reduces analog complexity compared to existing techniques.
[3] Simulation results and theoretical analysis are presented to support that the proposed architecture can provide similar performance
This document describes a novel design of ternary logic gates using carbon nanotube field-effect transistors (CNTFETs). The authors propose a CNTFET-based design for ternary logic gates that eliminates the need for large off-chip resistors used in previous designs. Simulation results show the proposed ternary logic gates consume significantly lower power and delay compared to previous resistive-load CNTFET gate implementations. When used in arithmetic circuits like a full adder and multiplier, the proposed ternary gates combined with binary gates can reduce power delay product by over 90%.
At this present scenario, the demand of the system capacity is very high in wireless network. MIMO
technology is used from the last decade to provide this requirement for wireless network antenna
technology. MIMO channels are mostly used for advanced antenna array technology. But it is most
important to control the error rate with enhanced system capacity in MIMO for present-day progressive
wireless communication. This paper explores the frame error rate with respect to different path gain of
MIMO channel. This work has been done in different fading scenario and produces a comparative analysis
of MIMO on the basis of those fading models in various conditions. Here, it is to be considered that
modulation technique as QPSK to observe these comparative evaluations for different Doppler frequencies.
From the comparative analysis, minimum amount of frame error rate is viewed for Rician distribution at
LOS path Doppler shift of 0 Hz. At last, this work is concluded with a comparative bit error rate study on
the basis of singular parameters at different SNR levels to produce the system performance for uncoded
QPSK modulation.
Performance Assessment of Polyphase Sequences Using Cyclic Algorithmrahulmonikasharma
Polyphase Sequences (known as P1, P2, Px, Frank) exist for a square integer length with good auto correlation properties are helpful in the several applications. Unlike the Barker and Binary Sequences which exist for certain length and exhibits a maximum of two digit merit factor. The Integrated Sidelobe level (ISL) is often used to define excellence of the autocorrelation properties of given Polyphase sequence. In this paper, we present the application of Cyclic Algorithm named CA which minimizes the ISL (Integrated Sidelobe Level) related metric which in turn improve the Merit factor to a greater extent is main thing in applications like RADAR, SONAR and communications. To illustrate the performance of the P1, P2, Px, Frank sequences when cyclic Algorithm is applied. we presented a number of examples for integer lengths. CA(Px) sequence exhibits the good Merit Factor among all the Polyphase sequences that are considered.
This document presents results from a lattice QCD calculation of the proton isovector scalar charge (gs) at two light quark masses. The calculation uses domain-wall fermions and Iwasaki gauge actions on a 323x64 lattice with a spacing of 0.144 fm. Ratios of three-point to two-point correlation functions are formed and fit to a plateau to extract gs. Values of gs are obtained for quark masses of 0.0042 and 0.001, and all-mode averaging is used for the lighter mass. Chiral perturbation theory will be used to extrapolate gs to the physical quark mass. Preliminary results for gs at the unphysical quark masses are reported in lattice units.
In this paper, a new algorithm for a high resolution
Direction Of Arrival (DOA) estimation method for multiple
wideband signals is proposed. The proposed method proceeds
in two steps. In the first step, the received signals data is
decomposed in a Toeplitz form using the first-order statistics.
In the second step, The QR decomposition is applied on the
constructed Toeplitz matrix. Compared with existing schemes,
the proposed scheme provides several advantages. First, it
requires computing the triangular matrix R or the orthogonal
matrix Q to find the DOA; these matrices can be computed
with O(n2) operation. However, most of the existing schemes
required eignvalue decomposition (EVD) for the covariance
matrix or singular value decomposition (SVD) for the data
matrix; using EVD or SVD requires much more complex
computational O(n3) operation. Second, the proposed scheme
is more suitable for high-speed communication since it
requires first-order statistics and a single snapshot. Third,
the proposed scheme can estimate the correlated wideband
signals without using spatial smoothing techniques; whereas,
already-existing schemes do not. Accuracy of the proposed
wideband DOA estimation method is evaluated through
computer simulation in comparison with a conventional
method.
