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The design and implementation of FFTW3
 Proceedings of the IEEE
, 2005
"... FFTW is an implementation of the discrete Fourier transform (DFT) that adapts to the hardware in order to maximize performance. This paper shows that such an approach can yield an implementation that is competitive with handoptimized libraries, and describes the software structure that makes our cu ..."
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Cited by 396 (6 self)
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FFTW is an implementation of the discrete Fourier transform (DFT) that adapts to the hardware in order to maximize performance. This paper shows that such an approach can yield an implementation that is competitive with handoptimized libraries, and describes the software structure that makes our current FFTW3 version flexible and adaptive. We further discuss a new algorithm for realdata DFTs of prime size, a new way of implementing DFTs by means of machinespecific singleinstruction, multipledata (SIMD) instructions, and how a specialpurpose compiler can derive optimized implementations of the discrete cosine and sine transforms automatically from a DFT algorithm. Keywords—Adaptive software, cosine transform, fast Fourier transform (FFT), Fourier transform, Hartley transform, I/O tensor.
Manipulation and Compositing of MCDCT Compressed Video
, 1994
"... Many advanced video applications require manipulations of compressed video signals. Popular video manipulation functions include overlap (opaque or semitransparent), translation, scaling, linear filtering, rotation, and pixel multiplication. In this paper, we propose algorithms to manipulate compre ..."
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Cited by 94 (16 self)
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Many advanced video applications require manipulations of compressed video signals. Popular video manipulation functions include overlap (opaque or semitransparent), translation, scaling, linear filtering, rotation, and pixel multiplication. In this paper, we propose algorithms to manipulate compressed video in the compressed domain. Specifically, we focus on compression algorithms using the Discrete Cosine Transform (DCT) with or without Motion Compensation (MC). Compression systems of such kind include JPEG, Motion JPEG, MPEG, and H.261. We derive a complete set of algorithms for all aforementioned manipulation functions in the transform domain, in which video signals are represented by quantized transform coefficients. Due to a much lower data rate and the elimination of decompression/compression conversion, the transformdomain approach has great potential in reducing the computational complexity. The actual computational speedup depends on the specific manipulation functions and ...
The Discrete Cosine Transform
 SIAM Review
, 1999
"... Each Discrete Cosine Transform uses N real basis vectors whose components are cosines. In the DCT4, for example, the jth component of v k is cos(j + 1 2 )(k + 1 2 ) ß N . These basis vectors are orthogonal and the transform is extremely useful in image processing. If the vector x gives the inten ..."
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Cited by 73 (2 self)
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Each Discrete Cosine Transform uses N real basis vectors whose components are cosines. In the DCT4, for example, the jth component of v k is cos(j + 1 2 )(k + 1 2 ) ß N . These basis vectors are orthogonal and the transform is extremely useful in image processing. If the vector x gives the intensities along a row of pixels, its cosine series P c k v k has the coefficients c k = (x; v k )=N . They are quickly computed from an FFT. But a direct proof of orthogonality, by calculating inner products, does not reveal how natural these cosine vectors are. We prove orthogonality in a different way. Each DCT basis contains the eigenvectors of a symmetric "second difference" matrix. By varying the boundary conditions we get the established transforms DCT1 through DCT4. Other combinations lead to four additional cosine transforms. The type of boundary condition (Dirichlet or Neumann, centered at a meshpoint or a midpoint) determines the applications that are appropriate for each transfor...
A Fast Algorithm for Deblurring Models with Neumann Boundary Conditions
, 1999
"... Blur removal is an important problem in signal and image processing. The blurring matrices obtained by using the zero boundary condition (corresponding to assuming dark background outside the scene) are Toeplitz matrices for 1dimensional problems and blockToeplitz Toeplitzblock matrices for 2dim ..."
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Cited by 68 (18 self)
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Blur removal is an important problem in signal and image processing. The blurring matrices obtained by using the zero boundary condition (corresponding to assuming dark background outside the scene) are Toeplitz matrices for 1dimensional problems and blockToeplitz Toeplitzblock matrices for 2dimensional cases. They are computationally intensive to invert especially in the block case. If the periodic boundary condition is used, the matrices become (block) circulant and can be diagonalized by discrete Fourier transform matrices. In this paper, we consider the use of the Neumann boundary condition (corresponding to a reflection of the original scene at the boundary). The resulting matrices are (block) Toeplitzplus Hankel matrices. We show that for symmetric blurring functions, these blurring matrices can always be diagonalized by discrete cosine transform matrices. Thus the cost of inversion is significantly lower than that of using the zero or periodic boundary conditions. We also s...
Algebraic Signal Processing Theory: 1D Space
, 2008
"... In our paper titled “Algebraic Signal Processing Theory: Foundation and 1D Time ” appearing in this issue of the IEEE TRANSACTIONS ON SIGNAL PROCESSING, we presented the algebraic signal processing theory, an axiomatic and general framework for linear signal processing. The basic concept in this ..."
