Reversible integer 2D Fourier transform
10 Feb 2009-Proceedings of SPIE (International Society for Optics and Photonics)-Vol. 7245, pp 724503
TL;DR: The 2-D reversible integer discrete Fourier transform (RiDFT), which is based on the concept of the paired representation of the pair of 1-D signals which carry the spectral information of the signal at disjoint sets of frequency-points, is described.
Abstract: This paper describes the 2-D reversible integer discrete Fourier transform (RiDFT), which is based on the concept
of the paired representation of the 2-D signal or image. The Fourier transform is split into a minimum set of
short transforms. By means of the paired transform, the 2-D signal is represented as a set of 1-D signals which
carry the spectral information of the signal at disjoint sets of frequency-points. The paired transform-based
2-D DFT involves a few operations of multiplication that can be approximated by integer transforms. Such
one-point transforms with one control bit are applied for calculating the 2-D DFT. 24 real multiplications and
24 control bits are required to perform the 8x8-point RiDFT, and 264 real multiplications and 168 control bits
for the 16 x 16-point 2-D RiDFT of real inputs. The computational complexity of the proposed 2-D RiDFTs is
comparative with the complexity of the fast 2-D DFT.
References
More filters
TL;DR: Two approaches to build integer to integer wavelet transforms are presented and the precoder of Laroiaet al., used in information transmission, is adapted and combined with expansion factors for the high and low pass band in subband filtering.
1,269 citations
Book•
31 Jul 2003
TL;DR: This reference presents a more efficient, flexible, and manageable approach to unitary transform calculation and examines novel concepts in the design, classification, and management of fast algorithms for different transforms in one-, two-, and multidimensional cases.
Abstract: This reference presents a more efficient, flexible, and manageable approach to unitary transform calculation and examines novel concepts in the design, classification, and management of fast algorithms for different transforms in one-, two-, and multidimensional cases. Illustrating methods to construct new unitary transforms for best algorithm selection and development in real-world applications, the book contains a wide range of examples to compare the efficacy of different algorithms in a variety of one-, two-, and three-dimensional cases. Multidimensional Discrete Unitary Transforms builds progressively from simple representative cases to higher levels of generalization.
97 citations
Additional excerpts
...(6) The signal fTp,s = {fp,s,0, fp,s,1, fp,s,2, ....
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TL;DR: A two-dimensional (2-D) integer discrete cosine transform is proposed, which needs only integer operations and shifts and is nonseparable and requires a far fewer number of operations than that used by the corresponding row-column 2-D integer discrete Cosine transform.
Abstract: A method is proposed to factor the type-II discrete cosine transform (DCT-II) into lifting steps and additions. After approximating the lifting matrices, we get a new type-II integer discrete cosine transform (IntDCT-II) that is float-point multiplication free. Based on the relationships among the various types of DCTs, we can generally factor any DCTs into lifting steps and additions and then get four types of integer DCTs, which need no float-point multiplications. By combining the polynomial transform and the one-dimensional (1-D) integer cosine transform, a two-dimensional (2-D) integer discrete cosine transform is proposed. The proposed transform needs only integer operations and shifts. Furthermore, it is nonseparable and requires a far fewer number of operations than that used by the corresponding row-column 2-D integer discrete cosine transform.
83 citations
TL;DR: Considering paired transforms, this work analyzes simultaneously the splitting of the multidimensional Fourier transform as well as the presentation of the processed multiddimensional signal in the form of the short one-dimensional "signals", that determine such splitting.
Abstract: A concept of multipaired unitary transforms is introduced. These kinds of transforms reveal the mathematical structure of Fourier transforms and can be considered intermediate unitary transforms when transferring processed data from the original real space of signals to the complex or frequency space of their images. Considering paired transforms, we analyze simultaneously the splitting of the multidimensional Fourier transform as well as the presentation of the processed multidimensional signal in the form of the short one-dimensional (1-D) "signals", that determine such splitting. The main properties of the orthogonal system of paired functions are described, and the matrix decompositions of the Fourier and Hadamard transforms via the paired transforms are given. The multiplicative complexity of the two-dimensional (2-D) 2/sup r//spl times/2/sup r/-point discrete Fourier transform by the paired transforms is 4/sup r//2(r-7/3)+8/3-12 (r>3), which shows the maximum splitting of the 5-D Fourier transform into the number of the short 1-D Fourier transforms. The 2-D paired transforms are not separable and represent themselves as frequency-time type wavelets for which two parameters are united: frequency and time. The decomposition of the signal is performed in a way that is different from the traditional Haar system of functions.
71 citations
Book•
20 Feb 2009TL;DR: Mixed transformations: continuous case Paired Transform-Based DecompositionDecomposition of 1D signals 2D paired representation Fourier Transform and Multiresolution Fourier transform Representation by frequency-time wavelets Time-frequency correlation analysis Givens-Haar transformations.
Abstract: Discrete Fourier Transform Properties of the discrete Fourier transform Fourier transform splitting Fast Fourier transform Codes for the paired FFT Paired and Haar transforms Integer Fourier Transform Reversible integer Fourier transform Lifting schemes for DFT One-point integer transform DFT in vector form Roots of the unit Codes for the block DFT General elliptic Fourier transforms Cosine Transform Partitioning the DCT Paired algorithm for the N-point DCT Codes for the paired transform Reversible integer DCT Method of nonlinear equations Canonical representation of the integer DCT Hadamard Transform The Walsh and Hadamard transform Mixed Hadamard transformation Generalized bit and transformations T-decomposition of Hadamard matrices Mixed Fourier transformations Mixed transformations: continuous case Paired Transform-Based Decomposition Decomposition of 1D signals 2D paired representation Fourier Transform and Multiresolution Fourier transform Representation by frequency-time wavelets Time-frequency correlation analysis Givens-Haar transformations References Index
58 citations
"Reversible integer 2D Fourier trans..." refers methods in this paper
...Together with the 16-paired transform χ ′ 16 and other χ ′ 2, these transforms require 30 + [2(9) + 2(3) + 0] + 2(2) = 58 additions....
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...(9) To remove the redundancy of the tensor representation, we consider a partitioning of the 2-D DFT....
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