IRJET- An Efficient Reverse Converter for the Three Non-Coprime Moduli Set {4...IRJET Journal
This paper proposes a new and efficient reverse converter for converting residue numbers to decimal numbers for the three moduli set {6, 10, 15} which shares the common factor of 5. The proposed converter replaces larger multipliers used in previous converters with smaller multipliers and adders, reducing the hardware requirements. The hardware implementation of the proposed converter is presented and compared to other state-of-the-art converters, showing it performs better with fewer adders and multipliers. The proposed converter efficiently implements reverse conversion for the non-coprime three moduli set while requiring less hardware than previous approaches.
IRJET- An Efficient Reverse Converter for the Three Non-Coprime Moduli Set {4...
Fx3111501156
1. Ayman Elnaggar, Mokhtar Aboelaze / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January -February 2013, pp.1150-1156
Embedded Reconfigurable Architectures for Multidimensional
Transforms
Ayman Elnaggar, Mokhtar Aboelaze
Department of Computer Science & Engineering, German University in Cairo, New Cairo City, Egypt
Department of Computer Science, York University, Toronto, Canada M3J 1P3
Abstract
This paper presents a general approach AB CD ( A B)(C D) (1)
for generating higher order (longer size)
multidimensional (m-d) architectures from 2
m
( A B) C A ( B C) (2)
lower order (shorter sizes) architectures. The If n n1n2 , then
objective of our work is to derive a unified
framework and a design methodology that allows An1 Bn2 Pn,n1 ( I n2 An1 ) Pn,n2 ( I n1 Bn2 ) (3)
direct mapping of the proposed algorithms into If n n1n2 n3 , then
embedded reconfigurable architectures such as
FPGAs. Our methodology is based on I n1 An2 I n3 Pn, n1n2 ( I n1n3 An2 ) Pn, n3 (4)
manipulating tensor product forms so that they If 2n n1n2 , then
can be mapped directly into modular parallel Pn, 2 Pn, n1 Pn, n2 (5)
architectures. The resulting circuits have very
simple modular structure and regular topology. Where denotes the tensor product, I n is
the identity matrix of size n, and Pn, s , the
Keywords – Reconfigurable Architectures,
Recursive algorithms, multidimensional transforms, permutation matrix, is n n binary matrix whose
tensor products, permutation matrices. entries are zeroes and ones, such that each row or
column of has a single 1 entry. If n rs then Pn, s
I.INTRODUCTION n
This paper proposes an efficient and cost- is an n n binary matrix specifying an -shuffle
effective general methodology for mapping s
multidimensional transforms onto efficient (or s-stride) permutation. The effect of the
reconfigurable architectures such as FPGAs. The permutation matrix Pn, s on an input vector X n of
main objective of this paper is to derive a design
methodology and recursive formulation for the length n is to shuffle the elements of X n by
multidimensional transforms which is useful for the grouping all the r elements separated by distance s
true modularization and parallelization of the together. The first r element will be
resulting computation. x0 , x s , x 2 s , , x( r 1) s , the next r elements are
Our methodology employs tensor product
(or Kronecker products) decompositions and x1 , x1 s , x1 2 s , , x1( r 1) s , and so on.