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Cited by 7 (4 self)
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In our paper titled “Algebraic Signal Processing Theory: Foundation and 1D Time ” appearing in this issue of the IEEE TRANSACTIONS ON SIGNAL PROCESSING, we presented the algebraic signal processing theory, an axiomatic and general framework for linear signal processing. The basic concept in this theory is the signal model defined as the triple ( 8), where is a chosen algebra of filters, an associatedmodule of signals, and 8 is a generalization of thetransform. Each signal model has its own associated set of basic SP concepts, including filtering, spectrum, and Fourier transform. Examples include infinite and finite discrete time where these notions take their wellknown forms. In this paper, we use the algebraic theory to develop infinite and finite space signal models. These models are based on a symmetric space shift operator, which is distinct from the standard time shift. We present the space signal processing concepts of filtering or convolution, “transform,” spectrum, and Fourier transform. For finite length space signals, we obtain 16 variants of space models, which have the 16 discrete cosine and sine transforms (DCTs/DSTs) as Fourier transforms. Using this novel derivation, we provide missing signal processing concepts associated with the DCTs/DSTs, establish them as precise analogs to the DFT, get deep insight into their origin, and enable the easy derivation of many of their properties including their fast algorithms.
A unified superresolution approach for optical and Synthetic Aperture Radar images
, 1998
"... It is easy to feel overwhelmed by the amount of time and effort required of the Ph.D. degree. These feelings were, often times, too familiar. With patience, persistence and the help of many individuals, my doctoral studies provided for a very memorable and fruitful four years. I would like to thank ..."
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Cited by 7 (2 self)
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It is easy to feel overwhelmed by the amount of time and effort required of the Ph.D. degree. These feelings were, often times, too familiar. With patience, persistence and the help of many individuals, my doctoral studies provided for a very memorable and fruitful four years. I would like to thank these people for their help and friendship throughout my studies. First, I wish to acknowledge and thank Dr. Jose C. Principe for his role as advisor and mentor throughout my doctoral studies. Our discussions and exchanging of ideas were fundamental in the shaping of this work. I would also like to thank him for providing a stimulating work environment in CNEL through which my engineering horizons were expanded. Thank you to my supervisory committee members: Dr. John M. M. Anderson, Dr. A. Antonio Arroyo, Dr. John G. Harris and Dr. Janice C. Honeyman – your interest and suggestions were appreciated. Dr. John G. Harris deserves additional thanks, particularly for our fruitful discussions and his suggestion to me to work on optical images. I would like to thank Dr. Leslie Novak of MIT Lincoln Laboratory for providing me the opportunity to work with him during the summer of 1997. It was a truly
Fast DCT Domain Filtering Using the DCT and the DST
 IEEE Trans. on Signal Processing
, 1995
"... A method for efficient spatial domain filtering, directly in the DCTIIe domain, is developed and proposed. It consists of using the discrete sine transform (DST), together with the discrete cosine transform (DCT), for transform domain processing, based on the recently derived convolutionmultiplica ..."
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Cited by 5 (0 self)
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A method for efficient spatial domain filtering, directly in the DCTIIe domain, is developed and proposed. It consists of using the discrete sine transform (DST), together with the discrete cosine transform (DCT), for transform domain processing, based on the recently derived convolutionmultiplication properties of discrete trigonometric transforms. The proposed scheme requires no zero padding of the input data, or kernel symmetry. It is demonstrated that, in typical applications, the proposed algorithm is significantly more efficient than the conventional spatial domain method. The method is applicable to any DCT based data compression standard, such as JPEG, MPEG, and H.261. Keywords: DCTdomain filtering, discrete sine transform, data compression. While on sabbatical leave at HewlettPackard Laboratories, 1501 Page Mill Road, Palo Alto, CA 94304, USA. y Address: HP Israel Science Center, Technion City, Haifa 32000, Israel. Email: [renato,merhav]@hp.technion.ac.il 1 Introduc...
A generic framework for filtering in subbanddomain
 in Proc. of the Ninth DSP Workshop (DSP2000
, 2000
"... Subband domain filtering is a beneficial method that can be advantageously used in many applications. This paper discusses a method of generating filters in subband domain of any critically sampled perfect reconstruction filter bank, equivalent to any rational (FIR or IIR) timedomain filter. As exam ..."
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Cited by 5 (0 self)
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Subband domain filtering is a beneficial method that can be advantageously used in many applications. This paper discusses a method of generating filters in subband domain of any critically sampled perfect reconstruction filter bank, equivalent to any rational (FIR or IIR) timedomain filter. As example, implementation for the particular case of MDCT filter bank is given. T 1.
A Fast Algorithm for DCT Domain Filtering
, 1996
"... A method is developed and proposed to efficiently implement spatial domain filtering directly on compressed digital video and images in the discrete cosine transform (DCT) domain. It is demonstrated that the computational complexity of this method is significantly smaller than that of the straigh ..."
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Cited by 3 (1 self)
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A method is developed and proposed to efficiently implement spatial domain filtering directly on compressed digital video and images in the discrete cosine transform (DCT) domain. It is demonstrated that the computational complexity of this method is significantly smaller than that of the straightforward approach, of converting back to the uncompressed domain, convolving in the spatial domain, and retransforming to the DCT domain. It is assumed that the impulse response of the two dimensional filter is symmetric and separable. The method is applicable to any DCT based data compression standard, such as JPEG, MPEG, and H.261. Address: HP Israel Science Center, Technion City, Haifa 32000, Israel. Email: merhav@hp.technion.ac.il y Address: HewlettPackard Laboratories, 1501 Page Mill Road, Palo Alto, CA 94304, U.S.A. Email: bhaskara@hpl.hp.com For HP Internal Use Only 1 Introduction The last few years have witnessed a rapidly growing interest in developing fast algorithms for ...