permutation matrices as the main tools for
expressing the general framework for The main result reported in this paper
multidimensional DSP transforms. We employ shows that a large two-dimensional (2-d)
several techniques to manipulate such computation for a given DSP transform on an n n
decompositions into suitable recursive expressions input array can be decomposed recursively into three
which can be mapped efficiently onto reconfigurable stages as shown in Fig. 1 for the case n 4 . The
FPGAs structures. middle stage is constructed recursively from 22
Our work is based on a non-trivial generalization of parallel (data-independent) blocks each realizing a
the one-dimensional DSP transforms. It has been smaller-size computation of the same DSP
shown that when coupled with stride permutation transform. The pre-additions and the post-
matrices, tensor products provide a unifying permutations stages serve as "glue" circuits that
framework for describing and programming a wide combine the 22 lower order blocks to construct the
range of fast recursive algorithms for various higher order architecture. We also show that the
transform. This unifying framework is suited for proposed unified approach can be extended such that
parallel processing machines and vector processing an m-d DSP transform can be constructed from 2m
machines [6], [10]. smaller size m-d ones. The objective of our work is
Some of the tensor product properties that will be to derive a unified framework and a design
used throughout this paper are [6], [10]: methodology that allows direct mapping of the
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2. Ayman Elnaggar, Mokhtar Aboelaze / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January -February 2013, pp.1150-1156
proposed algorithms into reconfigurable FPGAs
1 0
1 0 0
A 1 1 ,
architectures.
B 1 1 1
Observe that, we have drawn our networks
0 1 0 0 1
such that data flows from right to left. We chose this
convention to show the direct correspondence In this case j 3 (three parallel blocks of the
between the derived algorithms and the proposed convolution of smaller size n / 2) ). The realization
reconfigurable architecture. of the 1-d linear convolution is shown in Fig. 1.
I. A GENERAL FRAMEWORK FOR 1-D RECURSIVE
DSP TRANSFORMS
In this section, we present a general
framework to derive recursive formulations for
multidimensional transforms. Given a 1-d DSP
algorithm in a matrix-vector form
Ym Tm, n X n (6)
Where, Tm, n is the transform matrix, X n
and Ym are the input vector of size n and the
Fig. 1. The realization of the 1-d linear convolution
output vector of size m , respectively. Then, using
sparse matrix factorization approach [9], the matrix B. The 1-d DCT
Tm, n can be factorized so that
Tm, n T1 T2 Tk (7) The 1-d DCT Tn of size n can be written as [3], [7]
Where, each of the matrices T1 , T2 ,, Tk Tn Rn ( I 3 Tn / 2 ) Qn . (10)
is sparse. Sparseness implies that either most of the Where
elements of the matrix are zeros or the matrix in the Rn Pn, n / 2 ( I n / 2 Ln / 2 ) ,
block diagonal form. By applying tensor product
property (3) to the block diagonal matrices of Qn ( I 2 V 1 ) ( I n / 2 Cn / 2 ) ( F2 I n / 2 ) Vn .
n/2
equation (7), we have
1
C n diag ,
Tm, n ( Ri ) ( I j Tm / 2 , n / 2 ) (Qk ) (8) 2 cosn
n (4M 1), M 0, 1, , n 1,
Where, Q k and Ri are the pre- and post- 2n
processing glue structure that combine j blocks in 1 1 0 0 0 0
parallel of the lower-order transform of size 0 1 1 0 0 0
Tm / 2 , n / 2 .
0 0 1 1 0 0
Lm ,
A. The 1-d Linear Convolution 0
The 1-d linear convolution matrix C (n) of 0 0 0 0 1 0
size n 2 , where is an integer can be written 0 0 0 0 0 1
as [5], [6] Vn ( I n / 2 J n / 2 ) Pn,2 ,
C (n) Rn ( I 3 C (n / 2) ) Qn . (9) 1 1
F2 ,
1 1
Where,
I n / 2 is the identity matrix of dimension n / 2 ,
Qn ( P 1 2 ) 1[( I 1 A)P 1 ].
2 3,2 3 2 2 ,2 is the direct sum operator, J n / 2 is the exchange
Rn R k ( P B) P
3( 2 1),3 2 1 3( 2 1),( 2 1)
(I )
2
matrix of order n / 2 defined as
1151 | P a g e
3. Ayman Elnaggar, Mokhtar Aboelaze / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January -February 2013, pp.1150-1156
0 0 0 0 third stage) will be replaced by the single
0 0 1 0
permutation P8,2 as shown in Fig. 3 (b). Similarly,
equation (13) can be simplified to
Jn / 2 , k 1
Wn Pn , 2 ( I 2k 1 W2 ) (15)
0 1 0 0 i 0
1
0 0 0
Thus, Wn can be computed by the
cascaded product of k similar stages (independent of
In this case j 2 (two parallel blocks of the DCT of i) of double matrix products instead of the triple
matrix products in equation (8). Alternatively, we
smaller size n / 2) ). The realization of the 1-d DCT
can realize (15) by a single block of
is shown in Fig. 2.
Pn , 2 ( I 2k 1 W2 ) and take the output after k
iterations that allows a hardware saving without
slowing down the processing speed and reduction in
the hardware size as shown in Fig. 4 for the case
n 8.
It should be mentioned that we have applied
property (5) to reduce the shuffling inherited in the
original WHT algorithm to allow a uniform
hardware blocks as shown in Fig. 3 (b). We haven’t
modified the original complexity of the WHT that
are centered in the W2 blocks as shown in Fig. 3
and Fig. 4.
Fig. 2. The realization of the 1-d DCT
Applying property (1), equation (12) can be
C. The 1-d WHT
modified to
Our last example is the 1-d WHT. The
original 1-d WHT transform matrix is defined as [1], Wn W2 Wn / 2 I 2 W2 Wn / 2 I n / 2
[2] ( I 2 Wn / 2 ) (W2 I n / 2 ) (16)
W Wn / 2 1 1 ( I 2 Wn / 2 ) Qn
Wn n / 2
W , W2
1 1 ,
(11)
n / 2 Wn / 2
Where, Qn (W2 I n / 2 ) (17)
Where, W2 is the 2-point WHT. Let k log 2 n , we
Equation (16) represents the two-stage
can write equation (11) in the iterative tensor- recursive tensor product formulation of the 1-d WHT
product form
(in this case j 2 ) in which the first stage is the pre-
Wn W2 Wn / 2 W2 W2 W2
additions ( Qn ), followed by the second stage of the
k 1 (12)
( I i W2 I k i 1 )
2 2
core computation I 2 Wn / 2 that consists of a
i 0 parallel blocks of two identical smaller WHT
which using property (4), can be modified to
computations each of size n / 2 as shown in Fig. 5.
k 1
Wn Pn , 2i 1 ( I 2k 1 W2 ) Pn , 2k i 1 (13)
i 0 I. A GENERAL FRAMEWORK FOR 2-D RECURSIVE
As an example, we can express W8 as DSP TRANSFORMS
W8 P8 , 2 ( I 4 W2 ) P8 , 4 P8 , 4 ( I 4 W2 ) P8 , 2 . For a 2-d input data, X n1 , n 2 , of size n1 n2 , and
P8 , 8 ( I 4 W2 ) P8 ,1 a separable 2-d transform, Tn1 , n 2 , we can write the
(14) output, Yn1 , n 2 , in the form
The realization of W8 is shown in Fig. 3 (a).
Yn1 , n 2 = Yn1 , n 2 X n1 , n 2 (18)
Applying property (5) to equation (14)
and noting that now the permutations in two adjacent where, X n1 , n 2 and Yn1 , n 2 are the input and
stages can be grouped together into a single
output column-scanned vectors, respectively. For
permutation, the adjacent permutations P8,2 P8,8
separable matrices, the 2-d transform matrix
(from the first and the second stage) will be replaced Tn1 , n 2 can be written in the tensor product form as
by the single permutation P8,2 and the adjacent
[9]
permutations P8,4 P8,4 (from the second and the
1152 | P a g e
4. Ayman Elnaggar, Mokhtar Aboelaze / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January -February 2013, pp.1150-1156
Tn1 , n 2 Tn1 Tn 2 (19)
Where Tn1 and T n 2 are the row and column 1- Applying properties (1) to (4) to derive the
2-d recursive form
d transforms, respectively as defined in (8). By
replacing Tn1 and T n 2 by their corresponding ~ ~
Tn1, n2 ( Rn1, n2 ) ( I 2 Tn1 / 2, n2 / 2 ) (Qn1, n2 ) (20)
j
values from equation (8) and
1153 | P a g e
5. Ayman Elnaggar, Mokhtar Aboelaze / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January -February 2013, pp.1150-1156
~ ~ Since the convolution matrix C(n/2) is of dimension
Where, Qn , n and Rn , n are the 2-d pre- and
1 2 1 2 [(n 1) n / 2 ] , we can write C(n2 / 2) as
post-processing glue structure, respectively that
C(n2 / 2) I n2 1 .C(n2 / 2) . I n2 / 2 (28)
combine j 2 of the lower-order (smaller size) 2-d
Substituting (21) in (20) and applying property (1),
transform Tn1 / 2, n 2 / 2 of dimension ~
C n1 , n 2 ( P9( n1 1),3( n1 1) ( I 9 C (n1 / 2)
n1 / 2 n2 / 2 . P9( n1 / 2),3 ) ( I n 2 1.C (n 2 / 2). I n 2 / 2 )
A. The 2-d convolution ( P9( n1 1),3( n1 1) I n 2 1 )
Let n1 21 and n2 2 2 . Pratt [9] has ( I 9 C (n1 / 2) C (n 2 / 2))
shown that for an n1 n2 input data image, the 2-d ( P9( n1 / 2),3 I n 2 / 2 ).
convolution output is given by (29)
q Cn1, n2 f (21) Now, substituting (29) in (24) gives
~
~ ~
~
Where, Cn1 ,n2 is the 2-d convolution transform Cn1 ,n2 R ( I 9 Cn1 / 2,n2 / 2 ) Q , (30)
matrix; and q and f are the output and input column-
scanned vectors, respectively of size n n1 n2 . Where, Cn1 / 2, n2 / 2 C(n1 / 2) C(n2 / 2) is the
Pratt has also shown that, for separable transforms, lower order 2-d convolution matrix for an
the matrix Cn1, n2 can be decomposed into the n1 / 2 n2 / 2 input image,
~
~
tensor form Q ( P9( n1 / 2),3 I n2 / 2 ) (Q1 Q2 ) and
Cn1,n2 C ( n1)C ( n2 ) (22) ~
~
R ( R1 R2 ) ( P9(n1 1),3(n1 1) I n 2 1 ) are
Where, C (n1 ) and C (n 2 ) represent row and
column 1-d convolution operators on f, respectively, the new 2-d pre- and post-additions, respectively.
as defined in (8) and (9). From (9) and (22), we can Equation (30) represents the recursive 2-d
express the 2-d convolution matrix as a function of convolution algorithm. In this case we use 9 (
1-d convolutions as follows [4] j 2 32 9 ) of the lower-order C n1 / 2 , n2 / 2
C n1 , n2 [ R1 ( I 3 C (n1 / 2)Q1 ][ R2 ( I 3 C (n2 / 2)Q2 ]
convolution blocks in parallel to generate the higher
(23) order Cn1 , n2 convolution as shown before in Fig. 6.
Applying property (1), leads to
C n1 , n 2 [( R1 R2 )(( I 3 C (n1 / 2))
(( I 3 C ( n 2 / 2))(Q1 Q2 )], (24)
~~ ~
R C n1 , n 2 Q.
Where,
~
R ( R1 R2 ),
~
C n1 , n 2 ( I 3 C (n1 / 2))( I 3 C (n2 / 2)) ,
~
Q (Q1 Q2 )]. B. The 2-d DCT
(25) Since the DCT matrix is separable, the 2-d
~
Note that the matrix Cn1 ,n2 contains the 1-d DCT for an image of dimension n1 n 2 can be
convolutions matrices C(n1 / 2) and C(n2 / 2) in an computed by a stage of n 2 parallel 1-d DCT
involved tensor product expression. By applying computations on n1 points each, followed by
property (2), we can write (24) as
~ another stage of n1 parallel 1-d DCT computations
C n , n (( I 3 C (n1 / 2) I 3 ) C (n2 / 2)) , (26)
1 2
on n 2 points each. This can be represented by the
Applying property (4), yields to
~ matrix-vector form
C n1 , n 2 ( P9(n1 1),3(n1 1) X Tn1 , n 2 x , (31)
( I 9 C (n1 / 2) P9(n1 / 2),3 ) C (n2 / 2)) ,
Where Tn1 , n 2 is the 2-d DCT transform matrix for
(27)
an n1 n 2 image, X and x are the output and input
column-scanned vectors, respectively. By
1154 | P a g e
6. Ayman Elnaggar, Mokhtar Aboelaze / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January -February 2013, pp.1150-1156
substituting (10) in (31), we have
X (Tn1 Tn 2 ) x . (32) ˆ
m
ˆ
X [ R (I m Wni / 2 ) Q] x , (36)
By further manipulation of equation (32) in a similar 2 i1
way to that we did to (23) of the 2-d convolution, m
we can write (23) as [3] Where Wni / 2 is the lower order m-d WHT and
i 1
~ ~
X (Rn1 , n2 (I 4 Tn1 / 2, n2 / 2 ) Qn1, n2 ) x (33) m 1 i m
Where, Q Pu1 ,u 2 I u3 Qi ,
ˆ
k 1 i 1
~ i 1
Qn1 , n2 ( P2n1 ,2 I n2 / 2 ) Qn1 , n2 , and (37)
m 1 m 1
~ R Pw1 , w2 I w3 ,
ˆ
Rn1 , n2 Rn1 , n2 ( P2n1 , n1 I n2 / 2 ) . i 1
l i
Equation (33) represents the truly recursive 2-d m i 1 i
~ ~ u1 2 n j , u 2 2 , u 3 (n m j 1 ),
DCT in which Qn1 , n2 and Rn1 , n 2 are the pre- j 1 2 j 1
and post-processing glue structures, respectively, 1 i i 1 i
2 w1 (n k ), w2 (n k ), w3 (n j ) ,
that combine 2 ( j 2 2 2 4 ) identical lower- 2 k 1 k 1 2 j i 1
order 2-d DCT modules each of size
n1 / 2 n2 / 2 in parallel, to construct the higher Qi is the 1-d pre-processing as defined by (17).
Equation (37) extends our results by showing that a
order 2-d DCT of size n1 n 2 .
large m-d WHT can be computed from a single
stage of smaller m-d WHTs.
II. A GENERAL FRAMEWORK FOR M-D
RECURSIVE DSP TRANSFORMS III. CONCLUSIONS
We can extend the steps in deriving In this paper, we presented a general
recursive formulae of the 1-d and the 2-d transforms approach for decomposing higher order (longer size)
to the multidimensional case. For an m-d transform multidimensional (m-d) architectures from 2
m
Tni ,The general form will be lower order (shorter sizes) architectures. We have
m m shown several examples for the 1-d and 2-d
ˆ ˆ
Tn R ( I m Tni / 2 ) Q (34) common transforms such as linear convolution,
i 1 i j i 1 DCT, and WHT. We have extended our results to
ˆ ˆ
Where, Q and R are the m-d pre- and post-
cover the m-d case as well. The objective of our
work was to derive a unified framework and a
processing glue structures that combine j m design methodology that allows direct mapping of
the proposed algorithms into reconfigurable
parallel blocks of the lower-order m-d transforms of
architectures. The resulting circuits have very
size Tni / 2 . simple modular structure and regular topology that
can be mapped directly to FPGAs.
A. The m-d WHT
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